Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5883 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #63070 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #62492 | C2 | 1 | 10 | yes | maximal, complemented |
| 3 | #62373 | C2 | 1 | 5 | yes | maximal, complemented |
| 4 | #62242 | C2 | 1 | 20 | yes | maximal, complemented |
| 5 | #61876 | C2 | 1 | 10 | yes | maximal, complemented |
| 6 | #61479 | C2 | 1 | 10 | yes | maximal, complemented |
| 7 | #61376 | C2 | 1 | 5 | yes | maximal, complemented |
| 8 | #60332 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #3559 | — | 2 | 20 | no | — |
| 10 | #3559 | C22 | 1 | 20 | yes | — |
| 11 | #3548 | — | 2 | 10 | no | — |
| 12 | #3548 | C22 | 1 | 10 | yes | — |
| 13 | #3531 | — | 2 | 2 | no | — |
| 14 | #3531 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3517 | C22 | 1 | 20 | yes | — |
| 16 | #3504 | C22 | 1 | 10 | yes | — |
| 17 | #3391 | C22 | 1 | 10 | yes | — |
| 18 | #3389 | C22 | 1 | 20 | yes | — |
| 19 | #3388 | C22 | 1 | 40 | yes | — |
| 20 | #3383 | C22 | 1 | 10 | yes | — |
| 21 | #3380 | C22 | 1 | 20 | yes | — |
| 22 | #3368 | C22 | 1 | 20 | yes | — |
| 23 | #3366 | C22 | 1 | 40 | yes | — |
| 24 | #3355 | C22 | 1 | 20 | yes | — |
| 25 | #3345 | C22 | 1 | 40 | yes | — |
| 26 | #3339 | C22 | 1 | 20 | yes | — |
| 27 | #3334 | C22 | 1 | 10 | yes | — |
| 28 | #3332 | C22 | 1 | 20 | yes | — |
| 29 | #3309 | C22 | 1 | 40 | yes | — |
| 30 | #3287 | C22 | 1 | 10 | yes | — |
| 31 | #3259 | C22 | 1 | 20 | yes | — |
| 32 | #3253 | C22 | 1 | 10 | yes | — |
| 33 | #3241 | C22 | 1 | 40 | yes | — |
| 34 | #3234 | C22 | 1 | 20 | yes | — |
| 35 | #3176 | C22 | 1 | 20 | yes | — |
| 36 | #3169 | C22 | 1 | 10 | yes | — |
| 37 | #3164 | C22 | 1 | 10 | yes | — |
| 38 | C2×#499 | C22 | 1 | 5 | yes | — |
| 39 | #3119 | C22 | 1 | 40 | yes | — |
| 40 | #3115 | 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 | D4 | 1 | 1 | yes | characteristic |
| 51 | #563 | — | 2 | 20 | no | — |
| 52 | #563 | D4 | 1 | 10 | yes | — |
| 53 | #562 | — | 2 | 10 | no | — |
| 54 | #562 | — | 2 | 10 | no | — |
| 55 | #562 | — | 2 | 20 | no | — |
| 56 | #562 | D4 | 1 | 5 | yes | — |
| 57 | #557 | C23 | 1 | 20 | yes | — |
| 58 | #554 | — | 2 | 40 | no | — |
| 59 | #554 | C23 | 1 | 40 | yes | — |
| 60 | #553 | — | 2 | 40 | no | — |
| 61 | #553 | C23 | 1 | 40 | yes | — |
| 62 | #550 | — | 2 | 20 | no | — |
| 63 | #550 | C23 | 1 | 20 | yes | — |
| 64 | #550 | C23 | 1 | 20 | yes | — |
| 65 | #547 | — | 2 | 20 | no | — |
| 66 | #547 | — | 2 | 40 | no | — |
| 67 | #547 | — | 2 | 20 | no | — |
| 68 | #547 | C23 | 1 | 20 | yes | — |
| 69 | #544 | — | 2 | 20 | no | — |
| 70 | #544 | C23 | 1 | 20 | yes | — |
| 71 | #540 | — | 2 | 40 | no | — |
| 72 | #540 | C23 | 1 | 40 | yes | — |
| 73 | #539 | — | 2 | 40 | no | — |
| 74 | #539 | — | 2 | 20 | no | — |
| 75 | #537 | — | 2 | 40 | no | — |
| 76 | #537 | — | 2 | 20 | no | — |
| 77 | #533 | — | 2 | 10 | no | — |
| 78 | #533 | C23 | 1 | 5 | yes | — |
| 79 | #532 | — | 2 | 20 | no | — |
| 80 | #532 | C23 | 1 | 20 | yes | — |
| 81 | #529 | — | 2 | 20 | no | — |
| 82 | #529 | C23 | 1 | 10 | yes | — |
| 83 | C2×#125 | — | 2 | 10 | 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 | #493 | C23 | 1 | 10 | yes | — |
| 92 | #490 | C23 | 1 | 40 | yes | — |
| 93 | #486 | C23 | 1 | 20 | yes | — |
| 94 | #482 | C23 | 1 | 40 | yes | — |
| 95 | #481 | C23 | 1 | 40 | yes | — |
| 96 | #480 | C23 | 1 | 40 | yes | — |
| 97 | #479 | C23 | 1 | 40 | yes | — |
| 98 | #478 | C23 | 1 | 40 | yes | — |
| 99 | #478 | C23 | 1 | 40 | yes | — |
| 100 | #478 | C23 | 1 | 40 | yes | — |
| 101 | #476 | C23 | 1 | 40 | yes | — |
| 102 | #475 | C23 | 1 | 20 | yes | — |
| 103 | #474 | C23 | 1 | 20 | yes | — |
| 104 | #469 | C23 | 1 | 40 | yes | — |
| 105 | #466 | C23 | 1 | 20 | yes | — |
| 106 | #466 | C23 | 1 | 20 | yes | — |
| 107 | #460 | C23 | 1 | 20 | yes | — |
| 108 | #459 | C23 | 1 | 20 | yes | — |
| 109 | #459 | C23 | 1 | 40 | yes | — |
| 110 | #449 | C23 | 1 | 40 | yes | — |
| 111 | #446 | C23 | 1 | 20 | yes | — |
| 112 | #446 | C23 | 1 | 10 | yes | — |
| 113 | C8×Q8 | C23 | 1 | 10 | yes | — |
| 114 | #442 | C23 | 1 | 40 | yes | — |
| 115 | #440 | C23 | 1 | 40 | yes | — |
| 116 | C4×Q16 | C23 | 1 | 10 | yes | — |
| 117 | C4×SD16 | C23 | 1 | 20 | yes | — |
| 118 | C4×D8 | C23 | 1 | 10 | yes | — |
| 119 | C4×D8 | C23 | 1 | 20 | yes | — |
| 120 | C4×D8 | C23 | 1 | 20 | 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 | — | 2 | 20 | no | — |
| 137 | C2×#41 | — | 2 | 20 | no | — |
| 138 | C2×#41 | — | 2 | 20 | no | — |
| 139 | C2×#41 | C24 | 1 | 10 | yes | — |
| 140 | #135 | — | 2 | 20 | no | — |
| 141 | C2×Q16 | C24 | 1 | 1 | yes | characteristic |
| 142 | C2×SD16 | C24 | 1 | 10 | yes | — |
| 143 | C2×D8 | C24 | 1 | 5 | yes | — |
| 144 | C2×D8 | C24 | 1 | 20 | yes | — |
| 145 | C2×M16 | C24 | 1 | 10 | yes | — |
| 146 | C22×C8 | C24 | 1 | 5 | yes | — |
| 147 | C22×C8 | C24 | 1 | 5 | yes | — |
| 148 | C4⋊Q8 | — | 2 | 20 | no | — |
| 149 | C4⋊Q8 | — | 2 | 20 | no | — |
| 150 | C4⋊Q8 | C2×D4 | 1 | 10 | yes | — |
| 151 | C4⋊Q8 | C24 | 1 | 20 | yes | — |
| 152 | #127 | — | 2 | 20 | no | — |
| 153 | #127 | C24 | 1 | 20 | yes | — |
| 154 | #126 | — | 2 | 40 | no | — |
| 155 | #126 | C2×D4 | 1 | 20 | yes | — |
| 156 | #125 | — | 2 | 20 | no | — |
| 157 | #125 | — | 2 | 40 | no | — |
| 158 | #125 | — | 2 | 10 | no | — |
| 159 | #125 | — | 2 | 10 | no | — |
| 160 | #125 | C2×D4 | 1 | 5 | yes | — |
| 161 | #125 | C2×D4 | 1 | 10 | yes | — |
| 162 | #125 | C2×D4 | 1 | 5 | yes | — |
| 163 | #125 | C24 | 1 | 40 | yes | — |
| 164 | #124 | — | 2 | 20 | no | — |
| 165 | #124 | C2×D4 | 1 | 10 | yes | — |
| 166 | #123 | — | 2 | 40 | no | — |
| 167 | #123 | — | 2 | 40 | no | — |
| 168 | #123 | — | 2 | 40 | no | — |
| 169 | #123 | — | 2 | 40 | no | — |
| 170 | #123 | C2×D4 | 1 | 20 | yes | — |
| 171 | C22⋊Q8 | — | 2 | 40 | no | — |
| 172 | C22⋊Q8 | — | 2 | 20 | no | — |
| 173 | C22⋊Q8 | — | 2 | 40 | no | — |
| 174 | C22⋊Q8 | — | 2 | 40 | no | — |
| 175 | C22⋊Q8 | — | 2 | 40 | no | — |
| 176 | C22⋊Q8 | C2×D4 | 1 | 10 | yes | — |
| 177 | C22⋊Q8 | C2×D4 | 1 | 20 | yes | — |
| 178 | #121 | — | 2 | 40 | no | — |
| 179 | #121 | — | 2 | 20 | no | — |
| 180 | #121 | — | 2 | 20 | no | — |
| 181 | #121 | — | 2 | 40 | no | — |
| 182 | #121 | — | 2 | 40 | no | — |
| 183 | #121 | — | 2 | 40 | no | — |
| 184 | #121 | — | 2 | 40 | no | — |
| 185 | #121 | C2×D4 | 1 | 10 | yes | — |
| 186 | #121 | C24 | 1 | 40 | yes | — |
| 187 | C4×Q8 | — | 2 | 20 | no | — |
| 188 | C4×Q8 | C24 | 1 | 20 | 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 | — | 2 | 40 | no | — |
| 194 | C4×D4 | — | 2 | 20 | no | — |
| 195 | C4×D4 | C24 | 1 | 20 | yes | — |
| 196 | C4×D4 | C24 | 1 | 40 | yes | — |
| 197 | C4×D4 | C24 | 1 | 10 | yes | — |
| 198 | #117 | — | 2 | 20 | no | — |
| 199 | #117 | — | 2 | 10 | no | — |
| 200 | #117 | C24 | 1 | 10 | yes | — |
| 201 | #117 | C24 | 1 | 10 | yes | — |
| 202 | C2×C4⋊C4 | — | 2 | 20 | no | — |
| 203 | C2×C4⋊C4 | — | 2 | 20 | no | — |
| 204 | C2×C4⋊C4 | — | 2 | 40 | 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 | #102 | C24 | 1 | 20 | yes | — |
| 218 | #102 | C24 | 1 | 40 | yes | — |
| 219 | #102 | C24 | 1 | 40 | 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 | #41 | — | 16 | 20 | no | — |
| 227 | #41 | — | 16 | 20 | no | — |
| 228 | #41 | — | 16 | 20 | no | — |
| 229 | C2×Q8 | — | 2 | 20 | no | — |
| 230 | C2×Q8 | C22×D4 | 1 | 10 | yes | — |
| 231 | C2×Q8 | C25 | 1 | 2 | yes | — |
| 232 | C2×D4 | — | 2 | 10 | no | — |
| 233 | C2×D4 | — | 2 | 40 | no | — |
| 234 | C2×D4 | — | 2 | 20 | no | — |
| 235 | C2×D4 | — | 2 | 20 | no | — |
| 236 | C2×D4 | C22×D4 | 1 | 10 | yes | — |
| 237 | C2×D4 | C25 | 1 | 10 | yes | — |
| 238 | C22×C4 | — | 2 | 20 | no | — |
| 239 | C22×C4 | — | 2 | 10 | no | — |
| 240 | C22×C4 | — | 2 | 20 | no | — |
| 241 | C22×C4 | — | 2 | 40 | no | — |
| 242 | C22×C4 | C22×D4 | 1 | 10 | yes | — |
| 243 | C22×C4 | C25 | 1 | 5 | yes | — |
| 244 | Q16 | 2−1+4 | 1 | 4 | yes | — |
| 245 | SD16 | — | 2 | 20 | no | — |
| 246 | D8 | — | 2 | 10 | no | — |
| 247 | D8 | — | 4 | 20 | no | — |
| 248 | M16 | — | 4 | 10 | no | — |
| 249 | C2×C8 | — | 2 | 10 | no | — |
| 250 | C2×C8 | — | 4 | 5 | no | — |
| 251 | C2×C8 | C25 | 1 | 5 | yes | — |
| 252 | C2×C8 | C25 | 1 | 10 | yes | — |
| 253 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 254 | C4⋊C4 | — | 2 | 20 | no | — |
| 255 | C4⋊C4 | — | 2 | 20 | no | — |
| 256 | C4⋊C4 | — | 2 | 40 | no | — |
| 257 | C4⋊C4 | — | 2 | 40 | no | — |
| 258 | C4⋊C4 | — | 2 | 20 | no | — |
| 259 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 260 | C4⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 261 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 262 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 263 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 264 | C4⋊C4 | C25 | 1 | 20 | yes | — |
| 265 | C22⋊C4 | — | 2 | 10 | no | — |
| 266 | C22⋊C4 | — | 2 | 40 | no | — |
| 267 | C22⋊C4 | — | 2 | 20 | no | — |
| 268 | C22⋊C4 | — | 2 | 40 | no | — |
| 269 | C22⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 270 | C22⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 271 | C42 | — | 2 | 20 | no | — |
| 272 | C42 | — | 2 | 10 | no | — |
| 273 | C42 | C22×D4 | 1 | 5 | yes | — |
| 274 | C42 | C22×D4 | 1 | 5 | yes | — |
| 275 | C42 | C25 | 1 | 10 | yes | — |
| 276 | C23 | — | 2 | 10 | no | — |
| 277 | C23 | C23×D4 | 1 | 5 | yes | — |
| 278 | Q8 | — | 2 | 2 | no | — |
| 279 | Q8 | — | 4 | 10 | no | — |
| 280 | Q8 | — | 8 | 20 | no | — |
| 281 | Q8 | C2×2−1+4 | 1 | 4 | yes | — |
| 282 | D4 | — | 2 | 10 | no | — |
| 283 | D4 | — | 2 | 10 | no | — |
| 284 | D4 | — | 4 | 10 | no | — |
| 285 | D4 | — | 4 | 20 | no | — |
| 286 | D4 | — | 8 | 20 | no | — |
| 287 | D4 | — | 8 | 20 | no | — |
| 288 | D4 | — | 8 | 40 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 5 | no | — |
| 291 | C2×C4 | — | 2 | 5 | no | — |
| 292 | C2×C4 | — | 2 | 20 | no | — |
| 293 | C2×C4 | — | 4 | 10 | no | — |
| 294 | C2×C4 | — | 4 | 20 | no | — |
| 295 | C2×C4 | — | 8 | 40 | no | — |
| 296 | C2×C4 | — | 8 | 20 | no | — |
| 297 | C2×C4 | — | 8 | 20 | no | — |
| 298 | C2×C4 | C23×D4 | 1 | 5 | yes | — |
| 299 | C2×C4 | C23×D4 | 1 | 10 | yes | — |
| 300 | C2×C4 | C23×D4 | 1 | 10 | yes | — |
| 301 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 2 | 5 | no | — |
| 303 | C8 | — | 2 | 10 | no | — |
| 304 | C8 | C2×2−1+4 | 1 | 2 | yes | — |
| 305 | C22 | — | 2 | 5 | no | — |
| 306 | C22 | — | 2 | 10 | no | — |
| 307 | C22 | — | 4 | 10 | no | — |
| 308 | C22 | — | 4 | 20 | no | — |
| 309 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 2 | 5 | no | — |
| 311 | C4 | — | 2 | 4 | no | — |
| 312 | C4 | — | 2 | 10 | no | — |
| 313 | C4 | — | 2 | 10 | no | — |
| 314 | C4 | — | 4 | 20 | no | — |
| 315 | C4 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 316 | C4 | 2−1+6 | 1 | 1 | yes | characteristic |
| 317 | C2 | — | 4 | 5 | no | — |
| 318 | C2 | — | 8 | 10 | no | — |
| 319 | C2 | D4×2−1+4 | 1 | 1 | yes | characteristic |
| 320 | C2 | #63091 | 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. #63070 | C2 | semidirect |
| 2. #62492 | C2 | semidirect |
| 3. #62373 | C2 | semidirect |
| 4. #62242 | C2 | semidirect |
| 5. #61876 | C2 | semidirect |
| 6. #61479 | 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 |
|---|---|
| #63070 | 2 |
| #62492 | 2 |
| #62373 | 2 |
| #62242 | 2 |
| #61876 | 2 |
| #61479 | 2 |
| #61376 | 2 |
| #60332 | 2 |