← #219657
Subgroups — #219657
Order 512.
470 conjugacy classes of subgroups.
The subgroup lattice has 311 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #19712 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #17972 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #17970 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #9765 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #9764 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9285 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #9284 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #9282 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #9279 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #8798 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #8141 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3228 | C22 | 1 | 2 | yes | — |
| 13 | #3210 | C22 | 1 | 2 | yes | — |
| 14 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2985 | C22 | 1 | 2 | yes | — |
| 17 | #2981 | C22 | 1 | 2 | yes | — |
| 18 | #1746 | C22 | 1 | 1 | yes | characteristic |
| 19 | #1723 | C4 | 1 | 2 | yes | complemented |
| 20 | #1722 | C4 | 1 | 2 | yes | complemented |
| 21 | #1720 | C4 | 1 | 2 | yes | complemented |
| 22 | #1717 | C4 | 1 | 2 | yes | complemented |
| 23 | #1594 | — | 2 | 1 | no | — |
| 24 | #1593 | — | 2 | 1 | no | — |
| 25 | #1591 | — | 2 | 1 | no | — |
| 26 | #1589 | — | 2 | 1 | no | — |
| 27 | #1580 | — | 2 | 1 | no | — |
| 28 | #1579 | — | 2 | 1 | no | — |
| 29 | C8×Q16 | — | 2 | 1 | no | — |
| 30 | C8×D8 | — | 2 | 1 | no | — |
| 31 | #1568 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1568 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1531 | C22 | 1 | 2 | yes | — |
| 34 | #1530 | C22 | 1 | 2 | yes | — |
| 35 | #1506 | C22 | 1 | 2 | yes | — |
| 36 | #1505 | C22 | 1 | 2 | yes | — |
| 37 | #1497 | C22 | 1 | 2 | yes | — |
| 38 | #1496 | C22 | 1 | 2 | yes | — |
| 39 | #1452 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1287 | C22 | 1 | 2 | yes | — |
| 42 | #1287 | C22 | 1 | 2 | yes | — |
| 43 | #1275 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1275 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1274 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1274 | C22 | 1 | 1 | yes | characteristic |
| 47 | #533 | C23 | 1 | 2 | yes | — |
| 48 | #532 | C23 | 1 | 2 | yes | — |
| 49 | #500 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #500 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | #500 | C2×C4 | 1 | 2 | yes | — |
| 52 | #498 | C2×C4 | 1 | 2 | yes | — |
| 53 | #494 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #494 | C2×C4 | 1 | 2 | yes | — |
| 55 | #493 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | #492 | C2×C4 | 1 | 2 | yes | — |
| 57 | #487 | C2×C4 | 1 | 2 | yes | — |
| 58 | #487 | C2×C4 | 1 | 2 | yes | — |
| 59 | #486 | C2×C4 | 1 | 2 | yes | — |
| 60 | #486 | C2×C4 | 1 | 2 | yes | — |
| 61 | #486 | C2×C4 | 1 | 2 | yes | — |
| 62 | #486 | C2×C4 | 1 | 2 | yes | — |
| 63 | #482 | — | 4 | 2 | no | — |
| 64 | #478 | — | 4 | 2 | no | — |
| 65 | #467 | — | 2 | 1 | no | — |
| 66 | #467 | — | 2 | 1 | no | — |
| 67 | #466 | — | 2 | 1 | no | — |
| 68 | #466 | — | 2 | 1 | no | — |
| 69 | #464 | — | 4 | 2 | no | — |
| 70 | #452 | — | 4 | 2 | no | — |
| 71 | #446 | — | 2 | 2 | no | — |
| 72 | #446 | — | 4 | 2 | no | — |
| 73 | C8×Q8 | — | 2 | 2 | no | — |
| 74 | C4×Q16 | — | 2 | 1 | no | — |
| 75 | C4×Q16 | — | 2 | 1 | no | — |
| 76 | C4×D8 | — | 2 | 1 | no | — |
| 77 | C4×D8 | — | 2 | 1 | no | — |
| 78 | #436 | — | 2 | 2 | no | — |
| 79 | #435 | — | 4 | 2 | no | — |
| 80 | C8×D4 | — | 2 | 2 | no | — |
| 81 | #433 | — | 4 | 2 | no | — |
| 82 | #432 | C23 | 1 | 2 | yes | — |
| 83 | #432 | C23 | 1 | 2 | yes | — |
| 84 | #431 | C23 | 1 | 1 | yes | characteristic |
| 85 | #431 | C23 | 1 | 1 | yes | characteristic |
| 86 | #431 | C23 | 1 | 1 | yes | characteristic |
| 87 | #431 | C23 | 1 | 1 | yes | characteristic |
| 88 | #408 | — | 4 | 2 | no | — |
| 89 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 91 | #376 | C23 | 1 | 1 | yes | characteristic |
| 92 | #336 | — | 2 | 1 | no | — |
| 93 | #336 | — | 2 | 1 | no | — |
| 94 | #335 | — | 2 | 1 | no | — |
| 95 | #335 | — | 2 | 1 | no | — |
| 96 | #334 | — | 2 | 1 | no | — |
| 97 | #334 | — | 2 | 1 | no | — |
| 98 | #330 | — | 4 | 2 | no | — |
| 99 | #329 | — | 4 | 2 | no | — |
| 100 | #328 | — | 4 | 2 | no | — |
| 101 | #327 | — | 4 | 2 | no | — |
| 102 | Q8⋊C8 | — | 2 | 2 | no | — |
| 103 | Q8⋊C8 | — | 2 | 2 | no | — |
| 104 | D4⋊C8 | — | 2 | 2 | no | — |
| 105 | D4⋊C8 | — | 2 | 2 | no | — |
| 106 | #322 | C2×C4 | 1 | 2 | yes | — |
| 107 | #322 | C2×C4 | 1 | 2 | yes | — |
| 108 | #322 | D4 | 1 | 1 | yes | characteristic |
| 109 | #322 | D4 | 1 | 1 | yes | characteristic |
| 110 | C82 | D4 | 1 | 1 | yes | characteristic |
| 111 | C82 | D4 | 1 | 1 | yes | characteristic |
| 112 | C2×#41 | — | 2 | 2 | no | — |
| 113 | C2×Q16 | — | 2 | 1 | no | — |
| 114 | C2×Q16 | — | 2 | 1 | no | — |
| 115 | C2×Q16 | — | 4 | 2 | no | — |
| 116 | C2×SD16 | — | 4 | 2 | no | — |
| 117 | C2×D8 | — | 2 | 1 | no | — |
| 118 | C2×D8 | — | 2 | 1 | no | — |
| 119 | C2×M16 | — | 4 | 2 | no | — |
| 120 | C22×C8 | — | 2 | 1 | no | — |
| 121 | C22×C8 | — | 4 | 2 | no | — |
| 122 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 123 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 124 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 125 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 126 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 127 | #127 | C22×C4 | 1 | 2 | yes | — |
| 128 | #125 | — | 2 | 2 | no | — |
| 129 | #124 | — | 2 | 2 | no | — |
| 130 | C22⋊Q8 | — | 2 | 2 | no | — |
| 131 | C22⋊Q8 | — | 2 | 2 | no | — |
| 132 | #121 | — | 2 | 2 | no | — |
| 133 | #121 | — | 2 | 2 | no | — |
| 134 | C4×Q8 | — | 2 | 2 | no | — |
| 135 | C4×Q8 | — | 2 | 2 | no | — |
| 136 | C4×D4 | — | 2 | 2 | no | — |
| 137 | C4×D4 | — | 2 | 2 | no | — |
| 138 | #117 | — | 2 | 2 | no | — |
| 139 | C2×C42 | — | 2 | 1 | no | — |
| 140 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 141 | #107 | — | 2 | 1 | no | — |
| 142 | #107 | — | 2 | 1 | no | — |
| 143 | #107 | — | 2 | 1 | no | — |
| 144 | #107 | — | 2 | 1 | no | — |
| 145 | #107 | — | 4 | 2 | no | — |
| 146 | #106 | — | 4 | 2 | no | — |
| 147 | C4⋊C8 | — | 2 | 2 | no | — |
| 148 | C4⋊C8 | — | 2 | 2 | no | — |
| 149 | C4⋊C8 | — | 2 | 1 | no | — |
| 150 | C4⋊C8 | — | 2 | 1 | no | — |
| 151 | C4⋊C8 | — | 2 | 1 | no | — |
| 152 | C4⋊C8 | — | 2 | 1 | no | — |
| 153 | C4⋊C8 | — | 4 | 2 | no | — |
| 154 | C4⋊C8 | — | 4 | 2 | no | — |
| 155 | #103 | — | 2 | 2 | no | — |
| 156 | #103 | — | 2 | 2 | no | — |
| 157 | #103 | — | 4 | 2 | no | — |
| 158 | #102 | — | 2 | 2 | no | — |
| 159 | #102 | — | 2 | 2 | no | — |
| 160 | #102 | — | 4 | 2 | no | — |
| 161 | #102 | — | 4 | 2 | no | — |
| 162 | #102 | — | 4 | 2 | no | — |
| 163 | C22⋊C8 | — | 2 | 1 | no | — |
| 164 | C22⋊C8 | — | 2 | 1 | no | — |
| 165 | C22⋊C8 | — | 2 | 1 | no | — |
| 166 | C22⋊C8 | — | 2 | 1 | no | — |
| 167 | C22⋊C8 | — | 2 | 2 | no | — |
| 168 | C22⋊C8 | — | 2 | 2 | no | — |
| 169 | C22⋊C8 | — | 4 | 2 | no | — |
| 170 | C22⋊C8 | — | 4 | 2 | no | — |
| 171 | #97 | — | 2 | 1 | no | — |
| 172 | #97 | — | 2 | 2 | no | — |
| 173 | #97 | — | 2 | 1 | no | — |
| 174 | #97 | — | 2 | 2 | no | — |
| 175 | #97 | — | 4 | 2 | no | — |
| 176 | C4×C8 | — | 4 | 2 | no | — |
| 177 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 178 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 179 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 180 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 181 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 182 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 183 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 184 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 185 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 187 | #95 | — | 2 | 1 | no | — |
| 188 | #95 | — | 2 | 1 | no | — |
| 189 | #41 | — | 16 | 2 | no | — |
| 190 | C2×Q8 | — | 2 | 2 | no | — |
| 191 | C2×Q8 | — | 2 | 2 | no | — |
| 192 | C2×Q8 | — | 2 | 2 | no | — |
| 193 | C2×D4 | — | 2 | 2 | no | — |
| 194 | C2×D4 | — | 2 | 2 | no | — |
| 195 | C2×D4 | — | 2 | 2 | no | — |
| 196 | C2×D4 | — | 4 | 2 | no | — |
| 197 | C22×C4 | — | 4 | 2 | no | — |
| 198 | C22×C4 | — | 4 | 1 | no | — |
| 199 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 200 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 201 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 202 | Q16 | — | 4 | 2 | no | — |
| 203 | Q16 | — | 4 | 2 | no | — |
| 204 | Q16 | — | 16 | 2 | no | — |
| 205 | SD16 | — | 16 | 2 | no | — |
| 206 | D8 | — | 4 | 2 | no | — |
| 207 | D8 | — | 4 | 2 | no | — |
| 208 | M16 | — | 16 | 2 | no | — |
| 209 | C2×C8 | — | 2 | 2 | no | — |
| 210 | C2×C8 | — | 2 | 2 | no | — |
| 211 | C2×C8 | — | 2 | 1 | no | — |
| 212 | C2×C8 | — | 2 | 1 | no | — |
| 213 | C2×C8 | — | 2 | 1 | no | — |
| 214 | C2×C8 | — | 2 | 1 | no | — |
| 215 | C2×C8 | — | 4 | 1 | no | — |
| 216 | C2×C8 | — | 4 | 1 | no | — |
| 217 | C2×C8 | — | 4 | 1 | no | — |
| 218 | C2×C8 | — | 4 | 1 | no | — |
| 219 | C2×C8 | — | 4 | 2 | no | — |
| 220 | C2×C8 | — | 4 | 2 | no | — |
| 221 | C2×C8 | — | 4 | 1 | no | — |
| 222 | C2×C8 | — | 16 | 2 | no | — |
| 223 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 226 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 227 | C4⋊C4 | — | 2 | 2 | no | — |
| 228 | C4⋊C4 | — | 2 | 2 | no | — |
| 229 | C4⋊C4 | — | 2 | 2 | no | — |
| 230 | C4⋊C4 | — | 2 | 2 | no | — |
| 231 | C4⋊C4 | — | 2 | 2 | no | — |
| 232 | C4⋊C4 | — | 2 | 2 | no | — |
| 233 | C4⋊C4 | — | 4 | 2 | no | — |
| 234 | C4⋊C4 | — | 4 | 2 | no | — |
| 235 | C22⋊C4 | — | 4 | 2 | no | — |
| 236 | C22⋊C4 | — | 4 | 2 | no | — |
| 237 | C22⋊C4 | — | 4 | 2 | no | — |
| 238 | C42 | — | 4 | 1 | no | — |
| 239 | C42 | — | 4 | 1 | no | — |
| 240 | C42 | — | 4 | 2 | no | — |
| 241 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 242 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 243 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 244 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 245 | C23 | — | 4 | 2 | no | — |
| 246 | C23 | #540 | 1 | 1 | yes | characteristic |
| 247 | Q8 | — | 4 | 2 | no | — |
| 248 | Q8 | — | 4 | 2 | no | — |
| 249 | Q8 | — | 8 | 2 | no | — |
| 250 | Q8 | — | 8 | 2 | no | — |
| 251 | D4 | — | 4 | 2 | no | — |
| 252 | D4 | — | 4 | 2 | no | — |
| 253 | D4 | — | 8 | 2 | no | — |
| 254 | D4 | — | 8 | 2 | no | — |
| 255 | D4 | — | 16 | 2 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 1 | no | — |
| 260 | C2×C4 | — | 4 | 2 | no | — |
| 261 | C2×C4 | — | 4 | 2 | no | — |
| 262 | C2×C4 | — | 4 | 2 | no | — |
| 263 | C2×C4 | — | 8 | 2 | no | — |
| 264 | C2×C4 | — | 8 | 2 | no | — |
| 265 | C2×C4 | — | 8 | 1 | no | — |
| 266 | C2×C4 | — | 8 | 1 | no | — |
| 267 | C2×C4 | — | 8 | 1 | no | — |
| 268 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | #540 | 1 | 1 | yes | characteristic |
| 274 | C8 | — | 2 | 1 | no | — |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | — | 2 | 1 | no | — |
| 278 | C8 | — | 4 | 1 | no | — |
| 279 | C8 | — | 4 | 1 | no | — |
| 280 | C8 | — | 4 | 2 | no | — |
| 281 | C8 | — | 4 | 1 | no | — |
| 282 | C8 | — | 4 | 1 | no | — |
| 283 | C8 | — | 4 | 2 | no | — |
| 284 | C8 | — | 8 | 2 | no | — |
| 285 | C8 | — | 8 | 1 | no | — |
| 286 | C8 | — | 8 | 2 | no | — |
| 287 | C22 | — | 2 | 1 | no | — |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 2 | 1 | no | — |
| 290 | C22 | — | 8 | 2 | no | — |
| 291 | C22 | — | 8 | 2 | no | — |
| 292 | C22 | — | 8 | 2 | no | — |
| 293 | C22 | #2328 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 1 | no | — |
| 295 | C4 | — | 2 | 1 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 2 | 1 | no | — |
| 299 | C4 | — | 8 | 2 | no | — |
| 300 | C4 | — | 8 | 2 | no | — |
| 301 | C4 | — | 8 | 2 | no | — |
| 302 | C4 | — | 16 | 1 | no | — |
| 303 | C4 | #2955 | 1 | 1 | yes | characteristic |
| 304 | C4 | #2955 | 1 | 1 | yes | characteristic |
| 305 | C2 | — | 4 | 1 | no | — |
| 306 | C2 | — | 16 | 2 | no | — |
| 307 | C2 | #22281 | 1 | 1 | yes | characteristic |
| 308 | C2 | #22285 | 1 | 1 | yes | characteristic |
| 309 | C2 | #22286 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta