Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1504 conjugacy classes of subgroups.
The subgroup lattice has 344 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62487 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #20324 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #20019 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Q162 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20006 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #19957 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #19930 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C2×#1713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #19359 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #18685 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #17959 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #3544 | C22 | 1 | 4 | yes | — |
| 13 | #3385 | C22 | 1 | 8 | yes | — |
| 14 | Q8×Q16 | C22 | 1 | 4 | yes | — |
| 15 | #3380 | C22 | 1 | 4 | yes | — |
| 16 | #3357 | C22 | 1 | 8 | yes | — |
| 17 | #3343 | — | 2 | 8 | no | — |
| 18 | #3341 | C22 | 1 | 4 | yes | — |
| 19 | #3338 | C22 | 1 | 4 | yes | — |
| 20 | #3336 | — | 2 | 4 | no | — |
| 21 | #3334 | — | 2 | 4 | no | — |
| 22 | #3243 | — | 2 | 8 | no | — |
| 23 | #3238 | C22 | 1 | 8 | yes | — |
| 24 | #3175 | — | 2 | 4 | no | — |
| 25 | #3168 | C22 | 1 | 2 | yes | — |
| 26 | #3164 | — | 2 | 4 | no | — |
| 27 | C2×#500 | C22 | 1 | 4 | yes | — |
| 28 | C2×#494 | C22 | 1 | 2 | yes | — |
| 29 | #2966 | C22 | 1 | 4 | yes | — |
| 30 | #1713 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1713 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1713 | C22 | 1 | 2 | yes | — |
| 33 | #1713 | C22 | 1 | 4 | yes | — |
| 34 | #1696 | C22 | 1 | 8 | yes | — |
| 35 | #1695 | C22 | 1 | 8 | yes | — |
| 36 | #1676 | C22 | 1 | 4 | yes | — |
| 37 | #1676 | C22 | 1 | 4 | yes | — |
| 38 | #1675 | C22 | 1 | 4 | yes | — |
| 39 | #1675 | C22 | 1 | 4 | yes | — |
| 40 | #1634 | C22 | 1 | 4 | yes | — |
| 41 | #1634 | C22 | 1 | 4 | yes | — |
| 42 | #1633 | C22 | 1 | 4 | yes | — |
| 43 | #1633 | C22 | 1 | 4 | yes | — |
| 44 | #1632 | C22 | 1 | 8 | yes | — |
| 45 | #1632 | C22 | 1 | 8 | yes | — |
| 46 | #1591 | C22 | 1 | 2 | yes | — |
| 47 | #1590 | C22 | 1 | 4 | yes | — |
| 48 | #1589 | C22 | 1 | 2 | yes | — |
| 49 | C8×Q16 | C22 | 1 | 2 | yes | — |
| 50 | C8×SD16 | C22 | 1 | 4 | yes | — |
| 51 | C8×D8 | C22 | 1 | 2 | yes | — |
| 52 | #1568 | C22 | 1 | 2 | yes | — |
| 53 | #1558 | C22 | 1 | 4 | yes | — |
| 54 | #1494 | C22 | 1 | 4 | yes | — |
| 55 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 56 | C22×Q16 | — | 4 | 2 | no | — |
| 57 | #563 | — | 4 | 4 | no | — |
| 58 | Q82 | C23 | 1 | 4 | yes | — |
| 59 | #555 | C23 | 1 | 4 | yes | — |
| 60 | #554 | C23 | 1 | 8 | yes | — |
| 61 | #552 | — | 2 | 8 | no | — |
| 62 | #547 | — | 2 | 8 | no | — |
| 63 | #544 | — | 2 | 8 | no | — |
| 64 | #542 | — | 2 | 8 | no | — |
| 65 | #536 | — | 4 | 4 | no | — |
| 66 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 67 | #529 | C23 | 1 | 4 | yes | — |
| 68 | #501 | — | 2 | 8 | no | — |
| 69 | #501 | — | 2 | 8 | no | — |
| 70 | #500 | — | 2 | 4 | no | — |
| 71 | #500 | — | 2 | 4 | no | — |
| 72 | #500 | — | 2 | 4 | no | — |
| 73 | #500 | C23 | 1 | 4 | yes | — |
| 74 | #500 | C23 | 1 | 4 | yes | — |
| 75 | #500 | C23 | 1 | 8 | yes | — |
| 76 | #499 | — | 2 | 4 | no | — |
| 77 | #499 | D4 | 1 | 2 | yes | — |
| 78 | #495 | D4 | 1 | 2 | yes | — |
| 79 | #494 | — | 4 | 2 | no | — |
| 80 | #494 | C23 | 1 | 2 | yes | — |
| 81 | #494 | C23 | 1 | 4 | yes | — |
| 82 | #494 | C23 | 1 | 2 | yes | — |
| 83 | #489 | — | 2 | 8 | no | — |
| 84 | #488 | D4 | 1 | 4 | yes | — |
| 85 | #485 | — | 2 | 8 | no | — |
| 86 | #484 | — | 2 | 8 | no | — |
| 87 | #483 | — | 2 | 8 | no | — |
| 88 | #482 | — | 2 | 8 | no | — |
| 89 | #480 | — | 2 | 8 | no | — |
| 90 | #480 | — | 2 | 8 | no | — |
| 91 | #479 | — | 2 | 8 | no | — |
| 92 | #478 | — | 2 | 8 | no | — |
| 93 | #477 | — | 2 | 8 | no | — |
| 94 | #477 | C23 | 1 | 8 | yes | — |
| 95 | #477 | C23 | 1 | 8 | yes | — |
| 96 | #476 | C23 | 1 | 8 | yes | — |
| 97 | #476 | C23 | 1 | 8 | yes | — |
| 98 | #474 | — | 2 | 8 | no | — |
| 99 | #468 | — | 2 | 8 | no | — |
| 100 | #467 | — | 2 | 4 | no | — |
| 101 | #467 | D4 | 1 | 2 | yes | — |
| 102 | #466 | — | 2 | 4 | no | — |
| 103 | #466 | D4 | 1 | 2 | yes | — |
| 104 | #465 | D4 | 1 | 4 | yes | — |
| 105 | #446 | C23 | 1 | 4 | yes | — |
| 106 | C8×Q8 | C23 | 1 | 4 | yes | — |
| 107 | #440 | — | 2 | 8 | no | — |
| 108 | C4×Q16 | — | 2 | 4 | no | — |
| 109 | C4×Q16 | C23 | 1 | 2 | yes | — |
| 110 | C4×SD16 | C23 | 1 | 4 | yes | — |
| 111 | C4×D8 | — | 2 | 4 | no | — |
| 112 | C4×D8 | C23 | 1 | 2 | yes | — |
| 113 | #436 | C23 | 1 | 4 | yes | — |
| 114 | #435 | — | 2 | 4 | no | — |
| 115 | C8×D4 | — | 2 | 4 | no | — |
| 116 | C8×D4 | C23 | 1 | 4 | yes | — |
| 117 | #433 | — | 2 | 4 | no | — |
| 118 | #432 | C23 | 1 | 4 | yes | — |
| 119 | #428 | — | 2 | 8 | no | — |
| 120 | #427 | — | 2 | 4 | no | — |
| 121 | C2×#107 | — | 2 | 4 | no | — |
| 122 | C2×#107 | — | 2 | 4 | no | — |
| 123 | C2×#107 | — | 2 | 4 | no | — |
| 124 | #405 | — | 2 | 4 | no | — |
| 125 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 127 | #335 | C23 | 1 | 2 | yes | — |
| 128 | #335 | C23 | 1 | 2 | yes | — |
| 129 | #334 | C23 | 1 | 4 | yes | — |
| 130 | #333 | C23 | 1 | 8 | yes | — |
| 131 | #333 | C23 | 1 | 4 | yes | — |
| 132 | #333 | C23 | 1 | 4 | yes | — |
| 133 | Q8⋊C8 | C23 | 1 | 4 | yes | — |
| 134 | Q8⋊C8 | C23 | 1 | 4 | yes | — |
| 135 | D4⋊C8 | C23 | 1 | 4 | yes | — |
| 136 | D4⋊C8 | C23 | 1 | 4 | yes | — |
| 137 | C82 | C23 | 1 | 1 | yes | characteristic |
| 138 | C82 | C23 | 1 | 2 | yes | — |
| 139 | C82 | C23 | 1 | 1 | yes | characteristic |
| 140 | C2×#41 | — | 4 | 4 | no | — |
| 141 | C22×Q8 | — | 2 | 4 | no | — |
| 142 | C2×Q16 | — | 4 | 2 | no | — |
| 143 | C2×Q16 | — | 4 | 4 | no | — |
| 144 | C2×Q16 | — | 4 | 2 | no | — |
| 145 | C2×Q16 | — | 8 | 8 | no | — |
| 146 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 147 | C2×SD16 | C2×D4 | 1 | 4 | yes | — |
| 148 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 149 | C2×M16 | — | 2 | 4 | no | — |
| 150 | C22×C8 | — | 2 | 1 | no | — |
| 151 | C22×C8 | — | 2 | 4 | no | — |
| 152 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 153 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 154 | C4⋊Q8 | — | 2 | 8 | no | — |
| 155 | C4⋊Q8 | — | 2 | 8 | no | — |
| 156 | C4⋊Q8 | — | 2 | 4 | no | — |
| 157 | C4⋊Q8 | — | 4 | 4 | no | — |
| 158 | C4⋊Q8 | — | 4 | 4 | no | — |
| 159 | C4⋊Q8 | C24 | 1 | 4 | yes | — |
| 160 | C4⋊Q8 | C24 | 1 | 4 | yes | — |
| 161 | #126 | — | 2 | 4 | no | — |
| 162 | #126 | — | 4 | 8 | no | — |
| 163 | #125 | — | 2 | 8 | no | — |
| 164 | #125 | — | 4 | 8 | no | — |
| 165 | #125 | C2×D4 | 1 | 4 | yes | — |
| 166 | #124 | — | 4 | 4 | no | — |
| 167 | #124 | — | 4 | 4 | no | — |
| 168 | #124 | C2×D4 | 1 | 4 | yes | — |
| 169 | #123 | — | 2 | 4 | no | — |
| 170 | #123 | — | 4 | 8 | no | — |
| 171 | #123 | — | 4 | 8 | no | — |
| 172 | C22⋊Q8 | — | 2 | 8 | no | — |
| 173 | C22⋊Q8 | — | 4 | 8 | no | — |
| 174 | C22⋊Q8 | — | 4 | 8 | no | — |
| 175 | C22⋊Q8 | — | 4 | 8 | no | — |
| 176 | C22⋊Q8 | C2×D4 | 1 | 4 | yes | — |
| 177 | #121 | — | 2 | 8 | no | — |
| 178 | #121 | C2×D4 | 1 | 4 | yes | — |
| 179 | C4×Q8 | — | 2 | 8 | no | — |
| 180 | C4×Q8 | C24 | 1 | 4 | yes | — |
| 181 | C4×D4 | — | 2 | 4 | no | — |
| 182 | C4×D4 | — | 2 | 8 | no | — |
| 183 | C4×D4 | C24 | 1 | 4 | yes | — |
| 184 | #117 | — | 2 | 4 | no | — |
| 185 | #117 | — | 2 | 8 | no | — |
| 186 | #117 | — | 4 | 4 | no | — |
| 187 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 188 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 189 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 190 | #107 | — | 2 | 4 | no | — |
| 191 | #107 | — | 2 | 4 | no | — |
| 192 | #107 | — | 2 | 8 | no | — |
| 193 | #107 | — | 2 | 8 | no | — |
| 194 | #107 | — | 2 | 4 | no | — |
| 195 | #107 | — | 2 | 4 | no | — |
| 196 | #107 | — | 2 | 8 | no | — |
| 197 | #107 | — | 2 | 8 | no | — |
| 198 | #107 | C2×D4 | 1 | 2 | yes | — |
| 199 | #107 | C2×D4 | 1 | 2 | yes | — |
| 200 | #106 | — | 2 | 8 | no | — |
| 201 | #106 | — | 2 | 8 | no | — |
| 202 | #106 | C2×D4 | 1 | 4 | yes | — |
| 203 | C4⋊C8 | — | 2 | 4 | no | — |
| 204 | C4⋊C8 | — | 2 | 4 | no | — |
| 205 | C4⋊C8 | C24 | 1 | 4 | yes | — |
| 206 | C4⋊C8 | C24 | 1 | 4 | yes | — |
| 207 | #103 | — | 2 | 8 | no | — |
| 208 | #103 | — | 2 | 8 | no | — |
| 209 | #103 | C2×D4 | 1 | 4 | yes | — |
| 210 | #103 | C2×D4 | 1 | 4 | yes | — |
| 211 | #102 | — | 2 | 8 | no | — |
| 212 | #102 | — | 2 | 8 | no | — |
| 213 | #102 | C2×D4 | 1 | 4 | yes | — |
| 214 | #102 | C2×D4 | 1 | 4 | yes | — |
| 215 | C22⋊C8 | — | 2 | 4 | no | — |
| 216 | C22⋊C8 | — | 2 | 4 | no | — |
| 217 | C22⋊C8 | — | 2 | 4 | no | — |
| 218 | #97 | — | 2 | 4 | no | — |
| 219 | #97 | — | 2 | 4 | no | — |
| 220 | C4×C8 | — | 2 | 4 | no | — |
| 221 | C4×C8 | — | 2 | 1 | no | — |
| 222 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 223 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 224 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 225 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 226 | C4×C8 | C24 | 1 | 2 | yes | — |
| 227 | C4×C8 | C24 | 1 | 2 | yes | — |
| 228 | #41 | — | 16 | 8 | no | — |
| 229 | C2×Q8 | — | 2 | 4 | no | — |
| 230 | C2×Q8 | — | 2 | 4 | no | — |
| 231 | C2×Q8 | — | 4 | 4 | no | — |
| 232 | C2×Q8 | — | 4 | 4 | no | — |
| 233 | C2×Q8 | — | 8 | 8 | no | — |
| 234 | C2×Q8 | C22×D4 | 1 | 4 | yes | — |
| 235 | C2×D4 | — | 2 | 4 | no | — |
| 236 | C2×D4 | — | 4 | 4 | no | — |
| 237 | C2×D4 | C22×D4 | 1 | 4 | yes | — |
| 238 | C22×C4 | — | 2 | 4 | no | — |
| 239 | C22×C4 | — | 4 | 8 | no | — |
| 240 | C22×C4 | — | 4 | 4 | no | — |
| 241 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 242 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 243 | Q16 | — | 2 | 4 | no | — |
| 244 | Q16 | — | 8 | 4 | no | — |
| 245 | Q16 | — | 8 | 8 | no | — |
| 246 | Q16 | — | 8 | 4 | no | — |
| 247 | SD16 | — | 2 | 8 | no | — |
| 248 | D8 | — | 2 | 4 | no | — |
| 249 | M16 | — | 4 | 8 | no | — |
| 250 | C2×C8 | — | 2 | 4 | no | — |
| 251 | C2×C8 | — | 2 | 1 | no | — |
| 252 | C2×C8 | — | 2 | 1 | no | — |
| 253 | C2×C8 | — | 2 | 4 | no | — |
| 254 | C2×C8 | — | 2 | 4 | no | — |
| 255 | C2×C8 | — | 2 | 4 | no | — |
| 256 | C2×C8 | — | 2 | 4 | no | — |
| 257 | C2×C8 | — | 4 | 8 | no | — |
| 258 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 259 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 260 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 261 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 262 | C4⋊C4 | — | 2 | 4 | no | — |
| 263 | C4⋊C4 | — | 2 | 4 | no | — |
| 264 | C4⋊C4 | — | 2 | 4 | no | — |
| 265 | C4⋊C4 | — | 2 | 4 | no | — |
| 266 | C4⋊C4 | — | 2 | 8 | no | — |
| 267 | C4⋊C4 | — | 2 | 8 | no | — |
| 268 | C4⋊C4 | — | 4 | 8 | no | — |
| 269 | C4⋊C4 | — | 4 | 8 | no | — |
| 270 | C4⋊C4 | — | 4 | 8 | no | — |
| 271 | C4⋊C4 | — | 4 | 8 | no | — |
| 272 | C4⋊C4 | C22×D4 | 1 | 4 | yes | — |
| 273 | C4⋊C4 | C22×D4 | 1 | 4 | yes | — |
| 274 | C22⋊C4 | — | 2 | 4 | no | — |
| 275 | C22⋊C4 | — | 2 | 4 | no | — |
| 276 | C22⋊C4 | — | 2 | 4 | no | — |
| 277 | C22⋊C4 | — | 2 | 4 | no | — |
| 278 | C22⋊C4 | — | 2 | 4 | no | — |
| 279 | C22⋊C4 | — | 4 | 8 | no | — |
| 280 | C22⋊C4 | — | 4 | 8 | no | — |
| 281 | C42 | — | 2 | 4 | no | — |
| 282 | C42 | — | 4 | 4 | no | — |
| 283 | C42 | — | 4 | 4 | no | — |
| 284 | C42 | C22×D4 | 1 | 2 | yes | — |
| 285 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 286 | C42 | C25 | 1 | 1 | yes | characteristic |
| 287 | C23 | — | 2 | 4 | no | — |
| 288 | C23 | D42 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 2 | 4 | no | — |
| 290 | Q8 | — | 2 | 4 | no | — |
| 291 | Q8 | — | 8 | 4 | no | — |
| 292 | Q8 | — | 8 | 4 | no | — |
| 293 | Q8 | — | 16 | 4 | no | — |
| 294 | Q8 | — | 16 | 4 | no | — |
| 295 | D4 | — | 2 | 4 | no | — |
| 296 | D4 | — | 2 | 4 | no | — |
| 297 | D4 | — | 4 | 8 | no | — |
| 298 | D4 | — | 16 | 4 | no | — |
| 299 | C2×C4 | — | 2 | 2 | no | — |
| 300 | C2×C4 | — | 2 | 2 | no | — |
| 301 | C2×C4 | — | 2 | 2 | no | — |
| 302 | C2×C4 | — | 2 | 4 | no | — |
| 303 | C2×C4 | — | 2 | 4 | no | — |
| 304 | C2×C4 | — | 2 | 4 | no | — |
| 305 | C2×C4 | — | 4 | 4 | no | — |
| 306 | C2×C4 | — | 4 | 4 | no | — |
| 307 | C2×C4 | — | 4 | 4 | no | — |
| 308 | C2×C4 | — | 4 | 4 | no | — |
| 309 | C2×C4 | — | 8 | 8 | no | — |
| 310 | C2×C4 | — | 16 | 8 | no | — |
| 311 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 312 | C2×C4 | D42 | 1 | 2 | yes | — |
| 313 | C2×C4 | C23×D4 | 1 | 2 | yes | — |
| 314 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 315 | C8 | — | 2 | 2 | no | — |
| 316 | C8 | — | 2 | 2 | no | — |
| 317 | C8 | — | 2 | 2 | no | — |
| 318 | C8 | — | 2 | 2 | no | — |
| 319 | C8 | — | 4 | 4 | no | — |
| 320 | C8 | — | 4 | 4 | no | — |
| 321 | C8 | #578 | 1 | 4 | yes | — |
| 322 | C8 | #578 | 1 | 4 | yes | — |
| 323 | C22 | — | 2 | 2 | no | — |
| 324 | C22 | — | 4 | 4 | no | — |
| 325 | C22 | — | 4 | 4 | no | — |
| 326 | C22 | — | 4 | 4 | no | — |
| 327 | C22 | #3292 | 1 | 2 | yes | — |
| 328 | C22 | C2×D42 | 1 | 1 | yes | characteristic |
| 329 | C4 | — | 2 | 2 | no | — |
| 330 | C4 | — | 2 | 1 | no | — |
| 331 | C4 | — | 2 | 1 | no | — |
| 332 | C4 | — | 4 | 4 | no | — |
| 333 | C4 | — | 4 | 4 | no | — |
| 334 | C4 | — | 4 | 4 | no | — |
| 335 | C4 | — | 8 | 4 | no | — |
| 336 | C4 | — | 8 | 4 | no | — |
| 337 | C4 | C2×#578 | 1 | 2 | yes | — |
| 338 | C4 | C2×#578 | 1 | 2 | yes | — |
| 339 | C2 | — | 2 | 2 | no | — |
| 340 | C2 | — | 8 | 4 | no | — |
| 341 | C2 | #61091 | 1 | 2 | yes | — |
| 342 | C2 | C2×#3292 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62487 | C2 | semidirect |
| 2. #20324 | C2 | semidirect |
| 3. #20019 | C2 | semidirect |
| 4. Q162 | C2 | semidirect |
| 5. #20006 | C2 | semidirect |
| 6. #19957 | C2 | semidirect |
| 7. #19930 | C2 | semidirect |
| 8. C2×#1713 | C2 | semidirect |
| 9. #19359 | C2 | semidirect |
| 10. #18685 | C2 | semidirect |
| 11. #17959 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62487 | 2 |
| #20324 | 2 |
| #20019 | 2 |
| Q162 | 2 |
| #20006 | 2 |
| #19957 | 2 |
| #19930 | 2 |
| C2×#1713 | 2 |
| #19359 | 2 |
| #18685 | 2 |
| #17959 | 2 |