← #7623933
Subgroups — #7623933
Order 512.
1248 conjugacy classes of subgroups.
The subgroup lattice has 339 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #61373 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33954 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #33951 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #33948 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | Q8×SD32 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #33730 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×#2254 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #33634 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #33629 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #33474 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #33460 | C2 | 1 | 2 | yes | maximal |
| 12 | C4×#508 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #3503 | — | 2 | 2 | no | — |
| 14 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3386 | — | 2 | 4 | no | — |
| 16 | #3385 | — | 2 | 4 | no | — |
| 17 | #3383 | C22 | 1 | 4 | yes | — |
| 18 | Q8×Q16 | C22 | 1 | 4 | yes | — |
| 19 | #3380 | C22 | 1 | 4 | yes | — |
| 20 | Q8×D8 | C22 | 1 | 4 | yes | — |
| 21 | #3164 | C22 | 1 | 4 | yes | — |
| 22 | C2×#500 | C22 | 1 | 2 | yes | — |
| 23 | #3083 | — | 2 | 4 | no | — |
| 24 | #3079 | — | 2 | 2 | no | — |
| 25 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 26 | #2951 | C22 | 1 | 4 | yes | — |
| 27 | C4×#135 | C22 | 1 | 2 | yes | — |
| 28 | #2255 | C22 | 1 | 8 | yes | — |
| 29 | #2254 | C22 | 1 | 4 | yes | — |
| 30 | #2254 | C22 | 1 | 4 | yes | — |
| 31 | #2254 | C22 | 1 | 4 | yes | — |
| 32 | #2253 | C22 | 1 | 4 | yes | — |
| 33 | #2229 | C22 | 1 | 8 | yes | — |
| 34 | #2228 | C22 | 1 | 8 | yes | — |
| 35 | #2227 | C22 | 1 | 4 | yes | — |
| 36 | #2226 | C22 | 1 | 4 | yes | — |
| 37 | #2226 | C22 | 1 | 8 | yes | — |
| 38 | #2225 | C22 | 1 | 4 | yes | — |
| 39 | #2225 | C22 | 1 | 8 | yes | — |
| 40 | #2224 | C22 | 1 | 4 | yes | — |
| 41 | #2183 | C22 | 1 | 4 | yes | — |
| 42 | Q8×C16 | C22 | 1 | 4 | yes | — |
| 43 | #2177 | C22 | 1 | 4 | yes | — |
| 44 | #2176 | C22 | 1 | 4 | yes | — |
| 45 | C4×Q32 | C22 | 1 | 2 | yes | — |
| 46 | C4×SD32 | C22 | 1 | 2 | yes | — |
| 47 | C4×SD32 | C22 | 1 | 8 | yes | — |
| 48 | C4×SD32 | C22 | 1 | 2 | yes | — |
| 49 | C4×D16 | C22 | 1 | 2 | yes | — |
| 50 | #2159 | C22 | 1 | 4 | yes | — |
| 51 | #2158 | C22 | 1 | 1 | yes | characteristic |
| 52 | C2×#367 | C22 | 1 | 2 | yes | — |
| 53 | C2×#367 | C22 | 1 | 2 | yes | — |
| 54 | #2151 | C22 | 1 | 2 | yes | — |
| 55 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 56 | #2139 | C22 | 1 | 4 | yes | — |
| 57 | #2138 | C22 | 1 | 2 | yes | — |
| 58 | #2108 | C22 | 1 | 2 | yes | — |
| 59 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 60 | C2×#135 | C23 | 1 | 2 | yes | — |
| 61 | Q82 | — | 2 | 4 | no | — |
| 62 | #557 | — | 2 | 4 | no | — |
| 63 | #554 | — | 2 | 4 | no | — |
| 64 | #554 | — | 4 | 4 | no | — |
| 65 | D4×Q8 | — | 2 | 4 | no | — |
| 66 | D4×Q8 | — | 4 | 4 | no | — |
| 67 | #544 | — | 2 | 4 | no | — |
| 68 | #544 | — | 4 | 4 | no | — |
| 69 | #533 | — | 4 | 2 | no | — |
| 70 | #533 | D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 72 | C2×C4⋊Q8 | — | 4 | 2 | no | — |
| 73 | #520 | — | 2 | 4 | no | — |
| 74 | C4×#41 | — | 2 | 2 | no | — |
| 75 | #508 | Q8 | 1 | 8 | yes | complemented |
| 76 | C2×Q32 | C23 | 1 | 1 | yes | characteristic |
| 77 | C2×SD32 | C23 | 1 | 2 | yes | — |
| 78 | C2×D16 | C23 | 1 | 1 | yes | characteristic |
| 79 | C2×M32 | C23 | 1 | 2 | yes | — |
| 80 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 81 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 82 | #501 | — | 2 | 8 | no | — |
| 83 | #500 | — | 2 | 2 | no | — |
| 84 | #500 | — | 2 | 2 | no | — |
| 85 | #500 | C23 | 1 | 8 | yes | — |
| 86 | #500 | C23 | 1 | 4 | yes | — |
| 87 | #500 | C23 | 1 | 4 | yes | — |
| 88 | #499 | C23 | 1 | 8 | yes | — |
| 89 | #479 | — | 2 | 8 | no | — |
| 90 | #478 | — | 2 | 8 | no | — |
| 91 | #477 | — | 2 | 4 | no | — |
| 92 | #477 | — | 2 | 8 | no | — |
| 93 | #476 | — | 2 | 4 | no | — |
| 94 | #475 | — | 2 | 4 | no | — |
| 95 | #474 | — | 2 | 4 | no | — |
| 96 | #474 | — | 2 | 8 | no | — |
| 97 | #446 | D4 | 1 | 4 | yes | — |
| 98 | C8×Q8 | C23 | 1 | 4 | yes | — |
| 99 | #440 | — | 2 | 8 | no | — |
| 100 | C4×Q16 | C23 | 1 | 2 | yes | — |
| 101 | C4×Q16 | C23 | 1 | 8 | yes | — |
| 102 | C4×Q16 | C23 | 1 | 2 | yes | — |
| 103 | C4×SD16 | — | 2 | 2 | no | — |
| 104 | C4×SD16 | — | 2 | 2 | no | — |
| 105 | C4×D8 | C23 | 1 | 2 | yes | — |
| 106 | C4×D8 | C23 | 1 | 8 | yes | — |
| 107 | C4×D8 | C23 | 1 | 2 | yes | — |
| 108 | #428 | — | 2 | 4 | no | — |
| 109 | #427 | C23 | 1 | 2 | yes | — |
| 110 | C2×#107 | — | 2 | 2 | no | — |
| 111 | C2×#107 | C23 | 1 | 4 | yes | — |
| 112 | #424 | D4 | 1 | 2 | yes | — |
| 113 | #423 | D4 | 1 | 1 | yes | characteristic |
| 114 | #417 | — | 2 | 4 | no | — |
| 115 | #416 | — | 2 | 2 | no | — |
| 116 | #405 | C23 | 1 | 2 | yes | — |
| 117 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 118 | #367 | C23 | 1 | 4 | yes | — |
| 119 | #367 | C23 | 1 | 4 | yes | — |
| 120 | #367 | C23 | 1 | 8 | yes | — |
| 121 | #367 | C23 | 1 | 1 | yes | characteristic |
| 122 | #367 | C23 | 1 | 1 | yes | characteristic |
| 123 | #367 | C23 | 1 | 8 | yes | — |
| 124 | #366 | C23 | 1 | 2 | yes | — |
| 125 | #366 | C23 | 1 | 8 | yes | — |
| 126 | C4⋊C16 | C23 | 1 | 4 | yes | — |
| 127 | C4⋊C16 | C23 | 1 | 8 | yes | — |
| 128 | #358 | C23 | 1 | 2 | yes | — |
| 129 | #358 | C23 | 1 | 2 | yes | — |
| 130 | #358 | C23 | 1 | 8 | yes | — |
| 131 | #357 | C23 | 1 | 2 | yes | — |
| 132 | #357 | C23 | 1 | 2 | yes | — |
| 133 | #357 | C23 | 1 | 8 | yes | — |
| 134 | #346 | C23 | 1 | 4 | yes | — |
| 135 | C4×C16 | C23 | 1 | 2 | yes | — |
| 136 | C4×C16 | C23 | 1 | 2 | yes | — |
| 137 | C4×C16 | C23 | 1 | 4 | yes | — |
| 138 | C2×#41 | — | 2 | 2 | no | — |
| 139 | C22×Q8 | — | 4 | 4 | no | — |
| 140 | C22×Q8 | — | 4 | 2 | no | — |
| 141 | #135 | — | 2 | 4 | no | — |
| 142 | #135 | C2×Q8 | 1 | 8 | yes | complemented |
| 143 | C2×Q16 | C24 | 1 | 2 | yes | — |
| 144 | C2×SD16 | — | 2 | 2 | no | — |
| 145 | C2×D8 | C24 | 1 | 2 | yes | — |
| 146 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 147 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 148 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 149 | C4⋊Q8 | — | 2 | 4 | no | — |
| 150 | C4⋊Q8 | — | 2 | 4 | no | — |
| 151 | C4⋊Q8 | — | 2 | 8 | no | — |
| 152 | C4⋊Q8 | — | 4 | 4 | no | — |
| 153 | C4⋊Q8 | — | 4 | 4 | no | — |
| 154 | C4⋊Q8 | — | 4 | 4 | no | — |
| 155 | C4⋊Q8 | — | 4 | 8 | no | — |
| 156 | C4⋊Q8 | — | 4 | 2 | no | — |
| 157 | C4⋊Q8 | — | 4 | 2 | no | — |
| 158 | C4⋊Q8 | D8 | 1 | 2 | yes | — |
| 159 | #125 | — | 2 | 8 | no | — |
| 160 | #125 | — | 4 | 4 | no | — |
| 161 | #125 | — | 4 | 8 | no | — |
| 162 | #125 | D8 | 1 | 2 | yes | — |
| 163 | C22⋊Q8 | — | 4 | 4 | no | — |
| 164 | C22⋊Q8 | — | 4 | 4 | no | — |
| 165 | C22⋊Q8 | — | 4 | 8 | no | — |
| 166 | C22⋊Q8 | — | 4 | 8 | no | — |
| 167 | C22⋊Q8 | — | 4 | 8 | no | — |
| 168 | C22⋊Q8 | D8 | 1 | 4 | yes | — |
| 169 | C4×Q8 | — | 2 | 2 | no | — |
| 170 | C4×Q8 | — | 2 | 2 | no | — |
| 171 | C4×Q8 | — | 2 | 8 | no | — |
| 172 | C4×Q8 | — | 4 | 4 | no | — |
| 173 | C4×Q8 | — | 4 | 4 | no | — |
| 174 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 175 | C4×D4 | — | 2 | 2 | no | — |
| 176 | C4×D4 | — | 2 | 2 | no | — |
| 177 | C4×D4 | — | 2 | 8 | no | — |
| 178 | C4×D4 | — | 4 | 2 | no | — |
| 179 | C4×D4 | — | 4 | 2 | no | — |
| 180 | C4×D4 | — | 4 | 8 | no | — |
| 181 | #117 | — | 2 | 2 | no | — |
| 182 | #117 | — | 4 | 4 | no | — |
| 183 | #117 | — | 4 | 2 | no | — |
| 184 | #117 | C2×D4 | 1 | 2 | yes | — |
| 185 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 186 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 187 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 188 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 189 | C2×C42 | — | 4 | 2 | no | — |
| 190 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 191 | Q32 | C2×Q8 | 1 | 4 | yes | complemented |
| 192 | SD32 | C2×Q8 | 1 | 8 | yes | complemented |
| 193 | D16 | C2×Q8 | 1 | 4 | yes | complemented |
| 194 | M32 | — | 2 | 4 | no | — |
| 195 | C2×C16 | — | 2 | 2 | no | — |
| 196 | C2×C16 | C2×Q8 | 1 | 4 | yes | complemented |
| 197 | C2×C16 | C24 | 1 | 2 | yes | — |
| 198 | C2×C16 | C24 | 1 | 4 | yes | — |
| 199 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 200 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 201 | #107 | — | 2 | 4 | no | — |
| 202 | #107 | — | 2 | 4 | no | — |
| 203 | #107 | — | 2 | 4 | no | — |
| 204 | #107 | — | 2 | 8 | no | — |
| 205 | #107 | C24 | 1 | 2 | yes | — |
| 206 | #107 | C24 | 1 | 8 | yes | — |
| 207 | #107 | C24 | 1 | 2 | yes | — |
| 208 | #106 | — | 2 | 2 | no | — |
| 209 | #106 | — | 2 | 8 | no | — |
| 210 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 211 | C4⋊C8 | C2×D4 | 1 | 8 | yes | — |
| 212 | #103 | — | 2 | 2 | no | — |
| 213 | #103 | — | 2 | 2 | no | — |
| 214 | #103 | — | 2 | 8 | no | — |
| 215 | #102 | — | 2 | 2 | no | — |
| 216 | #102 | — | 2 | 2 | no | — |
| 217 | #102 | — | 2 | 8 | no | — |
| 218 | #97 | C2×D4 | 1 | 4 | yes | — |
| 219 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 220 | C4×C8 | C24 | 1 | 2 | yes | — |
| 221 | C4×C8 | C24 | 1 | 4 | yes | — |
| 222 | #41 | — | 2 | 8 | no | — |
| 223 | #41 | — | 4 | 4 | no | — |
| 224 | C2×Q8 | — | 2 | 2 | no | — |
| 225 | C2×Q8 | — | 4 | 4 | no | — |
| 226 | C2×Q8 | — | 4 | 8 | no | — |
| 227 | C2×Q8 | — | 4 | 4 | no | — |
| 228 | C2×Q8 | — | 16 | 4 | no | — |
| 229 | C2×Q8 | — | 16 | 8 | no | — |
| 230 | C2×Q8 | C2×D8 | 1 | 4 | yes | — |
| 231 | C2×D4 | — | 2 | 2 | no | — |
| 232 | C2×D4 | — | 4 | 2 | no | — |
| 233 | C22×C4 | — | 4 | 2 | no | — |
| 234 | C22×C4 | — | 4 | 4 | no | — |
| 235 | C22×C4 | — | 4 | 2 | no | — |
| 236 | C22×C4 | — | 4 | 2 | no | — |
| 237 | C22×C4 | — | 4 | 8 | no | — |
| 238 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 239 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 240 | Q16 | — | 2 | 2 | no | — |
| 241 | Q16 | C22×Q8 | 1 | 4 | yes | — |
| 242 | SD16 | — | 2 | 8 | no | — |
| 243 | D8 | — | 2 | 2 | no | — |
| 244 | D8 | C22×Q8 | 1 | 4 | yes | — |
| 245 | M16 | — | 2 | 4 | no | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 248 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 249 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C22×Q8 | 1 | 2 | yes | — |
| 251 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 252 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 253 | C4⋊C4 | — | 2 | 2 | no | — |
| 254 | C4⋊C4 | — | 2 | 8 | no | — |
| 255 | C4⋊C4 | — | 2 | 2 | no | — |
| 256 | C4⋊C4 | — | 4 | 4 | no | — |
| 257 | C4⋊C4 | — | 4 | 4 | no | — |
| 258 | C4⋊C4 | — | 4 | 8 | no | — |
| 259 | C4⋊C4 | — | 4 | 8 | no | — |
| 260 | C4⋊C4 | — | 4 | 4 | no | — |
| 261 | C4⋊C4 | — | 4 | 2 | no | — |
| 262 | C4⋊C4 | — | 4 | 2 | no | — |
| 263 | C4⋊C4 | — | 4 | 8 | no | — |
| 264 | C4⋊C4 | — | 4 | 8 | no | — |
| 265 | C4⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 266 | C4⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 267 | C4⋊C4 | C2×D8 | 1 | 8 | yes | — |
| 268 | C22⋊C4 | — | 4 | 2 | no | — |
| 269 | C22⋊C4 | — | 4 | 2 | no | — |
| 270 | C22⋊C4 | — | 4 | 2 | no | — |
| 271 | C22⋊C4 | — | 4 | 8 | no | — |
| 272 | C22⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 273 | C42 | — | 4 | 2 | no | — |
| 274 | C42 | — | 4 | 2 | no | — |
| 275 | C42 | — | 4 | 2 | no | — |
| 276 | C42 | — | 4 | 8 | no | — |
| 277 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 278 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 279 | C42 | C22×D4 | 1 | 2 | yes | — |
| 280 | C42 | C22×D4 | 1 | 4 | yes | — |
| 281 | C16 | — | 2 | 2 | no | — |
| 282 | C16 | — | 2 | 4 | no | — |
| 283 | C16 | C22×Q8 | 1 | 2 | yes | — |
| 284 | C16 | C22×Q8 | 1 | 2 | yes | — |
| 285 | C23 | — | 4 | 2 | no | — |
| 286 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 287 | Q8 | — | 2 | 8 | no | — |
| 288 | Q8 | — | 2 | 4 | no | — |
| 289 | Q8 | — | 4 | 2 | no | — |
| 290 | Q8 | — | 8 | 8 | no | — |
| 291 | Q8 | — | 16 | 4 | no | — |
| 292 | Q8 | — | 16 | 8 | no | — |
| 293 | D4 | — | 2 | 4 | no | — |
| 294 | D4 | — | 4 | 8 | no | — |
| 295 | D4 | — | 4 | 2 | no | — |
| 296 | C2×C4 | — | 2 | 4 | no | — |
| 297 | C2×C4 | — | 4 | 2 | no | — |
| 298 | C2×C4 | — | 4 | 8 | no | — |
| 299 | C2×C4 | — | 4 | 2 | no | — |
| 300 | C2×C4 | — | 4 | 2 | no | — |
| 301 | C2×C4 | — | 4 | 8 | no | — |
| 302 | C2×C4 | — | 8 | 4 | no | — |
| 303 | C2×C4 | — | 8 | 4 | no | — |
| 304 | C2×C4 | — | 16 | 8 | no | — |
| 305 | C2×C4 | — | 16 | 4 | no | — |
| 306 | C2×C4 | D4×Q8 | 1 | 2 | yes | — |
| 307 | C2×C4 | C22×D8 | 1 | 8 | yes | — |
| 308 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 309 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 311 | C2×C4 | #578 | 1 | 2 | yes | — |
| 312 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 313 | C8 | — | 2 | 2 | no | — |
| 314 | C8 | — | 2 | 4 | no | — |
| 315 | C8 | D4×Q8 | 1 | 2 | yes | — |
| 316 | C8 | C23×Q8 | 1 | 1 | yes | characteristic |
| 317 | C8 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 318 | C22 | — | 2 | 2 | no | — |
| 319 | C22 | — | 4 | 4 | no | — |
| 320 | C22 | — | 8 | 2 | no | — |
| 321 | C22 | — | 8 | 2 | no | — |
| 322 | C22 | Q8×D8 | 1 | 2 | yes | — |
| 323 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 324 | C4 | — | 2 | 2 | no | — |
| 325 | C4 | — | 2 | 8 | no | — |
| 326 | C4 | — | 4 | 4 | no | — |
| 327 | C4 | — | 8 | 2 | no | — |
| 328 | C4 | — | 8 | 2 | no | — |
| 329 | C4 | — | 8 | 8 | no | — |
| 330 | C4 | Q8×D8 | 1 | 2 | yes | — |
| 331 | C4 | #3416 | 1 | 4 | yes | — |
| 332 | C4 | C2×D4×Q8 | 1 | 1 | yes | characteristic |
| 333 | C4 | C2×#578 | 1 | 1 | yes | characteristic |
| 334 | C2 | — | 2 | 2 | no | — |
| 335 | C2 | — | 8 | 4 | no | — |
| 336 | C2 | C2×Q8×D8 | 1 | 1 | yes | characteristic |
| 337 | C2 | C2×#3416 | 1 | 2 | yes | — |
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