← #26225133
Subgroups — #26225133
Order 12032.
654 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20253900 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20252327 | C2 | 1 | 2 | yes | maximal |
| 3 | #20244367 | C2 | 1 | 2 | yes | maximal |
| 4 | #20244141 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20244133 | C2 | 1 | 2 | yes | maximal |
| 6 | #20243558 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #20243275 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #20243274 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #20243266 | C2 | 1 | 2 | yes | maximal |
| 10 | #20243076 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C47×#1946 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #15689676 | C22 | 1 | 2 | yes | — |
| 13 | #15689527 | C4 | 1 | 2 | yes | complemented |
| 14 | #15689524 | C4 | 1 | 2 | yes | complemented |
| 15 | #15689523 | C4 | 1 | 2 | yes | complemented |
| 16 | #15689520 | C4 | 1 | 2 | yes | complemented |
| 17 | #15689252 | — | 2 | 4 | no | — |
| 18 | #15689250 | — | 2 | 4 | no | — |
| 19 | C2×#13093960 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#13093959 | C22 | 1 | 2 | yes | — |
| 21 | #15689161 | C22 | 1 | 1 | yes | characteristic |
| 22 | #15689159 | C22 | 1 | 2 | yes | — |
| 23 | #15689149 | C22 | 1 | 2 | yes | — |
| 24 | #15689146 | C22 | 1 | 2 | yes | — |
| 25 | #15689081 | C22 | 1 | 2 | yes | — |
| 26 | #15689079 | C22 | 1 | 2 | yes | — |
| 27 | C2×#13093934 | C22 | 1 | 1 | yes | characteristic |
| 28 | C2×#13093934 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×C47⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 30 | C2×C8×Dic47 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×C8×Dic47 | C22 | 1 | 1 | yes | characteristic |
| 32 | #15688969 | C22 | 1 | 1 | yes | characteristic |
| 33 | #15688884 | C22 | 1 | 2 | yes | — |
| 34 | #15688883 | C22 | 1 | 2 | yes | — |
| 35 | #15688882 | C22 | 1 | 2 | yes | — |
| 36 | C2×#108×C47 | C22 | 1 | 1 | yes | characteristic |
| 37 | C47×#427 | C22 | 1 | 1 | yes | characteristic |
| 38 | C47×#424 | C22 | 1 | 2 | yes | — |
| 39 | C2×#97×C47 | C22 | 1 | 1 | yes | characteristic |
| 40 | C47×#339 | C22 | 1 | 2 | yes | — |
| 41 | C2×#10625329 | C23 | 1 | 2 | yes | — |
| 42 | C22×C47⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C22×C47⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | #13094023 | C2×C4 | 1 | 4 | yes | — |
| 45 | #13094015 | C2×C4 | 1 | 4 | yes | — |
| 46 | #13094001 | — | 2 | 4 | no | — |
| 47 | C4×Dic94 | — | 2 | 4 | no | — |
| 48 | #13093983 | C23 | 1 | 2 | yes | — |
| 49 | C2×C4×Dic47 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×Dic47 | C23 | 1 | 1 | yes | characteristic |
| 51 | #13093960 | — | 2 | 2 | no | — |
| 52 | #13093959 | — | 2 | 4 | no | — |
| 53 | #13093958 | C2×C4 | 1 | 2 | yes | — |
| 54 | #13093957 | C2×C4 | 1 | 4 | yes | — |
| 55 | #13093956 | C2×C4 | 1 | 2 | yes | — |
| 56 | #13093955 | C2×C4 | 1 | 4 | yes | — |
| 57 | Dic47⋊C8 | — | 2 | 4 | no | — |
| 58 | Dic47⋊C8 | — | 2 | 4 | no | — |
| 59 | #13093935 | — | 2 | 2 | no | — |
| 60 | #13093934 | — | 2 | 2 | no | — |
| 61 | #13093934 | D4 | 1 | 4 | yes | — |
| 62 | #13093934 | Q8 | 1 | 4 | yes | complemented |
| 63 | C4×C47⋊C8 | — | 2 | 2 | no | — |
| 64 | C4×C47⋊C8 | Q8 | 1 | 4 | yes | complemented |
| 65 | C8×Dic47 | — | 2 | 2 | no | — |
| 66 | C8×Dic47 | — | 2 | 2 | no | — |
| 67 | C8×Dic47 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | C8×Dic47 | C2×C4 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic47 | C2×C4 | 1 | 1 | yes | characteristic |
| 70 | C8×Dic47 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C47⋊#95 | — | 2 | 2 | no | — |
| 72 | C2×M16×C47 | C23 | 1 | 2 | yes | — |
| 73 | C22×C8×C47 | C23 | 1 | 1 | yes | characteristic |
| 74 | C22×C8×C47 | C23 | 1 | 1 | yes | characteristic |
| 75 | #117×C47 | C23 | 1 | 2 | yes | — |
| 76 | C2×C42×C47 | C23 | 1 | 1 | yes | characteristic |
| 77 | #108×C47 | — | 2 | 2 | no | — |
| 78 | #107×C47 | C2×C4 | 1 | 2 | yes | — |
| 79 | #106×C47 | C2×C4 | 1 | 2 | yes | — |
| 80 | C4⋊C8×C47 | — | 2 | 4 | no | — |
| 81 | #97×C47 | — | 2 | 2 | no | — |
| 82 | C22×C4×C47 | — | 2 | 1 | no | — |
| 83 | C22×C4×C47 | C24 | 1 | 1 | yes | characteristic |
| 84 | C22×Dic47 | — | 2 | 1 | no | — |
| 85 | C22×Dic47 | #41 | 1 | 2 | yes | — |
| 86 | C2×Dic94 | — | 2 | 4 | no | — |
| 87 | M16×C47 | — | 2 | 2 | no | — |
| 88 | M16×C47 | — | 2 | 2 | no | — |
| 89 | C2×C8×C47 | — | 2 | 1 | no | — |
| 90 | C2×C8×C47 | — | 2 | 1 | no | — |
| 91 | C2×C8×C47 | — | 2 | 2 | no | — |
| 92 | C2×C8×C47 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C47 | C22×C4 | 1 | 1 | yes | characteristic |
| 94 | C2×C8×C47 | C2×D4 | 1 | 2 | yes | — |
| 95 | C2×C8×C47 | C2×Q8 | 1 | 2 | yes | — |
| 96 | C2×C8×C47 | #41 | 1 | 2 | yes | — |
| 97 | C4⋊C4×C47 | C22×C4 | 1 | 4 | yes | — |
| 98 | C22⋊C4×C47 | — | 2 | 2 | no | — |
| 99 | C42×C47 | — | 2 | 2 | no | — |
| 100 | C42×C47 | — | 2 | 1 | no | — |
| 101 | C42×C47 | C2×Q8 | 1 | 2 | yes | — |
| 102 | C22⋊Dic47 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic47 | — | 2 | 4 | no | — |
| 104 | C4⋊Dic47 | C22×C4 | 1 | 4 | yes | — |
| 105 | Dic47⋊C4 | — | 2 | 4 | no | — |
| 106 | Dic47⋊C4 | — | 2 | 4 | no | — |
| 107 | C4×Dic47 | — | 2 | 2 | no | — |
| 108 | C4×Dic47 | — | 2 | 1 | no | — |
| 109 | C4×Dic47 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C4×Dic47 | C22×C4 | 1 | 1 | yes | characteristic |
| 111 | C4×Dic47 | C2×D4 | 1 | 2 | yes | — |
| 112 | C4×Dic47 | C2×Q8 | 1 | 2 | yes | — |
| 113 | #10625329 | — | 2 | 2 | no | — |
| 114 | #10625329 | — | 2 | 2 | no | — |
| 115 | C2×C47⋊C8 | — | 2 | 2 | no | — |
| 116 | C2×C47⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 117 | C2×C47⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C47⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 119 | C2×C47⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 120 | C2×C47⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 121 | C2×C47⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 122 | C2×C47⋊C8 | #41 | 1 | 2 | yes | — |
| 123 | C23×C47 | C2×#41 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C47 | — | 2 | 4 | no | — |
| 125 | C2×C4×C47 | — | 2 | 2 | no | — |
| 126 | C2×C4×C47 | — | 4 | 2 | no | — |
| 127 | C2×C4×C47 | C23×C4 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C47 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C47 | C22×Q8 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C47 | C22×Q8 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C47 | C2×#41 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C47 | C2×#41 | 1 | 1 | yes | characteristic |
| 133 | C2×Dic47 | — | 2 | 4 | no | — |
| 134 | C2×Dic47 | — | 2 | 2 | no | — |
| 135 | C2×Dic47 | — | 4 | 2 | no | — |
| 136 | C2×Dic47 | — | 4 | 2 | no | — |
| 137 | C2×Dic47 | C4×D4 | 1 | 2 | yes | — |
| 138 | C2×Dic47 | C4×D4 | 1 | 2 | yes | — |
| 139 | Dic94 | — | 8 | 4 | no | — |
| 140 | C8×C47 | — | 2 | 2 | no | — |
| 141 | C8×C47 | — | 4 | 2 | no | — |
| 142 | C8×C47 | C4×Q8 | 1 | 2 | yes | — |
| 143 | C8×C47 | C4×Q8 | 1 | 2 | yes | — |
| 144 | C47⋊C8 | — | 4 | 2 | no | — |
| 145 | C47⋊C8 | C4×Q8 | 1 | 2 | yes | — |
| 146 | C47⋊C8 | C4×Q8 | 1 | 2 | yes | — |
| 147 | C47⋊C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 148 | #21733 | — | 47 | 1 | no | maximal, Sylow |
| 149 | C22×C47 | — | 2 | 2 | no | — |
| 150 | C22×C47 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 151 | C22×C47 | #533 | 1 | 1 | yes | characteristic |
| 152 | C22×C47 | #533 | 1 | 1 | yes | characteristic |
| 153 | C4×C47 | — | 4 | 2 | no | — |
| 154 | C4×C47 | — | 4 | 4 | no | — |
| 155 | C4×C47 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 156 | C4×C47 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 157 | C4×C47 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 158 | C4×C47 | #532 | 1 | 1 | yes | characteristic |
| 159 | Dic47 | — | 2 | 2 | no | — |
| 160 | Dic47 | — | 2 | 2 | no | — |
| 161 | Dic47 | — | 4 | 2 | no | — |
| 162 | Dic47 | — | 4 | 4 | no | — |
| 163 | #3160 | — | 47 | 1 | no | — |
| 164 | #2961 | — | 47 | 2 | no | — |
| 165 | #1995 | — | 47 | 2 | no | — |
| 166 | #1946 | — | 47 | 1 | no | — |
| 167 | #1946 | D47 | 1 | 1 | yes | characteristic |
| 168 | #1945 | — | 47 | 2 | no | — |
| 169 | #1848 | — | 47 | 1 | no | — |
| 170 | #1773 | — | 47 | 1 | no | — |
| 171 | #1773 | — | 47 | 1 | no | — |
| 172 | #1772 | — | 47 | 2 | no | — |
| 173 | #1726 | — | 47 | 1 | no | — |
| 174 | C2×C47 | — | 2 | 1 | no | — |
| 175 | C2×C47 | — | 2 | 1 | no | — |
| 176 | C2×C47 | C4×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 177 | C2×C47 | #2957 | 1 | 2 | yes | — |
| 178 | #533 | — | 47 | 2 | no | — |
| 179 | #500 | — | 47 | 2 | no | — |
| 180 | #499 | — | 47 | 2 | no | — |
| 181 | #499 | — | 47 | 2 | no | — |
| 182 | #498 | — | 47 | 2 | no | — |
| 183 | #446 | — | 94 | 4 | no | — |
| 184 | #446 | — | 94 | 4 | no | — |
| 185 | C2×#108 | — | 47 | 2 | no | — |
| 186 | C2×#108 | — | 47 | 1 | no | — |
| 187 | C2×#108 | C2×D47 | 1 | 1 | yes | characteristic |
| 188 | #427 | — | 47 | 1 | no | — |
| 189 | #427 | — | 47 | 2 | no | — |
| 190 | #427 | C2×D47 | 1 | 1 | yes | characteristic |
| 191 | #424 | — | 47 | 2 | no | — |
| 192 | #424 | C2×D47 | 1 | 2 | yes | — |
| 193 | #423 | — | 47 | 2 | no | — |
| 194 | #405 | — | 47 | 2 | no | — |
| 195 | #405 | — | 47 | 2 | no | — |
| 196 | C2×#97 | — | 47 | 1 | no | — |
| 197 | C2×#97 | — | 47 | 1 | no | — |
| 198 | C2×#97 | C2×D47 | 1 | 1 | yes | characteristic |
| 199 | C2×C4×C8 | — | 47 | 1 | no | — |
| 200 | C2×C4×C8 | — | 47 | 1 | no | — |
| 201 | C2×C4×C8 | — | 47 | 1 | no | — |
| 202 | #376 | — | 47 | 1 | no | — |
| 203 | #339 | — | 47 | 2 | no | — |
| 204 | #339 | — | 47 | 2 | no | — |
| 205 | #339 | — | 47 | 2 | no | — |
| 206 | #339 | C2×D47 | 1 | 2 | yes | — |
| 207 | C47 | #21733 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 208 | C2×M16 | — | 47 | 2 | no | — |
| 209 | C2×M16 | C22×D47 | 1 | 2 | yes | — |
| 210 | C22×C8 | — | 47 | 1 | no | — |
| 211 | C22×C8 | — | 47 | 1 | no | — |
| 212 | C22×C8 | C22×D47 | 1 | 1 | yes | characteristic |
| 213 | C22×C8 | C22×D47 | 1 | 1 | yes | characteristic |
| 214 | C4⋊Q8 | — | 47 | 4 | no | — |
| 215 | #125 | — | 47 | 4 | no | — |
| 216 | C22⋊Q8 | — | 94 | 4 | no | — |
| 217 | C4×Q8 | — | 94 | 4 | no | — |
| 218 | #117 | — | 47 | 2 | no | — |
| 219 | #117 | C22×D47 | 1 | 2 | yes | — |
| 220 | C2×C42 | — | 47 | 1 | no | — |
| 221 | C2×C42 | — | 47 | 1 | no | — |
| 222 | C2×C42 | C22×D47 | 1 | 1 | yes | characteristic |
| 223 | #108 | — | 2 | 2 | no | — |
| 224 | #108 | — | 94 | 4 | no | — |
| 225 | #108 | — | 94 | 2 | no | — |
| 226 | #107 | — | 47 | 2 | no | — |
| 227 | #107 | — | 47 | 4 | no | — |
| 228 | #107 | C4×D47 | 1 | 2 | yes | — |
| 229 | #106 | — | 47 | 2 | no | — |
| 230 | #106 | — | 47 | 4 | no | — |
| 231 | #106 | C4×D47 | 1 | 2 | yes | — |
| 232 | C4⋊C8 | — | 2 | 4 | no | — |
| 233 | C4⋊C8 | — | 94 | 4 | no | — |
| 234 | C4⋊C8 | — | 94 | 4 | no | — |
| 235 | #97 | — | 2 | 2 | no | — |
| 236 | #97 | — | 47 | 4 | no | — |
| 237 | #97 | — | 47 | 4 | no | — |
| 238 | #97 | — | 94 | 2 | no | — |
| 239 | #97 | — | 94 | 2 | no | — |
| 240 | C4×C8 | — | 47 | 1 | no | — |
| 241 | C4×C8 | — | 47 | 1 | no | — |
| 242 | C4×C8 | — | 47 | 1 | no | — |
| 243 | C4×C8 | — | 47 | 1 | no | — |
| 244 | C4×C8 | — | 47 | 4 | no | — |
| 245 | C4×C8 | — | 94 | 2 | no | — |
| 246 | C4×C8 | — | 94 | 2 | no | — |
| 247 | C4×C8 | — | 94 | 2 | no | — |
| 248 | #95 | — | 94 | 2 | no | — |
| 249 | C2×Q8 | — | 94 | 4 | no | — |
| 250 | C22×C4 | — | 2 | 1 | no | — |
| 251 | C22×C4 | — | 47 | 2 | no | — |
| 252 | C22×C4 | — | 94 | 1 | no | — |
| 253 | C22×C4 | C23×D47 | 1 | 1 | yes | characteristic |
| 254 | M16 | — | 2 | 2 | no | — |
| 255 | M16 | — | 2 | 2 | no | — |
| 256 | M16 | — | 94 | 2 | no | — |
| 257 | M16 | — | 94 | 2 | no | — |
| 258 | C2×C8 | — | 2 | 1 | no | — |
| 259 | C2×C8 | — | 2 | 1 | no | — |
| 260 | C2×C8 | — | 2 | 2 | no | — |
| 261 | C2×C8 | — | 47 | 1 | no | — |
| 262 | C2×C8 | — | 47 | 1 | no | — |
| 263 | C2×C8 | — | 47 | 2 | no | — |
| 264 | C2×C8 | — | 47 | 2 | no | — |
| 265 | C2×C8 | — | 47 | 2 | no | — |
| 266 | C2×C8 | — | 47 | 2 | no | — |
| 267 | C2×C8 | — | 47 | 2 | no | — |
| 268 | C2×C8 | — | 94 | 2 | no | — |
| 269 | C2×C8 | C2×C4×D47 | 1 | 1 | yes | characteristic |
| 270 | C2×C8 | C2×C4×D47 | 1 | 1 | yes | characteristic |
| 271 | C2×C8 | D4×D47 | 1 | 2 | yes | — |
| 272 | C2×C8 | #10625352 | 1 | 2 | yes | — |
| 273 | C2×C8 | Q8×D47 | 1 | 2 | yes | — |
| 274 | C4⋊C4 | — | 47 | 4 | no | — |
| 275 | C4⋊C4 | — | 94 | 4 | no | — |
| 276 | C4⋊C4 | — | 94 | 4 | no | — |
| 277 | C4⋊C4 | — | 94 | 4 | no | — |
| 278 | C4⋊C4 | C2×C4×D47 | 1 | 4 | yes | — |
| 279 | C22⋊C4 | — | 2 | 2 | no | — |
| 280 | C22⋊C4 | — | 94 | 2 | no | — |
| 281 | C42 | — | 2 | 2 | no | — |
| 282 | C42 | — | 2 | 1 | no | — |
| 283 | C42 | — | 47 | 1 | no | — |
| 284 | C42 | — | 47 | 1 | no | — |
| 285 | C42 | — | 47 | 2 | no | — |
| 286 | C42 | — | 47 | 2 | no | — |
| 287 | C42 | — | 94 | 2 | no | — |
| 288 | C42 | — | 94 | 1 | no | — |
| 289 | C42 | Q8×D47 | 1 | 2 | yes | — |
| 290 | C23 | #41×D47 | 1 | 1 | yes | characteristic |
| 291 | Q8 | — | 376 | 4 | no | — |
| 292 | C2×C4 | — | 2 | 4 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 47 | 2 | no | — |
| 296 | C2×C4 | — | 47 | 2 | no | — |
| 297 | C2×C4 | — | 94 | 4 | no | — |
| 298 | C2×C4 | — | 94 | 2 | no | — |
| 299 | C2×C4 | — | 188 | 2 | no | — |
| 300 | C2×C4 | — | 188 | 2 | no | — |
| 301 | C2×C4 | C22×C4×D47 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×D4×D47 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C2×Q8×D47 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×Q8×D47 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #41×D47 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | C2×#10625352 | 1 | 1 | yes | characteristic |
| 307 | C8 | — | 2 | 2 | no | — |
| 308 | C8 | — | 4 | 2 | no | — |
| 309 | C8 | — | 47 | 2 | no | — |
| 310 | C8 | — | 47 | 2 | no | — |
| 311 | C8 | — | 47 | 4 | no | — |
| 312 | C8 | — | 188 | 2 | no | — |
| 313 | C8 | #13093989 | 1 | 2 | yes | — |
| 314 | C8 | #13093989 | 1 | 2 | yes | — |
| 315 | C22 | — | 2 | 2 | no | — |
| 316 | C22 | C4×D4×D47 | 1 | 1 | yes | characteristic |
| 317 | C22 | #15689675 | 1 | 1 | yes | characteristic |
| 318 | C22 | #15689679 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 4 | 2 | no | — |
| 320 | C4 | — | 4 | 4 | no | — |
| 321 | C4 | — | 94 | 2 | no | — |
| 322 | C4 | — | 94 | 2 | no | — |
| 323 | C4 | — | 188 | 2 | no | — |
| 324 | C4 | — | 188 | 4 | no | — |
| 325 | C4 | C4×Q8×D47 | 1 | 1 | yes | characteristic |
| 326 | C4 | C2×#13093989 | 1 | 1 | yes | characteristic |
| 327 | C4 | C4⋊Q8×D47 | 1 | 1 | yes | characteristic |
| 328 | C4 | #15689668 | 1 | 1 | yes | characteristic |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | #20245959 | 1 | 1 | yes | characteristic |
| 332 | C2 | #20252302 | 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