Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1567 conjugacy classes of subgroups.
The subgroup lattice has 619 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60798 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #34210 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #21416 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #17741 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #17585 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #9299 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #9295 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #7961 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #7856 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3444 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×#127 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#518 | — | 2 | 1 | no | — |
| 13 | #3217 | — | 2 | 2 | no | — |
| 14 | #3217 | — | 2 | 2 | no | — |
| 15 | #3208 | C4 | 1 | 2 | yes | complemented |
| 16 | #3208 | C22 | 1 | 2 | yes | — |
| 17 | #3205 | C4 | 1 | 2 | yes | complemented |
| 18 | #3205 | C22 | 1 | 2 | yes | — |
| 19 | #3151 | C4 | 1 | 2 | yes | complemented |
| 20 | #3138 | C4 | 1 | 2 | yes | complemented |
| 21 | #3038 | — | 2 | 2 | no | — |
| 22 | #3003 | — | 2 | 2 | no | — |
| 23 | #2920 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2920 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2920 | C22 | 1 | 1 | yes | characteristic |
| 26 | #2366 | — | 2 | 2 | no | — |
| 27 | #2366 | — | 2 | 2 | no | — |
| 28 | D4×C22⋊C4 | — | 2 | 2 | no | — |
| 29 | D4×C22⋊C4 | — | 2 | 2 | no | — |
| 30 | #2328 | C22 | 1 | 4 | yes | — |
| 31 | #2309 | — | 2 | 1 | no | — |
| 32 | C4×#127 | — | 2 | 2 | no | — |
| 33 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 34 | C4×#117 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1995 | C22 | 1 | 2 | yes | complemented |
| 36 | #1963 | C22 | 1 | 2 | yes | complemented |
| 37 | #1897 | — | 2 | 1 | no | — |
| 38 | #1897 | — | 2 | 1 | no | — |
| 39 | C4×#104 | — | 2 | 2 | no | — |
| 40 | #1555 | — | 2 | 2 | no | — |
| 41 | #1545 | — | 2 | 2 | no | — |
| 42 | #1545 | C22 | 1 | 4 | yes | — |
| 43 | #1518 | C22 | 1 | 2 | yes | complemented |
| 44 | #1517 | C22 | 1 | 2 | yes | complemented |
| 45 | #1513 | — | 2 | 2 | no | — |
| 46 | #1509 | — | 2 | 2 | no | — |
| 47 | #1286 | C22 | 1 | 4 | yes | complemented |
| 48 | #1274 | C22 | 1 | 4 | yes | complemented |
| 49 | C23×D4 | — | 2 | 1 | no | — |
| 50 | C23×D4 | — | 2 | 1 | no | — |
| 51 | C23×D4 | D4 | 1 | 2 | yes | — |
| 52 | C2×AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 53 | #533 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 54 | #533 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 55 | #533 | C23 | 1 | 1 | yes | characteristic |
| 56 | #532 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 57 | #532 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 58 | #532 | C23 | 1 | 1 | yes | characteristic |
| 59 | C2×#127 | — | 2 | 1 | no | — |
| 60 | C2×#127 | — | 2 | 1 | no | — |
| 61 | C2×#127 | — | 2 | 1 | no | — |
| 62 | C2×#127 | — | 2 | 1 | no | — |
| 63 | C2×#127 | — | 2 | 1 | no | — |
| 64 | C2×#127 | — | 2 | 1 | no | — |
| 65 | C2×#127 | C2×C4 | 1 | 2 | yes | — |
| 66 | C2×#127 | C2×C4 | 1 | 2 | yes | — |
| 67 | C2×#127 | D4 | 1 | 2 | yes | — |
| 68 | C2×#127 | D4 | 1 | 2 | yes | — |
| 69 | C2×#127 | D4 | 1 | 2 | yes | — |
| 70 | C2×#127 | C23 | 1 | 2 | yes | — |
| 71 | #529 | — | 2 | 1 | no | — |
| 72 | #529 | — | 2 | 1 | no | — |
| 73 | #529 | — | 2 | 1 | no | — |
| 74 | #529 | — | 2 | 1 | no | — |
| 75 | #529 | — | 2 | 1 | no | — |
| 76 | #529 | — | 2 | 1 | no | — |
| 77 | #525 | — | 2 | 1 | no | — |
| 78 | #525 | — | 2 | 1 | no | — |
| 79 | #525 | — | 2 | 1 | no | — |
| 80 | #518 | — | 4 | 8 | no | — |
| 81 | C4×#41 | — | 2 | 1 | no | — |
| 82 | C4×#41 | — | 2 | 1 | no | — |
| 83 | C4×#41 | — | 2 | 1 | no | — |
| 84 | C2×C4×D4 | — | 2 | 4 | no | — |
| 85 | C2×C4×D4 | — | 2 | 4 | no | — |
| 86 | C2×C4×D4 | — | 2 | 4 | no | — |
| 87 | C2×C4×D4 | — | 2 | 4 | no | — |
| 88 | C2×#117 | — | 2 | 1 | no | — |
| 89 | C2×#117 | D4 | 1 | 1 | yes | characteristic |
| 90 | C2×#117 | D4 | 1 | 1 | yes | characteristic |
| 91 | C22×C22⋊C4 | — | 2 | 4 | no | — |
| 92 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 93 | #496 | — | 2 | 4 | no | — |
| 94 | #493 | C2×C4 | 1 | 4 | yes | complemented |
| 95 | #492 | C2×C4 | 1 | 4 | yes | complemented |
| 96 | #490 | — | 2 | 4 | no | — |
| 97 | #486 | C2×C4 | 1 | 4 | yes | complemented |
| 98 | #486 | C2×C4 | 1 | 4 | yes | complemented |
| 99 | #460 | — | 2 | 4 | no | — |
| 100 | #460 | — | 2 | 4 | no | — |
| 101 | #459 | — | 2 | 4 | no | — |
| 102 | #459 | — | 2 | 4 | no | — |
| 103 | #450 | — | 2 | 4 | no | — |
| 104 | #450 | — | 2 | 4 | no | — |
| 105 | #447 | — | 2 | 4 | no | — |
| 106 | #447 | — | 2 | 4 | no | — |
| 107 | #432 | — | 2 | 1 | no | — |
| 108 | #432 | — | 2 | 1 | no | — |
| 109 | #432 | — | 2 | 1 | no | — |
| 110 | #432 | — | 2 | 1 | no | — |
| 111 | #432 | — | 2 | 1 | no | — |
| 112 | #432 | — | 2 | 1 | no | — |
| 113 | #431 | — | 2 | 1 | no | — |
| 114 | #431 | — | 2 | 1 | no | — |
| 115 | #431 | — | 2 | 1 | no | — |
| 116 | #431 | C2×C4 | 1 | 2 | yes | complemented |
| 117 | #431 | C23 | 1 | 2 | yes | — |
| 118 | #431 | C23 | 1 | 2 | yes | — |
| 119 | #423 | — | 2 | 1 | no | — |
| 120 | #423 | — | 2 | 1 | no | — |
| 121 | #423 | C23 | 1 | 2 | yes | — |
| 122 | C2×#104 | — | 2 | 2 | no | — |
| 123 | C2×#104 | — | 2 | 2 | no | — |
| 124 | #418 | — | 2 | 4 | no | — |
| 125 | #407 | — | 2 | 1 | no | — |
| 126 | #407 | — | 2 | 1 | no | — |
| 127 | #407 | — | 2 | 1 | no | — |
| 128 | C4×M16 | — | 2 | 1 | no | — |
| 129 | C4×M16 | — | 2 | 1 | no | — |
| 130 | C4×M16 | C2×C4 | 1 | 2 | yes | complemented |
| 131 | #390 | — | 2 | 4 | no | — |
| 132 | #390 | — | 2 | 4 | no | — |
| 133 | #390 | — | 2 | 4 | no | — |
| 134 | #390 | — | 2 | 4 | no | — |
| 135 | #386 | — | 2 | 8 | no | — |
| 136 | #386 | — | 2 | 8 | no | — |
| 137 | #384 | — | 2 | 1 | no | — |
| 138 | #382 | C23 | 1 | 2 | yes | — |
| 139 | #380 | — | 2 | 1 | no | — |
| 140 | #380 | D4 | 1 | 2 | yes | — |
| 141 | #379 | — | 2 | 4 | no | — |
| 142 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 143 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 144 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 145 | C4×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 146 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 147 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 148 | #376 | C23 | 1 | 2 | yes | — |
| 149 | C43 | D4 | 1 | 2 | yes | — |
| 150 | #339 | — | 2 | 2 | no | — |
| 151 | #339 | — | 2 | 2 | no | — |
| 152 | #331 | — | 2 | 8 | no | — |
| 153 | #328 | — | 2 | 4 | no | — |
| 154 | #327 | — | 2 | 4 | no | — |
| 155 | C25 | — | 2 | 2 | no | — |
| 156 | C2×#41 | — | 2 | 1 | no | — |
| 157 | C2×#41 | — | 2 | 1 | no | — |
| 158 | C2×#41 | — | 2 | 1 | no | — |
| 159 | C22×D4 | — | 2 | 2 | no | — |
| 160 | C22×D4 | — | 2 | 2 | no | — |
| 161 | C22×D4 | — | 2 | 2 | no | — |
| 162 | C22×D4 | — | 2 | 2 | no | — |
| 163 | C22×D4 | — | 2 | 2 | no | — |
| 164 | C22×D4 | — | 2 | 2 | no | — |
| 165 | C22×D4 | — | 2 | 2 | no | — |
| 166 | C22×D4 | — | 2 | 2 | no | — |
| 167 | C22×D4 | — | 2 | 2 | no | — |
| 168 | C22×D4 | — | 2 | 2 | no | — |
| 169 | C22×D4 | — | 2 | 2 | no | — |
| 170 | C22×D4 | — | 2 | 2 | no | — |
| 171 | C22×D4 | — | 2 | 2 | no | — |
| 172 | C22×D4 | — | 2 | 2 | no | — |
| 173 | C22×D4 | — | 2 | 2 | no | — |
| 174 | C22×D4 | — | 4 | 4 | no | — |
| 175 | C22×D4 | — | 4 | 4 | no | — |
| 176 | C22×D4 | — | 8 | 4 | no | — |
| 177 | C22×D4 | — | 8 | 4 | no | — |
| 178 | C22×D4 | C22⋊C4 | 1 | 4 | yes | — |
| 179 | C22×D4 | C22⋊C4 | 1 | 4 | yes | — |
| 180 | C22×D4 | C2×D4 | 1 | 4 | yes | — |
| 181 | C23×C4 | — | 2 | 8 | no | — |
| 182 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 183 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 184 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 185 | AGL(1,ℤ/8) | — | 8 | 8 | no | — |
| 186 | C2×SD16 | — | 2 | 8 | no | — |
| 187 | C2×D8 | — | 2 | 8 | no | — |
| 188 | C2×M16 | — | 2 | 2 | no | — |
| 189 | C2×M16 | — | 2 | 2 | no | — |
| 190 | C2×M16 | — | 4 | 1 | no | — |
| 191 | C4⋊Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 192 | C4⋊Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 193 | C4⋊Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 194 | #127 | — | 2 | 2 | no | — |
| 195 | #127 | — | 2 | 2 | no | — |
| 196 | #127 | — | 2 | 2 | no | — |
| 197 | #127 | — | 2 | 2 | no | — |
| 198 | #127 | — | 2 | 2 | no | — |
| 199 | #127 | — | 2 | 2 | no | — |
| 200 | #127 | — | 2 | 2 | no | — |
| 201 | #127 | — | 2 | 2 | no | — |
| 202 | #127 | — | 2 | 2 | no | — |
| 203 | #127 | — | 2 | 2 | no | — |
| 204 | #127 | — | 2 | 2 | no | — |
| 205 | #127 | — | 2 | 2 | no | — |
| 206 | #127 | C22⋊C4 | 1 | 4 | yes | — |
| 207 | #127 | C22⋊C4 | 1 | 4 | yes | — |
| 208 | #127 | C22×C4 | 1 | 1 | yes | characteristic |
| 209 | #127 | C22×C4 | 1 | 4 | yes | — |
| 210 | #127 | C2×D4 | 1 | 4 | yes | — |
| 211 | #126 | — | 4 | 1 | no | — |
| 212 | #126 | — | 4 | 1 | no | — |
| 213 | #125 | — | 2 | 1 | no | — |
| 214 | #125 | — | 2 | 1 | no | — |
| 215 | #125 | — | 2 | 1 | no | — |
| 216 | #124 | — | 2 | 1 | no | — |
| 217 | #124 | — | 2 | 1 | no | — |
| 218 | #124 | — | 2 | 1 | no | — |
| 219 | #123 | — | 4 | 1 | no | — |
| 220 | #123 | — | 4 | 1 | no | — |
| 221 | #123 | — | 4 | 1 | no | — |
| 222 | C22⋊Q8 | — | 2 | 1 | no | — |
| 223 | C22⋊Q8 | — | 2 | 1 | no | — |
| 224 | C22⋊Q8 | — | 2 | 1 | no | — |
| 225 | C22⋊Q8 | — | 2 | 1 | no | — |
| 226 | C22⋊Q8 | — | 2 | 1 | no | — |
| 227 | C22⋊Q8 | — | 2 | 1 | no | — |
| 228 | #121 | — | 2 | 1 | no | — |
| 229 | #121 | — | 2 | 1 | no | — |
| 230 | #121 | — | 2 | 1 | no | — |
| 231 | #121 | — | 2 | 1 | no | — |
| 232 | #121 | — | 2 | 1 | no | — |
| 233 | #121 | — | 2 | 1 | no | — |
| 234 | #120 | — | 4 | 1 | no | — |
| 235 | C4×Q8 | — | 2 | 1 | no | — |
| 236 | C4×Q8 | — | 2 | 1 | no | — |
| 237 | C4×Q8 | — | 2 | 1 | no | — |
| 238 | C4×Q8 | — | 2 | 1 | no | — |
| 239 | C4×Q8 | — | 2 | 1 | no | — |
| 240 | C4×Q8 | — | 2 | 1 | no | — |
| 241 | C4×D4 | — | 2 | 1 | no | — |
| 242 | C4×D4 | — | 2 | 1 | no | — |
| 243 | C4×D4 | — | 2 | 1 | no | — |
| 244 | C4×D4 | — | 2 | 1 | no | — |
| 245 | C4×D4 | — | 2 | 1 | no | — |
| 246 | C4×D4 | — | 2 | 1 | no | — |
| 247 | C4×D4 | — | 4 | 8 | no | — |
| 248 | C4×D4 | — | 4 | 16 | no | — |
| 249 | C4×D4 | — | 4 | 1 | no | — |
| 250 | C4×D4 | — | 4 | 1 | no | — |
| 251 | C4×D4 | — | 4 | 1 | no | — |
| 252 | C4×D4 | — | 4 | 1 | no | — |
| 253 | C4×D4 | — | 4 | 1 | no | — |
| 254 | C4×D4 | — | 4 | 1 | no | — |
| 255 | C4×D4 | — | 4 | 8 | no | — |
| 256 | C4×D4 | — | 8 | 8 | no | — |
| 257 | C4×D4 | — | 8 | 8 | no | — |
| 258 | #117 | — | 2 | 4 | no | — |
| 259 | #117 | — | 2 | 1 | no | — |
| 260 | #117 | — | 2 | 1 | no | — |
| 261 | #117 | — | 2 | 1 | no | — |
| 262 | #117 | — | 4 | 2 | no | — |
| 263 | #117 | — | 4 | 4 | no | — |
| 264 | #117 | — | 4 | 1 | no | — |
| 265 | #117 | — | 4 | 1 | no | — |
| 266 | #117 | — | 4 | 1 | no | — |
| 267 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 268 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 269 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 270 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 271 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 272 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 273 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 274 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 275 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 276 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 277 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 278 | C2×C22⋊C4 | — | 4 | 16 | no | — |
| 279 | C2×C42 | — | 2 | 2 | no | — |
| 280 | C2×C42 | — | 2 | 2 | no | — |
| 281 | C2×C42 | — | 2 | 2 | no | — |
| 282 | C2×C42 | — | 4 | 1 | no | — |
| 283 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 284 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 285 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 286 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 287 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 288 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 289 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 290 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 291 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 292 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 293 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 294 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 295 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 296 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 297 | C4⋊C8 | — | 2 | 2 | no | — |
| 298 | C4⋊C8 | — | 2 | 2 | no | — |
| 299 | C4⋊C8 | — | 2 | 2 | no | — |
| 300 | C4⋊C8 | — | 2 | 2 | no | — |
| 301 | C4⋊C8 | — | 2 | 2 | no | — |
| 302 | C4⋊C8 | — | 4 | 1 | no | — |
| 303 | #104 | — | 4 | 4 | no | — |
| 304 | #104 | — | 4 | 4 | no | — |
| 305 | #102 | — | 2 | 8 | no | — |
| 306 | #102 | — | 2 | 8 | no | — |
| 307 | C22⋊C8 | — | 2 | 2 | no | — |
| 308 | C22⋊C8 | — | 2 | 2 | no | — |
| 309 | C22⋊C8 | — | 2 | 2 | no | — |
| 310 | C22⋊C8 | — | 4 | 1 | no | — |
| 311 | C22⋊C8 | — | 4 | 1 | no | — |
| 312 | C22⋊C8 | — | 4 | 1 | no | — |
| 313 | #97 | — | 2 | 2 | no | — |
| 314 | #97 | — | 4 | 1 | no | — |
| 315 | #97 | — | 4 | 1 | no | — |
| 316 | C4×C8 | — | 2 | 2 | no | — |
| 317 | C4×C8 | — | 4 | 1 | no | — |
| 318 | C4×C8 | C22×C4 | 1 | 4 | yes | — |
| 319 | #95 | — | 2 | 2 | no | — |
| 320 | #95 | — | 2 | 2 | no | — |
| 321 | #95 | — | 2 | 2 | no | — |
| 322 | #95 | C2×D4 | 1 | 4 | yes | — |
| 323 | C24 | — | 2 | 4 | no | — |
| 324 | C24 | — | 2 | 4 | no | — |
| 325 | C24 | — | 2 | 4 | no | — |
| 326 | C24 | — | 4 | 4 | no | — |
| 327 | C24 | — | 4 | 4 | no | — |
| 328 | C24 | — | 8 | 4 | no | — |
| 329 | C24 | — | 8 | 4 | no | — |
| 330 | C24 | #120 | 1 | 1 | yes | characteristic |
| 331 | #41 | — | 4 | 4 | no | — |
| 332 | #41 | — | 4 | 2 | no | — |
| 333 | #41 | — | 4 | 2 | no | — |
| 334 | #41 | — | 8 | 2 | no | — |
| 335 | C2×Q8 | — | 2 | 1 | no | — |
| 336 | C2×Q8 | — | 2 | 1 | no | — |
| 337 | C2×Q8 | — | 2 | 1 | no | — |
| 338 | C2×D4 | — | 2 | 1 | no | — |
| 339 | C2×D4 | — | 2 | 4 | no | — |
| 340 | C2×D4 | — | 2 | 1 | no | — |
| 341 | C2×D4 | — | 2 | 4 | no | — |
| 342 | C2×D4 | — | 2 | 1 | no | — |
| 343 | C2×D4 | — | 2 | 4 | no | — |
| 344 | C2×D4 | — | 2 | 4 | no | — |
| 345 | C2×D4 | — | 2 | 4 | no | — |
| 346 | C2×D4 | — | 2 | 4 | no | — |
| 347 | C2×D4 | — | 2 | 4 | no | — |
| 348 | C2×D4 | — | 2 | 4 | no | — |
| 349 | C2×D4 | — | 2 | 4 | no | — |
| 350 | C2×D4 | — | 2 | 4 | no | — |
| 351 | C2×D4 | — | 2 | 4 | no | — |
| 352 | C2×D4 | — | 4 | 4 | no | — |
| 353 | C2×D4 | — | 4 | 8 | no | — |
| 354 | C2×D4 | — | 4 | 8 | no | — |
| 355 | C2×D4 | — | 4 | 4 | no | — |
| 356 | C2×D4 | — | 4 | 8 | no | — |
| 357 | C2×D4 | — | 4 | 8 | no | — |
| 358 | C2×D4 | — | 4 | 8 | no | — |
| 359 | C2×D4 | — | 4 | 8 | no | — |
| 360 | C2×D4 | — | 4 | 8 | no | — |
| 361 | C2×D4 | — | 4 | 1 | no | — |
| 362 | C2×D4 | — | 4 | 1 | no | — |
| 363 | C2×D4 | — | 4 | 1 | no | — |
| 364 | C2×D4 | — | 4 | 4 | no | — |
| 365 | C2×D4 | — | 4 | 4 | no | — |
| 366 | C2×D4 | — | 8 | 4 | no | — |
| 367 | C2×D4 | — | 8 | 4 | no | — |
| 368 | C2×D4 | — | 8 | 4 | no | — |
| 369 | C2×D4 | — | 8 | 4 | no | — |
| 370 | C2×D4 | — | 8 | 4 | no | — |
| 371 | C2×D4 | — | 8 | 4 | no | — |
| 372 | C2×D4 | — | 8 | 4 | no | — |
| 373 | C2×D4 | — | 8 | 4 | no | — |
| 374 | C2×D4 | — | 8 | 4 | no | — |
| 375 | C2×D4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 376 | C22×C4 | — | 2 | 2 | no | — |
| 377 | C22×C4 | — | 2 | 2 | no | — |
| 378 | C22×C4 | — | 2 | 2 | no | — |
| 379 | C22×C4 | — | 2 | 2 | no | — |
| 380 | C22×C4 | — | 2 | 4 | no | — |
| 381 | C22×C4 | — | 4 | 2 | no | — |
| 382 | C22×C4 | — | 4 | 2 | no | — |
| 383 | C22×C4 | — | 4 | 16 | no | — |
| 384 | C22×C4 | — | 4 | 1 | no | — |
| 385 | C22×C4 | — | 4 | 1 | no | — |
| 386 | C22×C4 | — | 4 | 1 | no | — |
| 387 | C22×C4 | — | 8 | 8 | no | — |
| 388 | C22×C4 | — | 8 | 8 | no | — |
| 389 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 390 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 391 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 392 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 393 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 394 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 395 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 396 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 397 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 398 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 399 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 400 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 401 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 402 | C22×C4 | #123 | 1 | 1 | yes | characteristic |
| 403 | C22×C4 | #123 | 1 | 1 | yes | characteristic |
| 404 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 405 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 406 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 407 | SD16 | — | 8 | 8 | no | — |
| 408 | D8 | — | 8 | 8 | no | — |
| 409 | M16 | — | 4 | 4 | no | — |
| 410 | M16 | — | 4 | 2 | no | — |
| 411 | M16 | — | 4 | 2 | no | — |
| 412 | M16 | — | 8 | 2 | no | — |
| 413 | C2×C8 | — | 2 | 4 | no | — |
| 414 | C2×C8 | — | 4 | 2 | no | — |
| 415 | C2×C8 | — | 4 | 2 | no | — |
| 416 | C4⋊C4 | — | 2 | 4 | no | — |
| 417 | C4⋊C4 | — | 2 | 1 | no | — |
| 418 | C4⋊C4 | — | 2 | 1 | no | — |
| 419 | C4⋊C4 | — | 2 | 1 | no | — |
| 420 | C4⋊C4 | — | 2 | 1 | no | — |
| 421 | C4⋊C4 | — | 2 | 1 | no | — |
| 422 | C4⋊C4 | — | 2 | 1 | no | — |
| 423 | C4⋊C4 | — | 4 | 2 | no | — |
| 424 | C4⋊C4 | — | 4 | 4 | no | — |
| 425 | C4⋊C4 | — | 4 | 4 | no | — |
| 426 | C4⋊C4 | — | 4 | 1 | no | — |
| 427 | C4⋊C4 | — | 4 | 1 | no | — |
| 428 | C4⋊C4 | — | 4 | 4 | no | — |
| 429 | C4⋊C4 | — | 4 | 1 | no | — |
| 430 | C4⋊C4 | — | 4 | 1 | no | — |
| 431 | C4⋊C4 | — | 4 | 1 | no | — |
| 432 | C4⋊C4 | — | 4 | 1 | no | — |
| 433 | C22⋊C4 | — | 2 | 4 | no | — |
| 434 | C22⋊C4 | — | 4 | 16 | no | — |
| 435 | C22⋊C4 | — | 4 | 4 | no | — |
| 436 | C22⋊C4 | — | 4 | 2 | no | — |
| 437 | C22⋊C4 | — | 4 | 1 | no | — |
| 438 | C22⋊C4 | — | 4 | 1 | no | — |
| 439 | C22⋊C4 | — | 4 | 1 | no | — |
| 440 | C22⋊C4 | — | 4 | 1 | no | — |
| 441 | C22⋊C4 | — | 4 | 1 | no | — |
| 442 | C22⋊C4 | — | 4 | 1 | no | — |
| 443 | C22⋊C4 | — | 4 | 1 | no | — |
| 444 | C22⋊C4 | — | 4 | 1 | no | — |
| 445 | C22⋊C4 | — | 4 | 1 | no | — |
| 446 | C22⋊C4 | — | 8 | 8 | no | — |
| 447 | C22⋊C4 | — | 8 | 8 | no | — |
| 448 | C42 | — | 2 | 2 | no | — |
| 449 | C42 | — | 2 | 2 | no | — |
| 450 | C42 | — | 2 | 2 | no | — |
| 451 | C42 | — | 4 | 4 | no | — |
| 452 | C42 | — | 4 | 4 | no | — |
| 453 | C42 | — | 4 | 4 | no | — |
| 454 | C42 | — | 4 | 1 | no | — |
| 455 | C42 | — | 4 | 1 | no | — |
| 456 | C42 | — | 4 | 1 | no | — |
| 457 | C42 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 458 | C42 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 459 | C42 | C4×D4 | 1 | 1 | yes | characteristic |
| 460 | C42 | C4×D4 | 1 | 1 | yes | characteristic |
| 461 | C42 | C4×D4 | 1 | 1 | yes | characteristic |
| 462 | C42 | C4×D4 | 1 | 1 | yes | characteristic |
| 463 | C42 | #121 | 1 | 1 | yes | characteristic |
| 464 | C42 | #121 | 1 | 1 | yes | characteristic |
| 465 | C42 | #121 | 1 | 1 | yes | characteristic |
| 466 | C42 | #121 | 1 | 1 | yes | characteristic |
| 467 | C42 | #123 | 1 | 1 | yes | characteristic |
| 468 | C42 | #123 | 1 | 1 | yes | characteristic |
| 469 | C42 | AGL(1,ℤ/8) | 1 | 4 | yes | complemented |
| 470 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 471 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 472 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 473 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 474 | C23 | — | 2 | 1 | no | — |
| 475 | C23 | — | 2 | 1 | no | — |
| 476 | C23 | — | 2 | 8 | no | — |
| 477 | C23 | — | 4 | 1 | no | — |
| 478 | C23 | — | 4 | 1 | no | — |
| 479 | C23 | — | 4 | 8 | no | — |
| 480 | C23 | — | 4 | 8 | no | — |
| 481 | C23 | — | 4 | 8 | no | — |
| 482 | C23 | — | 4 | 8 | no | — |
| 483 | C23 | — | 4 | 4 | no | — |
| 484 | C23 | — | 4 | 4 | no | — |
| 485 | C23 | — | 4 | 1 | no | — |
| 486 | C23 | — | 8 | 4 | no | — |
| 487 | C23 | — | 8 | 4 | no | — |
| 488 | C23 | — | 8 | 4 | no | — |
| 489 | C23 | — | 8 | 4 | no | — |
| 490 | C23 | — | 16 | 8 | no | — |
| 491 | C23 | — | 16 | 8 | no | — |
| 492 | C23 | #386 | 1 | 1 | yes | characteristic |
| 493 | C23 | #386 | 1 | 1 | yes | characteristic |
| 494 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 495 | Q8 | — | 4 | 2 | no | — |
| 496 | Q8 | — | 4 | 2 | no | — |
| 497 | Q8 | — | 8 | 1 | no | — |
| 498 | D4 | — | 4 | 8 | no | — |
| 499 | D4 | — | 4 | 8 | no | — |
| 500 | D4 | — | 4 | 8 | no | — |
| 501 | D4 | — | 4 | 8 | no | — |
| 502 | D4 | — | 4 | 8 | no | — |
| 503 | D4 | — | 4 | 2 | no | — |
| 504 | D4 | — | 4 | 8 | no | — |
| 505 | D4 | — | 4 | 8 | no | — |
| 506 | D4 | — | 4 | 8 | no | — |
| 507 | D4 | — | 4 | 2 | no | — |
| 508 | D4 | — | 4 | 8 | no | — |
| 509 | D4 | — | 8 | 4 | no | — |
| 510 | D4 | — | 8 | 4 | no | — |
| 511 | D4 | — | 8 | 4 | no | — |
| 512 | D4 | — | 8 | 1 | no | — |
| 513 | D4 | — | 8 | 2 | no | — |
| 514 | D4 | — | 8 | 2 | no | — |
| 515 | D4 | — | 8 | 1 | no | — |
| 516 | D4 | — | 8 | 1 | no | — |
| 517 | C2×C4 | — | 2 | 1 | no | — |
| 518 | C2×C4 | — | 2 | 1 | no | — |
| 519 | C2×C4 | — | 2 | 2 | no | — |
| 520 | C2×C4 | — | 2 | 2 | no | — |
| 521 | C2×C4 | — | 2 | 2 | no | — |
| 522 | C2×C4 | — | 2 | 2 | no | — |
| 523 | C2×C4 | — | 2 | 2 | no | — |
| 524 | C2×C4 | — | 2 | 2 | no | — |
| 525 | C2×C4 | — | 2 | 2 | no | — |
| 526 | C2×C4 | — | 2 | 2 | no | — |
| 527 | C2×C4 | — | 2 | 2 | no | — |
| 528 | C2×C4 | — | 2 | 2 | no | — |
| 529 | C2×C4 | — | 2 | 4 | no | — |
| 530 | C2×C4 | — | 4 | 1 | no | — |
| 531 | C2×C4 | — | 4 | 1 | no | — |
| 532 | C2×C4 | — | 4 | 1 | no | — |
| 533 | C2×C4 | — | 4 | 1 | no | — |
| 534 | C2×C4 | — | 4 | 1 | no | — |
| 535 | C2×C4 | — | 4 | 1 | no | — |
| 536 | C2×C4 | — | 4 | 2 | no | — |
| 537 | C2×C4 | — | 4 | 2 | no | — |
| 538 | C2×C4 | — | 4 | 2 | no | — |
| 539 | C2×C4 | — | 4 | 2 | no | — |
| 540 | C2×C4 | — | 4 | 4 | no | — |
| 541 | C2×C4 | — | 4 | 4 | no | — |
| 542 | C2×C4 | — | 4 | 4 | no | — |
| 543 | C2×C4 | — | 4 | 2 | no | — |
| 544 | C2×C4 | — | 4 | 1 | no | — |
| 545 | C2×C4 | — | 4 | 1 | no | — |
| 546 | C2×C4 | — | 4 | 1 | no | — |
| 547 | C2×C4 | — | 8 | 2 | no | — |
| 548 | C2×C4 | — | 8 | 2 | no | — |
| 549 | C2×C4 | — | 8 | 2 | no | — |
| 550 | C2×C4 | — | 16 | 16 | no | — |
| 551 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 552 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 553 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 554 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 555 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 556 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 557 | C2×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 558 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 559 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 560 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 561 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 562 | C2×C4 | C2×#123 | 1 | 1 | yes | characteristic |
| 563 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 564 | C2×C4 | #575 | 1 | 2 | yes | — |
| 565 | C8 | — | 4 | 4 | no | — |
| 566 | C8 | — | 8 | 2 | no | — |
| 567 | C8 | — | 8 | 2 | no | — |
| 568 | C22 | — | 2 | 1 | no | — |
| 569 | C22 | — | 2 | 1 | no | — |
| 570 | C22 | — | 2 | 1 | no | — |
| 571 | C22 | — | 2 | 1 | no | — |
| 572 | C22 | — | 2 | 1 | no | — |
| 573 | C22 | — | 2 | 1 | no | — |
| 574 | C22 | — | 2 | 1 | no | — |
| 575 | C22 | — | 2 | 1 | no | — |
| 576 | C22 | — | 4 | 2 | no | — |
| 577 | C22 | — | 4 | 2 | no | — |
| 578 | C22 | — | 4 | 8 | no | — |
| 579 | C22 | — | 4 | 8 | no | — |
| 580 | C22 | — | 4 | 8 | no | — |
| 581 | C22 | — | 8 | 8 | no | — |
| 582 | C22 | — | 8 | 8 | no | — |
| 583 | C22 | — | 8 | 1 | no | — |
| 584 | C22 | — | 8 | 1 | no | — |
| 585 | C22 | — | 8 | 1 | no | — |
| 586 | C22 | — | 16 | 8 | no | — |
| 587 | C22 | — | 16 | 8 | no | — |
| 588 | C22 | C2×#386 | 1 | 1 | yes | characteristic |
| 589 | C22 | #3003 | 1 | 1 | yes | characteristic |
| 590 | C22 | #3005 | 1 | 1 | yes | characteristic |
| 591 | C4 | — | 2 | 1 | no | — |
| 592 | C4 | — | 2 | 1 | no | — |
| 593 | C4 | — | 2 | 1 | no | — |
| 594 | C4 | — | 2 | 1 | no | — |
| 595 | C4 | — | 2 | 1 | no | — |
| 596 | C4 | — | 2 | 1 | no | — |
| 597 | C4 | — | 2 | 1 | no | — |
| 598 | C4 | — | 2 | 1 | no | — |
| 599 | C4 | — | 4 | 4 | no | — |
| 600 | C4 | — | 8 | 2 | no | — |
| 601 | C4 | — | 8 | 4 | no | — |
| 602 | C4 | — | 8 | 1 | no | — |
| 603 | C4 | — | 8 | 1 | no | — |
| 604 | C4 | — | 8 | 1 | no | — |
| 605 | C4 | #1904 | 1 | 2 | yes | — |
| 606 | C4 | #1911 | 1 | 2 | yes | — |
| 607 | C4 | C4×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 608 | C4 | #3036 | 1 | 2 | yes | — |
| 609 | C2 | — | 2 | 1 | no | — |
| 610 | C2 | — | 2 | 1 | no | — |
| 611 | C2 | — | 4 | 1 | no | — |
| 612 | C2 | — | 4 | 1 | no | — |
| 613 | C2 | — | 8 | 8 | no | — |
| 614 | C2 | — | 16 | 1 | no | — |
| 615 | C2 | #20925 | 1 | 1 | yes | characteristic |
| 616 | C2 | C2×#1904 | 1 | 1 | yes | characteristic |
| 617 | C2 | C2×#1911 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #34210 | C2 | semidirect |
| 3. #21416 | C2 | semidirect |
| 4. #17741 | C2 | semidirect |
| 5. #17585 | C2 | semidirect |
| 6. #9299 | C2 | semidirect |
| 7. #9295 | C2 | semidirect |
| 8. #7961 | C2 | semidirect |
| 9. #7856 | C2 | semidirect |
| 15. #3208 | C4 | semidirect |
| 17. #3205 | C4 | semidirect |
| 19. #3151 | C4 | semidirect |
| 20. #3138 | C4 | semidirect |
| 35. #1995 | C22 | semidirect |
| 36. #1963 | C22 | semidirect |
| 43. #1518 | C22 | semidirect |
| 44. #1517 | C22 | semidirect |
| 47. #1286 | C22 | semidirect |
| 48. #1274 | C22 | semidirect |
| 53. #533 | C2×C4 | semidirect |
| 54. #533 | C2×C4 | semidirect |
| 56. #532 | C2×C4 | semidirect |
| 57. #532 | C2×C4 | semidirect |
| 94. #493 | C2×C4 | semidirect |
| 95. #492 | C2×C4 | semidirect |
| 97. #486 | C2×C4 | semidirect |
| 98. #486 | C2×C4 | semidirect |
| 116. #431 | C2×C4 | semidirect |
| 130. C4×M16 | C2×C4 | semidirect |
| 469. C42 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60798 | 2 |
| #34210 | 2 |
| #21416 | 2 |
| #17741 | 2 |
| #17585 | 2 |
| #9299 | 2 |
| #9295 | 2 |
| #7961 | 2 |
| #7856 | 2 |