Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7638 conjugacy classes of subgroups.
The subgroup lattice has 628 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3498 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #62793 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #62588 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #62453 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #62001 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | #61999 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #61881 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #61880 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #61582 | C2 | 1 | 12 | yes | maximal, complemented |
| 10 | #60951 | C2 | 1 | 6 | yes | maximal, complemented |
| 11 | #60798 | C2 | 1 | 6 | yes | maximal, complemented |
| 12 | #60284 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 13 | C22×H(5,𝔽2) | — | 2 | 1 | no | — |
| 14 | #3524 | — | 2 | 4 | no | — |
| 15 | #3518 | C22 | 1 | 4 | yes | complemented |
| 16 | #3517 | C22 | 1 | 4 | yes | complemented |
| 17 | #3506 | C22 | 1 | 6 | yes | complemented |
| 18 | #3502 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | #3498 | — | 2 | 4 | no | — |
| 20 | #3498 | — | 2 | 4 | no | — |
| 21 | #3498 | C22 | 1 | 8 | yes | — |
| 22 | #3498 | C22 | 1 | 8 | yes | — |
| 23 | #3497 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #3493 | — | 2 | 3 | no | — |
| 25 | #3487 | — | 2 | 1 | no | — |
| 26 | #3483 | C22 | 1 | 4 | yes | complemented |
| 27 | #3482 | C22 | 1 | 4 | yes | complemented |
| 28 | #3480 | — | 2 | 4 | no | — |
| 29 | C2×#550 | — | 2 | 1 | no | — |
| 30 | C2×#550 | C22 | 1 | 2 | yes | — |
| 31 | C2×D42 | — | 2 | 3 | no | — |
| 32 | C2×D42 | C22 | 1 | 6 | yes | — |
| 33 | #3460 | — | 2 | 3 | no | — |
| 34 | C2×#535 | — | 2 | 3 | no | — |
| 35 | #3444 | C22 | 1 | 6 | yes | complemented |
| 36 | C2×#532 | C22 | 1 | 3 | yes | — |
| 37 | C22×#127 | C22 | 1 | 3 | yes | — |
| 38 | C4×2−1+4 | — | 2 | 1 | no | — |
| 39 | C2×C4×#41 | C22 | 1 | 1 | yes | characteristic |
| 40 | #3392 | C22 | 1 | 16 | yes | complemented |
| 41 | #3391 | C22 | 1 | 16 | yes | complemented |
| 42 | #3362 | C22 | 1 | 16 | yes | complemented |
| 43 | #3358 | C22 | 1 | 16 | yes | complemented |
| 44 | #3325 | C22 | 1 | 48 | yes | complemented |
| 45 | #3324 | C22 | 1 | 48 | yes | complemented |
| 46 | #3274 | C22 | 1 | 48 | yes | complemented |
| 47 | #3272 | C22 | 1 | 48 | yes | complemented |
| 48 | #3254 | C22 | 1 | 12 | yes | complemented |
| 49 | #3253 | C22 | 1 | 12 | yes | complemented |
| 50 | #3238 | C22 | 1 | 4 | yes | complemented |
| 51 | #3235 | C22 | 1 | 12 | yes | complemented |
| 52 | #3217 | C22 | 1 | 12 | yes | complemented |
| 53 | #3217 | C22 | 1 | 24 | yes | complemented |
| 54 | #3214 | C22 | 1 | 12 | yes | complemented |
| 55 | #3208 | C22 | 1 | 12 | yes | complemented |
| 56 | #3208 | C22 | 1 | 12 | yes | complemented |
| 57 | #3205 | C22 | 1 | 12 | yes | complemented |
| 58 | #3205 | C22 | 1 | 12 | yes | complemented |
| 59 | #3194 | C22 | 1 | 24 | yes | complemented |
| 60 | #3171 | C22 | 1 | 24 | yes | complemented |
| 61 | #3151 | C22 | 1 | 12 | yes | complemented |
| 62 | #3138 | C22 | 1 | 12 | yes | complemented |
| 63 | #3106 | C22 | 1 | 24 | yes | complemented |
| 64 | #3101 | C22 | 1 | 12 | yes | complemented |
| 65 | #3100 | C22 | 1 | 12 | yes | complemented |
| 66 | #3038 | C22 | 1 | 24 | yes | complemented |
| 67 | #3003 | C22 | 1 | 24 | yes | complemented |
| 68 | #2991 | C22 | 1 | 8 | yes | complemented |
| 69 | #2982 | C22 | 1 | 2 | yes | complemented |
| 70 | #2975 | C22 | 1 | 6 | yes | complemented |
| 71 | #2971 | C22 | 1 | 8 | yes | complemented |
| 72 | #2920 | C22 | 1 | 6 | yes | complemented |
| 73 | C2×2−1+4 | — | 2 | 1 | no | — |
| 74 | C2×H(5,𝔽2) | — | 2 | 8 | no | — |
| 75 | C2×H(5,𝔽2) | — | 2 | 8 | no | — |
| 76 | C2×H(5,𝔽2) | — | 4 | 16 | no | — |
| 77 | C22×#41 | — | 2 | 2 | no | — |
| 78 | C22×#41 | — | 2 | 3 | no | — |
| 79 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 80 | C23×D4 | — | 2 | 6 | no | — |
| 81 | C23×D4 | — | 2 | 3 | no | — |
| 82 | C23×D4 | — | 2 | 3 | no | — |
| 83 | C23×D4 | C23 | 1 | 6 | yes | — |
| 84 | C2×AGL(1,ℤ/8) | C23 | 1 | 48 | yes | complemented |
| 85 | #559 | — | 2 | 12 | no | — |
| 86 | Q82 | C23 | 1 | 4 | yes | complemented |
| 87 | #557 | — | 2 | 4 | no | — |
| 88 | #557 | C23 | 1 | 4 | yes | complemented |
| 89 | #556 | — | 2 | 12 | no | — |
| 90 | #556 | C23 | 1 | 12 | yes | complemented |
| 91 | #555 | — | 2 | 12 | no | — |
| 92 | #555 | C23 | 1 | 12 | yes | complemented |
| 93 | #553 | — | 2 | 12 | no | — |
| 94 | #553 | — | 2 | 12 | no | — |
| 95 | #552 | — | 2 | 12 | no | — |
| 96 | #550 | — | 2 | 4 | no | — |
| 97 | #550 | — | 2 | 4 | no | — |
| 98 | #550 | — | 2 | 8 | no | — |
| 99 | #550 | — | 2 | 8 | no | — |
| 100 | #550 | — | 2 | 8 | no | — |
| 101 | #550 | C23 | 1 | 16 | yes | — |
| 102 | #550 | C23 | 1 | 12 | yes | complemented |
| 103 | #550 | C23 | 1 | 4 | yes | complemented |
| 104 | D4×Q8 | C23 | 1 | 4 | yes | complemented |
| 105 | #548 | — | 2 | 4 | no | — |
| 106 | #548 | — | 2 | 4 | no | — |
| 107 | #548 | — | 2 | 4 | no | — |
| 108 | #547 | C23 | 1 | 12 | yes | complemented |
| 109 | D42 | — | 2 | 24 | no | — |
| 110 | D42 | — | 2 | 24 | no | — |
| 111 | D42 | — | 2 | 24 | no | — |
| 112 | D42 | C23 | 1 | 48 | yes | — |
| 113 | #544 | — | 2 | 3 | no | — |
| 114 | #542 | — | 2 | 12 | no | — |
| 115 | #540 | — | 2 | 12 | no | — |
| 116 | #538 | — | 2 | 12 | no | — |
| 117 | #537 | — | 2 | 3 | no | — |
| 118 | #535 | — | 2 | 6 | no | — |
| 119 | #535 | — | 2 | 24 | no | — |
| 120 | #535 | — | 2 | 24 | no | — |
| 121 | #533 | C23 | 1 | 6 | yes | — |
| 122 | #533 | C23 | 1 | 6 | yes | complemented |
| 123 | #533 | C23 | 1 | 6 | yes | complemented |
| 124 | #532 | — | 2 | 12 | no | — |
| 125 | #532 | — | 2 | 6 | no | — |
| 126 | #532 | — | 2 | 6 | no | — |
| 127 | #532 | C23 | 1 | 6 | yes | — |
| 128 | #532 | C23 | 1 | 12 | yes | — |
| 129 | #532 | C23 | 1 | 12 | yes | — |
| 130 | #532 | C23 | 1 | 6 | yes | complemented |
| 131 | #532 | C23 | 1 | 6 | yes | complemented |
| 132 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 133 | C2×#127 | — | 2 | 3 | no | — |
| 134 | C2×#127 | — | 2 | 3 | no | — |
| 135 | C2×#127 | — | 2 | 6 | no | — |
| 136 | C2×#127 | — | 2 | 6 | no | — |
| 137 | C2×#127 | C23 | 1 | 12 | yes | — |
| 138 | C2×#127 | C23 | 1 | 3 | yes | — |
| 139 | C2×#127 | C23 | 1 | 6 | yes | — |
| 140 | C2×#127 | C23 | 1 | 12 | yes | — |
| 141 | #529 | — | 2 | 12 | no | — |
| 142 | #529 | — | 2 | 6 | no | — |
| 143 | #529 | — | 2 | 6 | no | — |
| 144 | #529 | — | 2 | 6 | no | — |
| 145 | #529 | — | 2 | 6 | no | — |
| 146 | #529 | C23 | 1 | 4 | yes | — |
| 147 | #529 | C23 | 1 | 12 | yes | complemented |
| 148 | C2×#124 | — | 2 | 3 | no | — |
| 149 | #525 | — | 2 | 6 | no | — |
| 150 | #525 | — | 2 | 6 | no | — |
| 151 | #525 | C23 | 1 | 12 | yes | — |
| 152 | C2×#121 | — | 2 | 3 | no | — |
| 153 | C2×#121 | — | 2 | 6 | no | — |
| 154 | C2×#121 | — | 2 | 6 | no | — |
| 155 | C2×#121 | C23 | 1 | 6 | yes | — |
| 156 | C2×#120 | — | 2 | 12 | no | — |
| 157 | #520 | — | 2 | 4 | no | — |
| 158 | #520 | — | 2 | 6 | no | — |
| 159 | #519 | — | 2 | 1 | no | — |
| 160 | #519 | — | 2 | 4 | no | — |
| 161 | C4×#41 | — | 2 | 4 | no | — |
| 162 | C4×#41 | — | 2 | 6 | no | — |
| 163 | C4×#41 | — | 2 | 6 | no | — |
| 164 | C4×#41 | D4 | 1 | 4 | yes | — |
| 165 | C4×#41 | D4 | 1 | 4 | yes | — |
| 166 | C4×#41 | C23 | 1 | 4 | yes | — |
| 167 | C4×#41 | C23 | 1 | 4 | yes | — |
| 168 | C4×#41 | C23 | 1 | 12 | yes | complemented |
| 169 | C2×C4×Q8 | — | 2 | 4 | no | — |
| 170 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 171 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 172 | C2×C4×D4 | — | 2 | 3 | no | — |
| 173 | C2×C4×D4 | D4 | 1 | 3 | yes | — |
| 174 | C2×C4×D4 | C23 | 1 | 6 | yes | — |
| 175 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 176 | C2×#117 | D4 | 1 | 3 | yes | — |
| 177 | C22×C42 | C23 | 1 | 3 | yes | — |
| 178 | #501 | C23 | 1 | 8 | yes | complemented |
| 179 | #496 | C23 | 1 | 24 | yes | complemented |
| 180 | #493 | C23 | 1 | 24 | yes | complemented |
| 181 | #492 | C23 | 1 | 24 | yes | complemented |
| 182 | #490 | C23 | 1 | 24 | yes | complemented |
| 183 | #489 | C23 | 1 | 8 | yes | complemented |
| 184 | #486 | C23 | 1 | 24 | yes | complemented |
| 185 | #486 | C23 | 1 | 24 | yes | complemented |
| 186 | #483 | C23 | 1 | 8 | yes | — |
| 187 | #482 | C23 | 1 | 48 | yes | complemented |
| 188 | #480 | C23 | 1 | 8 | yes | — |
| 189 | #477 | C23 | 1 | 8 | yes | complemented |
| 190 | #476 | C23 | 1 | 24 | yes | complemented |
| 191 | #476 | C23 | 1 | 8 | yes | complemented |
| 192 | #474 | C23 | 1 | 24 | yes | complemented |
| 193 | #469 | C23 | 1 | 48 | yes | complemented |
| 194 | #468 | C23 | 1 | 48 | yes | complemented |
| 195 | #461 | C23 | 1 | 24 | yes | complemented |
| 196 | #460 | C23 | 1 | 24 | yes | complemented |
| 197 | #460 | C23 | 1 | 48 | yes | complemented |
| 198 | #459 | C23 | 1 | 24 | yes | complemented |
| 199 | #459 | C23 | 1 | 48 | yes | complemented |
| 200 | #459 | C23 | 1 | 24 | yes | complemented |
| 201 | #450 | C23 | 1 | 24 | yes | — |
| 202 | #450 | C23 | 1 | 48 | yes | complemented |
| 203 | #449 | C23 | 1 | 48 | yes | complemented |
| 204 | #447 | C23 | 1 | 24 | yes | — |
| 205 | #447 | C23 | 1 | 48 | yes | complemented |
| 206 | C8×Q8 | C23 | 1 | 16 | yes | complemented |
| 207 | C4×SD16 | C23 | 1 | 48 | yes | complemented |
| 208 | C4×D8 | C23 | 1 | 48 | yes | complemented |
| 209 | #436 | C23 | 1 | 48 | yes | complemented |
| 210 | #432 | C23 | 1 | 4 | yes | — |
| 211 | #432 | C23 | 1 | 12 | yes | complemented |
| 212 | #431 | C23 | 1 | 12 | yes | — |
| 213 | #431 | C23 | 1 | 12 | yes | complemented |
| 214 | #423 | C23 | 1 | 12 | yes | complemented |
| 215 | #418 | C23 | 1 | 48 | yes | complemented |
| 216 | #407 | C23 | 1 | 12 | yes | — |
| 217 | C4×M16 | C23 | 1 | 12 | yes | complemented |
| 218 | C25 | — | 2 | 6 | no | — |
| 219 | 2−1+4 | — | 4 | 8 | no | — |
| 220 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 221 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 222 | C2×#41 | — | 2 | 8 | no | — |
| 223 | C2×#41 | — | 2 | 4 | no | — |
| 224 | C2×#41 | — | 2 | 16 | no | — |
| 225 | C2×#41 | — | 2 | 6 | no | — |
| 226 | C2×#41 | — | 2 | 12 | no | — |
| 227 | C2×#41 | — | 2 | 12 | no | — |
| 228 | C2×#41 | — | 2 | 6 | no | — |
| 229 | C2×#41 | — | 4 | 24 | no | — |
| 230 | C2×#41 | — | 4 | 16 | no | — |
| 231 | C2×#41 | C2×D4 | 1 | 4 | yes | — |
| 232 | C2×#41 | C2×D4 | 1 | 4 | yes | — |
| 233 | C2×#41 | C24 | 1 | 12 | yes | — |
| 234 | C22×Q8 | — | 2 | 4 | no | — |
| 235 | C22×Q8 | — | 2 | 1 | no | — |
| 236 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 237 | C22×D4 | — | 2 | 6 | no | — |
| 238 | C22×D4 | — | 2 | 48 | no | — |
| 239 | C22×D4 | — | 2 | 3 | no | — |
| 240 | C22×D4 | — | 2 | 6 | no | — |
| 241 | C22×D4 | — | 2 | 3 | no | — |
| 242 | C22×D4 | — | 2 | 12 | no | — |
| 243 | C22×D4 | — | 2 | 12 | no | — |
| 244 | C22×D4 | — | 2 | 6 | no | — |
| 245 | C22×D4 | — | 2 | 6 | no | — |
| 246 | C22×D4 | — | 2 | 12 | no | — |
| 247 | C22×D4 | — | 2 | 12 | no | — |
| 248 | C22×D4 | — | 2 | 12 | no | — |
| 249 | C22×D4 | — | 4 | 24 | no | — |
| 250 | C22×D4 | — | 4 | 48 | no | — |
| 251 | C22×D4 | — | 8 | 12 | no | — |
| 252 | C22×D4 | — | 8 | 12 | no | — |
| 253 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 254 | C22×D4 | C24 | 1 | 6 | yes | — |
| 255 | C22×D4 | C24 | 1 | 24 | yes | — |
| 256 | C23×C4 | — | 2 | 6 | no | — |
| 257 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 258 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 259 | C23×C4 | C24 | 1 | 3 | yes | — |
| 260 | AGL(1,ℤ/8) | — | 8 | 48 | no | — |
| 261 | C2×SD16 | C24 | 1 | 48 | yes | — |
| 262 | C2×D8 | C24 | 1 | 48 | yes | — |
| 263 | C2×M16 | C24 | 1 | 24 | yes | — |
| 264 | C4⋊Q8 | — | 2 | 12 | no | — |
| 265 | C4⋊Q8 | — | 2 | 1 | no | — |
| 266 | C4⋊Q8 | — | 2 | 1 | no | — |
| 267 | C4⋊Q8 | C24 | 1 | 6 | yes | — |
| 268 | C4⋊Q8 | C24 | 1 | 2 | yes | — |
| 269 | C4⋊Q8 | C24 | 1 | 2 | yes | — |
| 270 | C4⋊Q8 | C24 | 1 | 6 | yes | — |
| 271 | C4⋊Q8 | C24 | 1 | 6 | yes | — |
| 272 | #127 | — | 2 | 12 | no | — |
| 273 | #127 | — | 2 | 3 | no | — |
| 274 | #127 | — | 2 | 3 | no | — |
| 275 | #127 | — | 2 | 12 | no | — |
| 276 | #127 | — | 2 | 12 | no | — |
| 277 | #127 | — | 2 | 12 | no | — |
| 278 | #127 | — | 2 | 12 | no | — |
| 279 | #127 | — | 2 | 12 | no | — |
| 280 | #127 | C24 | 1 | 6 | yes | — |
| 281 | #127 | C24 | 1 | 6 | yes | — |
| 282 | #127 | C24 | 1 | 24 | yes | — |
| 283 | #127 | C24 | 1 | 6 | yes | — |
| 284 | #126 | — | 2 | 24 | no | — |
| 285 | #126 | — | 2 | 4 | no | — |
| 286 | #126 | — | 2 | 12 | no | — |
| 287 | #125 | — | 2 | 12 | no | — |
| 288 | #125 | — | 2 | 6 | no | — |
| 289 | #125 | — | 2 | 6 | no | — |
| 290 | #125 | C24 | 1 | 4 | yes | — |
| 291 | #124 | — | 2 | 12 | no | — |
| 292 | #124 | — | 2 | 24 | no | — |
| 293 | #124 | — | 2 | 12 | no | — |
| 294 | #124 | — | 2 | 12 | no | — |
| 295 | #124 | — | 2 | 6 | no | — |
| 296 | #124 | — | 2 | 6 | no | — |
| 297 | #124 | C24 | 1 | 4 | yes | — |
| 298 | #123 | — | 2 | 4 | no | — |
| 299 | #123 | — | 2 | 12 | no | — |
| 300 | #123 | — | 2 | 12 | no | — |
| 301 | #123 | — | 2 | 24 | no | — |
| 302 | C22⋊Q8 | — | 2 | 6 | no | — |
| 303 | C22⋊Q8 | — | 2 | 6 | no | — |
| 304 | C22⋊Q8 | — | 2 | 6 | no | — |
| 305 | C22⋊Q8 | — | 2 | 6 | no | — |
| 306 | C22⋊Q8 | — | 2 | 12 | no | — |
| 307 | C22⋊Q8 | C24 | 1 | 4 | yes | — |
| 308 | C22⋊Q8 | C24 | 1 | 12 | yes | — |
| 309 | #121 | — | 2 | 12 | no | — |
| 310 | #121 | — | 2 | 12 | no | — |
| 311 | #121 | — | 2 | 12 | no | — |
| 312 | #121 | — | 2 | 12 | no | — |
| 313 | #121 | — | 2 | 12 | no | — |
| 314 | #121 | — | 2 | 6 | no | — |
| 315 | #121 | — | 2 | 6 | no | — |
| 316 | #121 | — | 2 | 6 | no | — |
| 317 | #121 | — | 2 | 6 | no | — |
| 318 | #121 | — | 2 | 48 | no | — |
| 319 | #121 | — | 2 | 48 | no | — |
| 320 | #121 | — | 2 | 12 | no | — |
| 321 | #121 | — | 2 | 24 | no | — |
| 322 | #121 | C24 | 1 | 24 | yes | — |
| 323 | #121 | C24 | 1 | 4 | yes | — |
| 324 | #121 | C24 | 1 | 12 | yes | — |
| 325 | #120 | — | 2 | 12 | no | — |
| 326 | #120 | — | 2 | 96 | no | — |
| 327 | C4×Q8 | — | 2 | 4 | no | — |
| 328 | C4×Q8 | — | 2 | 4 | no | — |
| 329 | C4×Q8 | — | 2 | 12 | no | — |
| 330 | C4×Q8 | — | 2 | 6 | no | — |
| 331 | C4×Q8 | — | 2 | 6 | no | — |
| 332 | C4×Q8 | — | 2 | 6 | no | — |
| 333 | C4×Q8 | — | 2 | 6 | no | — |
| 334 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 335 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 336 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 337 | C4×Q8 | C24 | 1 | 4 | yes | — |
| 338 | C4×Q8 | C24 | 1 | 12 | yes | — |
| 339 | C4×Q8 | C24 | 1 | 4 | yes | — |
| 340 | C4×D4 | — | 2 | 4 | no | — |
| 341 | C4×D4 | — | 2 | 12 | no | — |
| 342 | C4×D4 | — | 2 | 12 | no | — |
| 343 | C4×D4 | — | 2 | 12 | no | — |
| 344 | C4×D4 | — | 2 | 12 | no | — |
| 345 | C4×D4 | — | 2 | 12 | no | — |
| 346 | C4×D4 | — | 2 | 12 | no | — |
| 347 | C4×D4 | — | 2 | 12 | no | — |
| 348 | C4×D4 | — | 2 | 12 | no | — |
| 349 | C4×D4 | — | 2 | 12 | no | — |
| 350 | C4×D4 | — | 2 | 6 | no | — |
| 351 | C4×D4 | — | 2 | 6 | no | — |
| 352 | C4×D4 | — | 2 | 6 | no | — |
| 353 | C4×D4 | — | 2 | 6 | no | — |
| 354 | C4×D4 | C2×D4 | 1 | 12 | yes | — |
| 355 | C4×D4 | C2×D4 | 1 | 12 | yes | — |
| 356 | C4×D4 | C2×D4 | 1 | 12 | yes | — |
| 357 | C4×D4 | C24 | 1 | 24 | yes | — |
| 358 | C4×D4 | C24 | 1 | 4 | yes | — |
| 359 | C4×D4 | C24 | 1 | 12 | yes | — |
| 360 | C4×D4 | C24 | 1 | 12 | yes | — |
| 361 | #117 | — | 2 | 6 | no | — |
| 362 | #117 | — | 2 | 6 | no | — |
| 363 | #117 | — | 2 | 12 | no | — |
| 364 | #117 | — | 2 | 4 | no | — |
| 365 | #117 | — | 2 | 12 | no | — |
| 366 | #117 | — | 2 | 12 | no | — |
| 367 | #117 | C2×D4 | 1 | 12 | yes | — |
| 368 | #117 | C2×D4 | 1 | 12 | yes | — |
| 369 | #117 | C24 | 1 | 12 | yes | — |
| 370 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 371 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 372 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 373 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 374 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 375 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 376 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 377 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 378 | C2×C42 | — | 2 | 12 | no | — |
| 379 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 380 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 381 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 382 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 383 | C2×C42 | C24 | 1 | 3 | yes | — |
| 384 | C2×C42 | C24 | 1 | 6 | yes | — |
| 385 | C2×C42 | C24 | 1 | 6 | yes | — |
| 386 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 387 | #107 | C24 | 1 | 16 | yes | — |
| 388 | #106 | C24 | 1 | 16 | yes | — |
| 389 | C4⋊C8 | C24 | 1 | 8 | yes | — |
| 390 | C4⋊C8 | C24 | 1 | 24 | yes | — |
| 391 | #103 | C24 | 1 | 16 | yes | — |
| 392 | #102 | C24 | 1 | 48 | yes | — |
| 393 | #102 | C24 | 1 | 48 | yes | — |
| 394 | #102 | C24 | 1 | 16 | yes | — |
| 395 | C22⋊C8 | C24 | 1 | 8 | yes | — |
| 396 | C22⋊C8 | C24 | 1 | 24 | yes | — |
| 397 | #97 | C24 | 1 | 8 | yes | — |
| 398 | C4×C8 | C24 | 1 | 24 | yes | — |
| 399 | C24 | — | 2 | 6 | no | — |
| 400 | C24 | — | 2 | 24 | no | — |
| 401 | C24 | — | 4 | 24 | no | — |
| 402 | C24 | — | 8 | 12 | no | — |
| 403 | C24 | — | 8 | 12 | no | — |
| 404 | C24 | C22×D4 | 1 | 3 | yes | — |
| 405 | #41 | — | 2 | 16 | no | — |
| 406 | #41 | — | 2 | 16 | no | — |
| 407 | #41 | — | 4 | 48 | no | — |
| 408 | #41 | — | 4 | 48 | no | — |
| 409 | #41 | — | 4 | 24 | no | — |
| 410 | #41 | — | 4 | 64 | no | — |
| 411 | #41 | — | 4 | 16 | no | — |
| 412 | #41 | — | 4 | 24 | no | — |
| 413 | #41 | — | 8 | 12 | no | — |
| 414 | C2×Q8 | — | 2 | 6 | no | — |
| 415 | C2×Q8 | — | 2 | 2 | no | — |
| 416 | C2×Q8 | — | 2 | 2 | no | — |
| 417 | C2×Q8 | — | 2 | 6 | no | — |
| 418 | C2×Q8 | — | 4 | 4 | no | — |
| 419 | C2×Q8 | — | 4 | 16 | no | — |
| 420 | C2×Q8 | AGL(1,ℤ/8) | 1 | 8 | yes | complemented |
| 421 | C2×Q8 | C22×D4 | 1 | 4 | yes | — |
| 422 | C2×Q8 | C25 | 1 | 4 | yes | — |
| 423 | C2×D4 | — | 2 | 4 | no | — |
| 424 | C2×D4 | — | 2 | 24 | no | — |
| 425 | C2×D4 | — | 2 | 12 | no | — |
| 426 | C2×D4 | — | 2 | 12 | no | — |
| 427 | C2×D4 | — | 2 | 12 | no | — |
| 428 | C2×D4 | — | 2 | 6 | no | — |
| 429 | C2×D4 | — | 2 | 6 | no | — |
| 430 | C2×D4 | — | 2 | 6 | no | — |
| 431 | C2×D4 | — | 2 | 6 | no | — |
| 432 | C2×D4 | — | 2 | 24 | no | — |
| 433 | C2×D4 | — | 2 | 6 | no | — |
| 434 | C2×D4 | — | 2 | 6 | no | — |
| 435 | C2×D4 | — | 2 | 6 | no | — |
| 436 | C2×D4 | — | 2 | 24 | no | — |
| 437 | C2×D4 | — | 2 | 24 | no | — |
| 438 | C2×D4 | — | 4 | 12 | no | — |
| 439 | C2×D4 | — | 4 | 48 | no | — |
| 440 | C2×D4 | — | 4 | 12 | no | — |
| 441 | C2×D4 | — | 4 | 48 | no | — |
| 442 | C2×D4 | — | 4 | 48 | no | — |
| 443 | C2×D4 | — | 4 | 12 | no | — |
| 444 | C2×D4 | — | 4 | 192 | no | — |
| 445 | C2×D4 | — | 4 | 24 | no | — |
| 446 | C2×D4 | — | 4 | 24 | no | — |
| 447 | C2×D4 | — | 8 | 24 | no | — |
| 448 | C2×D4 | — | 8 | 24 | no | — |
| 449 | C2×D4 | — | 8 | 24 | no | — |
| 450 | C2×D4 | — | 8 | 12 | no | — |
| 451 | C2×D4 | — | 8 | 12 | no | — |
| 452 | C2×D4 | C22×D4 | 1 | 12 | yes | — |
| 453 | C2×D4 | C25 | 1 | 12 | yes | — |
| 454 | C2×D4 | C25 | 1 | 4 | yes | — |
| 455 | C22×C4 | — | 2 | 12 | no | — |
| 456 | C22×C4 | — | 2 | 2 | no | — |
| 457 | C22×C4 | — | 2 | 4 | no | — |
| 458 | C22×C4 | — | 2 | 12 | no | — |
| 459 | C22×C4 | — | 2 | 24 | no | — |
| 460 | C22×C4 | — | 2 | 12 | no | — |
| 461 | C22×C4 | — | 4 | 6 | no | — |
| 462 | C22×C4 | — | 4 | 6 | no | — |
| 463 | C22×C4 | — | 4 | 24 | no | — |
| 464 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 465 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 466 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 467 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 468 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 469 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 470 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 471 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 472 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 473 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 474 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 475 | C22×C4 | C25 | 1 | 6 | yes | — |
| 476 | SD16 | — | 4 | 48 | no | — |
| 477 | D8 | — | 4 | 48 | no | — |
| 478 | M16 | — | 4 | 24 | no | — |
| 479 | C2×C8 | C25 | 1 | 16 | yes | — |
| 480 | C4⋊C4 | — | 2 | 6 | no | — |
| 481 | C4⋊C4 | — | 2 | 12 | no | — |
| 482 | C4⋊C4 | — | 2 | 2 | no | — |
| 483 | C4⋊C4 | — | 2 | 2 | no | — |
| 484 | C4⋊C4 | — | 2 | 2 | no | — |
| 485 | C4⋊C4 | — | 2 | 4 | no | — |
| 486 | C4⋊C4 | — | 2 | 6 | no | — |
| 487 | C4⋊C4 | — | 2 | 4 | no | — |
| 488 | C4⋊C4 | — | 2 | 4 | no | — |
| 489 | C4⋊C4 | — | 2 | 4 | no | — |
| 490 | C4⋊C4 | — | 2 | 6 | no | — |
| 491 | C4⋊C4 | — | 2 | 6 | no | — |
| 492 | C4⋊C4 | — | 2 | 12 | no | — |
| 493 | C4⋊C4 | C22×D4 | 1 | 12 | yes | — |
| 494 | C4⋊C4 | C22×D4 | 1 | 12 | yes | — |
| 495 | C4⋊C4 | C25 | 1 | 4 | yes | — |
| 496 | C4⋊C4 | C25 | 1 | 4 | yes | — |
| 497 | C4⋊C4 | C25 | 1 | 4 | yes | — |
| 498 | C22⋊C4 | — | 2 | 4 | no | — |
| 499 | C22⋊C4 | — | 2 | 12 | no | — |
| 500 | C22⋊C4 | — | 2 | 12 | no | — |
| 501 | C22⋊C4 | — | 2 | 24 | no | — |
| 502 | C22⋊C4 | — | 2 | 24 | no | — |
| 503 | C22⋊C4 | — | 2 | 4 | no | — |
| 504 | C22⋊C4 | — | 2 | 12 | no | — |
| 505 | C22⋊C4 | — | 2 | 4 | no | — |
| 506 | C22⋊C4 | — | 2 | 4 | no | — |
| 507 | C22⋊C4 | — | 2 | 4 | no | — |
| 508 | C22⋊C4 | — | 2 | 12 | no | — |
| 509 | C22⋊C4 | C22×D4 | 1 | 24 | yes | — |
| 510 | C42 | — | 2 | 4 | no | — |
| 511 | C42 | — | 2 | 12 | no | — |
| 512 | C42 | — | 2 | 12 | no | — |
| 513 | C42 | C22×D4 | 1 | 6 | yes | — |
| 514 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 515 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 516 | C42 | C22×D4 | 1 | 6 | yes | — |
| 517 | C42 | C22×D4 | 1 | 6 | yes | — |
| 518 | C42 | C22×D4 | 1 | 6 | yes | — |
| 519 | C42 | C22×D4 | 1 | 6 | yes | — |
| 520 | C42 | C25 | 1 | 6 | yes | — |
| 521 | C42 | C25 | 1 | 1 | yes | characteristic |
| 522 | C42 | C25 | 1 | 1 | yes | characteristic |
| 523 | C23 | — | 2 | 6 | no | — |
| 524 | C23 | — | 2 | 4 | no | — |
| 525 | C23 | — | 2 | 12 | no | — |
| 526 | C23 | — | 4 | 3 | no | — |
| 527 | C23 | — | 4 | 3 | no | — |
| 528 | C23 | — | 4 | 12 | no | — |
| 529 | C23 | — | 4 | 12 | no | — |
| 530 | C23 | — | 4 | 48 | no | — |
| 531 | C23 | — | 4 | 12 | no | — |
| 532 | C23 | — | 8 | 24 | no | — |
| 533 | C23 | — | 8 | 24 | no | — |
| 534 | C23 | — | 16 | 48 | no | — |
| 535 | C23 | C23×D4 | 1 | 6 | yes | — |
| 536 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 537 | Q8 | — | 2 | 8 | no | — |
| 538 | Q8 | — | 4 | 4 | no | — |
| 539 | Q8 | — | 4 | 4 | no | — |
| 540 | Q8 | — | 4 | 12 | no | — |
| 541 | Q8 | — | 8 | 6 | no | — |
| 542 | Q8 | C2×AGL(1,ℤ/8) | 1 | 16 | yes | — |
| 543 | D4 | — | 2 | 8 | no | — |
| 544 | D4 | — | 2 | 24 | no | — |
| 545 | D4 | — | 2 | 24 | no | — |
| 546 | D4 | — | 4 | 12 | no | — |
| 547 | D4 | — | 4 | 12 | no | — |
| 548 | D4 | — | 4 | 12 | no | — |
| 549 | D4 | — | 4 | 12 | no | — |
| 550 | D4 | — | 4 | 48 | no | — |
| 551 | D4 | — | 4 | 8 | no | — |
| 552 | D4 | — | 4 | 12 | no | — |
| 553 | D4 | — | 4 | 48 | no | — |
| 554 | D4 | — | 4 | 12 | no | — |
| 555 | D4 | — | 8 | 24 | no | — |
| 556 | D4 | — | 8 | 24 | no | — |
| 557 | D4 | — | 8 | 6 | no | — |
| 558 | D4 | — | 8 | 12 | no | — |
| 559 | D4 | — | 8 | 12 | no | — |
| 560 | C2×C4 | — | 2 | 2 | no | — |
| 561 | C2×C4 | — | 2 | 2 | no | — |
| 562 | C2×C4 | — | 2 | 12 | no | — |
| 563 | C2×C4 | — | 2 | 12 | no | — |
| 564 | C2×C4 | — | 2 | 2 | no | — |
| 565 | C2×C4 | — | 2 | 12 | no | — |
| 566 | C2×C4 | — | 2 | 12 | no | — |
| 567 | C2×C4 | — | 2 | 4 | no | — |
| 568 | C2×C4 | — | 2 | 4 | no | — |
| 569 | C2×C4 | — | 2 | 4 | no | — |
| 570 | C2×C4 | — | 2 | 4 | no | — |
| 571 | C2×C4 | — | 4 | 6 | no | — |
| 572 | C2×C4 | — | 4 | 3 | no | — |
| 573 | C2×C4 | — | 4 | 6 | no | — |
| 574 | C2×C4 | — | 4 | 3 | no | — |
| 575 | C2×C4 | — | 4 | 4 | no | — |
| 576 | C2×C4 | — | 4 | 8 | no | — |
| 577 | C2×C4 | — | 4 | 48 | no | — |
| 578 | C2×C4 | — | 4 | 24 | no | — |
| 579 | C2×C4 | — | 8 | 12 | no | — |
| 580 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 12 | yes | — |
| 581 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 582 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 583 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 584 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 585 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 586 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 587 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 588 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 589 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 590 | C8 | — | 2 | 16 | no | — |
| 591 | C22 | — | 2 | 1 | no | — |
| 592 | C22 | — | 2 | 1 | no | — |
| 593 | C22 | — | 2 | 6 | no | — |
| 594 | C22 | — | 2 | 6 | no | — |
| 595 | C22 | — | 2 | 6 | no | — |
| 596 | C22 | — | 4 | 12 | no | — |
| 597 | C22 | — | 4 | 4 | no | — |
| 598 | C22 | — | 4 | 12 | no | — |
| 599 | C22 | — | 4 | 4 | no | — |
| 600 | C22 | — | 4 | 12 | no | — |
| 601 | C22 | — | 4 | 4 | no | — |
| 602 | C22 | — | 4 | 12 | no | — |
| 603 | C22 | — | 8 | 24 | no | — |
| 604 | C22 | — | 16 | 48 | no | — |
| 605 | C22 | C22×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 606 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 607 | C4 | — | 2 | 1 | no | — |
| 608 | C4 | — | 2 | 1 | no | — |
| 609 | C4 | — | 2 | 1 | no | — |
| 610 | C4 | — | 2 | 6 | no | — |
| 611 | C4 | — | 2 | 1 | no | — |
| 612 | C4 | — | 2 | 6 | no | — |
| 613 | C4 | — | 4 | 4 | no | — |
| 614 | C4 | — | 4 | 4 | no | — |
| 615 | C4 | — | 4 | 4 | no | — |
| 616 | C4 | — | 4 | 4 | no | — |
| 617 | C4 | C22×AGL(1,ℤ/8) | 1 | 12 | yes | — |
| 618 | C4 | #3585 | 1 | 2 | yes | — |
| 619 | C4 | H(7,𝔽2) | 1 | 2 | yes | — |
| 620 | C2 | — | 2 | 2 | no | — |
| 621 | C2 | — | 4 | 6 | no | — |
| 622 | C2 | — | 8 | 4 | no | — |
| 623 | C2 | — | 8 | 12 | no | — |
| 624 | C2 | #62852 | 1 | 1 | yes | characteristic |
| 625 | C2 | C23×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 626 | C2 | C2×#3585 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3498 | C2 | semidirect |
| 2. #62793 | C2 | semidirect |
| 3. #62588 | C2 | semidirect |
| 4. #62453 | C2 | semidirect |
| 5. #62001 | C2 | semidirect |
| 6. #61999 | C2 | semidirect |
| 7. #61881 | C2 | semidirect |
| 8. #61880 | C2 | semidirect |
| 9. #61582 | C2 | semidirect |
| 10. #60951 | C2 | semidirect |
| 11. #60798 | C2 | semidirect |
| 12. #60284 | C2 | semidirect |
| 15. #3518 | C22 | semidirect |
| 16. #3517 | C22 | semidirect |
| 17. #3506 | C22 | semidirect |
| 18. #3502 | C22 | semidirect |
| 23. #3497 | C22 | semidirect |
| 26. #3483 | C22 | semidirect |
| 27. #3482 | C22 | semidirect |
| 35. #3444 | C22 | semidirect |
| 40. #3392 | C22 | semidirect |
| 41. #3391 | C22 | semidirect |
| 42. #3362 | C22 | semidirect |
| 43. #3358 | C22 | semidirect |
| 44. #3325 | C22 | semidirect |
| 45. #3324 | C22 | semidirect |
| 46. #3274 | C22 | semidirect |
| 47. #3272 | C22 | semidirect |
| 48. #3254 | C22 | semidirect |
| 49. #3253 | C22 | semidirect |
| 50. #3238 | C22 | semidirect |
| 51. #3235 | C22 | semidirect |
| 52. #3217 | C22 | semidirect |
| 53. #3217 | C22 | semidirect |
| 54. #3214 | C22 | semidirect |
| 55. #3208 | C22 | semidirect |
| 56. #3208 | C22 | semidirect |
| 57. #3205 | C22 | semidirect |
| 58. #3205 | C22 | semidirect |
| 59. #3194 | C22 | semidirect |
| 60. #3171 | C22 | semidirect |
| 61. #3151 | C22 | semidirect |
| 62. #3138 | C22 | semidirect |
| 63. #3106 | C22 | semidirect |
| 64. #3101 | C22 | semidirect |
| 65. #3100 | C22 | semidirect |
| 66. #3038 | C22 | semidirect |
| 67. #3003 | C22 | semidirect |
| 68. #2991 | C22 | semidirect |
| 69. #2982 | C22 | semidirect |
| 70. #2975 | C22 | semidirect |
| 71. #2971 | C22 | semidirect |
| 72. #2920 | C22 | semidirect |
| 84. C2×AGL(1,ℤ/8) | C23 | semidirect |
| 86. Q82 | C23 | semidirect |
| 88. #557 | C23 | semidirect |
| 90. #556 | C23 | semidirect |
| 92. #555 | C23 | semidirect |
| 102. #550 | C23 | semidirect |
| 103. #550 | C23 | semidirect |
| 104. D4×Q8 | C23 | semidirect |
| 108. #547 | C23 | semidirect |
| 122. #533 | C23 | semidirect |
| 123. #533 | C23 | semidirect |
| 130. #532 | C23 | semidirect |
| 131. #532 | C23 | semidirect |
| 147. #529 | C23 | semidirect |
| 168. C4×#41 | C23 | semidirect |
| 178. #501 | C23 | semidirect |
| 179. #496 | C23 | semidirect |
| 180. #493 | C23 | semidirect |
| 181. #492 | C23 | semidirect |
| 182. #490 | C23 | semidirect |
| 183. #489 | C23 | semidirect |
| 184. #486 | C23 | semidirect |
| 185. #486 | C23 | semidirect |
| 187. #482 | C23 | semidirect |
| 189. #477 | C23 | semidirect |
| 190. #476 | C23 | semidirect |
| 191. #476 | C23 | semidirect |
| 192. #474 | C23 | semidirect |
| 193. #469 | C23 | semidirect |
| 194. #468 | C23 | semidirect |
| 195. #461 | C23 | semidirect |
| 196. #460 | C23 | semidirect |
| 197. #460 | C23 | semidirect |
| 198. #459 | C23 | semidirect |
| 199. #459 | C23 | semidirect |
| 200. #459 | C23 | semidirect |
| 202. #450 | C23 | semidirect |
| 203. #449 | C23 | semidirect |
| 205. #447 | C23 | semidirect |
| 206. C8×Q8 | C23 | semidirect |
| 207. C4×SD16 | C23 | semidirect |
| 208. C4×D8 | C23 | semidirect |
| 209. #436 | C23 | semidirect |
| 211. #432 | C23 | semidirect |
| 213. #431 | C23 | semidirect |
| 214. #423 | C23 | semidirect |
| 215. #418 | C23 | semidirect |
| 217. C4×M16 | C23 | semidirect |
| 420. C2×Q8 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3498 | 2 |
| #62793 | 2 |
| #62588 | 2 |
| #62453 | 2 |
| #62001 | 2 |
| #61999 | 2 |
| #61881 | 2 |
| #61880 | 2 |
| #61582 | 2 |
| #60951 | 2 |
| #60798 | 2 |
| #60284 | 2 |