← #72225506
Subgroups — #72225506
Order 2048.
471 conjugacy classes of subgroups.
The subgroup lattice has 351 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72225484 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72225468 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72225453 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72225427 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #72225405 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #72225314 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #72216311 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #513082 | C22 | 1 | 1 | yes | characteristic |
| 9 | #483943 | — | 2 | 1 | no | — |
| 10 | #153582 | C22 | 1 | 1 | yes | characteristic |
| 11 | #153582 | C22 | 1 | 1 | yes | characteristic |
| 12 | #151998 | C22 | 1 | 1 | yes | characteristic |
| 13 | #151129 | — | 2 | 1 | no | — |
| 14 | #151129 | C4 | 1 | 2 | yes | complemented |
| 15 | #151085 | — | 2 | 1 | no | — |
| 16 | #151085 | C4 | 1 | 2 | yes | complemented |
| 17 | #151055 | C22 | 1 | 1 | yes | characteristic |
| 18 | #151035 | C22 | 1 | 1 | yes | characteristic |
| 19 | #150160 | — | 2 | 1 | no | — |
| 20 | #149495 | C22 | 1 | 1 | yes | characteristic |
| 21 | #129143 | — | 2 | 1 | no | — |
| 22 | #94743 | C4 | 1 | 2 | yes | complemented |
| 23 | #94738 | C4 | 1 | 2 | yes | complemented |
| 24 | #94696 | C4 | 1 | 2 | yes | — |
| 25 | #94694 | C4 | 1 | 2 | yes | — |
| 26 | #94509 | — | 2 | 1 | no | — |
| 27 | #94506 | — | 2 | 1 | no | — |
| 28 | #94054 | — | 2 | 1 | no | — |
| 29 | #33733 | — | 2 | 1 | no | — |
| 30 | #17956 | — | 4 | 1 | no | — |
| 31 | #13733 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | #13725 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | #13719 | — | 2 | 1 | no | — |
| 34 | #13697 | — | 2 | 1 | no | — |
| 35 | #13697 | C2×C4 | 1 | 2 | yes | — |
| 36 | #13654 | — | 2 | 1 | no | — |
| 37 | #13654 | C2×C4 | 1 | 2 | yes | — |
| 38 | #13638 | — | 2 | 1 | no | — |
| 39 | C4×M64 | D4 | 1 | 1 | yes | characteristic |
| 40 | C4×M64 | C23 | 1 | 1 | yes | characteristic |
| 41 | #12670 | — | 2 | 1 | no | — |
| 42 | #12664 | — | 4 | 1 | no | — |
| 43 | #12086 | — | 2 | 2 | no | — |
| 44 | #12086 | — | 4 | 1 | no | — |
| 45 | #12045 | — | 2 | 2 | no | — |
| 46 | #12045 | — | 4 | 1 | no | — |
| 47 | #12023 | — | 2 | 1 | no | — |
| 48 | C2×#1373 | — | 2 | 1 | no | — |
| 49 | #11671 | — | 4 | 1 | no | — |
| 50 | C2×#1311 | D4 | 1 | 1 | yes | characteristic |
| 51 | #7544 | — | 2 | 1 | no | — |
| 52 | #7544 | C2×C4 | 1 | 2 | yes | — |
| 53 | #7513 | — | 2 | 1 | no | — |
| 54 | #7513 | C2×C4 | 1 | 2 | yes | — |
| 55 | #7511 | — | 2 | 1 | no | — |
| 56 | #7511 | — | 2 | 1 | no | — |
| 57 | #7476 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | #7475 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | #7463 | — | 2 | 1 | no | — |
| 60 | #7463 | C2×C4 | 1 | 2 | yes | — |
| 61 | #7462 | — | 2 | 1 | no | — |
| 62 | #7462 | C2×C4 | 1 | 2 | yes | — |
| 63 | #7385 | — | 2 | 2 | no | — |
| 64 | #7385 | — | 2 | 2 | no | — |
| 65 | #7330 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | #7330 | C2×C4 | 1 | 1 | yes | characteristic |
| 67 | #7330 | D4 | 1 | 1 | yes | characteristic |
| 68 | #7330 | D4 | 1 | 1 | yes | characteristic |
| 69 | #7179 | — | 4 | 1 | no | — |
| 70 | #3169 | — | 4 | 1 | no | — |
| 71 | #2931 | — | 8 | 1 | no | — |
| 72 | C2×D32 | — | 2 | 1 | no | — |
| 73 | C2×D32 | C42 | 1 | 2 | yes | — |
| 74 | C2×M64 | — | 2 | 1 | no | — |
| 75 | C2×M64 | C22×C4 | 1 | 1 | yes | characteristic |
| 76 | C2×M64 | #41 | 1 | 1 | yes | characteristic |
| 77 | #2252 | — | 2 | 1 | no | — |
| 78 | #2246 | — | 2 | 1 | no | — |
| 79 | #2240 | — | 4 | 1 | no | — |
| 80 | #2215 | — | 2 | 2 | no | — |
| 81 | #2215 | — | 4 | 1 | no | — |
| 82 | C4×D16 | — | 2 | 2 | no | — |
| 83 | C4×D16 | — | 4 | 1 | no | — |
| 84 | C2×#368 | — | 2 | 1 | no | — |
| 85 | #2158 | — | 2 | 1 | no | — |
| 86 | C2×#364 | — | 4 | 1 | no | — |
| 87 | #2108 | — | 8 | 1 | no | — |
| 88 | C2×C4×C16 | D8 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | #1772 | — | 4 | 1 | no | — |
| 91 | #1589 | — | 4 | 2 | no | — |
| 92 | #1589 | — | 8 | 1 | no | — |
| 93 | C8×D8 | — | 4 | 2 | no | — |
| 94 | C8×D8 | — | 8 | 1 | no | — |
| 95 | #1567 | — | 4 | 1 | no | — |
| 96 | #1492 | — | 8 | 1 | no | — |
| 97 | C2×C82 | D8 | 1 | 1 | yes | characteristic |
| 98 | M128 | — | 2 | 2 | no | — |
| 99 | M128 | — | 2 | 2 | no | — |
| 100 | #1426 | — | 2 | 4 | no | — |
| 101 | #1424 | — | 2 | 1 | no | — |
| 102 | #1424 | C42 | 1 | 2 | yes | — |
| 103 | #1415 | — | 2 | 2 | no | — |
| 104 | #1415 | — | 2 | 2 | no | — |
| 105 | #1397 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 106 | #1397 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 107 | #1397 | C22×C4 | 1 | 1 | yes | characteristic |
| 108 | #1397 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C4×C32 | — | 2 | 1 | no | — |
| 110 | C4×C32 | C22×C4 | 1 | 1 | yes | characteristic |
| 111 | C4×C32 | #41 | 1 | 1 | yes | characteristic |
| 112 | #1382 | — | 4 | 1 | no | — |
| 113 | #1379 | — | 8 | 1 | no | — |
| 114 | #1373 | — | 2 | 2 | no | — |
| 115 | #1373 | — | 4 | 1 | no | — |
| 116 | #1372 | — | 2 | 2 | no | — |
| 117 | #1372 | — | 4 | 1 | no | — |
| 118 | #1331 | — | 4 | 2 | no | — |
| 119 | #1331 | — | 4 | 2 | no | — |
| 120 | #1311 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 121 | #1311 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 122 | #1311 | SD16 | 1 | 1 | yes | characteristic |
| 123 | #1311 | SD16 | 1 | 1 | yes | characteristic |
| 124 | C2×#131 | — | 16 | 1 | no | — |
| 125 | #532 | — | 8 | 1 | no | — |
| 126 | C2×D16 | — | 2 | 2 | no | — |
| 127 | C2×D16 | — | 4 | 1 | no | — |
| 128 | C2×M32 | — | 8 | 1 | no | — |
| 129 | C22×C16 | — | 2 | 1 | no | — |
| 130 | C22×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 131 | C22×C16 | #135 | 1 | 1 | yes | characteristic |
| 132 | #500 | — | 4 | 1 | no | — |
| 133 | #493 | — | 4 | 1 | no | — |
| 134 | #486 | — | 8 | 1 | no | — |
| 135 | #466 | — | 4 | 2 | no | — |
| 136 | #466 | — | 8 | 1 | no | — |
| 137 | C4×D8 | — | 4 | 2 | no | — |
| 138 | C4×D8 | — | 8 | 1 | no | — |
| 139 | #436 | — | 8 | 2 | no | — |
| 140 | #436 | — | 16 | 1 | no | — |
| 141 | C8×D4 | — | 8 | 2 | no | — |
| 142 | C8×D4 | — | 16 | 1 | no | — |
| 143 | C2×#108 | — | 4 | 1 | no | — |
| 144 | #427 | — | 4 | 1 | no | — |
| 145 | #423 | — | 8 | 1 | no | — |
| 146 | #408 | — | 16 | 1 | no | — |
| 147 | #405 | — | 8 | 1 | no | — |
| 148 | C2×C4×C8 | — | 2 | 1 | no | — |
| 149 | C2×C4×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 150 | D32 | — | 4 | 2 | no | — |
| 151 | D32 | — | 4 | 2 | no | — |
| 152 | M64 | — | 2 | 2 | no | — |
| 153 | M64 | — | 4 | 1 | no | — |
| 154 | M64 | — | 4 | 2 | no | — |
| 155 | C2×C32 | — | 2 | 1 | no | — |
| 156 | C2×C32 | — | 2 | 1 | no | — |
| 157 | C2×C32 | C2×C42 | 1 | 1 | yes | characteristic |
| 158 | C2×C32 | #117 | 1 | 1 | yes | characteristic |
| 159 | C2×C32 | C4×D4 | 1 | 1 | yes | characteristic |
| 160 | C2×C32 | C4×D4 | 1 | 1 | yes | characteristic |
| 161 | #368 | — | 2 | 2 | no | — |
| 162 | #368 | — | 4 | 1 | no | — |
| 163 | #367 | — | 2 | 2 | no | — |
| 164 | #366 | — | 2 | 2 | no | — |
| 165 | #366 | — | 4 | 1 | no | — |
| 166 | #366 | — | 4 | 1 | no | — |
| 167 | #364 | — | 4 | 2 | no | — |
| 168 | #364 | — | 8 | 1 | no | — |
| 169 | #357 | — | 4 | 2 | no | — |
| 170 | #357 | — | 4 | 2 | no | — |
| 171 | #346 | — | 16 | 1 | no | — |
| 172 | C4×C16 | — | 2 | 1 | no | — |
| 173 | C4×C16 | — | 16 | 1 | no | — |
| 174 | C4×C16 | #102 | 1 | 1 | yes | characteristic |
| 175 | C4×C16 | #102 | 1 | 1 | yes | characteristic |
| 176 | C4×C16 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 177 | C4×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 178 | C4×C16 | C2×SD16 | 1 | 1 | yes | characteristic |
| 179 | C4×C16 | #135 | 1 | 1 | yes | characteristic |
| 180 | #339 | — | 8 | 1 | no | — |
| 181 | #335 | — | 4 | 2 | no | — |
| 182 | #335 | — | 8 | 1 | no | — |
| 183 | D4⋊C8 | — | 8 | 2 | no | — |
| 184 | D4⋊C8 | — | 8 | 2 | no | — |
| 185 | C82 | — | 2 | 1 | no | — |
| 186 | C82 | #102 | 1 | 1 | yes | characteristic |
| 187 | C82 | #102 | 1 | 1 | yes | characteristic |
| 188 | C64 | — | 4 | 2 | no | — |
| 189 | C64 | — | 4 | 2 | no | — |
| 190 | C2×#41 | — | 16 | 1 | no | — |
| 191 | C2×D8 | — | 4 | 2 | no | — |
| 192 | C2×D8 | — | 8 | 1 | no | — |
| 193 | #131 | — | 32 | 4 | no | — |
| 194 | C2×M16 | — | 8 | 1 | no | — |
| 195 | C2×M16 | — | 16 | 2 | no | — |
| 196 | C2×M16 | — | 16 | 1 | no | — |
| 197 | C22×C8 | — | 4 | 1 | no | — |
| 198 | C22×C8 | — | 16 | 2 | no | — |
| 199 | C22×C8 | C4×D8 | 1 | 1 | yes | characteristic |
| 200 | C22×C8 | #509 | 1 | 1 | yes | characteristic |
| 201 | C4⋊Q8 | — | 8 | 1 | no | — |
| 202 | #127 | — | 8 | 1 | no | — |
| 203 | #124 | — | 16 | 1 | no | — |
| 204 | #121 | — | 8 | 2 | no | — |
| 205 | #121 | — | 16 | 1 | no | — |
| 206 | C4×D4 | — | 8 | 2 | no | — |
| 207 | C4×D4 | — | 16 | 1 | no | — |
| 208 | #117 | — | 8 | 1 | no | — |
| 209 | C2×C42 | #509 | 1 | 1 | yes | characteristic |
| 210 | D16 | — | 4 | 2 | no | — |
| 211 | D16 | — | 4 | 2 | no | — |
| 212 | D16 | — | 8 | 2 | no | — |
| 213 | M32 | — | 8 | 2 | no | — |
| 214 | M32 | — | 16 | 1 | no | — |
| 215 | C2×C16 | — | 2 | 1 | no | — |
| 216 | C2×C16 | — | 2 | 1 | no | — |
| 217 | C2×C16 | — | 2 | 1 | no | — |
| 218 | C2×C16 | — | 2 | 1 | no | — |
| 219 | C2×C16 | — | 4 | 1 | no | — |
| 220 | C2×C16 | — | 4 | 1 | no | — |
| 221 | C2×C16 | — | 4 | 1 | no | — |
| 222 | C2×C16 | — | 16 | 1 | no | — |
| 223 | C2×C16 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 224 | C2×C16 | #416 | 1 | 1 | yes | characteristic |
| 225 | C2×C16 | C4×SD16 | 1 | 1 | yes | characteristic |
| 226 | C2×C16 | C4×SD16 | 1 | 1 | yes | characteristic |
| 227 | #108 | — | 4 | 2 | no | — |
| 228 | #108 | — | 8 | 1 | no | — |
| 229 | #107 | — | 4 | 2 | no | — |
| 230 | #107 | — | 4 | 2 | no | — |
| 231 | #107 | — | 8 | 1 | no | — |
| 232 | #106 | — | 8 | 1 | no | — |
| 233 | C4⋊C8 | — | 8 | 2 | no | — |
| 234 | C4⋊C8 | — | 16 | 1 | no | — |
| 235 | #102 | — | 8 | 2 | no | — |
| 236 | #102 | — | 8 | 2 | no | — |
| 237 | C22⋊C8 | — | 16 | 2 | no | — |
| 238 | C22⋊C8 | — | 16 | 2 | no | — |
| 239 | #97 | — | 16 | 1 | no | — |
| 240 | C4×C8 | — | 2 | 1 | no | — |
| 241 | C4×C8 | — | 2 | 1 | no | — |
| 242 | C4×C8 | — | 4 | 1 | no | — |
| 243 | C4×C8 | — | 16 | 1 | no | — |
| 244 | C4×C8 | C2×#102 | 1 | 1 | yes | characteristic |
| 245 | C4×C8 | C4×D8 | 1 | 1 | yes | characteristic |
| 246 | C4×C8 | #509 | 1 | 1 | yes | characteristic |
| 247 | C4×C8 | #510 | 1 | 1 | yes | characteristic |
| 248 | C32 | — | 2 | 2 | no | — |
| 249 | C32 | — | 2 | 1 | no | — |
| 250 | C32 | — | 2 | 1 | no | — |
| 251 | C32 | — | 2 | 1 | no | — |
| 252 | C32 | — | 4 | 1 | no | — |
| 253 | C32 | #443 | 1 | 2 | yes | — |
| 254 | #41 | — | 32 | 2 | no | — |
| 255 | #41 | — | 32 | 2 | no | — |
| 256 | C2×Q8 | — | 16 | 1 | no | — |
| 257 | C2×D4 | — | 8 | 2 | no | — |
| 258 | C2×D4 | — | 16 | 1 | no | — |
| 259 | C2×D4 | — | 16 | 1 | no | — |
| 260 | C2×D4 | — | 16 | 2 | no | — |
| 261 | C22×C4 | — | 2 | 1 | no | — |
| 262 | C22×C4 | — | 16 | 2 | no | — |
| 263 | C22×C4 | #2177 | 1 | 1 | yes | characteristic |
| 264 | D8 | — | 8 | 2 | no | — |
| 265 | D8 | — | 8 | 2 | no | — |
| 266 | D8 | — | 16 | 1 | no | — |
| 267 | D8 | — | 16 | 1 | no | — |
| 268 | M16 | — | 8 | 2 | no | — |
| 269 | M16 | — | 16 | 1 | no | — |
| 270 | M16 | — | 32 | 4 | no | — |
| 271 | M16 | — | 32 | 2 | no | — |
| 272 | C2×C8 | — | 2 | 1 | no | — |
| 273 | C2×C8 | — | 2 | 1 | no | — |
| 274 | C2×C8 | — | 2 | 1 | no | — |
| 275 | C2×C8 | — | 2 | 1 | no | — |
| 276 | C2×C8 | — | 4 | 1 | no | — |
| 277 | C2×C8 | — | 4 | 1 | no | — |
| 278 | C2×C8 | — | 8 | 2 | no | — |
| 279 | C2×C8 | — | 16 | 1 | no | — |
| 280 | C2×C8 | — | 16 | 2 | no | — |
| 281 | C2×C8 | — | 32 | 4 | no | — |
| 282 | C2×C8 | C4×#102 | 1 | 1 | yes | characteristic |
| 283 | C2×C8 | #2139 | 1 | 1 | yes | characteristic |
| 284 | C2×C8 | #2175 | 1 | 1 | yes | characteristic |
| 285 | C2×C8 | #2177 | 1 | 1 | yes | characteristic |
| 286 | C4⋊C4 | — | 8 | 2 | no | — |
| 287 | C4⋊C4 | — | 8 | 2 | no | — |
| 288 | C4⋊C4 | — | 16 | 1 | no | — |
| 289 | C22⋊C4 | — | 16 | 2 | no | — |
| 290 | C22⋊C4 | — | 16 | 1 | no | — |
| 291 | C22⋊C4 | — | 16 | 2 | no | — |
| 292 | C42 | — | 2 | 1 | no | — |
| 293 | C42 | — | 16 | 1 | no | — |
| 294 | C42 | #2140 | 1 | 1 | yes | characteristic |
| 295 | C42 | #2175 | 1 | 1 | yes | characteristic |
| 296 | C16 | — | 2 | 1 | no | — |
| 297 | C16 | — | 2 | 1 | no | — |
| 298 | C16 | — | 2 | 1 | no | — |
| 299 | C16 | — | 4 | 1 | no | — |
| 300 | C16 | — | 4 | 1 | no | — |
| 301 | C16 | — | 16 | 2 | no | — |
| 302 | C16 | #1757 | 1 | 1 | yes | characteristic |
| 303 | C16 | #1764 | 1 | 1 | yes | characteristic |
| 304 | C23 | — | 16 | 2 | no | — |
| 305 | C23 | #13658 | 1 | 1 | yes | characteristic |
| 306 | Q8 | — | 32 | 2 | no | — |
| 307 | D4 | — | 16 | 2 | no | — |
| 308 | D4 | — | 16 | 2 | no | — |
| 309 | D4 | — | 32 | 2 | no | — |
| 310 | D4 | — | 32 | 2 | no | — |
| 311 | D4 | — | 32 | 4 | no | — |
| 312 | C2×C4 | — | 2 | 1 | no | — |
| 313 | C2×C4 | — | 2 | 1 | no | — |
| 314 | C2×C4 | — | 2 | 1 | no | — |
| 315 | C2×C4 | — | 2 | 1 | no | — |
| 316 | C2×C4 | — | 4 | 2 | no | — |
| 317 | C2×C4 | — | 16 | 2 | no | — |
| 318 | C2×C4 | — | 16 | 2 | no | — |
| 319 | C2×C4 | — | 32 | 2 | no | — |
| 320 | C2×C4 | — | 32 | 2 | no | — |
| 321 | C2×C4 | #12637 | 1 | 1 | yes | characteristic |
| 322 | C2×C4 | #13658 | 1 | 1 | yes | characteristic |
| 323 | C8 | — | 2 | 1 | no | — |
| 324 | C8 | — | 2 | 1 | no | — |
| 325 | C8 | — | 8 | 1 | no | — |
| 326 | C8 | — | 8 | 1 | no | — |
| 327 | C8 | — | 16 | 2 | no | — |
| 328 | C8 | — | 32 | 2 | no | — |
| 329 | C8 | #8251 | 1 | 1 | yes | characteristic |
| 330 | C8 | #12646 | 1 | 1 | yes | characteristic |
| 331 | C8 | #13657 | 1 | 1 | yes | characteristic |
| 332 | C8 | #13658 | 1 | 1 | yes | characteristic |
| 333 | C22 | — | 2 | 1 | no | — |
| 334 | C22 | — | 4 | 1 | no | — |
| 335 | C22 | — | 32 | 2 | no | — |
| 336 | C22 | — | 32 | 2 | no | — |
| 337 | C22 | — | 32 | 2 | no | — |
| 338 | C22 | #151969 | 1 | 1 | yes | characteristic |
| 339 | C4 | — | 2 | 2 | no | — |
| 340 | C4 | — | 2 | 1 | no | — |
| 341 | C4 | — | 4 | 1 | no | — |
| 342 | C4 | — | 32 | 2 | no | — |
| 343 | C4 | — | 32 | 2 | no | — |
| 344 | C4 | #129001 | 1 | 1 | yes | characteristic |
| 345 | C4 | #151972 | 1 | 1 | yes | characteristic |
| 346 | C2 | — | 2 | 1 | no | — |
| 347 | C2 | — | 4 | 1 | no | — |
| 348 | C2 | — | 64 | 2 | no | — |
| 349 | C2 | #72225505 | 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