Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5681 conjugacy classes of subgroups.
The subgroup lattice has 315 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #63067 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #62493 | C2 | 1 | 10 | yes | maximal, complemented |
| 3 | #62374 | C2 | 1 | 5 | yes | maximal, complemented |
| 4 | #62237 | C2 | 1 | 20 | yes | maximal, complemented |
| 5 | #61877 | C2 | 1 | 10 | yes | maximal, complemented |
| 6 | #61477 | C2 | 1 | 10 | yes | maximal, complemented |
| 7 | #61376 | C2 | 1 | 5 | yes | maximal, complemented |
| 8 | #60332 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #3563 | — | 2 | 20 | no | — |
| 10 | #3563 | C22 | 1 | 20 | yes | — |
| 11 | #3556 | C22 | 1 | 10 | yes | — |
| 12 | #3553 | — | 2 | 10 | no | — |
| 13 | #3531 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#564 | — | 2 | 2 | no | — |
| 15 | #3515 | C22 | 1 | 20 | yes | — |
| 16 | #3504 | C22 | 1 | 10 | yes | — |
| 17 | #3395 | C22 | 1 | 10 | yes | — |
| 18 | #3390 | C22 | 1 | 20 | yes | — |
| 19 | #3388 | C22 | 1 | 40 | yes | — |
| 20 | #3383 | C22 | 1 | 20 | yes | — |
| 21 | #3380 | C22 | 1 | 10 | yes | — |
| 22 | #3374 | C22 | 1 | 40 | yes | — |
| 23 | #3372 | C22 | 1 | 20 | yes | — |
| 24 | #3352 | C22 | 1 | 20 | yes | — |
| 25 | #3344 | C22 | 1 | 40 | yes | — |
| 26 | #3338 | C22 | 1 | 10 | yes | — |
| 27 | #3336 | C22 | 1 | 20 | yes | — |
| 28 | #3331 | C22 | 1 | 20 | yes | — |
| 29 | #3315 | C22 | 1 | 40 | yes | — |
| 30 | #3283 | C22 | 1 | 10 | yes | — |
| 31 | #3261 | C22 | 1 | 20 | yes | — |
| 32 | #3255 | C22 | 1 | 10 | yes | — |
| 33 | #3242 | C22 | 1 | 40 | yes | — |
| 34 | #3236 | C22 | 1 | 20 | yes | — |
| 35 | #3177 | C22 | 1 | 20 | yes | — |
| 36 | #3170 | C22 | 1 | 10 | yes | — |
| 37 | #3164 | C22 | 1 | 10 | yes | — |
| 38 | C2×#499 | C22 | 1 | 5 | yes | — |
| 39 | #3120 | C22 | 1 | 40 | yes | — |
| 40 | #3116 | C22 | 1 | 20 | yes | — |
| 41 | #3084 | C22 | 1 | 20 | yes | — |
| 42 | #3080 | C22 | 1 | 10 | yes | — |
| 43 | #2995 | C22 | 1 | 10 | yes | — |
| 44 | #2987 | C22 | 1 | 5 | yes | — |
| 45 | #2966 | C22 | 1 | 10 | yes | — |
| 46 | #2960 | C22 | 1 | 5 | yes | — |
| 47 | #2951 | C22 | 1 | 10 | yes | — |
| 48 | C4×#135 | C22 | 1 | 5 | yes | — |
| 49 | C2×#135 | C23 | 1 | 5 | yes | — |
| 50 | #564 | — | 2 | 10 | no | — |
| 51 | #564 | — | 2 | 20 | no | — |
| 52 | #564 | — | 2 | 10 | no | — |
| 53 | #564 | D4 | 1 | 1 | yes | characteristic |
| 54 | #563 | — | 2 | 20 | no | — |
| 55 | #563 | D4 | 1 | 10 | yes | — |
| 56 | #562 | D4 | 1 | 5 | yes | — |
| 57 | #557 | — | 2 | 20 | no | — |
| 58 | #557 | C23 | 1 | 20 | yes | — |
| 59 | #557 | C23 | 1 | 40 | yes | — |
| 60 | #554 | — | 2 | 40 | no | — |
| 61 | #554 | C23 | 1 | 20 | yes | — |
| 62 | #554 | C23 | 1 | 20 | yes | — |
| 63 | #553 | — | 2 | 20 | no | — |
| 64 | #552 | — | 2 | 40 | no | — |
| 65 | #552 | — | 2 | 40 | no | — |
| 66 | #552 | — | 2 | 20 | no | — |
| 67 | #552 | C23 | 1 | 40 | yes | — |
| 68 | #547 | C23 | 1 | 20 | yes | — |
| 69 | #544 | — | 2 | 20 | no | — |
| 70 | #544 | — | 2 | 20 | no | — |
| 71 | #544 | C23 | 1 | 20 | yes | — |
| 72 | #541 | — | 2 | 40 | no | — |
| 73 | #541 | — | 2 | 40 | no | — |
| 74 | #541 | C23 | 1 | 40 | yes | — |
| 75 | #539 | — | 2 | 40 | no | — |
| 76 | #539 | — | 2 | 20 | no | — |
| 77 | #533 | — | 2 | 10 | no | — |
| 78 | #533 | C23 | 1 | 20 | yes | — |
| 79 | #533 | C23 | 1 | 5 | yes | — |
| 80 | #529 | — | 2 | 20 | no | — |
| 81 | #529 | C23 | 1 | 10 | yes | — |
| 82 | C2×#125 | — | 2 | 10 | no | — |
| 83 | C2×#125 | — | 2 | 20 | no | — |
| 84 | C2×#125 | C23 | 1 | 10 | yes | — |
| 85 | #520 | C23 | 1 | 20 | yes | — |
| 86 | C4×#41 | C23 | 1 | 10 | yes | — |
| 87 | #501 | C23 | 1 | 40 | yes | — |
| 88 | #500 | C23 | 1 | 10 | yes | — |
| 89 | #499 | C23 | 1 | 20 | yes | — |
| 90 | #499 | C23 | 1 | 20 | yes | — |
| 91 | #494 | C23 | 1 | 10 | yes | — |
| 92 | #491 | C23 | 1 | 40 | yes | — |
| 93 | #487 | C23 | 1 | 20 | yes | — |
| 94 | #485 | C23 | 1 | 40 | yes | — |
| 95 | #484 | C23 | 1 | 40 | yes | — |
| 96 | #483 | C23 | 1 | 40 | yes | — |
| 97 | #479 | C23 | 1 | 40 | yes | — |
| 98 | #479 | C23 | 1 | 40 | yes | — |
| 99 | #479 | C23 | 1 | 40 | yes | — |
| 100 | #478 | C23 | 1 | 40 | yes | — |
| 101 | #477 | C23 | 1 | 20 | yes | — |
| 102 | #476 | C23 | 1 | 20 | yes | — |
| 103 | #475 | C23 | 1 | 40 | yes | — |
| 104 | #470 | C23 | 1 | 40 | yes | — |
| 105 | #467 | C23 | 1 | 20 | yes | — |
| 106 | #467 | C23 | 1 | 20 | yes | — |
| 107 | #462 | C23 | 1 | 20 | yes | — |
| 108 | #462 | C23 | 1 | 40 | yes | — |
| 109 | #461 | C23 | 1 | 20 | yes | — |
| 110 | #452 | C23 | 1 | 40 | yes | — |
| 111 | #446 | C23 | 1 | 10 | yes | — |
| 112 | #446 | C23 | 1 | 20 | yes | — |
| 113 | C8×Q8 | C23 | 1 | 10 | yes | — |
| 114 | #441 | C23 | 1 | 40 | yes | — |
| 115 | #440 | C23 | 1 | 40 | yes | — |
| 116 | C4×Q16 | C23 | 1 | 20 | yes | — |
| 117 | C4×Q16 | C23 | 1 | 10 | yes | — |
| 118 | C4×Q16 | C23 | 1 | 20 | yes | — |
| 119 | C4×SD16 | C23 | 1 | 20 | yes | — |
| 120 | C4×D8 | C23 | 1 | 10 | yes | — |
| 121 | #436 | C23 | 1 | 10 | yes | — |
| 122 | #435 | C23 | 1 | 20 | yes | — |
| 123 | C8×D4 | C23 | 1 | 10 | yes | — |
| 124 | #433 | C23 | 1 | 20 | yes | — |
| 125 | #432 | C23 | 1 | 10 | yes | — |
| 126 | #428 | C23 | 1 | 20 | yes | — |
| 127 | #427 | C23 | 1 | 5 | yes | — |
| 128 | C2×#107 | C23 | 1 | 10 | yes | — |
| 129 | C2×#106 | C23 | 1 | 10 | yes | — |
| 130 | #424 | C23 | 1 | 10 | yes | — |
| 131 | #423 | C23 | 1 | 5 | yes | — |
| 132 | #417 | C23 | 1 | 20 | yes | — |
| 133 | #416 | C23 | 1 | 10 | yes | — |
| 134 | #405 | C23 | 1 | 10 | yes | — |
| 135 | C2×C4×C8 | C23 | 1 | 5 | yes | — |
| 136 | C2×#41 | C24 | 1 | 10 | yes | — |
| 137 | #135 | — | 2 | 20 | no | — |
| 138 | C2×Q16 | C24 | 1 | 5 | yes | — |
| 139 | C2×Q16 | C24 | 1 | 20 | yes | — |
| 140 | C2×SD16 | C24 | 1 | 10 | yes | — |
| 141 | C2×D8 | C24 | 1 | 1 | yes | characteristic |
| 142 | C2×M16 | C24 | 1 | 10 | yes | — |
| 143 | C22×C8 | C24 | 1 | 5 | yes | — |
| 144 | C22×C8 | C24 | 1 | 5 | yes | — |
| 145 | C4⋊Q8 | — | 2 | 40 | no | — |
| 146 | C4⋊Q8 | — | 2 | 20 | no | — |
| 147 | C4⋊Q8 | C2×D4 | 1 | 10 | yes | — |
| 148 | C4⋊Q8 | C24 | 1 | 20 | yes | — |
| 149 | C4⋊Q8 | C24 | 1 | 20 | yes | — |
| 150 | #126 | — | 2 | 40 | no | — |
| 151 | #126 | — | 2 | 40 | no | — |
| 152 | #126 | — | 2 | 40 | no | — |
| 153 | #126 | C2×D4 | 1 | 20 | yes | — |
| 154 | #125 | — | 2 | 40 | no | — |
| 155 | #125 | — | 2 | 40 | no | — |
| 156 | #125 | — | 2 | 20 | no | — |
| 157 | #125 | — | 2 | 20 | no | — |
| 158 | #125 | — | 2 | 40 | no | — |
| 159 | #125 | — | 2 | 20 | no | — |
| 160 | #125 | — | 2 | 40 | no | — |
| 161 | #125 | — | 2 | 40 | no | — |
| 162 | #125 | — | 2 | 40 | no | — |
| 163 | #125 | — | 2 | 10 | no | — |
| 164 | #125 | — | 2 | 20 | no | — |
| 165 | #125 | — | 2 | 10 | no | — |
| 166 | #125 | C2×D4 | 1 | 10 | yes | — |
| 167 | #125 | C2×D4 | 1 | 5 | yes | — |
| 168 | #125 | C2×D4 | 1 | 5 | yes | — |
| 169 | #125 | C24 | 1 | 40 | yes | — |
| 170 | #124 | C2×D4 | 1 | 10 | yes | — |
| 171 | #123 | — | 2 | 40 | no | — |
| 172 | #123 | — | 2 | 40 | no | — |
| 173 | #123 | C2×D4 | 1 | 20 | yes | — |
| 174 | C22⋊Q8 | — | 2 | 20 | no | — |
| 175 | C22⋊Q8 | — | 2 | 20 | no | — |
| 176 | C22⋊Q8 | — | 2 | 40 | no | — |
| 177 | C22⋊Q8 | — | 2 | 40 | no | — |
| 178 | C22⋊Q8 | — | 2 | 40 | no | — |
| 179 | C22⋊Q8 | C2×D4 | 1 | 10 | yes | — |
| 180 | C22⋊Q8 | C2×D4 | 1 | 20 | yes | — |
| 181 | C22⋊Q8 | C24 | 1 | 40 | yes | — |
| 182 | #121 | — | 2 | 20 | no | — |
| 183 | #121 | C2×D4 | 1 | 10 | yes | — |
| 184 | C4×Q8 | — | 2 | 20 | no | — |
| 185 | C4×Q8 | — | 2 | 40 | no | — |
| 186 | C4×Q8 | — | 2 | 20 | no | — |
| 187 | C4×Q8 | C24 | 1 | 20 | yes | — |
| 188 | C4×Q8 | C24 | 1 | 40 | yes | — |
| 189 | C4×Q8 | C24 | 1 | 10 | yes | — |
| 190 | C4×D4 | — | 2 | 20 | no | — |
| 191 | C4×D4 | — | 2 | 20 | no | — |
| 192 | C4×D4 | — | 2 | 40 | no | — |
| 193 | C4×D4 | C24 | 1 | 20 | yes | — |
| 194 | C4×D4 | C24 | 1 | 10 | yes | — |
| 195 | #117 | — | 2 | 20 | no | — |
| 196 | #117 | — | 2 | 20 | no | — |
| 197 | #117 | — | 2 | 40 | no | — |
| 198 | #117 | — | 2 | 10 | no | — |
| 199 | #117 | C24 | 1 | 10 | yes | — |
| 200 | #117 | C24 | 1 | 10 | yes | — |
| 201 | C2×C4⋊C4 | — | 2 | 20 | no | — |
| 202 | C2×C4⋊C4 | — | 2 | 20 | no | — |
| 203 | C2×C4⋊C4 | — | 2 | 20 | no | — |
| 204 | C2×C4⋊C4 | — | 2 | 20 | no | — |
| 205 | C2×C4⋊C4 | C24 | 1 | 20 | yes | — |
| 206 | C2×C42 | — | 2 | 10 | no | — |
| 207 | C2×C42 | C24 | 1 | 5 | yes | — |
| 208 | #107 | C24 | 1 | 10 | yes | — |
| 209 | #107 | C24 | 1 | 20 | yes | — |
| 210 | #107 | C24 | 1 | 20 | yes | — |
| 211 | #106 | C24 | 1 | 40 | yes | — |
| 212 | #106 | C24 | 1 | 10 | yes | — |
| 213 | C4⋊C8 | C24 | 1 | 20 | yes | — |
| 214 | C4⋊C8 | C24 | 1 | 10 | yes | — |
| 215 | C4⋊C8 | C24 | 1 | 10 | yes | — |
| 216 | #103 | C24 | 1 | 20 | yes | — |
| 217 | #103 | C24 | 1 | 40 | yes | — |
| 218 | #103 | C24 | 1 | 40 | yes | — |
| 219 | #102 | C24 | 1 | 20 | yes | — |
| 220 | C22⋊C8 | C24 | 1 | 20 | yes | — |
| 221 | #97 | C24 | 1 | 20 | yes | — |
| 222 | C4×C8 | C24 | 1 | 10 | yes | — |
| 223 | C4×C8 | C24 | 1 | 10 | yes | — |
| 224 | #41 | — | 2 | 20 | no | — |
| 225 | #41 | — | 4 | 10 | no | — |
| 226 | C2×Q8 | — | 2 | 20 | no | — |
| 227 | C2×Q8 | C22×D4 | 1 | 10 | yes | — |
| 228 | C2×Q8 | C25 | 1 | 10 | yes | — |
| 229 | C2×D4 | — | 2 | 10 | no | — |
| 230 | C2×D4 | C22×D4 | 1 | 10 | yes | — |
| 231 | C2×D4 | C25 | 1 | 2 | yes | — |
| 232 | C22×C4 | — | 2 | 20 | no | — |
| 233 | C22×C4 | — | 2 | 10 | no | — |
| 234 | C22×C4 | — | 2 | 20 | no | — |
| 235 | C22×C4 | C22×D4 | 1 | 10 | yes | — |
| 236 | C22×C4 | C25 | 1 | 5 | yes | — |
| 237 | Q16 | — | 2 | 10 | no | — |
| 238 | Q16 | — | 4 | 20 | no | — |
| 239 | SD16 | — | 2 | 20 | no | — |
| 240 | D8 | 2−1+4 | 1 | 4 | yes | — |
| 241 | M16 | — | 4 | 10 | no | — |
| 242 | C2×C8 | — | 2 | 10 | no | — |
| 243 | C2×C8 | — | 4 | 5 | no | — |
| 244 | C2×C8 | C25 | 1 | 5 | yes | — |
| 245 | C2×C8 | C25 | 1 | 10 | yes | — |
| 246 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 247 | C4⋊C4 | — | 2 | 40 | no | — |
| 248 | C4⋊C4 | — | 2 | 20 | no | — |
| 249 | C4⋊C4 | — | 2 | 20 | no | — |
| 250 | C4⋊C4 | — | 2 | 20 | no | — |
| 251 | C4⋊C4 | — | 2 | 40 | no | — |
| 252 | C4⋊C4 | — | 2 | 40 | no | — |
| 253 | C4⋊C4 | — | 2 | 40 | no | — |
| 254 | C4⋊C4 | — | 2 | 20 | no | — |
| 255 | C4⋊C4 | — | 2 | 20 | no | — |
| 256 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 257 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 258 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 259 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 260 | C4⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 261 | C4⋊C4 | C25 | 1 | 20 | yes | — |
| 262 | C22⋊C4 | — | 2 | 40 | no | — |
| 263 | C22⋊C4 | — | 2 | 10 | no | — |
| 264 | C22⋊C4 | — | 2 | 20 | no | — |
| 265 | C22⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 266 | C22⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 267 | C42 | — | 2 | 10 | no | — |
| 268 | C42 | — | 2 | 20 | no | — |
| 269 | C42 | — | 2 | 40 | no | — |
| 270 | C42 | C22×D4 | 1 | 5 | yes | — |
| 271 | C42 | C22×D4 | 1 | 5 | yes | — |
| 272 | C42 | C25 | 1 | 10 | yes | — |
| 273 | C23 | — | 2 | 2 | no | — |
| 274 | C23 | C23×D4 | 1 | 5 | yes | — |
| 275 | Q8 | — | 2 | 10 | no | — |
| 276 | Q8 | — | 2 | 10 | no | — |
| 277 | Q8 | — | 4 | 10 | no | — |
| 278 | Q8 | — | 8 | 20 | no | — |
| 279 | D4 | — | 2 | 2 | no | — |
| 280 | D4 | — | 4 | 10 | no | — |
| 281 | D4 | — | 4 | 20 | no | — |
| 282 | D4 | C2×2−1+4 | 1 | 4 | yes | — |
| 283 | C2×C4 | — | 2 | 5 | no | — |
| 284 | C2×C4 | — | 2 | 5 | no | — |
| 285 | C2×C4 | — | 2 | 10 | no | — |
| 286 | C2×C4 | — | 2 | 20 | no | — |
| 287 | C2×C4 | — | 4 | 10 | no | — |
| 288 | C2×C4 | — | 4 | 20 | no | — |
| 289 | C2×C4 | — | 8 | 20 | no | — |
| 290 | C2×C4 | — | 8 | 20 | no | — |
| 291 | C2×C4 | C23×D4 | 1 | 10 | yes | — |
| 292 | C2×C4 | C23×D4 | 1 | 5 | yes | — |
| 293 | C2×C4 | C23×D4 | 1 | 10 | yes | — |
| 294 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 5 | no | — |
| 296 | C8 | — | 2 | 10 | no | — |
| 297 | C8 | C2×2−1+4 | 1 | 2 | yes | — |
| 298 | C22 | — | 2 | 5 | no | — |
| 299 | C22 | — | 2 | 4 | no | — |
| 300 | C22 | — | 2 | 10 | no | — |
| 301 | C22 | — | 4 | 4 | no | — |
| 302 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 2 | 5 | no | — |
| 304 | C4 | — | 2 | 10 | no | — |
| 305 | C4 | — | 2 | 10 | no | — |
| 306 | C4 | — | 4 | 10 | no | — |
| 307 | C4 | — | 4 | 20 | no | — |
| 308 | C4 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 309 | C4 | 2−1+6 | 1 | 1 | yes | characteristic |
| 310 | C2 | — | 4 | 5 | no | — |
| 311 | C2 | — | 4 | 4 | no | — |
| 312 | C2 | D4×2−1+4 | 1 | 1 | yes | characteristic |
| 313 | C2 | #63093 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3531 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63067 | C2 | semidirect |
| 2. #62493 | C2 | semidirect |
| 3. #62374 | C2 | semidirect |
| 4. #62237 | C2 | semidirect |
| 5. #61877 | C2 | semidirect |
| 6. #61477 | C2 | semidirect |
| 7. #61376 | C2 | semidirect |
| 8. #60332 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63067 | 2 |
| #62493 | 2 |
| #62374 | 2 |
| #62237 | 2 |
| #61877 | 2 |
| #61477 | 2 |
| #61376 | 2 |
| #60332 | 2 |