← #26204259
Subgroups — #26204259
Order 12032.
878 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20254329 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20253974 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20253762 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20243026 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #20242993 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20242977 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C8×#10625329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C47×#493 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #15689680 | C22 | 1 | 2 | yes | — |
| 10 | #15689528 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #15689503 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #15689495 | C22 | 1 | 4 | yes | complemented |
| 13 | #15689487 | — | 2 | 4 | no | — |
| 14 | #15689474 | C22 | 1 | 2 | yes | complemented |
| 15 | #15689469 | C22 | 1 | 4 | yes | complemented |
| 16 | #15689469 | C22 | 1 | 4 | yes | complemented |
| 17 | #15689469 | C22 | 1 | 4 | yes | complemented |
| 18 | #15689364 | — | 2 | 2 | no | — |
| 19 | #15689320 | — | 2 | 4 | no | — |
| 20 | #15689274 | — | 2 | 4 | no | — |
| 21 | C4×Dic188 | — | 2 | 2 | no | — |
| 22 | #15689176 | C22 | 1 | 1 | yes | characteristic |
| 23 | #15689128 | — | 2 | 4 | no | — |
| 24 | C4×#10625329 | C22 | 1 | 1 | yes | characteristic |
| 25 | #15688845 | C22 | 1 | 2 | yes | complemented |
| 26 | C8×C47⋊C8 | C22 | 1 | 2 | yes | complemented |
| 27 | C22×D8×C47 | — | 2 | 2 | no | — |
| 28 | C2×#127×C47 | C22 | 1 | 2 | yes | — |
| 29 | C47×#493 | — | 2 | 2 | no | — |
| 30 | C47×#493 | C22 | 1 | 2 | yes | — |
| 31 | C47×#493 | C22 | 1 | 2 | yes | — |
| 32 | C2×C4×C8×C47 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×Dic188 | — | 2 | 2 | no | — |
| 34 | C2×#10625335 | — | 2 | 8 | no | — |
| 35 | C2×#10625329 | D4 | 1 | 2 | yes | — |
| 36 | #13094024 | C23 | 1 | 2 | yes | — |
| 37 | #13094024 | C23 | 1 | 2 | yes | — |
| 38 | #13094016 | — | 2 | 2 | no | — |
| 39 | #13094003 | — | 2 | 2 | no | — |
| 40 | #13094003 | — | 2 | 2 | no | — |
| 41 | C4×Dic94 | — | 2 | 2 | no | — |
| 42 | C4×Dic94 | — | 2 | 2 | no | — |
| 43 | #13093983 | — | 2 | 2 | no | — |
| 44 | #13093958 | — | 2 | 2 | no | — |
| 45 | C4⋊(C47⋊C8) | D4 | 1 | 2 | yes | — |
| 46 | #13093949 | — | 2 | 4 | no | — |
| 47 | #13093945 | — | 2 | 8 | no | — |
| 48 | #13093945 | — | 2 | 8 | no | — |
| 49 | #13093945 | — | 2 | 8 | no | — |
| 50 | C22⋊(C47⋊C8) | D4 | 1 | 2 | yes | — |
| 51 | #13093935 | D4 | 1 | 2 | yes | — |
| 52 | C4×C47⋊C8 | C23 | 1 | 2 | yes | — |
| 53 | C4×C47⋊C8 | C23 | 1 | 2 | yes | — |
| 54 | C22×D4×C47 | — | 2 | 2 | no | — |
| 55 | C22×D4×C47 | — | 2 | 2 | no | — |
| 56 | C22×D4×C47 | — | 2 | 2 | no | — |
| 57 | C2×D8×C47 | — | 2 | 4 | no | — |
| 58 | C2×D8×C47 | — | 2 | 4 | no | — |
| 59 | C2×D8×C47 | — | 4 | 8 | no | — |
| 60 | C22×C8×C47 | D4 | 1 | 2 | yes | — |
| 61 | #127×C47 | — | 2 | 2 | no | — |
| 62 | #127×C47 | — | 2 | 2 | no | — |
| 63 | #127×C47 | — | 2 | 2 | no | — |
| 64 | #127×C47 | C23 | 1 | 4 | yes | — |
| 65 | C2×C42×C47 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4×C8×C47 | D4 | 1 | 2 | yes | — |
| 67 | C4×C8×C47 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4×C8×C47 | C23 | 1 | 1 | yes | characteristic |
| 69 | C24×C47 | — | 4 | 2 | no | — |
| 70 | C2×D4×C47 | — | 2 | 4 | no | — |
| 71 | C2×D4×C47 | — | 2 | 4 | no | — |
| 72 | C2×D4×C47 | — | 2 | 4 | no | — |
| 73 | C2×D4×C47 | — | 4 | 8 | no | — |
| 74 | C2×D4×C47 | — | 4 | 2 | no | — |
| 75 | C2×D4×C47 | — | 4 | 2 | no | — |
| 76 | C2×D4×C47 | — | 4 | 2 | no | — |
| 77 | C2×D4×C47 | — | 4 | 8 | no | — |
| 78 | C2×D4×C47 | — | 8 | 4 | no | — |
| 79 | C22×C4×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C22×C4×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C22×C4×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×Dic94 | — | 2 | 2 | no | — |
| 83 | C2×Dic94 | — | 2 | 2 | no | — |
| 84 | D8×C47 | — | 4 | 8 | no | — |
| 85 | D8×C47 | — | 4 | 8 | no | — |
| 86 | C2×C8×C47 | — | 2 | 4 | no | — |
| 87 | C2×C8×C47 | C2×D4 | 1 | 2 | yes | — |
| 88 | C2×C8×C47 | C2×D4 | 1 | 2 | yes | — |
| 89 | C42×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C42×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C42×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C42×C47 | C24 | 1 | 1 | yes | characteristic |
| 93 | C22⋊Dic47 | — | 4 | 2 | no | — |
| 94 | #10625335 | — | 8 | 8 | no | — |
| 95 | C4⋊Dic47 | — | 2 | 2 | no | — |
| 96 | C4⋊Dic47 | — | 2 | 2 | no | — |
| 97 | C4⋊Dic47 | — | 2 | 2 | no | — |
| 98 | C4⋊Dic47 | — | 2 | 2 | no | — |
| 99 | Dic47⋊C4 | — | 4 | 2 | no | — |
| 100 | Dic47⋊C4 | — | 4 | 2 | no | — |
| 101 | C4×Dic47 | — | 4 | 2 | no | — |
| 102 | #10625329 | — | 2 | 4 | no | — |
| 103 | C2×C47⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 104 | C2×C47⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 105 | Dic188 | — | 4 | 4 | no | — |
| 106 | C23×C47 | — | 4 | 4 | no | — |
| 107 | C23×C47 | — | 8 | 4 | no | — |
| 108 | C23×C47 | — | 8 | 4 | no | — |
| 109 | C23×C47 | — | 8 | 4 | no | — |
| 110 | C23×C47 | #127 | 1 | 1 | yes | characteristic |
| 111 | D4×C47 | — | 4 | 8 | no | — |
| 112 | D4×C47 | — | 4 | 8 | no | — |
| 113 | D4×C47 | — | 4 | 8 | no | — |
| 114 | D4×C47 | — | 8 | 4 | no | — |
| 115 | D4×C47 | — | 8 | 4 | no | — |
| 116 | D4×C47 | — | 8 | 4 | no | — |
| 117 | C2×C4×C47 | — | 2 | 2 | no | — |
| 118 | C2×C4×C47 | — | 2 | 2 | no | — |
| 119 | C2×C4×C47 | #127 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C47 | #127 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C47 | #127 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C47 | #135 | 1 | 4 | yes | — |
| 123 | C2×C4×C47 | C22×D4 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C47 | C22×D4 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C47 | C22×D4 | 1 | 1 | yes | characteristic |
| 126 | C2×Dic47 | — | 4 | 2 | no | — |
| 127 | C2×Dic47 | — | 4 | 2 | no | — |
| 128 | Dic94 | — | 4 | 4 | no | — |
| 129 | Dic94 | — | 4 | 4 | no | — |
| 130 | C8×C47 | — | 2 | 2 | no | — |
| 131 | C8×C47 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 132 | C47⋊C8 | — | 2 | 4 | no | — |
| 133 | C47⋊C8 | — | 2 | 4 | no | — |
| 134 | #19635 | — | 47 | 1 | no | maximal, Sylow |
| 135 | C22×C47 | — | 2 | 1 | no | — |
| 136 | C22×C47 | — | 2 | 1 | no | — |
| 137 | C22×C47 | — | 8 | 4 | no | — |
| 138 | C22×C47 | — | 8 | 4 | no | — |
| 139 | C22×C47 | — | 8 | 4 | no | — |
| 140 | C22×C47 | — | 16 | 8 | no | — |
| 141 | C22×C47 | #495 | 1 | 2 | yes | — |
| 142 | C22×C47 | C2×#127 | 1 | 1 | yes | characteristic |
| 143 | C4×C47 | — | 2 | 1 | no | — |
| 144 | C4×C47 | — | 2 | 1 | no | — |
| 145 | C4×C47 | #495 | 1 | 2 | yes | — |
| 146 | C4×C47 | C2×#135 | 1 | 2 | yes | — |
| 147 | C4×C47 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 148 | C4×C47 | #576 | 1 | 2 | yes | — |
| 149 | Dic47 | — | 8 | 2 | no | — |
| 150 | Dic47 | — | 8 | 2 | no | — |
| 151 | #3208 | — | 47 | 2 | no | — |
| 152 | #3170 | — | 47 | 1 | no | — |
| 153 | C2×#493 | D47 | 1 | 1 | yes | characteristic |
| 154 | #3138 | — | 47 | 2 | no | — |
| 155 | #1717 | — | 47 | 4 | no | — |
| 156 | #1709 | — | 47 | 2 | no | — |
| 157 | #1706 | — | 47 | 2 | no | — |
| 158 | C8×M16 | — | 47 | 1 | no | — |
| 159 | C2×C47 | — | 2 | 2 | no | — |
| 160 | C2×C47 | — | 16 | 4 | no | — |
| 161 | C2×C47 | C2×#495 | 1 | 1 | yes | characteristic |
| 162 | C2×C47 | #3151 | 1 | 1 | yes | characteristic |
| 163 | C2×C47 | #3155 | 1 | 1 | yes | characteristic |
| 164 | C22×D8 | — | 2 | 2 | no | — |
| 165 | #533 | — | 47 | 2 | no | — |
| 166 | C2×#127 | C2×D47 | 1 | 2 | yes | — |
| 167 | #500 | — | 47 | 1 | no | — |
| 168 | #494 | — | 47 | 1 | no | — |
| 169 | #493 | — | 2 | 2 | no | — |
| 170 | #493 | C2×D47 | 1 | 2 | yes | — |
| 171 | #493 | C2×D47 | 1 | 2 | yes | — |
| 172 | #492 | — | 47 | 4 | no | — |
| 173 | #490 | — | 94 | 4 | no | — |
| 174 | #487 | — | 47 | 2 | no | — |
| 175 | #486 | — | 47 | 4 | no | — |
| 176 | #486 | — | 47 | 4 | no | — |
| 177 | #486 | — | 47 | 4 | no | — |
| 178 | #467 | — | 94 | 2 | no | — |
| 179 | #460 | — | 94 | 4 | no | — |
| 180 | #450 | — | 94 | 4 | no | — |
| 181 | C4×Q16 | — | 94 | 2 | no | — |
| 182 | #431 | — | 47 | 1 | no | — |
| 183 | #418 | — | 94 | 4 | no | — |
| 184 | C4×M16 | — | 47 | 1 | no | — |
| 185 | C2×C4×C8 | C2×D47 | 1 | 1 | yes | characteristic |
| 186 | #322 | — | 47 | 2 | no | — |
| 187 | C82 | — | 47 | 2 | no | — |
| 188 | C47 | #19635 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 189 | C22×D4 | — | 2 | 2 | no | — |
| 190 | C22×D4 | — | 2 | 2 | no | — |
| 191 | C22×D4 | — | 2 | 2 | no | — |
| 192 | C2×Q16 | — | 94 | 2 | no | — |
| 193 | C2×SD16 | — | 94 | 8 | no | — |
| 194 | C2×D8 | — | 2 | 4 | no | — |
| 195 | C2×D8 | — | 2 | 4 | no | — |
| 196 | C2×D8 | — | 4 | 8 | no | — |
| 197 | C2×M16 | — | 47 | 2 | no | — |
| 198 | C22×C8 | C22⋊D47 | 1 | 2 | yes | — |
| 199 | C4⋊Q8 | — | 47 | 2 | no | — |
| 200 | C4⋊Q8 | — | 47 | 2 | no | — |
| 201 | #127 | — | 2 | 2 | no | — |
| 202 | #127 | — | 2 | 2 | no | — |
| 203 | #127 | — | 2 | 2 | no | — |
| 204 | #127 | C22×D47 | 1 | 4 | yes | — |
| 205 | #125 | — | 94 | 2 | no | — |
| 206 | C22⋊Q8 | — | 94 | 2 | no | — |
| 207 | C22⋊Q8 | — | 94 | 2 | no | — |
| 208 | C4×Q8 | — | 94 | 2 | no | — |
| 209 | C4×Q8 | — | 94 | 2 | no | — |
| 210 | #117 | — | 94 | 2 | no | — |
| 211 | C2×C42 | C22×D47 | 1 | 1 | yes | characteristic |
| 212 | #107 | — | 94 | 2 | no | — |
| 213 | C4⋊C8 | — | 47 | 2 | no | — |
| 214 | #103 | — | 94 | 4 | no | — |
| 215 | #102 | — | 94 | 8 | no | — |
| 216 | #102 | — | 94 | 8 | no | — |
| 217 | #102 | — | 94 | 8 | no | — |
| 218 | C22⋊C8 | — | 47 | 2 | no | — |
| 219 | #97 | — | 47 | 2 | no | — |
| 220 | C4×C8 | — | 47 | 2 | no | — |
| 221 | C4×C8 | — | 47 | 2 | no | — |
| 222 | C4×C8 | C22⋊D47 | 1 | 2 | yes | — |
| 223 | C4×C8 | C22×D47 | 1 | 1 | yes | characteristic |
| 224 | C4×C8 | C22×D47 | 1 | 1 | yes | characteristic |
| 225 | C24 | — | 4 | 2 | no | — |
| 226 | C2×Q8 | — | 94 | 2 | no | — |
| 227 | C2×Q8 | — | 94 | 2 | no | — |
| 228 | C2×D4 | — | 2 | 4 | no | — |
| 229 | C2×D4 | — | 2 | 4 | no | — |
| 230 | C2×D4 | — | 2 | 4 | no | — |
| 231 | C2×D4 | — | 4 | 8 | no | — |
| 232 | C2×D4 | — | 4 | 2 | no | — |
| 233 | C2×D4 | — | 4 | 2 | no | — |
| 234 | C2×D4 | — | 4 | 2 | no | — |
| 235 | C2×D4 | — | 4 | 8 | no | — |
| 236 | C2×D4 | — | 8 | 4 | no | — |
| 237 | C22×C4 | D4×D47 | 1 | 1 | yes | characteristic |
| 238 | C22×C4 | D4×D47 | 1 | 1 | yes | characteristic |
| 239 | C22×C4 | C2×C22⋊D47 | 1 | 1 | yes | characteristic |
| 240 | Q16 | — | 188 | 4 | no | — |
| 241 | SD16 | — | 376 | 8 | no | — |
| 242 | D8 | — | 4 | 8 | no | — |
| 243 | D8 | — | 4 | 8 | no | — |
| 244 | M16 | — | 94 | 4 | no | — |
| 245 | C2×C8 | — | 2 | 4 | no | — |
| 246 | C2×C8 | — | 47 | 4 | no | — |
| 247 | C2×C8 | — | 47 | 4 | no | — |
| 248 | C2×C8 | C2×C22⋊D47 | 1 | 2 | yes | — |
| 249 | C2×C8 | C2×C22⋊D47 | 1 | 2 | yes | — |
| 250 | C4⋊C4 | — | 94 | 2 | no | — |
| 251 | C4⋊C4 | — | 94 | 2 | no | — |
| 252 | C4⋊C4 | — | 94 | 2 | no | — |
| 253 | C4⋊C4 | — | 94 | 2 | no | — |
| 254 | C4⋊C4 | — | 188 | 2 | no | — |
| 255 | C4⋊C4 | — | 188 | 2 | no | — |
| 256 | C22⋊C4 | — | 188 | 2 | no | — |
| 257 | C42 | — | 188 | 2 | no | — |
| 258 | C42 | D4×D47 | 1 | 1 | yes | characteristic |
| 259 | C42 | D4×D47 | 1 | 1 | yes | characteristic |
| 260 | C42 | C2×C22⋊D47 | 1 | 1 | yes | characteristic |
| 261 | C42 | C23×D47 | 1 | 1 | yes | characteristic |
| 262 | C23 | — | 4 | 4 | no | — |
| 263 | C23 | — | 8 | 4 | no | — |
| 264 | C23 | — | 8 | 4 | no | — |
| 265 | C23 | — | 8 | 4 | no | — |
| 266 | C23 | #13094020 | 1 | 1 | yes | characteristic |
| 267 | Q8 | — | 188 | 4 | no | — |
| 268 | Q8 | — | 188 | 4 | no | — |
| 269 | D4 | — | 4 | 8 | no | — |
| 270 | D4 | — | 4 | 8 | no | — |
| 271 | D4 | — | 4 | 8 | no | — |
| 272 | D4 | — | 8 | 4 | no | — |
| 273 | D4 | — | 8 | 4 | no | — |
| 274 | D4 | — | 8 | 4 | no | — |
| 275 | C2×C4 | — | 2 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 188 | 2 | no | — |
| 278 | C2×C4 | — | 188 | 2 | no | — |
| 279 | C2×C4 | #13094020 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | #13094020 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #13094020 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | #13094043 | 1 | 4 | yes | — |
| 283 | C2×C4 | C2×D4×D47 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C2×D4×D47 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C22×C22⋊D47 | 1 | 1 | yes | characteristic |
| 286 | C8 | — | 2 | 2 | no | — |
| 287 | C8 | — | 94 | 4 | no | — |
| 288 | C8 | — | 94 | 4 | no | — |
| 289 | C8 | #13094050 | 1 | 4 | yes | — |
| 290 | C22 | — | 2 | 1 | no | — |
| 291 | C22 | — | 2 | 1 | no | — |
| 292 | C22 | — | 8 | 4 | no | — |
| 293 | C22 | — | 8 | 4 | no | — |
| 294 | C22 | — | 8 | 4 | no | — |
| 295 | C22 | — | 16 | 8 | no | — |
| 296 | C22 | #15689506 | 1 | 2 | yes | — |
| 297 | C22 | C2×#13094020 | 1 | 1 | yes | characteristic |
| 298 | C4 | — | 2 | 1 | no | — |
| 299 | C4 | — | 2 | 1 | no | — |
| 300 | C4 | — | 376 | 2 | no | — |
| 301 | C4 | — | 376 | 2 | no | — |
| 302 | C4 | #15689506 | 1 | 2 | yes | — |
| 303 | C4 | C2×#13094043 | 1 | 2 | yes | — |
| 304 | C4 | C2×#13094050 | 1 | 2 | yes | — |
| 305 | C4 | #15689924 | 1 | 2 | yes | — |
| 306 | C2 | — | 2 | 2 | no | — |
| 307 | C2 | — | 16 | 4 | no | — |
| 308 | C2 | C2×#15689506 | 1 | 1 | yes | characteristic |
| 309 | C2 | #20253838 | 1 | 1 | yes | characteristic |
| 310 | C2 | #20253869 | 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