← #490624
Subgroups — #490624
Order 512.
441 conjugacy classes of subgroups.
The subgroup lattice has 321 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33697 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×Q8×C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19118 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19114 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #12478 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12477 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12476 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12475 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12008 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #11939 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #11929 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3104 | — | 2 | 1 | no | — |
| 15 | #3103 | — | 2 | 1 | no | — |
| 16 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2242 | — | 2 | 1 | no | — |
| 18 | #2242 | — | 2 | 1 | no | — |
| 19 | #2241 | C22 | 1 | 1 | yes | characteristic |
| 20 | #2241 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2240 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2240 | C22 | 1 | 1 | yes | characteristic |
| 23 | Q8×C16 | — | 2 | 1 | no | — |
| 24 | Q8×C16 | — | 2 | 1 | no | — |
| 25 | Q8×C16 | C22 | 1 | 2 | yes | — |
| 26 | Q8×C16 | C22 | 1 | 2 | yes | — |
| 27 | C2×C4⋊C16 | — | 2 | 1 | no | — |
| 28 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 29 | #2138 | — | 2 | 1 | no | — |
| 30 | #2138 | — | 2 | 1 | no | — |
| 31 | C2×C4×C16 | — | 2 | 1 | no | — |
| 32 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1676 | C22 | 1 | 2 | yes | — |
| 34 | #1674 | C22 | 1 | 2 | yes | — |
| 35 | #1672 | — | 2 | 2 | no | — |
| 36 | #1670 | C22 | 1 | 2 | yes | — |
| 37 | #1668 | C22 | 1 | 2 | yes | — |
| 38 | #1666 | — | 2 | 2 | no | — |
| 39 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1542 | — | 2 | 1 | no | — |
| 42 | #1540 | — | 2 | 1 | no | — |
| 43 | #1372 | C22 | 1 | 2 | yes | — |
| 44 | #1371 | C22 | 1 | 2 | yes | — |
| 45 | #1365 | C22 | 1 | 2 | yes | — |
| 46 | #1364 | C22 | 1 | 2 | yes | — |
| 47 | #1362 | C22 | 1 | 2 | yes | — |
| 48 | #1361 | C22 | 1 | 2 | yes | — |
| 49 | C2×#135 | — | 2 | 1 | no | — |
| 50 | C2×#135 | — | 2 | 1 | no | — |
| 51 | #533 | — | 2 | 1 | no | — |
| 52 | #532 | — | 2 | 1 | no | — |
| 53 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 54 | C22×C16 | — | 2 | 1 | no | — |
| 55 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 56 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 57 | #500 | C23 | 1 | 1 | yes | characteristic |
| 58 | #500 | C23 | 1 | 1 | yes | characteristic |
| 59 | #499 | — | 2 | 1 | no | — |
| 60 | #499 | — | 2 | 1 | no | — |
| 61 | #498 | — | 2 | 1 | no | — |
| 62 | #495 | — | 2 | 1 | no | — |
| 63 | #495 | — | 2 | 1 | no | — |
| 64 | #494 | C23 | 1 | 1 | yes | characteristic |
| 65 | #493 | C23 | 1 | 1 | yes | characteristic |
| 66 | #492 | — | 2 | 1 | no | — |
| 67 | #484 | — | 4 | 1 | no | — |
| 68 | #481 | — | 4 | 1 | no | — |
| 69 | #477 | — | 2 | 2 | no | — |
| 70 | #475 | — | 2 | 2 | no | — |
| 71 | #462 | — | 2 | 2 | no | — |
| 72 | #460 | — | 2 | 2 | no | — |
| 73 | #451 | — | 4 | 1 | no | — |
| 74 | #448 | — | 4 | 1 | no | — |
| 75 | C8×Q8 | — | 2 | 1 | no | — |
| 76 | C8×Q8 | — | 2 | 1 | no | — |
| 77 | C8×Q8 | D4 | 1 | 2 | yes | — |
| 78 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 79 | C4×Q16 | — | 4 | 2 | no | — |
| 80 | C4×SD16 | — | 4 | 2 | no | — |
| 81 | #431 | — | 2 | 1 | no | — |
| 82 | #431 | — | 2 | 1 | no | — |
| 83 | #427 | — | 2 | 1 | no | — |
| 84 | #427 | — | 2 | 1 | no | — |
| 85 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 86 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C8 | — | 2 | 1 | no | — |
| 88 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C4⋊C16 | — | 2 | 2 | no | — |
| 90 | C4⋊C16 | — | 2 | 2 | no | — |
| 91 | C4⋊C16 | — | 2 | 1 | no | — |
| 92 | C4⋊C16 | C23 | 1 | 2 | yes | — |
| 93 | #358 | — | 2 | 1 | no | — |
| 94 | #358 | — | 2 | 1 | no | — |
| 95 | #358 | — | 2 | 1 | no | — |
| 96 | #358 | — | 2 | 1 | no | — |
| 97 | #357 | — | 2 | 1 | no | — |
| 98 | #357 | — | 2 | 1 | no | — |
| 99 | #357 | — | 2 | 1 | no | — |
| 100 | #357 | — | 2 | 1 | no | — |
| 101 | C4×C16 | — | 2 | 2 | no | — |
| 102 | C4×C16 | — | 2 | 2 | no | — |
| 103 | C4×C16 | D4 | 1 | 1 | yes | characteristic |
| 104 | C4×C16 | D4 | 1 | 1 | yes | characteristic |
| 105 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 106 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 107 | #335 | C23 | 1 | 2 | yes | — |
| 108 | #335 | C23 | 1 | 2 | yes | — |
| 109 | #334 | — | 2 | 1 | no | — |
| 110 | #334 | — | 2 | 1 | no | — |
| 111 | #333 | — | 2 | 2 | no | — |
| 112 | #332 | — | 2 | 2 | no | — |
| 113 | #332 | — | 2 | 2 | no | — |
| 114 | #331 | — | 2 | 2 | no | — |
| 115 | C2×#41 | — | 4 | 1 | no | — |
| 116 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 117 | #135 | — | 4 | 2 | no | — |
| 118 | #135 | — | 4 | 2 | no | — |
| 119 | #135 | — | 8 | 2 | no | — |
| 120 | C2×Q16 | — | 2 | 1 | no | — |
| 121 | C2×Q16 | — | 2 | 1 | no | — |
| 122 | C2×Q16 | — | 4 | 2 | no | — |
| 123 | C2×SD16 | — | 4 | 2 | no | — |
| 124 | C2×SD16 | — | 4 | 1 | no | — |
| 125 | C2×SD16 | — | 4 | 1 | no | — |
| 126 | C2×D8 | — | 2 | 1 | no | — |
| 127 | C2×D8 | — | 2 | 1 | no | — |
| 128 | C22×C8 | — | 2 | 1 | no | — |
| 129 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 131 | C4⋊Q8 | — | 2 | 1 | no | — |
| 132 | C4⋊Q8 | — | 2 | 1 | no | — |
| 133 | C4⋊Q8 | — | 2 | 1 | no | — |
| 134 | #127 | — | 2 | 1 | no | — |
| 135 | #125 | — | 4 | 1 | no | — |
| 136 | #124 | — | 4 | 1 | no | — |
| 137 | C22⋊Q8 | — | 4 | 1 | no | — |
| 138 | C22⋊Q8 | — | 4 | 1 | no | — |
| 139 | #121 | — | 4 | 1 | no | — |
| 140 | #121 | — | 4 | 1 | no | — |
| 141 | C4×Q8 | — | 2 | 1 | no | — |
| 142 | C4×Q8 | — | 2 | 1 | no | — |
| 143 | C4×Q8 | — | 4 | 1 | no | — |
| 144 | C4×Q8 | — | 4 | 1 | no | — |
| 145 | C4×Q8 | D8 | 1 | 2 | yes | — |
| 146 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 147 | C4×D4 | — | 4 | 1 | no | — |
| 148 | C4×D4 | — | 4 | 1 | no | — |
| 149 | #117 | — | 4 | 1 | no | — |
| 150 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 151 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 152 | C2×C42 | — | 2 | 1 | no | — |
| 153 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 154 | C2×C16 | — | 2 | 2 | no | — |
| 155 | C2×C16 | — | 2 | 2 | no | — |
| 156 | C2×C16 | — | 2 | 2 | no | — |
| 157 | C2×C16 | — | 4 | 2 | no | — |
| 158 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 159 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 160 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 161 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 162 | #107 | — | 2 | 1 | no | — |
| 163 | #107 | — | 2 | 1 | no | — |
| 164 | #107 | — | 2 | 1 | no | — |
| 165 | #107 | — | 2 | 1 | no | — |
| 166 | #107 | — | 4 | 2 | no | — |
| 167 | #106 | — | 4 | 1 | no | — |
| 168 | #106 | — | 4 | 1 | no | — |
| 169 | #106 | — | 4 | 2 | no | — |
| 170 | C4⋊C8 | — | 2 | 2 | no | — |
| 171 | C4⋊C8 | — | 2 | 2 | no | — |
| 172 | C4⋊C8 | — | 2 | 1 | no | — |
| 173 | C4⋊C8 | — | 2 | 2 | no | — |
| 174 | C4⋊C8 | — | 2 | 2 | no | — |
| 175 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 176 | #103 | — | 4 | 2 | no | — |
| 177 | #103 | — | 4 | 2 | no | — |
| 178 | #103 | — | 4 | 2 | no | — |
| 179 | #102 | — | 4 | 2 | no | — |
| 180 | C22⋊C8 | — | 4 | 1 | no | — |
| 181 | C22⋊C8 | — | 4 | 1 | no | — |
| 182 | C4×C8 | — | 2 | 2 | no | — |
| 183 | C4×C8 | — | 2 | 2 | no | — |
| 184 | C4×C8 | — | 4 | 1 | no | — |
| 185 | C4×C8 | — | 4 | 1 | no | — |
| 186 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 188 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 189 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 190 | #41 | — | 8 | 2 | no | — |
| 191 | #41 | — | 8 | 2 | no | — |
| 192 | C2×Q8 | — | 2 | 4 | no | — |
| 193 | C2×Q8 | — | 2 | 1 | no | — |
| 194 | C2×Q8 | — | 4 | 1 | no | — |
| 195 | C2×Q8 | — | 4 | 1 | no | — |
| 196 | C2×Q8 | — | 4 | 1 | no | — |
| 197 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 198 | C2×D4 | — | 4 | 1 | no | — |
| 199 | C2×D4 | — | 4 | 1 | no | — |
| 200 | C2×D4 | — | 4 | 1 | no | — |
| 201 | C2×D4 | — | 8 | 1 | no | — |
| 202 | C22×C4 | — | 2 | 1 | no | — |
| 203 | C22×C4 | — | 2 | 1 | no | — |
| 204 | C22×C4 | — | 8 | 1 | no | — |
| 205 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 207 | C22×C4 | #135 | 1 | 1 | yes | characteristic |
| 208 | Q16 | — | 4 | 1 | no | — |
| 209 | Q16 | — | 4 | 1 | no | — |
| 210 | Q16 | — | 8 | 1 | no | — |
| 211 | Q16 | — | 8 | 4 | no | — |
| 212 | SD16 | — | 8 | 2 | no | — |
| 213 | SD16 | — | 8 | 4 | no | — |
| 214 | SD16 | — | 8 | 2 | no | — |
| 215 | D8 | — | 4 | 1 | no | — |
| 216 | D8 | — | 4 | 1 | no | — |
| 217 | D8 | — | 8 | 1 | no | — |
| 218 | C2×C8 | — | 2 | 2 | no | — |
| 219 | C2×C8 | — | 2 | 2 | no | — |
| 220 | C2×C8 | — | 4 | 2 | no | — |
| 221 | C2×C8 | — | 4 | 2 | no | — |
| 222 | C2×C8 | — | 4 | 2 | no | — |
| 223 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | #135 | 1 | 2 | yes | — |
| 226 | C2×C8 | #137 | 1 | 2 | yes | — |
| 227 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 228 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 229 | C4⋊C4 | — | 2 | 2 | no | — |
| 230 | C4⋊C4 | — | 2 | 2 | no | — |
| 231 | C4⋊C4 | — | 2 | 1 | no | — |
| 232 | C4⋊C4 | — | 4 | 1 | no | — |
| 233 | C4⋊C4 | — | 4 | 1 | no | — |
| 234 | C4⋊C4 | — | 4 | 1 | no | — |
| 235 | C4⋊C4 | — | 4 | 1 | no | — |
| 236 | C4⋊C4 | — | 4 | 1 | no | — |
| 237 | C4⋊C4 | — | 4 | 1 | no | — |
| 238 | C4⋊C4 | — | 8 | 1 | no | — |
| 239 | C4⋊C4 | — | 8 | 1 | no | — |
| 240 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 241 | C22⋊C4 | — | 8 | 1 | no | — |
| 242 | C22⋊C4 | — | 8 | 1 | no | — |
| 243 | C22⋊C4 | — | 8 | 1 | no | — |
| 244 | C42 | — | 2 | 2 | no | — |
| 245 | C42 | — | 2 | 2 | no | — |
| 246 | C42 | — | 8 | 1 | no | — |
| 247 | C42 | #121 | 1 | 1 | yes | characteristic |
| 248 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 249 | C42 | #135 | 1 | 1 | yes | characteristic |
| 250 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 251 | C16 | — | 2 | 1 | no | — |
| 252 | C16 | — | 2 | 1 | no | — |
| 253 | C16 | — | 2 | 1 | no | — |
| 254 | C16 | — | 2 | 1 | no | — |
| 255 | C16 | — | 4 | 2 | no | — |
| 256 | C23 | — | 8 | 1 | no | — |
| 257 | C23 | #466 | 1 | 1 | yes | characteristic |
| 258 | Q8 | — | 2 | 4 | no | — |
| 259 | Q8 | — | 2 | 2 | no | — |
| 260 | Q8 | — | 8 | 2 | no | — |
| 261 | Q8 | — | 8 | 1 | no | — |
| 262 | Q8 | — | 8 | 1 | no | — |
| 263 | Q8 | — | 16 | 1 | no | — |
| 264 | Q8 | #508 | 1 | 4 | yes | — |
| 265 | D4 | — | 8 | 2 | no | — |
| 266 | D4 | — | 8 | 1 | no | — |
| 267 | D4 | — | 8 | 1 | no | — |
| 268 | D4 | — | 16 | 1 | no | — |
| 269 | D4 | — | 16 | 2 | no | — |
| 270 | C2×C4 | — | 2 | 4 | no | — |
| 271 | C2×C4 | — | 2 | 2 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 2 | 2 | no | — |
| 274 | C2×C4 | — | 4 | 1 | no | — |
| 275 | C2×C4 | — | 4 | 2 | no | — |
| 276 | C2×C4 | — | 8 | 1 | no | — |
| 277 | C2×C4 | — | 8 | 1 | no | — |
| 278 | C2×C4 | — | 8 | 1 | no | — |
| 279 | C2×C4 | — | 16 | 2 | no | — |
| 280 | C2×C4 | #464 | 1 | 2 | yes | — |
| 281 | C2×C4 | #466 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | #466 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #466 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #471 | 1 | 2 | yes | — |
| 285 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 288 | C8 | — | 2 | 1 | no | — |
| 289 | C8 | — | 2 | 1 | no | — |
| 290 | C8 | — | 4 | 2 | no | — |
| 291 | C8 | — | 8 | 2 | no | — |
| 292 | C8 | — | 8 | 2 | no | — |
| 293 | C8 | #464 | 1 | 2 | yes | — |
| 294 | C8 | C2×#135 | 1 | 1 | yes | characteristic |
| 295 | C8 | C2×#137 | 1 | 1 | yes | characteristic |
| 296 | C22 | — | 2 | 1 | no | — |
| 297 | C22 | — | 2 | 1 | no | — |
| 298 | C22 | — | 16 | 1 | no | — |
| 299 | C22 | — | 16 | 1 | no | — |
| 300 | C22 | — | 16 | 1 | no | — |
| 301 | C22 | #1682 | 1 | 2 | yes | — |
| 302 | C22 | C2×#466 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 2 | 4 | no | — |
| 304 | C4 | — | 2 | 1 | no | — |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 2 | 1 | no | — |
| 307 | C4 | — | 4 | 2 | no | — |
| 308 | C4 | — | 16 | 1 | no | — |
| 309 | C4 | — | 16 | 1 | no | — |
| 310 | C4 | — | 16 | 1 | no | — |
| 311 | C4 | #1682 | 1 | 2 | yes | — |
| 312 | C4 | C2×#464 | 1 | 1 | yes | characteristic |
| 313 | C4 | C2×#471 | 1 | 1 | yes | characteristic |
| 314 | C4 | C2×#508 | 1 | 2 | yes | — |
| 315 | C2 | — | 2 | 2 | no | — |
| 316 | C2 | — | 32 | 1 | no | — |
| 317 | C2 | C2×#1682 | 1 | 1 | yes | characteristic |
| 318 | C2 | #33600 | 1 | 1 | yes | characteristic |
| 319 | C2 | #33617 | 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