Finite Group DatabasebetaFinite Group DatabasebetaOrder 2816.
2326 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #13075642 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #13074298 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #13073691 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #13072237 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #13071727 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #13071105 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C11×#2811 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#4343 | C22 | 1 | 3 | yes | — |
| 9 | C2×#67000 | C22 | 1 | 6 | yes | — |
| 10 | C2×#67000 | C22 | 1 | 6 | yes | — |
| 11 | C2×#66948 | C22 | 1 | 6 | yes | — |
| 12 | C2×#66945 | C22 | 1 | 3 | yes | — |
| 13 | C2×#66927 | C22 | 1 | 2 | yes | — |
| 14 | C2×C42×D11 | C22 | 1 | 1 | yes | characteristic |
| 15 | C11×#395 | C22 | 1 | 6 | yes | — |
| 16 | C11×#391 | C22 | 1 | 3 | yes | — |
| 17 | C11×#391 | C22 | 1 | 6 | yes | — |
| 18 | #10622844 | C22 | 1 | 2 | yes | — |
| 19 | #10622843 | C22 | 1 | 3 | yes | — |
| 20 | #10622604 | C22 | 1 | 6 | yes | — |
| 21 | #10622603 | C22 | 1 | 12 | yes | — |
| 22 | #10622601 | C22 | 1 | 3 | yes | — |
| 23 | #10622601 | C22 | 1 | 6 | yes | — |
| 24 | #10622552 | C22 | 1 | 3 | yes | — |
| 25 | C4×(C2×D11)⋊C4 | C22 | 1 | 6 | yes | — |
| 26 | #10622284 | C22 | 1 | 12 | yes | — |
| 27 | #10622284 | C22 | 1 | 6 | yes | — |
| 28 | #10622283 | C22 | 1 | 12 | yes | — |
| 29 | #10622279 | C22 | 1 | 12 | yes | — |
| 30 | #10622278 | C22 | 1 | 12 | yes | — |
| 31 | #10622278 | C22 | 1 | 12 | yes | — |
| 32 | #10622261 | C22 | 1 | 6 | yes | — |
| 33 | C22×Q8×C11 | C23 | 1 | 2 | yes | — |
| 34 | C22×Q8×C11 | C23 | 1 | 3 | yes | — |
| 35 | C2×#4343 | — | 2 | 12 | no | — |
| 36 | C2×#4343 | — | 4 | 6 | no | — |
| 37 | C2×#4343 | — | 4 | 6 | no | — |
| 38 | C22×D44 | C23 | 1 | 6 | yes | — |
| 39 | C22×D44 | C23 | 1 | 6 | yes | — |
| 40 | C22×D44 | C23 | 1 | 3 | yes | — |
| 41 | C22×D44 | C23 | 1 | 6 | yes | — |
| 42 | C22×C4×D11 | C23 | 1 | 3 | yes | — |
| 43 | C22×C4×D11 | C23 | 1 | 6 | yes | — |
| 44 | C2×C11×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 45 | C2×C11×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 46 | C2×C42×C11 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×C42×C11 | C23 | 1 | 2 | yes | — |
| 48 | #67000 | — | 2 | 24 | no | — |
| 49 | #67000 | — | 4 | 12 | no | — |
| 50 | C2×(C2×D11)⋊C4 | C23 | 1 | 12 | yes | — |
| 51 | C2×(C2×D11)⋊C4 | C23 | 1 | 12 | yes | — |
| 52 | C2×(C2×D11)⋊C4 | C23 | 1 | 12 | yes | — |
| 53 | C2×(C2×D11)⋊C4 | C23 | 1 | 6 | yes | — |
| 54 | C2×(C2×D11)⋊C4 | C23 | 1 | 12 | yes | — |
| 55 | C2×C4⋊Dic11 | C23 | 1 | 3 | yes | — |
| 56 | C2×C4×Dic11 | C23 | 1 | 6 | yes | — |
| 57 | C2×C4×Dic11 | C23 | 1 | 6 | yes | — |
| 58 | C2×C4×Dic11 | C23 | 1 | 3 | yes | — |
| 59 | #66948 | — | 4 | 12 | no | — |
| 60 | #66945 | — | 4 | 6 | no | — |
| 61 | #66927 | — | 4 | 4 | no | — |
| 62 | C42×D11 | — | 4 | 2 | no | — |
| 63 | C11×#95 | C23 | 1 | 6 | yes | — |
| 64 | C11×#95 | C23 | 1 | 12 | yes | — |
| 65 | C11⋊#95 | C23 | 1 | 12 | yes | — |
| 66 | C11⋊#95 | C23 | 1 | 6 | yes | — |
| 67 | #60048 | — | 11 | 1 | no | maximal, Sylow |
| 68 | C23×D11 | C24 | 1 | 6 | yes | — |
| 69 | C23×D11 | C24 | 1 | 1 | yes | characteristic |
| 70 | C2×Q8×C11 | — | 4 | 3 | no | — |
| 71 | C2×Q8×C11 | — | 4 | 3 | no | — |
| 72 | C2×Q8×C11 | — | 4 | 6 | no | — |
| 73 | C2×Q8×C11 | #41 | 1 | 12 | yes | complemented |
| 74 | C22×C4×C11 | C24 | 1 | 6 | yes | — |
| 75 | C22×C4×C11 | C24 | 1 | 6 | yes | — |
| 76 | C22×C4×C11 | C24 | 1 | 3 | yes | — |
| 77 | C22×Dic11 | C24 | 1 | 6 | yes | — |
| 78 | C22×Dic11 | C24 | 1 | 3 | yes | — |
| 79 | #4343 | — | 4 | 48 | no | — |
| 80 | C2×D44 | — | 2 | 12 | no | — |
| 81 | C2×D44 | — | 2 | 12 | no | — |
| 82 | C2×D44 | — | 2 | 6 | no | — |
| 83 | C2×D44 | — | 2 | 12 | no | — |
| 84 | C2×D44 | — | 4 | 3 | no | — |
| 85 | C2×D44 | — | 4 | 3 | no | — |
| 86 | C2×D44 | — | 4 | 6 | no | — |
| 87 | C2×D44 | — | 4 | 6 | no | — |
| 88 | C2×D44 | — | 4 | 6 | no | — |
| 89 | C2×D44 | — | 4 | 12 | no | — |
| 90 | C2×D44 | — | 4 | 6 | no | — |
| 91 | C2×C4×D11 | — | 2 | 6 | no | — |
| 92 | C2×C4×D11 | — | 2 | 12 | no | — |
| 93 | C2×C4×D11 | — | 2 | 6 | no | — |
| 94 | C2×C4×D11 | — | 2 | 6 | no | — |
| 95 | C2×C4×D11 | — | 2 | 12 | no | — |
| 96 | C2×C4×D11 | — | 4 | 6 | no | — |
| 97 | C11×C4⋊C4 | — | 4 | 3 | no | — |
| 98 | C11×C4⋊C4 | — | 4 | 6 | no | — |
| 99 | C42×C11 | — | 4 | 1 | no | — |
| 100 | C42×C11 | — | 4 | 2 | no | — |
| 101 | (C2×D11)⋊C4 | — | 2 | 24 | no | — |
| 102 | (C2×D11)⋊C4 | — | 2 | 24 | no | — |
| 103 | (C2×D11)⋊C4 | — | 2 | 24 | no | — |
| 104 | (C2×D11)⋊C4 | — | 4 | 6 | no | — |
| 105 | (C2×D11)⋊C4 | — | 4 | 12 | no | — |
| 106 | C4⋊Dic11 | — | 4 | 3 | no | — |
| 107 | C4×Dic11 | — | 2 | 12 | no | — |
| 108 | C4×Dic11 | — | 2 | 12 | no | — |
| 109 | C4×Dic11 | — | 4 | 3 | no | — |
| 110 | #2811 | D11 | 1 | 1 | yes | characteristic, complemented |
| 111 | #2733 | — | 11 | 3 | no | — |
| 112 | #2610 | — | 11 | 6 | no | — |
| 113 | #2555 | — | 11 | 6 | no | — |
| 114 | #2431 | — | 11 | 6 | no | — |
| 115 | #2372 | — | 11 | 6 | no | — |
| 116 | #2297 | — | 11 | 3 | no | — |
| 117 | C23×C11 | C25 | 1 | 1 | yes | characteristic |
| 118 | C22×D11 | — | 2 | 3 | no | — |
| 119 | C22×D11 | — | 2 | 3 | no | — |
| 120 | C22×D11 | — | 2 | 12 | no | — |
| 121 | C22×D11 | — | 2 | 6 | no | — |
| 122 | C22×D11 | — | 2 | 12 | no | — |
| 123 | C22×D11 | — | 2 | 6 | no | — |
| 124 | C22×D11 | — | 2 | 6 | no | — |
| 125 | C22×D11 | H(5,𝔽2) | 1 | 2 | yes | — |
| 126 | Q8×C11 | — | 4 | 12 | no | — |
| 127 | Q8×C11 | — | 8 | 4 | no | — |
| 128 | C2×C4×C11 | — | 2 | 3 | no | — |
| 129 | C2×C4×C11 | — | 2 | 3 | no | — |
| 130 | C2×C4×C11 | — | 2 | 6 | no | — |
| 131 | C2×C4×C11 | — | 2 | 6 | no | — |
| 132 | C2×C4×C11 | — | 2 | 6 | no | — |
| 133 | C2×C4×C11 | — | 2 | 12 | no | — |
| 134 | C2×C4×C11 | C2×#41 | 1 | 12 | yes | — |
| 135 | C2×C4×C11 | C2×#41 | 1 | 6 | yes | — |
| 136 | C2×Dic11 | — | 2 | 3 | no | — |
| 137 | C2×Dic11 | — | 2 | 3 | no | — |
| 138 | C2×Dic11 | — | 2 | 12 | no | — |
| 139 | C2×Dic11 | — | 2 | 6 | no | — |
| 140 | C2×Dic11 | H(5,𝔽2) | 1 | 6 | yes | — |
| 141 | D44 | — | 4 | 24 | no | — |
| 142 | D44 | — | 4 | 12 | no | — |
| 143 | D44 | — | 8 | 12 | no | — |
| 144 | D44 | — | 8 | 12 | no | — |
| 145 | C4×D11 | — | 2 | 24 | no | — |
| 146 | C4×D11 | — | 4 | 24 | no | — |
| 147 | C22×#41 | — | 11 | 3 | no | — |
| 148 | C2×#127 | — | 11 | 2 | no | — |
| 149 | C2×#124 | — | 11 | 6 | no | — |
| 150 | C2×#124 | — | 11 | 6 | no | — |
| 151 | C2×#121 | — | 11 | 6 | no | — |
| 152 | C2×#117 | — | 11 | 3 | no | — |
| 153 | C22×C42 | — | 11 | 1 | no | — |
| 154 | #395 | C2×D11 | 1 | 6 | yes | — |
| 155 | #394 | — | 11 | 6 | no | — |
| 156 | #393 | — | 11 | 2 | no | — |
| 157 | #392 | — | 11 | 6 | no | — |
| 158 | #392 | — | 11 | 12 | no | — |
| 159 | #391 | — | 11 | 3 | no | — |
| 160 | #391 | C2×D11 | 1 | 6 | yes | — |
| 161 | #391 | C2×D11 | 1 | 3 | yes | — |
| 162 | #390 | — | 11 | 6 | no | — |
| 163 | #390 | — | 11 | 3 | no | — |
| 164 | #390 | — | 11 | 3 | no | — |
| 165 | #388 | — | 11 | 12 | no | — |
| 166 | #388 | — | 11 | 12 | no | — |
| 167 | #386 | — | 11 | 12 | no | — |
| 168 | #386 | — | 11 | 12 | no | — |
| 169 | C4×C22⋊C4 | — | 11 | 6 | no | — |
| 170 | C4×C22⋊C4 | — | 11 | 12 | no | — |
| 171 | #376 | — | 11 | 6 | no | — |
| 172 | C22×C11 | C22×#41 | 1 | 3 | yes | — |
| 173 | C22×C11 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 174 | C22×C11 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 175 | C2×D11 | — | 2 | 6 | no | — |
| 176 | C2×D11 | — | 4 | 3 | no | — |
| 177 | C2×D11 | — | 4 | 1 | no | — |
| 178 | C2×D11 | — | 4 | 6 | no | — |
| 179 | C2×D11 | — | 4 | 12 | no | — |
| 180 | C2×D11 | — | 4 | 6 | no | — |
| 181 | C2×D11 | — | 4 | 12 | no | — |
| 182 | C2×D11 | — | 4 | 6 | no | — |
| 183 | C4×C11 | — | 2 | 6 | no | — |
| 184 | C4×C11 | — | 2 | 12 | no | — |
| 185 | C4×C11 | — | 4 | 6 | no | — |
| 186 | Dic11 | — | 2 | 6 | no | — |
| 187 | Dic11 | — | 4 | 6 | no | — |
| 188 | C2×#41 | — | 22 | 12 | no | — |
| 189 | C2×#41 | — | 44 | 6 | no | — |
| 190 | C2×#41 | — | 44 | 6 | no | — |
| 191 | C22×Q8 | C22×D11 | 1 | 2 | yes | — |
| 192 | C22×Q8 | C22×D11 | 1 | 3 | yes | — |
| 193 | C22×D4 | — | 11 | 6 | no | — |
| 194 | C22×D4 | — | 11 | 6 | no | — |
| 195 | C22×D4 | — | 11 | 3 | no | — |
| 196 | C22×D4 | — | 11 | 6 | no | — |
| 197 | C23×C4 | — | 11 | 3 | no | — |
| 198 | C23×C4 | — | 11 | 6 | no | — |
| 199 | #127 | — | 44 | 4 | no | — |
| 200 | #124 | — | 22 | 24 | no | — |
| 201 | #124 | — | 44 | 12 | no | — |
| 202 | #121 | — | 44 | 12 | no | — |
| 203 | #117 | — | 44 | 6 | no | — |
| 204 | C2×C4⋊C4 | — | 11 | 3 | no | — |
| 205 | C2×C4⋊C4 | C22×D11 | 1 | 3 | yes | — |
| 206 | C2×C4⋊C4 | C22×D11 | 1 | 6 | yes | — |
| 207 | C2×C22⋊C4 | — | 11 | 12 | no | — |
| 208 | C2×C22⋊C4 | — | 11 | 12 | no | — |
| 209 | C2×C22⋊C4 | — | 11 | 12 | no | — |
| 210 | C2×C22⋊C4 | — | 11 | 6 | no | — |
| 211 | C2×C22⋊C4 | — | 11 | 12 | no | — |
| 212 | C2×C42 | — | 11 | 6 | no | — |
| 213 | C2×C42 | — | 11 | 6 | no | — |
| 214 | C2×C42 | — | 11 | 3 | no | — |
| 215 | C2×C42 | — | 44 | 2 | no | — |
| 216 | C2×C42 | C22×D11 | 1 | 2 | yes | — |
| 217 | C2×C42 | C22×D11 | 1 | 1 | yes | characteristic |
| 218 | #95 | — | 11 | 12 | no | — |
| 219 | #95 | — | 11 | 6 | no | — |
| 220 | #95 | C22×D11 | 1 | 6 | yes | — |
| 221 | #95 | C22×D11 | 1 | 12 | yes | — |
| 222 | C2×C11 | #3483 | 1 | 3 | yes | — |
| 223 | C2×C11 | #3490 | 1 | 3 | yes | — |
| 224 | C2×C11 | #3549 | 1 | 1 | yes | characteristic |
| 225 | D11 | — | 4 | 2 | no | — |
| 226 | D11 | — | 8 | 6 | no | — |
| 227 | C24 | — | 11 | 1 | no | — |
| 228 | C24 | — | 11 | 6 | no | — |
| 229 | #41 | — | 44 | 48 | no | — |
| 230 | C2×Q8 | — | 4 | 3 | no | — |
| 231 | C2×Q8 | — | 4 | 3 | no | — |
| 232 | C2×Q8 | — | 4 | 6 | no | — |
| 233 | C2×Q8 | #4343 | 1 | 12 | yes | complemented |
| 234 | C2×D4 | — | 22 | 12 | no | — |
| 235 | C2×D4 | — | 22 | 12 | no | — |
| 236 | C2×D4 | — | 22 | 6 | no | — |
| 237 | C2×D4 | — | 22 | 12 | no | — |
| 238 | C2×D4 | — | 44 | 3 | no | — |
| 239 | C2×D4 | — | 44 | 3 | no | — |
| 240 | C2×D4 | — | 44 | 6 | no | — |
| 241 | C2×D4 | — | 44 | 6 | no | — |
| 242 | C2×D4 | — | 44 | 12 | no | — |
| 243 | C2×D4 | — | 44 | 6 | no | — |
| 244 | C2×D4 | — | 44 | 6 | no | — |
| 245 | C22×C4 | — | 11 | 3 | no | — |
| 246 | C22×C4 | — | 11 | 6 | no | — |
| 247 | C22×C4 | — | 22 | 6 | no | — |
| 248 | C22×C4 | — | 22 | 12 | no | — |
| 249 | C22×C4 | — | 22 | 6 | no | — |
| 250 | C22×C4 | — | 22 | 6 | no | — |
| 251 | C22×C4 | — | 22 | 12 | no | — |
| 252 | C22×C4 | — | 44 | 6 | no | — |
| 253 | C22×C4 | C23×D11 | 1 | 6 | yes | — |
| 254 | C22×C4 | C23×D11 | 1 | 6 | yes | — |
| 255 | C22×C4 | C23×D11 | 1 | 3 | yes | — |
| 256 | C4⋊C4 | — | 4 | 3 | no | — |
| 257 | C4⋊C4 | — | 4 | 6 | no | — |
| 258 | C4⋊C4 | — | 44 | 3 | no | — |
| 259 | C22⋊C4 | — | 22 | 24 | no | — |
| 260 | C22⋊C4 | — | 22 | 24 | no | — |
| 261 | C22⋊C4 | — | 22 | 24 | no | — |
| 262 | C22⋊C4 | — | 44 | 6 | no | — |
| 263 | C22⋊C4 | — | 44 | 12 | no | — |
| 264 | C42 | — | 4 | 1 | no | — |
| 265 | C42 | — | 4 | 2 | no | — |
| 266 | C42 | — | 22 | 12 | no | — |
| 267 | C42 | — | 22 | 12 | no | — |
| 268 | C42 | — | 44 | 3 | no | — |
| 269 | C11 | #60048 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 270 | C23 | — | 11 | 2 | no | — |
| 271 | C23 | — | 22 | 3 | no | — |
| 272 | C23 | — | 22 | 3 | no | — |
| 273 | C23 | — | 22 | 12 | no | — |
| 274 | C23 | — | 22 | 6 | no | — |
| 275 | C23 | — | 22 | 12 | no | — |
| 276 | C23 | — | 22 | 6 | no | — |
| 277 | C23 | — | 22 | 6 | no | — |
| 278 | C23 | C24×D11 | 1 | 1 | yes | characteristic |
| 279 | Q8 | — | 4 | 12 | no | — |
| 280 | Q8 | — | 8 | 4 | no | — |
| 281 | D4 | — | 44 | 24 | no | — |
| 282 | D4 | — | 44 | 12 | no | — |
| 283 | D4 | — | 88 | 12 | no | — |
| 284 | D4 | — | 88 | 12 | no | — |
| 285 | C2×C4 | — | 2 | 3 | no | — |
| 286 | C2×C4 | — | 2 | 6 | no | — |
| 287 | C2×C4 | — | 2 | 3 | no | — |
| 288 | C2×C4 | — | 2 | 6 | no | — |
| 289 | C2×C4 | — | 2 | 6 | no | — |
| 290 | C2×C4 | — | 2 | 12 | no | — |
| 291 | C2×C4 | — | 11 | 6 | no | — |
| 292 | C2×C4 | — | 22 | 24 | no | — |
| 293 | C2×C4 | — | 22 | 3 | no | — |
| 294 | C2×C4 | — | 22 | 3 | no | — |
| 295 | C2×C4 | — | 22 | 12 | no | — |
| 296 | C2×C4 | — | 22 | 6 | no | — |
| 297 | C2×C4 | — | 44 | 24 | no | — |
| 298 | C2×C4 | C2×#4343 | 1 | 12 | yes | — |
| 299 | C2×C4 | C2×#4343 | 1 | 6 | yes | — |
| 300 | C22 | — | 22 | 6 | no | — |
| 301 | C22 | — | 44 | 3 | no | — |
| 302 | C22 | — | 44 | 1 | no | — |
| 303 | C22 | — | 44 | 6 | no | — |
| 304 | C22 | — | 44 | 12 | no | — |
| 305 | C22 | — | 44 | 6 | no | — |
| 306 | C22 | — | 44 | 6 | no | — |
| 307 | C22 | — | 44 | 12 | no | — |
| 308 | C22 | C22×#4343 | 1 | 3 | yes | — |
| 309 | C22 | D11×H(5,𝔽2) | 1 | 3 | yes | — |
| 310 | C22 | D11×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 311 | C4 | — | 2 | 6 | no | — |
| 312 | C4 | — | 2 | 12 | no | — |
| 313 | C4 | — | 4 | 6 | no | — |
| 314 | C4 | — | 22 | 6 | no | — |
| 315 | C4 | — | 44 | 6 | no | — |
| 316 | C2 | — | 44 | 2 | no | — |
| 317 | C2 | — | 88 | 6 | no | — |
| 318 | C2 | #13082243 | 1 | 3 | yes | — |
| 319 | C2 | #13082306 | 1 | 3 | yes | — |
| 320 | C2 | #13083020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D11 |
| commutator, socle, cosocle | C23×C11 | C25 |
| Fitting | C11×#2811 | C2 |
| hypercenter | #2811 | D11 |
| hypocenter | C11 | #60048 |
| LCS | G ▷ C23×C11 ▷ C11 | |
| UCS | C1 ◁ C23 ◁ #2811 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13075642 | C2 | semidirect |
| 2. #13074298 | C2 | semidirect |
| 3. #13073691 | C2 | semidirect |
| 4. #13072237 | C2 | semidirect |
| 5. #13071727 | C2 | semidirect |
| 6. #13071105 | C2 | semidirect |
| 7. C11×#2811 | C2 | semidirect |
| 73. C2×Q8×C11 | #41 | semidirect |
| 110. #2811 | D11 | semidirect |
| 233. C2×Q8 | #4343 | semidirect |
| 269. C11 | #60048 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13075642 | 2 |
| #13074298 | 2 |
| #13073691 | 2 |
| #13072237 | 2 |
| #13071727 | 2 |
| #13071105 | 2 |
| C11×#2811 | 2 |
| #60048 | 11 |