← #496131
Subgroups — #496131
Order 512.
635 conjugacy classes of subgroups.
The subgroup lattice has 326 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33721 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33470 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #33465 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #33458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #22285 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13248 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13244 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13243 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #13216 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #12911 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #12542 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#510 | C22 | 1 | 2 | yes | — |
| 13 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 14 | C22×Q32 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3156 | C22 | 1 | 2 | yes | — |
| 16 | #2957 | C22 | 1 | 2 | yes | — |
| 17 | #2950 | C22 | 1 | 2 | yes | — |
| 18 | #2251 | C4 | 1 | 4 | yes | complemented |
| 19 | #2249 | C4 | 1 | 2 | yes | complemented |
| 20 | #2247 | C4 | 1 | 2 | yes | complemented |
| 21 | #2180 | — | 2 | 8 | no | — |
| 22 | #2179 | — | 2 | 2 | no | — |
| 23 | #2178 | — | 2 | 2 | no | — |
| 24 | #2176 | — | 2 | 2 | no | — |
| 25 | C4×Q32 | — | 2 | 2 | no | — |
| 26 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2158 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2155 | C22 | 1 | 2 | yes | — |
| 29 | C2×#362 | C22 | 1 | 2 | yes | — |
| 30 | #2145 | C22 | 1 | 2 | yes | — |
| 31 | #2143 | C22 | 1 | 2 | yes | — |
| 32 | #2141 | C22 | 1 | 2 | yes | — |
| 33 | #2110 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2110 | C22 | 1 | 1 | yes | characteristic |
| 35 | #2110 | C22 | 1 | 1 | yes | characteristic |
| 36 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 37 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 38 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1979 | C22 | 1 | 2 | yes | — |
| 40 | #1976 | C22 | 1 | 2 | yes | — |
| 41 | #1912 | — | 2 | 2 | no | — |
| 42 | #1909 | — | 2 | 2 | no | — |
| 43 | #1905 | — | 2 | 2 | no | — |
| 44 | #1902 | — | 2 | 2 | no | — |
| 45 | #1833 | C22 | 1 | 2 | yes | — |
| 46 | #1747 | C22 | 1 | 1 | yes | characteristic |
| 47 | C2×#137 | — | 2 | 2 | no | — |
| 48 | C2×#135 | C23 | 1 | 2 | yes | — |
| 49 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 50 | #568 | — | 2 | 2 | no | — |
| 51 | #536 | — | 2 | 2 | no | — |
| 52 | #519 | — | 2 | 2 | no | — |
| 53 | #510 | — | 2 | 4 | no | — |
| 54 | #510 | — | 2 | 4 | no | — |
| 55 | #508 | — | 2 | 1 | no | — |
| 56 | #508 | — | 2 | 1 | no | — |
| 57 | #508 | — | 2 | 1 | no | — |
| 58 | #508 | — | 2 | 1 | no | — |
| 59 | C2×Q32 | — | 2 | 1 | no | — |
| 60 | C2×Q32 | — | 2 | 1 | no | — |
| 61 | C2×Q32 | — | 2 | 1 | no | — |
| 62 | C2×Q32 | — | 2 | 1 | no | — |
| 63 | C2×Q32 | C2×C4 | 1 | 2 | yes | — |
| 64 | C2×Q32 | C2×C4 | 1 | 2 | yes | — |
| 65 | C2×Q32 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | C2×Q32 | C2×C4 | 1 | 4 | yes | — |
| 67 | C2×SD32 | C2×C4 | 1 | 2 | yes | — |
| 68 | C2×SD32 | C2×C4 | 1 | 4 | yes | — |
| 69 | C2×D16 | C2×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 71 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 72 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 73 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 74 | #497 | — | 2 | 4 | no | — |
| 75 | #495 | C2×C4 | 1 | 4 | yes | — |
| 76 | #494 | C2×C4 | 1 | 4 | yes | — |
| 77 | #444 | — | 2 | 4 | no | — |
| 78 | #443 | — | 2 | 4 | no | — |
| 79 | #441 | — | 2 | 4 | no | — |
| 80 | C4×Q16 | — | 2 | 4 | no | — |
| 81 | C2×#108 | C23 | 1 | 2 | yes | — |
| 82 | #427 | C23 | 1 | 2 | yes | — |
| 83 | #421 | — | 2 | 2 | no | — |
| 84 | #421 | — | 2 | 2 | no | — |
| 85 | #419 | — | 2 | 2 | no | — |
| 86 | #419 | — | 2 | 2 | no | — |
| 87 | C2×#101 | — | 2 | 2 | no | — |
| 88 | #410 | — | 2 | 2 | no | — |
| 89 | #405 | C23 | 1 | 1 | yes | characteristic |
| 90 | #405 | C23 | 1 | 1 | yes | characteristic |
| 91 | #405 | C23 | 1 | 1 | yes | characteristic |
| 92 | #368 | — | 2 | 1 | no | — |
| 93 | #368 | — | 2 | 1 | no | — |
| 94 | #367 | — | 2 | 1 | no | — |
| 95 | #366 | — | 2 | 1 | no | — |
| 96 | #365 | — | 2 | 4 | no | — |
| 97 | #362 | — | 2 | 4 | no | — |
| 98 | #360 | — | 2 | 4 | no | — |
| 99 | #359 | — | 2 | 4 | no | — |
| 100 | #358 | — | 2 | 4 | no | — |
| 101 | #347 | — | 2 | 2 | no | — |
| 102 | #347 | — | 2 | 2 | no | — |
| 103 | #347 | D4 | 1 | 4 | yes | — |
| 104 | #346 | — | 2 | 1 | no | — |
| 105 | #346 | C2×C4 | 1 | 2 | yes | — |
| 106 | #346 | D4 | 1 | 2 | yes | — |
| 107 | C4×C16 | — | 2 | 1 | no | — |
| 108 | C4×C16 | C2×C4 | 1 | 2 | yes | — |
| 109 | C4×C16 | D4 | 1 | 2 | yes | — |
| 110 | #344 | — | 2 | 2 | no | — |
| 111 | #338 | D4 | 1 | 1 | yes | characteristic |
| 112 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 113 | H(3,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 114 | H(3,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 115 | H(3,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 116 | C2×#41 | — | 2 | 2 | no | — |
| 117 | C22×Q8 | — | 2 | 2 | no | — |
| 118 | #137 | — | 8 | 2 | no | — |
| 119 | #137 | — | 8 | 2 | no | — |
| 120 | #135 | — | 2 | 2 | no | — |
| 121 | #135 | — | 2 | 2 | no | — |
| 122 | #135 | — | 2 | 2 | no | — |
| 123 | #135 | — | 2 | 2 | no | — |
| 124 | C2×Q16 | — | 2 | 4 | no | — |
| 125 | C2×Q16 | — | 2 | 2 | no | — |
| 126 | C2×Q16 | — | 2 | 2 | no | — |
| 127 | C2×Q16 | — | 2 | 2 | no | — |
| 128 | C2×Q16 | — | 2 | 2 | no | — |
| 129 | C2×Q16 | — | 2 | 2 | no | — |
| 130 | C2×Q16 | C22×C4 | 1 | 4 | yes | — |
| 131 | C2×Q16 | C22×C4 | 1 | 2 | yes | — |
| 132 | C2×SD16 | — | 2 | 4 | no | — |
| 133 | C2×SD16 | — | 2 | 2 | no | — |
| 134 | C2×D8 | C22×C4 | 1 | 2 | yes | — |
| 135 | #131 | — | 4 | 4 | no | — |
| 136 | #131 | — | 8 | 2 | no | — |
| 137 | C2×M16 | — | 2 | 2 | no | — |
| 138 | C2×M16 | — | 8 | 2 | no | — |
| 139 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 140 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 141 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 142 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 143 | C4⋊Q8 | — | 2 | 4 | no | — |
| 144 | #124 | — | 2 | 4 | no | — |
| 145 | #123 | — | 8 | 2 | no | — |
| 146 | C22⋊Q8 | — | 8 | 2 | no | — |
| 147 | C4×Q8 | — | 4 | 4 | no | — |
| 148 | C4×Q8 | — | 8 | 2 | no | — |
| 149 | #117 | — | 2 | 2 | no | — |
| 150 | #117 | — | 8 | 2 | no | — |
| 151 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 152 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 153 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 154 | Q32 | — | 2 | 2 | no | — |
| 155 | Q32 | — | 2 | 2 | no | — |
| 156 | Q32 | — | 2 | 2 | no | — |
| 157 | Q32 | — | 2 | 2 | no | — |
| 158 | Q32 | — | 2 | 1 | no | — |
| 159 | Q32 | — | 2 | 1 | no | — |
| 160 | Q32 | — | 2 | 8 | no | — |
| 161 | SD32 | — | 2 | 2 | no | — |
| 162 | SD32 | — | 2 | 2 | no | — |
| 163 | SD32 | — | 2 | 8 | no | — |
| 164 | D16 | — | 2 | 1 | no | — |
| 165 | D16 | — | 2 | 1 | no | — |
| 166 | M32 | — | 2 | 1 | no | — |
| 167 | M32 | — | 2 | 1 | no | — |
| 168 | M32 | — | 2 | 1 | no | — |
| 169 | M32 | C2×D4 | 1 | 2 | yes | — |
| 170 | M32 | C2×D4 | 1 | 2 | yes | — |
| 171 | M32 | #41 | 1 | 2 | yes | — |
| 172 | C2×C16 | — | 2 | 1 | no | — |
| 173 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 174 | C2×C16 | C22×C4 | 1 | 2 | yes | — |
| 175 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 176 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 177 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 178 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 179 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 180 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 181 | #108 | — | 2 | 2 | no | — |
| 182 | #108 | — | 2 | 2 | no | — |
| 183 | #107 | — | 2 | 2 | no | — |
| 184 | #106 | — | 2 | 2 | no | — |
| 185 | #104 | — | 4 | 4 | no | — |
| 186 | #104 | — | 8 | 2 | no | — |
| 187 | #103 | — | 4 | 4 | no | — |
| 188 | #103 | — | 8 | 2 | no | — |
| 189 | #101 | — | 8 | 2 | no | — |
| 190 | #99 | — | 8 | 2 | no | — |
| 191 | #97 | — | 2 | 1 | no | — |
| 192 | #97 | C2×D4 | 1 | 2 | yes | — |
| 193 | #97 | #41 | 1 | 2 | yes | — |
| 194 | C4×C8 | — | 2 | 1 | no | — |
| 195 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 196 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 197 | #41 | — | 4 | 2 | no | — |
| 198 | #41 | — | 4 | 2 | no | — |
| 199 | #41 | — | 4 | 2 | no | — |
| 200 | #41 | — | 4 | 2 | no | — |
| 201 | C2×Q8 | — | 2 | 4 | no | — |
| 202 | C2×Q8 | — | 2 | 2 | no | — |
| 203 | C2×Q8 | — | 4 | 2 | no | — |
| 204 | C2×Q8 | — | 4 | 2 | no | — |
| 205 | C2×Q8 | — | 4 | 2 | no | — |
| 206 | C2×Q8 | — | 4 | 2 | no | — |
| 207 | C2×Q8 | — | 4 | 2 | no | — |
| 208 | C2×D4 | — | 2 | 2 | no | — |
| 209 | C2×D4 | — | 4 | 2 | no | — |
| 210 | C22×C4 | — | 4 | 2 | no | — |
| 211 | C22×C4 | — | 4 | 2 | no | — |
| 212 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 213 | Q16 | — | 2 | 4 | no | — |
| 214 | Q16 | — | 2 | 2 | no | — |
| 215 | Q16 | — | 2 | 4 | no | — |
| 216 | Q16 | — | 2 | 2 | no | — |
| 217 | Q16 | — | 4 | 2 | no | — |
| 218 | Q16 | — | 4 | 2 | no | — |
| 219 | Q16 | — | 8 | 4 | no | — |
| 220 | SD16 | — | 4 | 2 | no | — |
| 221 | SD16 | — | 4 | 2 | no | — |
| 222 | SD16 | — | 8 | 4 | no | — |
| 223 | D8 | — | 2 | 2 | no | — |
| 224 | D8 | — | 2 | 2 | no | — |
| 225 | M16 | — | 2 | 1 | no | — |
| 226 | M16 | — | 2 | 1 | no | — |
| 227 | M16 | — | 2 | 1 | no | — |
| 228 | M16 | — | 2 | 1 | no | — |
| 229 | M16 | — | 2 | 1 | no | — |
| 230 | M16 | — | 2 | 1 | no | — |
| 231 | M16 | — | 4 | 2 | no | — |
| 232 | M16 | — | 4 | 2 | no | — |
| 233 | M16 | — | 8 | 4 | no | — |
| 234 | C2×C8 | — | 2 | 1 | no | — |
| 235 | C2×C8 | — | 4 | 2 | no | — |
| 236 | C2×C8 | — | 8 | 4 | no | — |
| 237 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 238 | C2×C8 | #127 | 1 | 2 | yes | — |
| 239 | C2×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 245 | C4⋊C4 | — | 2 | 1 | no | — |
| 246 | C4⋊C4 | — | 2 | 1 | no | — |
| 247 | C4⋊C4 | — | 2 | 1 | no | — |
| 248 | C4⋊C4 | — | 4 | 2 | no | — |
| 249 | C4⋊C4 | — | 8 | 4 | no | — |
| 250 | C4⋊C4 | — | 8 | 4 | no | — |
| 251 | C22⋊C4 | — | 2 | 1 | no | — |
| 252 | C22⋊C4 | — | 2 | 1 | no | — |
| 253 | C22⋊C4 | — | 2 | 1 | no | — |
| 254 | C22⋊C4 | — | 4 | 2 | no | — |
| 255 | C22⋊C4 | — | 8 | 4 | no | — |
| 256 | C42 | — | 2 | 1 | no | — |
| 257 | C42 | — | 4 | 2 | no | — |
| 258 | C42 | — | 8 | 4 | no | — |
| 259 | C42 | C4×D4 | 1 | 2 | yes | — |
| 260 | C42 | #127 | 1 | 2 | yes | — |
| 261 | C16 | — | 2 | 2 | no | — |
| 262 | C16 | — | 2 | 2 | no | — |
| 263 | C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 264 | C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 265 | C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 266 | C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 267 | C16 | C4×D4 | 1 | 4 | yes | — |
| 268 | C23 | — | 4 | 2 | no | — |
| 269 | C23 | #576 | 1 | 1 | yes | characteristic |
| 270 | Q8 | — | 4 | 2 | no | — |
| 271 | Q8 | — | 4 | 4 | no | — |
| 272 | Q8 | — | 4 | 4 | no | — |
| 273 | Q8 | — | 4 | 2 | no | — |
| 274 | Q8 | — | 8 | 2 | no | — |
| 275 | Q8 | — | 8 | 2 | no | — |
| 276 | D4 | — | 4 | 2 | no | — |
| 277 | D4 | — | 4 | 2 | no | — |
| 278 | D4 | — | 8 | 2 | no | — |
| 279 | D4 | — | 8 | 2 | no | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 4 | 1 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 8 | 2 | no | — |
| 286 | C2×C4 | — | 8 | 2 | no | — |
| 287 | C2×C4 | — | 8 | 2 | no | — |
| 288 | C2×C4 | — | 8 | 2 | no | — |
| 289 | C2×C4 | — | 8 | 2 | no | — |
| 290 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 2 | 2 | no | — |
| 297 | C8 | — | 2 | 2 | no | — |
| 298 | C8 | — | 4 | 1 | no | — |
| 299 | C8 | — | 8 | 2 | no | — |
| 300 | C8 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 301 | C8 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 302 | C8 | #532 | 1 | 1 | yes | characteristic |
| 303 | C8 | #532 | 1 | 1 | yes | characteristic |
| 304 | C22 | — | 4 | 1 | no | — |
| 305 | C22 | — | 8 | 2 | no | — |
| 306 | C22 | — | 16 | 2 | no | — |
| 307 | C22 | #2948 | 1 | 1 | yes | characteristic |
| 308 | C22 | #3155 | 1 | 1 | yes | characteristic |
| 309 | C22 | #3176 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 4 | 2 | no | — |
| 311 | C4 | — | 4 | 2 | no | — |
| 312 | C4 | — | 8 | 1 | no | — |
| 313 | C4 | — | 8 | 2 | no | — |
| 314 | C4 | — | 8 | 4 | no | — |
| 315 | C4 | — | 16 | 2 | no | — |
| 316 | C4 | C4×#127 | 1 | 1 | yes | characteristic |
| 317 | C4 | #2948 | 1 | 1 | yes | characteristic |
| 318 | C4 | #3155 | 1 | 1 | yes | characteristic |
| 319 | C4 | #3176 | 1 | 1 | yes | characteristic |
| 320 | C2 | — | 2 | 1 | no | — |
| 321 | C2 | — | 2 | 1 | no | — |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 16 | 2 | no | — |
| 324 | C2 | #21697 | 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