← #72323159
Subgroups — #72323159
Order 2048.
765 conjugacy classes of subgroups.
The subgroup lattice has 317 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72320557 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #72286471 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #72231932 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72231877 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72225240 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #72224639 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #72216311 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #513082 | C4 | 1 | 1 | yes | characteristic |
| 9 | #483943 | C22 | 1 | 2 | yes | — |
| 10 | #483711 | — | 2 | 2 | no | — |
| 11 | #151129 | C4 | 1 | 4 | yes | — |
| 12 | #151085 | C22 | 1 | 4 | yes | — |
| 13 | #151055 | C4 | 1 | 1 | yes | characteristic |
| 14 | #150160 | C4 | 1 | 2 | yes | — |
| 15 | C2×#7332 | C22 | 1 | 1 | yes | characteristic |
| 16 | #149495 | C22 | 1 | 1 | yes | characteristic |
| 17 | #122966 | — | 2 | 2 | no | — |
| 18 | #122966 | C22 | 1 | 4 | yes | — |
| 19 | #122963 | — | 2 | 2 | no | — |
| 20 | #122963 | C22 | 1 | 4 | yes | — |
| 21 | #122922 | C22 | 1 | 2 | yes | — |
| 22 | #122484 | — | 2 | 2 | no | — |
| 23 | #122214 | C22 | 1 | 1 | yes | characteristic |
| 24 | #93877 | C22 | 1 | 4 | yes | — |
| 25 | #93757 | C22 | 1 | 8 | yes | — |
| 26 | #93668 | C22 | 1 | 2 | yes | — |
| 27 | #93668 | C22 | 1 | 2 | yes | — |
| 28 | #33733 | C2×C4 | 1 | 2 | yes | — |
| 29 | #33495 | — | 4 | 2 | no | — |
| 30 | #17956 | — | 2 | 2 | no | — |
| 31 | #13733 | C8 | 1 | 1 | yes | characteristic |
| 32 | #13725 | C8 | 1 | 1 | yes | characteristic |
| 33 | #13719 | C8 | 1 | 2 | yes | — |
| 34 | #13697 | C8 | 1 | 4 | yes | — |
| 35 | #13654 | C2×C4 | 1 | 4 | yes | — |
| 36 | C4×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C2×#1397 | — | 2 | 1 | no | — |
| 39 | #12086 | — | 2 | 2 | no | — |
| 40 | #12086 | C2×C4 | 1 | 4 | yes | — |
| 41 | #12064 | — | 2 | 4 | no | — |
| 42 | #12062 | — | 2 | 4 | no | — |
| 43 | #12045 | — | 2 | 2 | no | — |
| 44 | #12045 | C23 | 1 | 4 | yes | — |
| 45 | #12039 | — | 4 | 2 | no | — |
| 46 | C16×D8 | — | 4 | 2 | no | — |
| 47 | #12023 | C2×C4 | 1 | 2 | yes | — |
| 48 | #12013 | — | 2 | 2 | no | — |
| 49 | #11671 | — | 2 | 2 | no | — |
| 50 | #11632 | — | 4 | 2 | no | — |
| 51 | #11410 | D4 | 1 | 2 | yes | — |
| 52 | C2×#1311 | C23 | 1 | 1 | yes | characteristic |
| 53 | #7463 | C2×C4 | 1 | 4 | yes | — |
| 54 | #7385 | C2×C4 | 1 | 8 | yes | — |
| 55 | #7332 | — | 2 | 1 | no | — |
| 56 | #7332 | C23 | 1 | 2 | yes | — |
| 57 | #7330 | C2×C4 | 1 | 2 | yes | — |
| 58 | #7330 | C23 | 1 | 2 | yes | — |
| 59 | #7099 | — | 2 | 2 | no | — |
| 60 | #7099 | C23 | 1 | 4 | yes | — |
| 61 | #7070 | — | 2 | 4 | no | — |
| 62 | #7070 | — | 2 | 4 | no | — |
| 63 | #7053 | D4 | 1 | 2 | yes | — |
| 64 | #7053 | C23 | 1 | 1 | yes | characteristic |
| 65 | #7053 | C23 | 1 | 1 | yes | characteristic |
| 66 | #3169 | — | 2 | 2 | no | — |
| 67 | #2931 | — | 4 | 2 | no | — |
| 68 | C2×D32 | C2×C8 | 1 | 4 | yes | — |
| 69 | C2×M64 | — | 2 | 1 | no | — |
| 70 | C2×M64 | C2×C8 | 1 | 2 | yes | — |
| 71 | C22×C32 | — | 4 | 1 | no | — |
| 72 | #2252 | C2×C8 | 1 | 2 | yes | — |
| 73 | #2246 | C2×C8 | 1 | 2 | yes | — |
| 74 | #2240 | — | 2 | 2 | no | — |
| 75 | #2215 | — | 2 | 2 | no | — |
| 76 | #2215 | C2×C8 | 1 | 4 | yes | — |
| 77 | C4×D16 | — | 2 | 2 | no | — |
| 78 | C4×D16 | C22×C4 | 1 | 4 | yes | — |
| 79 | #2169 | — | 4 | 4 | no | — |
| 80 | #2168 | — | 8 | 4 | no | — |
| 81 | D4×C16 | — | 4 | 4 | no | — |
| 82 | #2163 | — | 4 | 2 | no | — |
| 83 | #2162 | — | 2 | 2 | no | — |
| 84 | #2113 | — | 8 | 4 | no | — |
| 85 | C2×C4×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C16 | #41 | 1 | 1 | yes | characteristic |
| 87 | #1589 | — | 2 | 4 | no | — |
| 88 | #1589 | — | 4 | 2 | no | — |
| 89 | C8×D8 | — | 2 | 4 | no | — |
| 90 | C8×D8 | — | 4 | 2 | no | — |
| 91 | #1567 | — | 2 | 2 | no | — |
| 92 | #1492 | — | 4 | 2 | no | — |
| 93 | C2×C82 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | #1424 | C2×C8 | 1 | 4 | yes | — |
| 95 | #1415 | C2×C8 | 1 | 8 | yes | — |
| 96 | #1397 | — | 2 | 2 | no | — |
| 97 | #1397 | — | 4 | 1 | no | — |
| 98 | #1397 | C22×C4 | 1 | 2 | yes | — |
| 99 | #1397 | C22×C4 | 1 | 2 | yes | — |
| 100 | C4×C32 | — | 2 | 1 | no | — |
| 101 | C4×C32 | C2×C8 | 1 | 1 | yes | characteristic |
| 102 | C4×C32 | C2×C8 | 1 | 1 | yes | characteristic |
| 103 | #1372 | — | 2 | 2 | no | — |
| 104 | #1372 | C22×C4 | 1 | 4 | yes | — |
| 105 | #1370 | — | 4 | 2 | no | — |
| 106 | #1367 | — | 2 | 4 | no | — |
| 107 | #1331 | — | 2 | 4 | no | — |
| 108 | #1331 | — | 2 | 4 | no | — |
| 109 | D4⋊C16 | — | 4 | 4 | no | — |
| 110 | D4⋊C16 | — | 4 | 4 | no | — |
| 111 | #1312 | C2×D4 | 1 | 2 | yes | — |
| 112 | #1312 | C2×D4 | 1 | 2 | yes | — |
| 113 | #1311 | C22×C4 | 1 | 1 | yes | characteristic |
| 114 | #1311 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | #1311 | #41 | 1 | 1 | yes | characteristic |
| 116 | #1311 | C24 | 1 | 1 | yes | characteristic |
| 117 | C8×C16 | — | 2 | 2 | no | — |
| 118 | C2×#131 | — | 8 | 2 | no | — |
| 119 | #532 | — | 4 | 2 | no | — |
| 120 | C2×D16 | — | 2 | 2 | no | — |
| 121 | C2×D16 | C22×C8 | 1 | 4 | yes | — |
| 122 | C2×M32 | — | 8 | 4 | no | — |
| 123 | C22×C16 | — | 2 | 1 | no | — |
| 124 | C22×C16 | — | 8 | 4 | no | — |
| 125 | C22×C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 126 | C22×C16 | #131 | 1 | 1 | yes | characteristic |
| 127 | #500 | — | 2 | 2 | no | — |
| 128 | #493 | — | 2 | 2 | no | — |
| 129 | #486 | — | 4 | 2 | no | — |
| 130 | #466 | — | 2 | 4 | no | — |
| 131 | #466 | — | 4 | 2 | no | — |
| 132 | C4×D8 | — | 2 | 4 | no | — |
| 133 | C4×D8 | — | 4 | 2 | no | — |
| 134 | #436 | — | 4 | 4 | no | — |
| 135 | #436 | — | 8 | 2 | no | — |
| 136 | C8×D4 | — | 4 | 4 | no | — |
| 137 | C8×D4 | — | 8 | 2 | no | — |
| 138 | #423 | — | 4 | 2 | no | — |
| 139 | #408 | — | 8 | 2 | no | — |
| 140 | C2×C4×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 142 | D32 | — | 2 | 8 | no | — |
| 143 | M64 | — | 2 | 4 | no | — |
| 144 | M64 | — | 4 | 2 | no | — |
| 145 | C2×C32 | — | 2 | 2 | no | — |
| 146 | C2×C32 | — | 4 | 2 | no | — |
| 147 | C2×C32 | — | 8 | 2 | no | — |
| 148 | C2×C32 | C22×C8 | 1 | 2 | yes | — |
| 149 | C2×C32 | C22×C8 | 1 | 2 | yes | — |
| 150 | #366 | — | 2 | 2 | no | — |
| 151 | #366 | C22×C8 | 1 | 4 | yes | — |
| 152 | C4⋊C16 | — | 4 | 2 | no | — |
| 153 | C4⋊C16 | — | 4 | 4 | no | — |
| 154 | #357 | — | 2 | 4 | no | — |
| 155 | #357 | — | 2 | 4 | no | — |
| 156 | C22⋊C16 | — | 4 | 2 | no | — |
| 157 | C22⋊C16 | — | 8 | 4 | no | — |
| 158 | C22⋊C16 | — | 8 | 4 | no | — |
| 159 | C22⋊C16 | — | 8 | 2 | no | — |
| 160 | #346 | — | 8 | 2 | no | — |
| 161 | C4×C16 | — | 2 | 2 | no | — |
| 162 | C4×C16 | — | 2 | 2 | no | — |
| 163 | C4×C16 | — | 2 | 1 | no | — |
| 164 | C4×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 165 | C4×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 166 | C4×C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 167 | C4×C16 | #131 | 1 | 1 | yes | characteristic |
| 168 | C4×C16 | C23×C4 | 1 | 1 | yes | characteristic |
| 169 | C4×C16 | C2×#41 | 1 | 1 | yes | characteristic |
| 170 | #335 | — | 2 | 4 | no | — |
| 171 | #335 | — | 4 | 2 | no | — |
| 172 | D4⋊C8 | — | 4 | 4 | no | — |
| 173 | D4⋊C8 | — | 4 | 4 | no | — |
| 174 | C82 | C4×D4 | 1 | 1 | yes | characteristic |
| 175 | C82 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 176 | C82 | C22×D4 | 1 | 1 | yes | characteristic |
| 177 | C2×#41 | — | 8 | 2 | no | — |
| 178 | C2×D8 | — | 2 | 4 | no | — |
| 179 | C2×D8 | — | 4 | 2 | no | — |
| 180 | #131 | — | 32 | 4 | no | — |
| 181 | C2×M16 | — | 8 | 4 | no | — |
| 182 | C2×M16 | — | 8 | 2 | no | — |
| 183 | C22×C8 | — | 2 | 2 | no | — |
| 184 | C22×C8 | — | 8 | 4 | no | — |
| 185 | C22×C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 186 | C22×C8 | #444 | 1 | 1 | yes | characteristic |
| 187 | C4⋊Q8 | — | 4 | 2 | no | — |
| 188 | #127 | — | 4 | 2 | no | — |
| 189 | #124 | — | 8 | 2 | no | — |
| 190 | #121 | — | 4 | 4 | no | — |
| 191 | #121 | — | 8 | 2 | no | — |
| 192 | C4×D4 | — | 4 | 4 | no | — |
| 193 | C4×D4 | — | 8 | 2 | no | — |
| 194 | C2×C42 | #442 | 1 | 1 | yes | characteristic |
| 195 | D16 | — | 2 | 8 | no | — |
| 196 | D16 | — | 4 | 4 | no | — |
| 197 | M32 | — | 32 | 4 | no | — |
| 198 | C2×C16 | — | 2 | 2 | no | — |
| 199 | C2×C16 | — | 2 | 2 | no | — |
| 200 | C2×C16 | — | 2 | 1 | no | — |
| 201 | C2×C16 | — | 2 | 1 | no | — |
| 202 | C2×C16 | — | 4 | 2 | no | — |
| 203 | C2×C16 | — | 4 | 2 | no | — |
| 204 | C2×C16 | — | 4 | 2 | no | — |
| 205 | C2×C16 | — | 8 | 4 | no | — |
| 206 | C2×C16 | — | 32 | 4 | no | — |
| 207 | C2×C16 | C8×D4 | 1 | 1 | yes | characteristic |
| 208 | C2×C16 | C8×D4 | 1 | 1 | yes | characteristic |
| 209 | C2×C16 | C23×C8 | 1 | 1 | yes | characteristic |
| 210 | C2×C16 | C2×#131 | 1 | 1 | yes | characteristic |
| 211 | #107 | — | 2 | 4 | no | — |
| 212 | #107 | — | 4 | 2 | no | — |
| 213 | C4⋊C8 | — | 4 | 4 | no | — |
| 214 | C4⋊C8 | — | 8 | 2 | no | — |
| 215 | #102 | — | 4 | 4 | no | — |
| 216 | #102 | — | 4 | 4 | no | — |
| 217 | C22⋊C8 | — | 8 | 4 | no | — |
| 218 | C22⋊C8 | — | 8 | 4 | no | — |
| 219 | C4×C8 | — | 2 | 2 | no | — |
| 220 | C4×C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 221 | C4×C8 | #442 | 1 | 2 | yes | — |
| 222 | C4×C8 | #442 | 1 | 1 | yes | characteristic |
| 223 | C4×C8 | #444 | 1 | 1 | yes | characteristic |
| 224 | C4×C8 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 225 | C4×C8 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 226 | C32 | — | 2 | 2 | no | — |
| 227 | C32 | — | 4 | 2 | no | — |
| 228 | C32 | — | 8 | 2 | no | — |
| 229 | C32 | #504 | 1 | 4 | yes | — |
| 230 | #41 | — | 32 | 2 | no | — |
| 231 | #41 | — | 32 | 2 | no | — |
| 232 | C2×Q8 | — | 8 | 2 | no | — |
| 233 | C2×D4 | — | 4 | 4 | no | — |
| 234 | C2×D4 | — | 8 | 4 | no | — |
| 235 | C2×D4 | — | 8 | 2 | no | — |
| 236 | C2×D4 | — | 8 | 2 | no | — |
| 237 | C22×C4 | — | 8 | 4 | no | — |
| 238 | C22×C4 | #1580 | 1 | 1 | yes | characteristic |
| 239 | C22×C4 | AGL(1,ℤ/16) | 1 | 2 | yes | — |
| 240 | D8 | — | 4 | 4 | no | — |
| 241 | D8 | — | 4 | 4 | no | — |
| 242 | D8 | — | 8 | 4 | no | — |
| 243 | M16 | — | 32 | 4 | no | — |
| 244 | M16 | — | 32 | 2 | no | — |
| 245 | C2×C8 | — | 2 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 1 | no | — |
| 247 | C2×C8 | — | 2 | 1 | no | — |
| 248 | C2×C8 | — | 2 | 2 | no | — |
| 249 | C2×C8 | — | 2 | 2 | no | — |
| 250 | C2×C8 | — | 8 | 2 | no | — |
| 251 | C2×C8 | — | 8 | 4 | no | — |
| 252 | C2×C8 | — | 16 | 4 | no | — |
| 253 | C2×C8 | — | 16 | 4 | no | — |
| 254 | C2×C8 | #1580 | 1 | 1 | yes | characteristic |
| 255 | C2×C8 | #1580 | 1 | 1 | yes | characteristic |
| 256 | C2×C8 | C2×C8×D4 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | C2×#444 | 1 | 1 | yes | characteristic |
| 258 | C4⋊C4 | — | 4 | 4 | no | — |
| 259 | C4⋊C4 | — | 8 | 2 | no | — |
| 260 | C22⋊C4 | — | 8 | 4 | no | — |
| 261 | C22⋊C4 | — | 8 | 4 | no | — |
| 262 | C42 | #1580 | 1 | 1 | yes | characteristic |
| 263 | C42 | AGL(1,ℤ/16) | 1 | 2 | yes | — |
| 264 | C42 | C2×#442 | 1 | 1 | yes | characteristic |
| 265 | C16 | — | 2 | 1 | no | — |
| 266 | C16 | — | 2 | 1 | no | — |
| 267 | C16 | — | 4 | 1 | no | — |
| 268 | C16 | — | 4 | 1 | no | — |
| 269 | C16 | — | 8 | 2 | no | — |
| 270 | C16 | — | 8 | 2 | no | — |
| 271 | C16 | — | 16 | 4 | no | — |
| 272 | C16 | #2171 | 1 | 2 | yes | — |
| 273 | C16 | C2×#504 | 1 | 2 | yes | — |
| 274 | C23 | — | 8 | 4 | no | — |
| 275 | C23 | #12067 | 1 | 1 | yes | characteristic |
| 276 | Q8 | — | 32 | 2 | no | — |
| 277 | D4 | — | 8 | 4 | no | — |
| 278 | D4 | — | 8 | 4 | no | — |
| 279 | D4 | — | 32 | 2 | no | — |
| 280 | D4 | — | 32 | 2 | no | — |
| 281 | D4 | — | 32 | 4 | no | — |
| 282 | C2×C4 | — | 2 | 1 | no | — |
| 283 | C2×C4 | — | 2 | 1 | no | — |
| 284 | C2×C4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 8 | 4 | no | — |
| 286 | C2×C4 | — | 16 | 4 | no | — |
| 287 | C2×C4 | — | 16 | 4 | no | — |
| 288 | C2×C4 | #12067 | 1 | 2 | yes | — |
| 289 | C2×C4 | #12067 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C2×#1580 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C2×AGL(1,ℤ/16) | 1 | 2 | yes | — |
| 292 | C8 | — | 2 | 1 | no | — |
| 293 | C8 | — | 2 | 1 | no | — |
| 294 | C8 | — | 4 | 2 | no | — |
| 295 | C8 | — | 4 | 2 | no | — |
| 296 | C8 | — | 16 | 4 | no | — |
| 297 | C8 | #12068 | 1 | 2 | yes | — |
| 298 | C8 | #33434 | 1 | 1 | yes | characteristic |
| 299 | C8 | C2×#2171 | 1 | 1 | yes | characteristic |
| 300 | C22 | — | 2 | 2 | no | — |
| 301 | C22 | — | 2 | 1 | no | — |
| 302 | C22 | — | 16 | 4 | no | — |
| 303 | C22 | — | 16 | 4 | no | — |
| 304 | C22 | — | 16 | 4 | no | — |
| 305 | C22 | C2×#12067 | 1 | 1 | yes | characteristic |
| 306 | C4 | — | 2 | 2 | no | — |
| 307 | C4 | — | 2 | 2 | no | — |
| 308 | C4 | — | 2 | 1 | no | — |
| 309 | C4 | — | 16 | 4 | no | — |
| 310 | C4 | #483583 | 1 | 1 | yes | characteristic |
| 311 | C4 | C2×#12068 | 1 | 1 | yes | characteristic |
| 312 | C2 | — | 4 | 1 | no | — |
| 313 | C2 | — | 32 | 4 | no | — |
| 314 | C2 | #72225075 | 1 | 2 | yes | — |
| 315 | C2 | #72225221 | 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