← #72227112
Subgroups — #72227112
Order 1024.
392 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 | #483327 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #151895 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#7457 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#7457 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #134661 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #94031 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #94031 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #33401 | — | 2 | 1 | no | — |
| 9 | #33326 | C22 | 1 | 1 | yes | characteristic |
| 10 | #13558 | C4 | 1 | 1 | yes | characteristic |
| 11 | #13558 | C4 | 1 | 1 | yes | characteristic |
| 12 | C2×#1398 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#1398 | C22 | 1 | 1 | yes | characteristic |
| 14 | #12002 | — | 2 | 2 | no | — |
| 15 | #12002 | — | 2 | 1 | no | — |
| 16 | C2×#1369 | — | 2 | 2 | no | — |
| 17 | C2×#1369 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#1369 | C22 | 1 | 1 | yes | characteristic |
| 19 | #11999 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#1368 | C22 | 1 | 1 | yes | characteristic |
| 21 | #8604 | — | 2 | 1 | no | — |
| 22 | #7469 | C4 | 1 | 1 | yes | characteristic |
| 23 | #7469 | C4 | 1 | 1 | yes | characteristic |
| 24 | #7457 | — | 2 | 2 | no | — |
| 25 | #7457 | — | 2 | 2 | no | — |
| 26 | #7158 | — | 2 | 1 | no | — |
| 27 | #3407 | — | 4 | 1 | no | — |
| 28 | C2×#504 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×#504 | Q8 | 1 | 1 | yes | characteristic |
| 30 | C4×#131 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | #2153 | — | 4 | 2 | no | — |
| 36 | #2153 | — | 4 | 1 | no | — |
| 37 | C2×#364 | — | 2 | 4 | no | — |
| 38 | C2×#364 | — | 2 | 1 | no | — |
| 39 | C2×#364 | — | 2 | 1 | no | — |
| 40 | #2108 | — | 2 | 1 | no | — |
| 41 | #2108 | — | 2 | 1 | no | — |
| 42 | #2108 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #2108 | Q8 | 1 | 1 | yes | characteristic |
| 44 | C4×M32 | — | 2 | 1 | no | — |
| 45 | C4×M32 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×#346 | — | 2 | 1 | no | — |
| 47 | C2×#346 | C23 | 1 | 1 | yes | characteristic |
| 48 | #1564 | — | 2 | 1 | no | — |
| 49 | C2×#334 | — | 2 | 1 | no | — |
| 50 | #1477 | — | 4 | 1 | no | — |
| 51 | #1401 | — | 2 | 2 | no | — |
| 52 | #1401 | C8 | 1 | 4 | yes | complemented |
| 53 | #1398 | — | 2 | 1 | no | — |
| 54 | #1398 | — | 2 | 1 | no | — |
| 55 | #1398 | — | 2 | 1 | no | — |
| 56 | #1398 | — | 2 | 1 | no | — |
| 57 | #1369 | — | 2 | 4 | no | — |
| 58 | #1369 | — | 2 | 1 | no | — |
| 59 | #1369 | — | 2 | 1 | no | — |
| 60 | #1369 | D4 | 1 | 2 | yes | — |
| 61 | #1369 | D4 | 1 | 2 | yes | — |
| 62 | #1368 | — | 2 | 1 | no | — |
| 63 | #1368 | — | 2 | 1 | no | — |
| 64 | #1368 | — | 2 | 1 | no | — |
| 65 | #1368 | — | 2 | 1 | no | — |
| 66 | #1283 | — | 4 | 1 | no | — |
| 67 | C2×#131 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C2×#131 | SD16 | 1 | 1 | yes | characteristic |
| 69 | C4×#41 | SD16 | 1 | 1 | yes | characteristic |
| 70 | #504 | — | 2 | 1 | no | — |
| 71 | #504 | — | 2 | 1 | no | — |
| 72 | #504 | — | 2 | 2 | no | — |
| 73 | #504 | — | 4 | 1 | no | — |
| 74 | #504 | — | 8 | 2 | no | — |
| 75 | #504 | — | 8 | 2 | no | — |
| 76 | #504 | C2×C8 | 1 | 2 | yes | — |
| 77 | #504 | M16 | 1 | 2 | yes | — |
| 78 | C2×M32 | — | 2 | 1 | no | — |
| 79 | C2×M32 | — | 2 | 1 | no | — |
| 80 | C2×M32 | — | 4 | 4 | no | — |
| 81 | C2×M32 | — | 4 | 1 | no | — |
| 82 | C2×M32 | — | 4 | 1 | no | — |
| 83 | C2×M32 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C2×M32 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C22×C16 | — | 2 | 1 | no | — |
| 86 | C22×C16 | — | 2 | 1 | no | — |
| 87 | C22×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C16 | C2×Q8 | 1 | 1 | yes | characteristic |
| 89 | #431 | — | 4 | 1 | no | — |
| 90 | C2×C4⋊C8 | — | 4 | 1 | no | — |
| 91 | #405 | — | 2 | 1 | no | — |
| 92 | #405 | — | 2 | 1 | no | — |
| 93 | #405 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 94 | #405 | SD16 | 1 | 1 | yes | characteristic |
| 95 | C4×M16 | — | 2 | 1 | no | — |
| 96 | C4×M16 | SD16 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C8 | — | 2 | 1 | no | — |
| 98 | C2×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | M64 | — | 2 | 1 | no | — |
| 100 | M64 | — | 2 | 1 | no | — |
| 101 | M64 | — | 2 | 1 | no | — |
| 102 | M64 | — | 2 | 1 | no | — |
| 103 | M64 | — | 2 | 1 | no | — |
| 104 | M64 | — | 2 | 1 | no | — |
| 105 | M64 | — | 2 | 1 | no | — |
| 106 | M64 | C2×C8 | 1 | 2 | yes | — |
| 107 | C2×C32 | — | 2 | 1 | no | — |
| 108 | C2×C32 | — | 2 | 1 | no | — |
| 109 | C2×C32 | — | 2 | 1 | no | — |
| 110 | C2×C32 | C2×C8 | 1 | 2 | yes | — |
| 111 | #364 | — | 2 | 2 | no | — |
| 112 | #364 | — | 2 | 2 | no | — |
| 113 | #364 | — | 4 | 4 | no | — |
| 114 | #364 | — | 4 | 1 | no | — |
| 115 | #364 | — | 4 | 1 | no | — |
| 116 | #346 | — | 2 | 1 | no | — |
| 117 | #346 | — | 2 | 1 | no | — |
| 118 | #346 | — | 2 | 1 | no | — |
| 119 | #346 | C2×D4 | 1 | 1 | yes | characteristic |
| 120 | #346 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | #346 | #41 | 1 | 1 | yes | characteristic |
| 122 | #346 | #41 | 1 | 1 | yes | characteristic |
| 123 | C4×C16 | — | 2 | 1 | no | — |
| 124 | C4×C16 | — | 2 | 1 | no | — |
| 125 | C4×C16 | — | 2 | 1 | no | — |
| 126 | #335 | — | 4 | 1 | no | — |
| 127 | #334 | — | 4 | 1 | no | — |
| 128 | #334 | — | 4 | 1 | no | — |
| 129 | #334 | — | 4 | 1 | no | — |
| 130 | Q8⋊C8 | — | 8 | 1 | no | — |
| 131 | D4⋊C8 | — | 8 | 1 | no | — |
| 132 | C2×#41 | #106 | 1 | 1 | yes | characteristic |
| 133 | #131 | — | 2 | 2 | no | — |
| 134 | #131 | — | 2 | 2 | no | — |
| 135 | #131 | — | 2 | 1 | no | — |
| 136 | #131 | — | 2 | 1 | no | — |
| 137 | #131 | C4⋊C8 | 1 | 2 | yes | — |
| 138 | #131 | #108 | 1 | 2 | yes | — |
| 139 | C2×M16 | — | 2 | 1 | no | — |
| 140 | C2×M16 | — | 2 | 1 | no | — |
| 141 | C2×M16 | #106 | 1 | 1 | yes | characteristic |
| 142 | C2×M16 | #106 | 1 | 1 | yes | characteristic |
| 143 | C22×C8 | — | 2 | 1 | no | — |
| 144 | C22×C8 | — | 2 | 1 | no | — |
| 145 | C22×C8 | — | 8 | 1 | no | — |
| 146 | C22×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C22×C8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 148 | C4×Q8 | — | 2 | 1 | no | — |
| 149 | C4×D4 | — | 2 | 1 | no | — |
| 150 | C4×D4 | — | 4 | 1 | no | — |
| 151 | #117 | — | 2 | 1 | no | — |
| 152 | #117 | #106 | 1 | 1 | yes | characteristic |
| 153 | C2×C42 | — | 2 | 1 | no | — |
| 154 | C2×C42 | C2×SD16 | 1 | 1 | yes | characteristic |
| 155 | M32 | — | 2 | 1 | no | — |
| 156 | M32 | — | 2 | 2 | no | — |
| 157 | M32 | — | 2 | 2 | no | — |
| 158 | M32 | — | 2 | 1 | no | — |
| 159 | M32 | — | 2 | 1 | no | — |
| 160 | M32 | — | 4 | 2 | no | — |
| 161 | M32 | — | 4 | 2 | no | — |
| 162 | M32 | — | 4 | 2 | no | — |
| 163 | M32 | — | 8 | 4 | no | — |
| 164 | M32 | — | 8 | 1 | no | — |
| 165 | M32 | — | 8 | 1 | no | — |
| 166 | M32 | C4⋊C8 | 1 | 2 | yes | — |
| 167 | C2×C16 | — | 2 | 1 | no | — |
| 168 | C2×C16 | — | 2 | 1 | no | — |
| 169 | C2×C16 | — | 2 | 1 | no | — |
| 170 | C2×C16 | — | 2 | 2 | no | — |
| 171 | C2×C16 | — | 2 | 2 | no | — |
| 172 | C2×C16 | — | 2 | 1 | no | — |
| 173 | C2×C16 | — | 4 | 2 | no | — |
| 174 | C2×C16 | — | 8 | 4 | no | — |
| 175 | C2×C16 | — | 8 | 1 | no | — |
| 176 | C2×C16 | — | 8 | 1 | no | — |
| 177 | C2×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 178 | C2×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 179 | C2×C16 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 180 | C2×C16 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 181 | C2×C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 182 | C2×C16 | C2×M16 | 1 | 1 | yes | characteristic |
| 183 | C2×C16 | #131 | 1 | 1 | yes | characteristic |
| 184 | C2×C16 | #131 | 1 | 1 | yes | characteristic |
| 185 | C4⋊C8 | — | 8 | 1 | no | — |
| 186 | C4⋊C8 | — | 8 | 1 | no | — |
| 187 | C4⋊C8 | — | 8 | 1 | no | — |
| 188 | C22⋊C8 | — | 8 | 1 | no | — |
| 189 | #97 | — | 2 | 1 | no | — |
| 190 | #97 | — | 2 | 1 | no | — |
| 191 | #97 | — | 2 | 1 | no | — |
| 192 | C4×C8 | — | 2 | 1 | no | — |
| 193 | C4×C8 | — | 2 | 1 | no | — |
| 194 | C4×C8 | — | 2 | 1 | no | — |
| 195 | C4×C8 | — | 8 | 1 | no | — |
| 196 | C4×C8 | #120 | 1 | 1 | yes | characteristic |
| 197 | C4×C8 | #123 | 1 | 1 | yes | characteristic |
| 198 | C4×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 199 | C4×C8 | #137 | 1 | 1 | yes | characteristic |
| 200 | C32 | — | 2 | 2 | no | — |
| 201 | C32 | — | 2 | 2 | no | — |
| 202 | C32 | — | 4 | 1 | no | — |
| 203 | C32 | — | 4 | 1 | no | — |
| 204 | #41 | — | 2 | 1 | no | — |
| 205 | #41 | — | 2 | 1 | no | — |
| 206 | #41 | #334 | 1 | 2 | yes | — |
| 207 | #41 | #365 | 1 | 2 | yes | — |
| 208 | C2×Q8 | #365 | 1 | 1 | yes | characteristic |
| 209 | C2×D4 | — | 2 | 1 | no | — |
| 210 | C2×D4 | #365 | 1 | 1 | yes | characteristic |
| 211 | C22×C4 | — | 2 | 1 | no | — |
| 212 | C22×C4 | — | 2 | 1 | no | — |
| 213 | C22×C4 | — | 4 | 1 | no | — |
| 214 | C22×C4 | C2×#106 | 1 | 1 | yes | characteristic |
| 215 | M16 | — | 2 | 4 | no | — |
| 216 | M16 | — | 2 | 1 | no | — |
| 217 | M16 | — | 2 | 1 | no | — |
| 218 | M16 | — | 2 | 2 | no | — |
| 219 | M16 | — | 2 | 2 | no | — |
| 220 | M16 | #334 | 1 | 2 | yes | — |
| 221 | M16 | #365 | 1 | 2 | yes | — |
| 222 | C2×C8 | — | 2 | 4 | no | — |
| 223 | C2×C8 | — | 2 | 1 | no | — |
| 224 | C2×C8 | — | 2 | 1 | no | — |
| 225 | C2×C8 | — | 2 | 1 | no | — |
| 226 | C2×C8 | — | 2 | 1 | no | — |
| 227 | C2×C8 | — | 2 | 2 | no | — |
| 228 | C2×C8 | — | 2 | 2 | no | — |
| 229 | C2×C8 | — | 16 | 1 | no | — |
| 230 | C2×C8 | — | 16 | 1 | no | — |
| 231 | C2×C8 | — | 32 | 1 | no | — |
| 232 | C2×C8 | #385 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C2×C4⋊C8 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | #424 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | #428 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | C2×#108 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | #430 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | #450 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | #483 | 1 | 1 | yes | characteristic |
| 240 | C4⋊C4 | — | 2 | 1 | no | — |
| 241 | C4⋊C4 | — | 4 | 1 | no | — |
| 242 | C22⋊C4 | — | 2 | 1 | no | — |
| 243 | C22⋊C4 | — | 4 | 1 | no | — |
| 244 | C42 | — | 2 | 1 | no | — |
| 245 | C42 | — | 2 | 1 | no | — |
| 246 | C42 | — | 4 | 1 | no | — |
| 247 | C42 | #448 | 1 | 1 | yes | characteristic |
| 248 | C42 | #481 | 1 | 1 | yes | characteristic |
| 249 | C16 | — | 2 | 2 | no | — |
| 250 | C16 | — | 2 | 1 | no | — |
| 251 | C16 | — | 2 | 1 | no | — |
| 252 | C16 | — | 4 | 1 | no | — |
| 253 | C16 | — | 8 | 2 | no | — |
| 254 | C16 | — | 8 | 2 | no | — |
| 255 | C16 | C8×D4 | 1 | 1 | yes | characteristic |
| 256 | C16 | C8×D4 | 1 | 1 | yes | characteristic |
| 257 | C16 | #435 | 1 | 1 | yes | characteristic |
| 258 | C16 | #435 | 1 | 1 | yes | characteristic |
| 259 | C23 | — | 2 | 1 | no | — |
| 260 | C23 | #2155 | 1 | 1 | yes | characteristic |
| 261 | Q8 | — | 2 | 1 | no | — |
| 262 | Q8 | #1368 | 1 | 2 | yes | — |
| 263 | D4 | — | 2 | 2 | no | — |
| 264 | D4 | — | 2 | 2 | no | — |
| 265 | D4 | — | 2 | 1 | no | — |
| 266 | D4 | #1368 | 1 | 2 | yes | — |
| 267 | C2×C4 | — | 2 | 1 | no | — |
| 268 | C2×C4 | — | 2 | 2 | no | — |
| 269 | C2×C4 | — | 2 | 2 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 4 | 1 | no | — |
| 273 | C2×C4 | — | 8 | 1 | no | — |
| 274 | C2×C4 | — | 16 | 1 | no | — |
| 275 | C2×C4 | C2×#334 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | #1570 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | #1853 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | C2×#365 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | C2×#365 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | C2×#365 | 1 | 1 | yes | characteristic |
| 281 | C8 | — | 2 | 2 | no | — |
| 282 | C8 | — | 2 | 1 | no | — |
| 283 | C8 | — | 2 | 2 | no | — |
| 284 | C8 | — | 2 | 1 | no | — |
| 285 | C8 | — | 32 | 1 | no | — |
| 286 | C8 | — | 32 | 1 | no | — |
| 287 | C8 | #1565 | 1 | 1 | yes | characteristic |
| 288 | C8 | #1573 | 1 | 1 | yes | characteristic |
| 289 | C8 | #1850 | 1 | 1 | yes | characteristic |
| 290 | C8 | #1855 | 1 | 1 | yes | characteristic |
| 291 | C22 | — | 2 | 2 | no | — |
| 292 | C22 | — | 4 | 1 | no | — |
| 293 | C22 | — | 8 | 1 | no | — |
| 294 | C22 | #12000 | 1 | 1 | yes | characteristic |
| 295 | C22 | #12028 | 1 | 1 | yes | characteristic |
| 296 | C22 | #12928 | 1 | 1 | yes | characteristic |
| 297 | C4 | — | 2 | 2 | no | — |
| 298 | C4 | — | 4 | 1 | no | — |
| 299 | C4 | — | 4 | 1 | no | — |
| 300 | C4 | — | 8 | 1 | no | — |
| 301 | C4 | #8830 | 1 | 1 | yes | characteristic |
| 302 | C4 | C2×#1368 | 1 | 1 | yes | characteristic |
| 303 | C4 | #12030 | 1 | 1 | yes | characteristic |
| 304 | C4 | #12929 | 1 | 1 | yes | characteristic |
| 305 | C2 | — | 2 | 1 | no | — |
| 306 | C2 | — | 2 | 1 | no | — |
| 307 | C2 | — | 2 | 1 | no | — |
| 308 | C2 | — | 4 | 2 | no | — |
| 309 | C2 | #135826 | 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