Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
751 conjugacy classes of subgroups.
The subgroup lattice has 451 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62417 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33815 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33774 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19126 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #12385 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12373 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12333 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12321 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C2×#1361 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #11646 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #11621 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3508 | — | 2 | 1 | no | — |
| 13 | #3396 | — | 2 | 1 | no | — |
| 14 | #3396 | C22 | 1 | 2 | yes | — |
| 15 | #3287 | — | 2 | 1 | no | — |
| 16 | #3287 | C22 | 1 | 2 | yes | — |
| 17 | #3280 | — | 2 | 2 | no | — |
| 18 | #3280 | — | 2 | 2 | no | — |
| 19 | #3208 | — | 2 | 1 | no | — |
| 20 | #3207 | — | 2 | 2 | no | — |
| 21 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 22 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 23 | #3169 | — | 2 | 1 | no | — |
| 24 | #3147 | — | 2 | 1 | no | — |
| 25 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 26 | #3076 | — | 2 | 1 | no | — |
| 27 | C8×#41 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2248 | C22 | 1 | 2 | yes | — |
| 29 | #2240 | C22 | 1 | 2 | yes | — |
| 30 | #2227 | — | 2 | 1 | no | — |
| 31 | #2227 | C22 | 1 | 2 | yes | complemented |
| 32 | #2216 | — | 2 | 2 | no | — |
| 33 | #2209 | — | 2 | 1 | no | — |
| 34 | #2209 | C22 | 1 | 2 | yes | complemented |
| 35 | #2207 | — | 2 | 2 | no | — |
| 36 | #2187 | — | 2 | 2 | no | — |
| 37 | #2162 | C22 | 1 | 1 | yes | characteristic |
| 38 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1684 | — | 2 | 1 | no | — |
| 40 | #1683 | — | 2 | 1 | no | — |
| 41 | #1676 | C22 | 1 | 1 | yes | characteristic, complemented |
| 42 | #1675 | C22 | 1 | 1 | yes | characteristic, complemented |
| 43 | #1670 | C22 | 1 | 1 | yes | characteristic, complemented |
| 44 | #1669 | C22 | 1 | 1 | yes | characteristic, complemented |
| 45 | C2×#335 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1542 | — | 2 | 1 | no | — |
| 47 | #1361 | C22 | 1 | 2 | yes | — |
| 48 | #1361 | C22 | 1 | 2 | yes | — |
| 49 | Q8⋊C16 | C22 | 1 | 2 | yes | complemented |
| 50 | #1332 | C22 | 1 | 2 | yes | complemented |
| 51 | #1332 | C22 | 1 | 2 | yes | complemented |
| 52 | D4⋊C16 | C22 | 1 | 2 | yes | complemented |
| 53 | C2×#135 | — | 2 | 2 | no | — |
| 54 | C2×#135 | — | 2 | 2 | no | — |
| 55 | C22×D8 | — | 2 | 1 | no | — |
| 56 | C22×D8 | D4 | 1 | 2 | yes | — |
| 57 | #559 | — | 2 | 2 | no | — |
| 58 | #559 | — | 4 | 1 | no | — |
| 59 | #550 | — | 2 | 2 | no | — |
| 60 | #550 | — | 4 | 1 | no | — |
| 61 | #546 | — | 2 | 4 | no | — |
| 62 | #546 | — | 4 | 2 | no | — |
| 63 | #540 | — | 4 | 2 | no | — |
| 64 | #535 | — | 4 | 1 | no | — |
| 65 | #533 | — | 2 | 1 | no | — |
| 66 | #533 | — | 2 | 1 | no | — |
| 67 | #532 | — | 2 | 2 | no | — |
| 68 | #532 | — | 2 | 2 | no | — |
| 69 | C2×#127 | — | 2 | 1 | no | — |
| 70 | #518 | — | 4 | 1 | no | — |
| 71 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 72 | C2×Q32 | — | 2 | 2 | no | — |
| 73 | C2×SD32 | — | 2 | 4 | no | — |
| 74 | C22×C16 | — | 2 | 1 | no | — |
| 75 | #500 | D4 | 1 | 1 | yes | characteristic |
| 76 | #500 | C23 | 1 | 1 | yes | characteristic |
| 77 | #495 | — | 2 | 2 | no | — |
| 78 | #495 | — | 2 | 2 | no | — |
| 79 | #494 | D4 | 1 | 1 | yes | characteristic |
| 80 | #494 | C23 | 1 | 1 | yes | characteristic |
| 81 | #493 | — | 2 | 1 | no | — |
| 82 | #493 | — | 2 | 1 | no | — |
| 83 | #493 | D4 | 1 | 2 | yes | — |
| 84 | #493 | C23 | 1 | 2 | yes | — |
| 85 | #492 | — | 2 | 2 | no | — |
| 86 | #487 | — | 2 | 1 | no | — |
| 87 | #487 | — | 2 | 1 | no | — |
| 88 | #486 | — | 2 | 2 | no | — |
| 89 | #479 | — | 4 | 1 | no | — |
| 90 | #478 | — | 4 | 1 | no | — |
| 91 | #477 | — | 2 | 1 | no | — |
| 92 | #476 | — | 2 | 1 | no | — |
| 93 | #475 | — | 2 | 1 | no | — |
| 94 | #474 | — | 2 | 1 | no | — |
| 95 | #467 | — | 2 | 1 | no | — |
| 96 | #467 | — | 2 | 1 | no | — |
| 97 | #467 | — | 2 | 1 | no | — |
| 98 | #467 | D4 | 1 | 2 | yes | — |
| 99 | #466 | — | 2 | 2 | no | — |
| 100 | #466 | — | 2 | 2 | no | — |
| 101 | #463 | — | 2 | 4 | no | — |
| 102 | #463 | — | 2 | 4 | no | — |
| 103 | #460 | — | 2 | 2 | no | — |
| 104 | #460 | — | 2 | 2 | no | — |
| 105 | #450 | — | 2 | 4 | no | — |
| 106 | #449 | — | 2 | 4 | no | — |
| 107 | #449 | — | 2 | 4 | no | — |
| 108 | C8×Q8 | D4 | 1 | 1 | yes | characteristic |
| 109 | C8×Q8 | C23 | 1 | 1 | yes | characteristic |
| 110 | C4×Q16 | — | 2 | 1 | no | — |
| 111 | C4×Q16 | — | 2 | 1 | no | — |
| 112 | C4×Q16 | — | 2 | 1 | no | — |
| 113 | C4×Q16 | C23 | 1 | 2 | yes | — |
| 114 | C4×D8 | — | 2 | 2 | no | — |
| 115 | C4×D8 | — | 2 | 2 | no | — |
| 116 | C8×D4 | — | 2 | 1 | no | — |
| 117 | C8×D4 | — | 2 | 1 | no | — |
| 118 | C8×D4 | D4 | 1 | 1 | yes | characteristic |
| 119 | C8×D4 | C23 | 1 | 1 | yes | characteristic |
| 120 | #431 | — | 2 | 1 | no | — |
| 121 | #431 | D4 | 1 | 1 | yes | characteristic |
| 122 | #427 | — | 4 | 1 | no | — |
| 123 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 124 | #416 | — | 4 | 1 | no | — |
| 125 | C2×C4×C8 | — | 2 | 1 | no | — |
| 126 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 127 | #366 | — | 2 | 2 | no | — |
| 128 | C4⋊C16 | — | 2 | 1 | no | — |
| 129 | C4⋊C16 | — | 2 | 1 | no | — |
| 130 | C4⋊C16 | C23 | 1 | 2 | yes | — |
| 131 | #358 | — | 2 | 2 | no | — |
| 132 | #358 | — | 2 | 2 | no | — |
| 133 | #357 | — | 2 | 4 | no | — |
| 134 | C22⋊C16 | — | 2 | 1 | no | — |
| 135 | C4×C16 | C23 | 1 | 2 | yes | — |
| 136 | #335 | — | 2 | 1 | no | — |
| 137 | #335 | C23 | 1 | 1 | yes | characteristic |
| 138 | #335 | C23 | 1 | 1 | yes | characteristic |
| 139 | #333 | — | 2 | 1 | no | — |
| 140 | #333 | — | 2 | 1 | no | — |
| 141 | #332 | — | 2 | 1 | no | — |
| 142 | #332 | — | 2 | 1 | no | — |
| 143 | C2×#41 | — | 2 | 4 | no | — |
| 144 | C2×#41 | — | 4 | 2 | no | — |
| 145 | C2×#41 | D8 | 1 | 1 | yes | characteristic |
| 146 | C22×D4 | — | 2 | 2 | no | — |
| 147 | C22×D4 | — | 4 | 1 | no | — |
| 148 | C22×D4 | — | 4 | 1 | no | — |
| 149 | #135 | — | 8 | 2 | no | — |
| 150 | #135 | — | 8 | 2 | no | — |
| 151 | #135 | — | 8 | 4 | no | — |
| 152 | C2×Q16 | — | 2 | 1 | no | — |
| 153 | C2×Q16 | — | 2 | 1 | no | — |
| 154 | C2×Q16 | — | 2 | 1 | no | — |
| 155 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 156 | C2×SD16 | — | 2 | 4 | no | — |
| 157 | C2×SD16 | — | 2 | 4 | no | — |
| 158 | C2×D8 | — | 2 | 2 | no | — |
| 159 | C2×D8 | — | 2 | 2 | no | — |
| 160 | C2×D8 | — | 2 | 2 | no | — |
| 161 | C2×D8 | — | 8 | 1 | no | — |
| 162 | C2×D8 | — | 8 | 1 | no | — |
| 163 | C2×D8 | — | 8 | 2 | no | — |
| 164 | C2×D8 | C2×D4 | 1 | 4 | yes | — |
| 165 | C22×C8 | — | 2 | 1 | no | — |
| 166 | C22×C8 | — | 2 | 1 | no | — |
| 167 | C22×C8 | — | 4 | 1 | no | — |
| 168 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 170 | C4⋊Q8 | — | 2 | 1 | no | — |
| 171 | C4⋊Q8 | — | 2 | 1 | no | — |
| 172 | C4⋊Q8 | — | 2 | 1 | no | — |
| 173 | C4⋊Q8 | — | 2 | 1 | no | — |
| 174 | #127 | — | 2 | 2 | no | — |
| 175 | #127 | — | 2 | 2 | no | — |
| 176 | #127 | — | 4 | 2 | no | — |
| 177 | #127 | — | 4 | 1 | no | — |
| 178 | #127 | — | 4 | 1 | no | — |
| 179 | #125 | — | 4 | 1 | no | — |
| 180 | #125 | — | 4 | 1 | no | — |
| 181 | #124 | — | 2 | 4 | no | — |
| 182 | #124 | — | 4 | 2 | no | — |
| 183 | #124 | — | 4 | 2 | no | — |
| 184 | #123 | — | 4 | 4 | no | — |
| 185 | #123 | — | 4 | 4 | no | — |
| 186 | C22⋊Q8 | — | 2 | 2 | no | — |
| 187 | C22⋊Q8 | — | 4 | 1 | no | — |
| 188 | C22⋊Q8 | — | 4 | 1 | no | — |
| 189 | C22⋊Q8 | — | 4 | 1 | no | — |
| 190 | #121 | — | 2 | 4 | no | — |
| 191 | #121 | — | 4 | 4 | no | — |
| 192 | #121 | — | 4 | 4 | no | — |
| 193 | #121 | — | 4 | 2 | no | — |
| 194 | #121 | — | 4 | 2 | no | — |
| 195 | #121 | — | 4 | 2 | no | — |
| 196 | #120 | — | 4 | 4 | no | — |
| 197 | C4×Q8 | — | 2 | 2 | no | — |
| 198 | C4×Q8 | — | 4 | 1 | no | — |
| 199 | C4×Q8 | — | 4 | 1 | no | — |
| 200 | C4×Q8 | — | 4 | 1 | no | — |
| 201 | C4×Q8 | D8 | 1 | 1 | yes | characteristic |
| 202 | C4×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 203 | C4×D4 | — | 2 | 1 | no | — |
| 204 | C4×D4 | — | 2 | 1 | no | — |
| 205 | C4×D4 | — | 2 | 4 | no | — |
| 206 | C4×D4 | — | 4 | 2 | no | — |
| 207 | C4×D4 | — | 4 | 2 | no | — |
| 208 | C4×D4 | — | 4 | 2 | no | — |
| 209 | C4×D4 | — | 4 | 2 | no | — |
| 210 | C4×D4 | — | 4 | 2 | no | — |
| 211 | C4×D4 | D8 | 1 | 1 | yes | characteristic |
| 212 | C4×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 213 | #117 | — | 2 | 1 | no | — |
| 214 | #117 | — | 4 | 1 | no | — |
| 215 | #117 | — | 4 | 1 | no | — |
| 216 | #117 | D8 | 1 | 1 | yes | characteristic |
| 217 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 218 | C2×C42 | — | 2 | 1 | no | — |
| 219 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 220 | Q32 | — | 8 | 2 | no | — |
| 221 | SD32 | — | 8 | 4 | no | — |
| 222 | C2×C16 | — | 2 | 2 | no | — |
| 223 | C2×C16 | — | 2 | 2 | no | — |
| 224 | C2×C16 | — | 8 | 1 | no | — |
| 225 | #107 | — | 2 | 1 | no | — |
| 226 | #107 | — | 2 | 1 | no | — |
| 227 | #107 | — | 2 | 1 | no | — |
| 228 | #107 | — | 4 | 1 | no | — |
| 229 | #107 | — | 4 | 1 | no | — |
| 230 | #107 | C2×D4 | 1 | 2 | yes | — |
| 231 | #106 | — | 4 | 1 | no | — |
| 232 | #106 | — | 4 | 1 | no | — |
| 233 | C4⋊C8 | — | 2 | 1 | no | — |
| 234 | C4⋊C8 | — | 2 | 1 | no | — |
| 235 | C4⋊C8 | — | 2 | 1 | no | — |
| 236 | C4⋊C8 | — | 2 | 1 | no | — |
| 237 | C4⋊C8 | — | 4 | 1 | no | — |
| 238 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 239 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 240 | #103 | — | 2 | 2 | no | — |
| 241 | #103 | — | 2 | 2 | no | — |
| 242 | #103 | — | 4 | 1 | no | — |
| 243 | #103 | — | 4 | 1 | no | — |
| 244 | #103 | — | 4 | 1 | no | — |
| 245 | #103 | — | 4 | 1 | no | — |
| 246 | #102 | — | 2 | 4 | no | — |
| 247 | #102 | — | 2 | 4 | no | — |
| 248 | #102 | — | 4 | 1 | no | — |
| 249 | #102 | — | 4 | 1 | no | — |
| 250 | C22⋊C8 | — | 2 | 1 | no | — |
| 251 | C22⋊C8 | — | 2 | 1 | no | — |
| 252 | C22⋊C8 | — | 2 | 1 | no | — |
| 253 | C4×C8 | — | 2 | 1 | no | — |
| 254 | C4×C8 | — | 2 | 1 | no | — |
| 255 | C4×C8 | — | 2 | 1 | no | — |
| 256 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 257 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 258 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 259 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 260 | C24 | — | 4 | 2 | no | — |
| 261 | #41 | — | 2 | 2 | no | — |
| 262 | #41 | — | 2 | 2 | no | — |
| 263 | #41 | — | 8 | 4 | no | — |
| 264 | #41 | — | 8 | 4 | no | — |
| 265 | #41 | — | 16 | 4 | no | — |
| 266 | C2×Q8 | — | 2 | 2 | no | — |
| 267 | C2×Q8 | — | 4 | 1 | no | — |
| 268 | C2×Q8 | — | 4 | 1 | no | — |
| 269 | C2×Q8 | — | 4 | 1 | no | — |
| 270 | C2×Q8 | C2×D8 | 1 | 1 | yes | characteristic |
| 271 | C2×D4 | — | 2 | 1 | no | — |
| 272 | C2×D4 | — | 2 | 4 | no | — |
| 273 | C2×D4 | — | 4 | 4 | no | — |
| 274 | C2×D4 | — | 4 | 4 | no | — |
| 275 | C2×D4 | — | 4 | 2 | no | — |
| 276 | C2×D4 | — | 4 | 2 | no | — |
| 277 | C2×D4 | — | 4 | 2 | no | — |
| 278 | C2×D4 | — | 4 | 2 | no | — |
| 279 | C2×D4 | — | 4 | 2 | no | — |
| 280 | C2×D4 | — | 8 | 2 | no | — |
| 281 | C2×D4 | — | 8 | 2 | no | — |
| 282 | C2×D4 | — | 8 | 2 | no | — |
| 283 | C2×D4 | — | 8 | 2 | no | — |
| 284 | C2×D4 | — | 16 | 2 | no | — |
| 285 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 286 | C22×C4 | — | 2 | 1 | no | — |
| 287 | C22×C4 | — | 2 | 1 | no | — |
| 288 | C22×C4 | — | 2 | 1 | no | — |
| 289 | C22×C4 | — | 4 | 4 | no | — |
| 290 | C22×C4 | — | 4 | 4 | no | — |
| 291 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 292 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 293 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 294 | Q16 | — | 2 | 2 | no | — |
| 295 | Q16 | — | 2 | 2 | no | — |
| 296 | Q16 | — | 4 | 2 | no | — |
| 297 | Q16 | — | 4 | 2 | no | — |
| 298 | Q16 | — | 4 | 1 | no | — |
| 299 | Q16 | — | 4 | 1 | no | — |
| 300 | SD16 | — | 8 | 4 | no | — |
| 301 | SD16 | — | 8 | 4 | no | — |
| 302 | D8 | — | 4 | 4 | no | — |
| 303 | D8 | — | 4 | 4 | no | — |
| 304 | D8 | — | 4 | 2 | no | — |
| 305 | D8 | — | 4 | 2 | no | — |
| 306 | D8 | — | 8 | 2 | no | — |
| 307 | C2×C8 | — | 2 | 1 | no | — |
| 308 | C2×C8 | — | 2 | 1 | no | — |
| 309 | C2×C8 | — | 2 | 1 | no | — |
| 310 | C2×C8 | — | 2 | 1 | no | — |
| 311 | C2×C8 | — | 4 | 1 | no | — |
| 312 | C2×C8 | — | 4 | 2 | no | — |
| 313 | C2×C8 | — | 4 | 1 | no | — |
| 314 | C2×C8 | — | 4 | 1 | no | — |
| 315 | C2×C8 | — | 4 | 1 | no | — |
| 316 | C2×C8 | — | 4 | 1 | no | — |
| 317 | C2×C8 | — | 8 | 2 | no | — |
| 318 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 319 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 320 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 321 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 322 | C4⋊C4 | — | 2 | 1 | no | — |
| 323 | C4⋊C4 | — | 2 | 1 | no | — |
| 324 | C4⋊C4 | — | 2 | 2 | no | — |
| 325 | C4⋊C4 | — | 4 | 2 | no | — |
| 326 | C4⋊C4 | — | 4 | 1 | no | — |
| 327 | C4⋊C4 | — | 4 | 1 | no | — |
| 328 | C4⋊C4 | — | 4 | 1 | no | — |
| 329 | C4⋊C4 | — | 4 | 2 | no | — |
| 330 | C4⋊C4 | — | 4 | 1 | no | — |
| 331 | C4⋊C4 | — | 4 | 1 | no | — |
| 332 | C4⋊C4 | — | 4 | 1 | no | — |
| 333 | C4⋊C4 | — | 4 | 1 | no | — |
| 334 | C4⋊C4 | — | 8 | 1 | no | — |
| 335 | C4⋊C4 | — | 8 | 1 | no | — |
| 336 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 337 | C4⋊C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 338 | C22⋊C4 | — | 2 | 1 | no | — |
| 339 | C22⋊C4 | — | 2 | 1 | no | — |
| 340 | C22⋊C4 | — | 2 | 1 | no | — |
| 341 | C22⋊C4 | — | 4 | 4 | no | — |
| 342 | C22⋊C4 | — | 4 | 4 | no | — |
| 343 | C22⋊C4 | — | 4 | 4 | no | — |
| 344 | C22⋊C4 | — | 4 | 4 | no | — |
| 345 | C22⋊C4 | — | 4 | 2 | no | — |
| 346 | C22⋊C4 | — | 8 | 1 | no | — |
| 347 | C42 | — | 2 | 1 | no | — |
| 348 | C42 | — | 2 | 1 | no | — |
| 349 | C42 | — | 2 | 1 | no | — |
| 350 | C42 | — | 4 | 2 | no | — |
| 351 | C42 | — | 8 | 1 | no | — |
| 352 | C42 | #120 | 1 | 1 | yes | characteristic |
| 353 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 354 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 355 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 356 | C16 | — | 4 | 2 | no | — |
| 357 | C16 | — | 4 | 2 | no | — |
| 358 | C23 | — | 2 | 1 | no | — |
| 359 | C23 | — | 4 | 4 | no | — |
| 360 | C23 | — | 8 | 4 | no | — |
| 361 | C23 | — | 16 | 2 | no | — |
| 362 | C23 | — | 16 | 2 | no | — |
| 363 | C23 | #447 | 1 | 1 | yes | characteristic |
| 364 | Q8 | — | 2 | 1 | no | — |
| 365 | Q8 | — | 2 | 1 | no | — |
| 366 | Q8 | — | 4 | 2 | no | — |
| 367 | Q8 | — | 4 | 2 | no | — |
| 368 | Q8 | — | 8 | 2 | no | — |
| 369 | Q8 | — | 8 | 1 | no | — |
| 370 | Q8 | — | 8 | 1 | no | — |
| 371 | Q8 | — | 16 | 1 | no | — |
| 372 | D4 | — | 2 | 1 | no | — |
| 373 | D4 | — | 2 | 1 | no | — |
| 374 | D4 | — | 4 | 2 | no | — |
| 375 | D4 | — | 4 | 4 | no | — |
| 376 | D4 | — | 4 | 4 | no | — |
| 377 | D4 | — | 8 | 4 | no | — |
| 378 | D4 | — | 8 | 2 | no | — |
| 379 | D4 | — | 8 | 2 | no | — |
| 380 | D4 | — | 8 | 4 | no | — |
| 381 | D4 | — | 16 | 2 | no | — |
| 382 | D4 | — | 16 | 2 | no | — |
| 383 | D4 | — | 16 | 4 | no | — |
| 384 | D4 | — | 16 | 4 | no | — |
| 385 | C2×C4 | — | 2 | 1 | no | — |
| 386 | C2×C4 | — | 2 | 1 | no | — |
| 387 | C2×C4 | — | 2 | 1 | no | — |
| 388 | C2×C4 | — | 2 | 1 | no | — |
| 389 | C2×C4 | — | 2 | 1 | no | — |
| 390 | C2×C4 | — | 4 | 1 | no | — |
| 391 | C2×C4 | — | 4 | 2 | no | — |
| 392 | C2×C4 | — | 4 | 2 | no | — |
| 393 | C2×C4 | — | 4 | 1 | no | — |
| 394 | C2×C4 | — | 4 | 1 | no | — |
| 395 | C2×C4 | — | 4 | 2 | no | — |
| 396 | C2×C4 | — | 4 | 2 | no | — |
| 397 | C2×C4 | — | 8 | 1 | no | — |
| 398 | C2×C4 | — | 8 | 1 | no | — |
| 399 | C2×C4 | — | 16 | 4 | no | — |
| 400 | C2×C4 | — | 16 | 4 | no | — |
| 401 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 402 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 403 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 404 | C2×C4 | #508 | 1 | 2 | yes | — |
| 405 | C2×C4 | #510 | 1 | 2 | yes | — |
| 406 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 407 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 408 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 409 | C8 | — | 2 | 1 | no | — |
| 410 | C8 | — | 2 | 1 | no | — |
| 411 | C8 | — | 2 | 1 | no | — |
| 412 | C8 | — | 4 | 1 | no | — |
| 413 | C8 | — | 4 | 1 | no | — |
| 414 | C8 | — | 8 | 1 | no | — |
| 415 | C8 | — | 8 | 1 | no | — |
| 416 | C8 | #576 | 1 | 1 | yes | characteristic |
| 417 | C8 | #577 | 1 | 1 | yes | characteristic |
| 418 | C22 | — | 2 | 1 | no | — |
| 419 | C22 | — | 2 | 1 | no | — |
| 420 | C22 | — | 4 | 1 | no | — |
| 421 | C22 | — | 4 | 1 | no | — |
| 422 | C22 | — | 4 | 1 | no | — |
| 423 | C22 | — | 8 | 4 | no | — |
| 424 | C22 | — | 8 | 4 | no | — |
| 425 | C22 | — | 8 | 4 | no | — |
| 426 | C22 | — | 32 | 4 | no | — |
| 427 | C22 | #2189 | 1 | 2 | yes | — |
| 428 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 429 | C4 | — | 2 | 1 | no | — |
| 430 | C4 | — | 2 | 1 | no | — |
| 431 | C4 | — | 2 | 1 | no | — |
| 432 | C4 | — | 4 | 1 | no | — |
| 433 | C4 | — | 4 | 1 | no | — |
| 434 | C4 | — | 4 | 1 | no | — |
| 435 | C4 | — | 8 | 2 | no | — |
| 436 | C4 | — | 8 | 2 | no | — |
| 437 | C4 | — | 16 | 1 | no | — |
| 438 | C4 | — | 16 | 1 | no | — |
| 439 | C4 | #2189 | 1 | 2 | yes | — |
| 440 | C4 | #3012 | 1 | 1 | yes | characteristic |
| 441 | C4 | #3019 | 1 | 1 | yes | characteristic |
| 442 | C4 | C2×#508 | 1 | 1 | yes | characteristic |
| 443 | C4 | C2×#510 | 1 | 1 | yes | characteristic |
| 444 | C2 | — | 2 | 2 | no | — |
| 445 | C2 | — | 8 | 1 | no | — |
| 446 | C2 | — | 16 | 4 | no | — |
| 447 | C2 | #18870 | 1 | 1 | yes | characteristic |
| 448 | C2 | C2×#2189 | 1 | 1 | yes | characteristic |
| 449 | C2 | #33539 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #62417 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33815 | C2 | semidirect |
| 3. #33774 | C2 | semidirect |
| 4. #19126 | C2 | semidirect |
| 5. #12385 | C2 | semidirect |
| 6. #12373 | C2 | semidirect |
| 7. #12333 | C2 | semidirect |
| 8. #12321 | C2 | semidirect |
| 9. C2×#1361 | C2 | semidirect |
| 10. #11646 | C2 | semidirect |
| 11. #11621 | C2 | semidirect |
| 31. #2227 | C22 | semidirect |
| 34. #2209 | C22 | semidirect |
| 41. #1676 | C22 | semidirect |
| 42. #1675 | C22 | semidirect |
| 43. #1670 | C22 | semidirect |
| 44. #1669 | C22 | semidirect |
| 49. Q8⋊C16 | C22 | semidirect |
| 50. #1332 | C22 | semidirect |
| 51. #1332 | C22 | semidirect |
| 52. D4⋊C16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62417 | 2 |
| #33815 | 2 |
| #33774 | 2 |
| #19126 | 2 |
| #12385 | 2 |
| #12373 | 2 |
| #12333 | 2 |
| #12321 | 2 |
| C2×#1361 | 2 |
| #11646 | 2 |
| #11621 | 2 |