← #504010
Subgroups — #504010
Order 512.
551 conjugacy classes of subgroups.
The subgroup lattice has 328 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 | #33460 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #26271 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13446 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13412 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13244 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13242 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #13240 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13172 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #13023 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #12665 | 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 | #3097 | — | 2 | 2 | no | — |
| 17 | #2951 | C22 | 1 | 2 | yes | — |
| 18 | #2251 | — | 2 | 2 | no | — |
| 19 | #2249 | — | 2 | 1 | no | — |
| 20 | #2247 | — | 2 | 1 | no | — |
| 21 | #2177 | — | 2 | 1 | no | — |
| 22 | #2176 | — | 2 | 1 | no | — |
| 23 | C4×SD32 | — | 2 | 2 | no | — |
| 24 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#367 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#362 | C22 | 1 | 2 | yes | — |
| 28 | C2×#359 | C22 | 1 | 2 | yes | — |
| 29 | #2139 | C22 | 1 | 2 | yes | — |
| 30 | C2×#358 | C22 | 1 | 2 | yes | — |
| 31 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 32 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2052 | — | 2 | 2 | no | — |
| 35 | #2038 | — | 2 | 2 | no | — |
| 36 | #1979 | C22 | 1 | 2 | yes | — |
| 37 | #1974 | C22 | 1 | 2 | yes | — |
| 38 | #1969 | C22 | 1 | 2 | yes | — |
| 39 | #1952 | C22 | 1 | 2 | yes | — |
| 40 | #1882 | — | 2 | 2 | no | — |
| 41 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 42 | #1387 | C22 | 1 | 2 | yes | — |
| 43 | #1379 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1379 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×#137 | — | 2 | 2 | no | — |
| 46 | C2×#135 | C23 | 1 | 2 | yes | — |
| 47 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 48 | #536 | — | 2 | 2 | no | — |
| 49 | #520 | — | 2 | 2 | no | — |
| 50 | #510 | — | 2 | 4 | no | — |
| 51 | #510 | — | 4 | 2 | no | — |
| 52 | #508 | — | 2 | 2 | no | — |
| 53 | #508 | — | 2 | 2 | no | — |
| 54 | C2×Q32 | — | 2 | 2 | no | — |
| 55 | C2×Q32 | — | 2 | 1 | no | — |
| 56 | C2×Q32 | — | 2 | 1 | no | — |
| 57 | C2×Q32 | — | 2 | 2 | no | — |
| 58 | C2×Q32 | — | 2 | 2 | no | — |
| 59 | C2×Q32 | D4 | 1 | 1 | yes | characteristic |
| 60 | C2×SD32 | — | 2 | 2 | no | — |
| 61 | C2×SD32 | D4 | 1 | 2 | yes | — |
| 62 | C2×D16 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 65 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 66 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 67 | #497 | — | 4 | 2 | no | — |
| 68 | #495 | — | 2 | 2 | no | — |
| 69 | #494 | — | 2 | 2 | no | — |
| 70 | #485 | — | 4 | 2 | no | — |
| 71 | #485 | — | 4 | 2 | no | — |
| 72 | #484 | — | 4 | 2 | no | — |
| 73 | #483 | — | 4 | 2 | no | — |
| 74 | #440 | — | 4 | 2 | no | — |
| 75 | C4×Q16 | — | 2 | 2 | no | — |
| 76 | C4×D8 | — | 2 | 2 | no | — |
| 77 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 78 | C2×#108 | C23 | 1 | 2 | yes | — |
| 79 | #428 | — | 2 | 2 | no | — |
| 80 | #427 | D4 | 1 | 1 | yes | characteristic |
| 81 | C2×#107 | C23 | 1 | 2 | yes | — |
| 82 | C2×#104 | — | 2 | 2 | no | — |
| 83 | #419 | — | 2 | 2 | no | — |
| 84 | #417 | — | 2 | 2 | no | — |
| 85 | C2×#103 | — | 2 | 2 | no | — |
| 86 | C2×#101 | — | 2 | 2 | no | — |
| 87 | #408 | — | 2 | 2 | no | — |
| 88 | #405 | C23 | 1 | 1 | yes | characteristic |
| 89 | #405 | C23 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 91 | #391 | — | 2 | 2 | no | — |
| 92 | #368 | — | 2 | 2 | no | — |
| 93 | #367 | — | 2 | 1 | no | — |
| 94 | #367 | — | 2 | 1 | no | — |
| 95 | #364 | — | 2 | 2 | no | — |
| 96 | #362 | — | 4 | 2 | no | — |
| 97 | #359 | — | 4 | 2 | no | — |
| 98 | #358 | — | 2 | 2 | no | — |
| 99 | #358 | — | 4 | 2 | no | — |
| 100 | #357 | — | 2 | 2 | no | — |
| 101 | #346 | — | 2 | 1 | no | — |
| 102 | #346 | — | 2 | 1 | no | — |
| 103 | #346 | — | 2 | 2 | no | — |
| 104 | C4×C16 | — | 2 | 1 | no | — |
| 105 | C4×C16 | — | 2 | 1 | no | — |
| 106 | #343 | — | 2 | 2 | no | — |
| 107 | #339 | D4 | 1 | 1 | yes | characteristic |
| 108 | #339 | D4 | 1 | 1 | yes | characteristic |
| 109 | C2×#41 | — | 2 | 2 | no | — |
| 110 | C22×Q8 | — | 2 | 2 | no | — |
| 111 | #137 | — | 4 | 4 | no | — |
| 112 | #137 | — | 8 | 2 | no | — |
| 113 | #135 | — | 2 | 2 | no | — |
| 114 | #135 | — | 2 | 2 | no | — |
| 115 | #135 | — | 2 | 2 | no | — |
| 116 | #135 | — | 2 | 2 | no | — |
| 117 | C2×Q16 | — | 2 | 2 | no | — |
| 118 | C2×Q16 | — | 2 | 2 | no | — |
| 119 | C2×Q16 | — | 2 | 2 | no | — |
| 120 | C2×Q16 | — | 2 | 2 | no | — |
| 121 | C2×Q16 | — | 2 | 2 | no | — |
| 122 | C2×Q16 | — | 2 | 2 | no | — |
| 123 | C2×Q16 | — | 4 | 2 | no | — |
| 124 | C2×Q16 | C2×D4 | 1 | 2 | yes | — |
| 125 | C2×SD16 | — | 2 | 2 | no | — |
| 126 | C2×SD16 | — | 4 | 2 | no | — |
| 127 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 128 | C2×M16 | — | 2 | 2 | no | — |
| 129 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 131 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 132 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 133 | C4⋊Q8 | — | 4 | 2 | no | — |
| 134 | #124 | — | 4 | 2 | no | — |
| 135 | #123 | — | 8 | 2 | no | — |
| 136 | C22⋊Q8 | — | 4 | 2 | no | — |
| 137 | C22⋊Q8 | — | 4 | 2 | no | — |
| 138 | C4×Q8 | — | 4 | 2 | no | — |
| 139 | C4×D4 | — | 4 | 2 | no | — |
| 140 | C4×D4 | — | 8 | 2 | no | — |
| 141 | #117 | — | 4 | 2 | no | — |
| 142 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 143 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 144 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 145 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 146 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 147 | Q32 | — | 2 | 2 | no | — |
| 148 | Q32 | — | 4 | 4 | no | — |
| 149 | Q32 | — | 4 | 2 | no | — |
| 150 | Q32 | — | 4 | 2 | no | — |
| 151 | SD32 | — | 2 | 4 | no | — |
| 152 | SD32 | — | 4 | 4 | no | — |
| 153 | D16 | — | 2 | 2 | no | — |
| 154 | M32 | — | 2 | 1 | no | — |
| 155 | M32 | — | 2 | 1 | no | — |
| 156 | M32 | — | 2 | 1 | no | — |
| 157 | M32 | #41 | 1 | 2 | yes | — |
| 158 | C2×C16 | — | 2 | 1 | no | — |
| 159 | C2×C16 | — | 2 | 1 | no | — |
| 160 | C2×C16 | — | 2 | 1 | no | — |
| 161 | C2×C16 | — | 2 | 1 | no | — |
| 162 | C2×C16 | — | 2 | 1 | no | — |
| 163 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 164 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 165 | C2×C16 | #41 | 1 | 2 | yes | — |
| 166 | C2×C16 | #41 | 1 | 2 | yes | — |
| 167 | #108 | — | 2 | 2 | no | — |
| 168 | #108 | — | 2 | 2 | no | — |
| 169 | #108 | — | 2 | 2 | no | — |
| 170 | #107 | — | 2 | 2 | no | — |
| 171 | #107 | — | 2 | 2 | no | — |
| 172 | #107 | — | 4 | 2 | no | — |
| 173 | #107 | D8 | 1 | 2 | yes | — |
| 174 | #106 | — | 4 | 2 | no | — |
| 175 | #106 | D8 | 1 | 2 | yes | — |
| 176 | #104 | — | 8 | 2 | no | — |
| 177 | #103 | — | 4 | 2 | no | — |
| 178 | #103 | — | 4 | 2 | no | — |
| 179 | #103 | — | 4 | 2 | no | — |
| 180 | #103 | — | 4 | 2 | no | — |
| 181 | #103 | — | 4 | 2 | no | — |
| 182 | #102 | — | 4 | 2 | no | — |
| 183 | #101 | — | 8 | 2 | no | — |
| 184 | C22⋊C8 | — | 4 | 2 | no | — |
| 185 | C22⋊C8 | — | 4 | 2 | no | — |
| 186 | #97 | — | 2 | 1 | no | — |
| 187 | #97 | — | 2 | 1 | no | — |
| 188 | C4×C8 | — | 2 | 1 | no | — |
| 189 | C4×C8 | — | 2 | 1 | no | — |
| 190 | C4×C8 | — | 2 | 1 | no | — |
| 191 | C4×C8 | — | 2 | 1 | no | — |
| 192 | #95 | — | 8 | 2 | no | — |
| 193 | #41 | — | 4 | 2 | no | — |
| 194 | #41 | — | 4 | 2 | no | — |
| 195 | #41 | — | 4 | 2 | no | — |
| 196 | #41 | — | 4 | 2 | no | — |
| 197 | C2×Q8 | — | 2 | 2 | no | — |
| 198 | C2×Q8 | — | 4 | 2 | no | — |
| 199 | C2×Q8 | — | 4 | 2 | no | — |
| 200 | C2×Q8 | — | 4 | 2 | no | — |
| 201 | C2×Q8 | — | 4 | 2 | no | — |
| 202 | C2×Q8 | — | 4 | 2 | no | — |
| 203 | C2×Q8 | — | 4 | 2 | no | — |
| 204 | C2×D4 | — | 2 | 2 | no | — |
| 205 | C2×D4 | — | 4 | 2 | no | — |
| 206 | C22×C4 | — | 2 | 1 | no | — |
| 207 | C22×C4 | — | 4 | 2 | no | — |
| 208 | C22×C4 | — | 4 | 2 | no | — |
| 209 | C22×C4 | — | 4 | 2 | no | — |
| 210 | C22×C4 | — | 8 | 2 | no | — |
| 211 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 212 | Q16 | — | 2 | 2 | no | — |
| 213 | Q16 | — | 2 | 2 | no | — |
| 214 | Q16 | — | 4 | 2 | no | — |
| 215 | Q16 | — | 4 | 2 | no | — |
| 216 | Q16 | — | 4 | 2 | no | — |
| 217 | Q16 | — | 4 | 2 | no | — |
| 218 | Q16 | — | 8 | 4 | no | — |
| 219 | SD16 | — | 4 | 2 | no | — |
| 220 | SD16 | — | 4 | 2 | no | — |
| 221 | SD16 | — | 8 | 4 | no | — |
| 222 | D8 | — | 2 | 2 | no | — |
| 223 | D8 | — | 2 | 2 | no | — |
| 224 | M16 | — | 2 | 1 | no | — |
| 225 | M16 | — | 2 | 1 | no | — |
| 226 | M16 | — | 2 | 1 | no | — |
| 227 | M16 | — | 4 | 2 | no | — |
| 228 | M16 | — | 4 | 2 | no | — |
| 229 | M16 | #135 | 1 | 2 | yes | — |
| 230 | C2×C8 | — | 2 | 1 | no | — |
| 231 | C2×C8 | — | 2 | 1 | no | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 1 | no | — |
| 234 | C2×C8 | — | 4 | 2 | no | — |
| 235 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 243 | C4⋊C4 | — | 4 | 2 | no | — |
| 244 | C4⋊C4 | — | 4 | 2 | no | — |
| 245 | C4⋊C4 | — | 8 | 2 | no | — |
| 246 | C4⋊C4 | — | 8 | 2 | no | — |
| 247 | C4⋊C4 | — | 8 | 2 | no | — |
| 248 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 249 | C4⋊C4 | C2×D8 | 1 | 2 | yes | — |
| 250 | C22⋊C4 | — | 2 | 1 | no | — |
| 251 | C22⋊C4 | — | 2 | 1 | no | — |
| 252 | C22⋊C4 | — | 8 | 2 | no | — |
| 253 | C22⋊C4 | — | 8 | 2 | no | — |
| 254 | C22⋊C4 | — | 8 | 2 | no | — |
| 255 | C42 | — | 2 | 1 | no | — |
| 256 | C42 | — | 2 | 1 | no | — |
| 257 | C42 | — | 2 | 1 | no | — |
| 258 | C42 | — | 2 | 1 | no | — |
| 259 | C42 | — | 8 | 2 | no | — |
| 260 | C16 | — | 2 | 2 | no | — |
| 261 | C16 | — | 4 | 1 | no | — |
| 262 | C16 | — | 4 | 1 | no | — |
| 263 | C16 | #123 | 1 | 2 | yes | — |
| 264 | C16 | #123 | 1 | 2 | yes | — |
| 265 | C23 | — | 4 | 2 | no | — |
| 266 | C23 | #576 | 1 | 1 | yes | characteristic |
| 267 | Q8 | — | 4 | 2 | no | — |
| 268 | Q8 | — | 4 | 2 | no | — |
| 269 | Q8 | — | 8 | 2 | no | — |
| 270 | Q8 | — | 8 | 2 | no | — |
| 271 | Q8 | — | 8 | 2 | no | — |
| 272 | Q8 | — | 8 | 2 | no | — |
| 273 | D4 | — | 4 | 2 | no | — |
| 274 | D4 | — | 4 | 2 | no | — |
| 275 | D4 | — | 8 | 2 | no | — |
| 276 | D4 | — | 8 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 4 | 2 | no | — |
| 280 | C2×C4 | — | 4 | 1 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 8 | 2 | no | — |
| 283 | C2×C4 | — | 8 | 2 | no | — |
| 284 | C2×C4 | — | 8 | 2 | no | — |
| 285 | C2×C4 | — | 8 | 2 | no | — |
| 286 | C2×C4 | — | 8 | 2 | no | — |
| 287 | C2×C4 | — | 8 | 2 | no | — |
| 288 | C2×C4 | — | 16 | 2 | no | — |
| 289 | C2×C4 | — | 32 | 2 | no | — |
| 290 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #550 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 2 | 1 | no | — |
| 297 | C8 | — | 2 | 1 | no | — |
| 298 | C8 | — | 2 | 2 | no | — |
| 299 | C8 | — | 4 | 1 | no | — |
| 300 | C8 | — | 8 | 2 | no | — |
| 301 | C8 | C2×#123 | 1 | 1 | yes | characteristic |
| 302 | C8 | #540 | 1 | 1 | yes | characteristic |
| 303 | C8 | #546 | 1 | 1 | yes | characteristic |
| 304 | C8 | #559 | 1 | 1 | yes | characteristic |
| 305 | C22 | — | 2 | 2 | no | — |
| 306 | C22 | — | 8 | 2 | no | — |
| 307 | C22 | — | 8 | 2 | no | — |
| 308 | C22 | — | 8 | 2 | no | — |
| 309 | C22 | #3176 | 1 | 1 | yes | characteristic |
| 310 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 311 | C22 | #3396 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 4 | 2 | no | — |
| 313 | C4 | — | 4 | 2 | no | — |
| 314 | C4 | — | 8 | 1 | no | — |
| 315 | C4 | — | 8 | 2 | no | — |
| 316 | C4 | — | 16 | 2 | no | — |
| 317 | C4 | — | 16 | 2 | no | — |
| 318 | C4 | #2445 | 1 | 1 | yes | characteristic |
| 319 | C4 | #3169 | 1 | 1 | yes | characteristic |
| 320 | C4 | #3280 | 1 | 1 | yes | characteristic |
| 321 | C4 | #3391 | 1 | 1 | yes | characteristic |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | — | 16 | 2 | no | — |
| 325 | C2 | #26254 | 1 | 1 | yes | characteristic |
| 326 | C2 | #33855 | 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