← #10948846
Subgroups — #10948846
Order 768.
460 conjugacy classes of subgroups.
The subgroup lattice has 278 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #78542 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#2164 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #69883 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #69282 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #69280 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #69094 | C2 | 1 | 2 | yes | maximal |
| 7 | #69092 | C2 | 1 | 2 | yes | maximal |
| 8 | #69068 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C16×Dic6 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #33500 | — | 3 | 1 | no | maximal, Sylow |
| 11 | #5517 | C4 | 1 | 1 | yes | characteristic |
| 12 | C3×#405 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5119 | C4 | 1 | 1 | yes | characteristic |
| 14 | #5118 | C4 | 1 | 1 | yes | characteristic |
| 15 | #5101 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 17 | #4999 | C4 | 1 | 1 | yes | characteristic |
| 18 | #4701 | C4 | 1 | 2 | yes | — |
| 19 | #4701 | C4 | 1 | 2 | yes | — |
| 20 | C8×Dic6 | C22 | 1 | 2 | yes | — |
| 21 | C8×Dic6 | C22 | 1 | 2 | yes | — |
| 22 | C3×C4⋊C16 | C22 | 1 | 2 | yes | — |
| 23 | C3×C22⋊C16 | C22 | 1 | 2 | yes | — |
| 24 | C3×#346 | C22 | 1 | 1 | yes | characteristic |
| 25 | C3×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 26 | #4549 | C22 | 1 | 2 | yes | — |
| 27 | #4502 | C22 | 1 | 2 | yes | — |
| 28 | Dic3⋊C16 | C22 | 1 | 2 | yes | — |
| 29 | Dic3⋊C16 | C22 | 1 | 2 | yes | — |
| 30 | Dic3⋊C16 | C22 | 1 | 2 | yes | — |
| 31 | Dic3⋊C16 | C22 | 1 | 2 | yes | — |
| 32 | Dic3×C16 | C22 | 1 | 2 | yes | — |
| 33 | Dic3×C16 | C22 | 1 | 2 | yes | — |
| 34 | Dic3×C16 | C22 | 1 | 2 | yes | — |
| 35 | C4⋊(C3⋊C16) | C22 | 1 | 2 | yes | — |
| 36 | #4461 | C22 | 1 | 1 | yes | characteristic |
| 37 | C4×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 38 | #2960 | — | 3 | 1 | no | — |
| 39 | #2183 | — | 3 | 2 | no | — |
| 40 | #2183 | — | 3 | 2 | no | — |
| 41 | Q8×C16 | — | 3 | 2 | no | — |
| 42 | Q8×C16 | — | 3 | 2 | no | — |
| 43 | #2164 | — | 3 | 1 | no | — |
| 44 | #2164 | — | 3 | 2 | no | — |
| 45 | #2164 | D3 | 1 | 1 | yes | characteristic |
| 46 | #2162 | — | 3 | 2 | no | — |
| 47 | C2×C3×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 49 | C3×#117 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #926 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C2×#208 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C22×C3⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #890 | C8 | 1 | 2 | yes | — |
| 55 | #888 | C8 | 1 | 2 | yes | — |
| 56 | #878 | C8 | 1 | 2 | yes | — |
| 57 | #878 | C8 | 1 | 2 | yes | — |
| 58 | C4×Dic6 | C2×C4 | 1 | 2 | yes | — |
| 59 | C4×Dic6 | C2×C4 | 1 | 2 | yes | — |
| 60 | C2×C3×C16 | C23 | 1 | 2 | yes | — |
| 61 | C2×C3×C16 | C23 | 1 | 2 | yes | — |
| 62 | C3×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C3×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 65 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 66 | #815 | C2×C4 | 1 | 2 | yes | — |
| 67 | Dic3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 68 | Dic3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 69 | Dic3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 70 | Dic3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 71 | C8×Dic3 | C2×C4 | 1 | 2 | yes | — |
| 72 | C8×Dic3 | C23 | 1 | 2 | yes | — |
| 73 | C8×Dic3 | C23 | 1 | 2 | yes | — |
| 74 | C2×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 75 | C2×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 76 | C4⋊(C3⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 77 | C4⋊(C3⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 78 | #533 | — | 3 | 1 | no | — |
| 79 | #446 | — | 3 | 2 | no | — |
| 80 | #446 | — | 3 | 2 | no | — |
| 81 | C8×Q8 | — | 3 | 2 | no | — |
| 82 | C8×Q8 | — | 3 | 2 | no | — |
| 83 | #424 | — | 3 | 1 | no | — |
| 84 | #424 | — | 3 | 1 | no | — |
| 85 | #423 | — | 3 | 1 | no | — |
| 86 | #405 | — | 3 | 1 | no | — |
| 87 | #405 | C2×D3 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C8 | — | 3 | 1 | no | — |
| 89 | C4⋊C16 | — | 3 | 2 | no | — |
| 90 | C4⋊C16 | — | 3 | 2 | no | — |
| 91 | C4⋊C16 | — | 3 | 2 | no | — |
| 92 | C4⋊C16 | — | 3 | 2 | no | — |
| 93 | C4⋊C16 | — | 3 | 2 | no | — |
| 94 | C4⋊C16 | C2×D3 | 1 | 2 | yes | — |
| 95 | C22⋊C16 | — | 3 | 2 | no | — |
| 96 | C22⋊C16 | C2×D3 | 1 | 2 | yes | — |
| 97 | #346 | — | 3 | 2 | no | — |
| 98 | #346 | — | 3 | 1 | no | — |
| 99 | #346 | C2×D3 | 1 | 1 | yes | characteristic |
| 100 | C4×C16 | — | 3 | 2 | no | — |
| 101 | C4×C16 | — | 3 | 2 | no | — |
| 102 | C4×C16 | — | 3 | 2 | no | — |
| 103 | C4×C16 | — | 3 | 1 | no | — |
| 104 | C4×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 105 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 106 | C22×Dic3 | C2×C8 | 1 | 2 | yes | — |
| 107 | C2×Dic6 | C2×C8 | 1 | 2 | yes | — |
| 108 | C2×Dic6 | C2×C8 | 1 | 2 | yes | — |
| 109 | C3×M16 | — | 4 | 1 | no | — |
| 110 | C2×C3×C8 | — | 2 | 2 | no | — |
| 111 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 112 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 113 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 114 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 115 | C3×C4⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 116 | C3×C22⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 117 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 118 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 119 | #217 | C2×C8 | 1 | 1 | yes | characteristic |
| 120 | #217 | C2×C8 | 1 | 1 | yes | characteristic |
| 121 | C4⋊Dic3 | C2×C8 | 1 | 2 | yes | — |
| 122 | C4⋊Dic3 | C2×C8 | 1 | 2 | yes | — |
| 123 | C4⋊Dic3 | C2×C8 | 1 | 2 | yes | — |
| 124 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 125 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 126 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 127 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 128 | C4×Dic3 | C2×C8 | 1 | 1 | yes | characteristic |
| 129 | C4×Dic3 | C2×C8 | 1 | 1 | yes | characteristic |
| 130 | C4×Dic3 | C22×C4 | 1 | 2 | yes | — |
| 131 | C4×Dic3 | C22×C4 | 1 | 2 | yes | — |
| 132 | #208 | — | 4 | 1 | no | — |
| 133 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 134 | C2×C3⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 135 | C2×C3⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 136 | C3×C16 | — | 2 | 2 | no | — |
| 137 | C3×C16 | — | 2 | 2 | no | — |
| 138 | C3⋊C16 | — | 2 | 2 | no | — |
| 139 | C3⋊C16 | — | 2 | 2 | no | — |
| 140 | C2×M16 | — | 3 | 1 | no | — |
| 141 | C2×M16 | C4×D3 | 1 | 1 | yes | characteristic |
| 142 | C22×C8 | — | 3 | 1 | no | — |
| 143 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 144 | C4⋊Q8 | — | 3 | 2 | no | — |
| 145 | #125 | — | 3 | 2 | no | — |
| 146 | C22⋊Q8 | — | 3 | 2 | no | — |
| 147 | C22⋊Q8 | — | 3 | 2 | no | — |
| 148 | C4×Q8 | — | 3 | 2 | no | — |
| 149 | C4×Q8 | — | 3 | 2 | no | — |
| 150 | #117 | — | 3 | 1 | no | — |
| 151 | #117 | C4×D3 | 1 | 1 | yes | characteristic |
| 152 | C2×C42 | — | 3 | 1 | no | — |
| 153 | C2×C16 | — | 3 | 2 | no | — |
| 154 | C2×C16 | — | 3 | 2 | no | — |
| 155 | C2×C16 | C22×D3 | 1 | 2 | yes | — |
| 156 | C2×C16 | C22×D3 | 1 | 2 | yes | — |
| 157 | C4⋊C8 | — | 3 | 2 | no | — |
| 158 | C4⋊C8 | — | 3 | 2 | no | — |
| 159 | C4⋊C8 | — | 3 | 2 | no | — |
| 160 | C4⋊C8 | — | 3 | 2 | no | — |
| 161 | C4⋊C8 | — | 3 | 2 | no | — |
| 162 | C4⋊C8 | — | 3 | 2 | no | — |
| 163 | #97 | — | 3 | 2 | no | — |
| 164 | #97 | C4×D3 | 1 | 1 | yes | characteristic |
| 165 | #97 | C4×D3 | 1 | 1 | yes | characteristic |
| 166 | C4×C8 | — | 3 | 2 | no | — |
| 167 | C4×C8 | — | 3 | 2 | no | — |
| 168 | C4×C8 | — | 3 | 2 | no | — |
| 169 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 170 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 171 | C23×C3 | C22×C8 | 1 | 1 | yes | characteristic |
| 172 | C2×C3×C4 | — | 2 | 1 | no | — |
| 173 | C2×C3×C4 | — | 2 | 1 | no | — |
| 174 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | C22×C8 | 1 | 2 | yes | — |
| 176 | C2×C3×C4 | C22×C8 | 1 | 2 | yes | — |
| 177 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 178 | C2×Dic3 | — | 2 | 4 | no | — |
| 179 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 180 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 181 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 182 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 183 | Dic6 | — | 2 | 4 | no | — |
| 184 | Dic6 | — | 2 | 4 | no | — |
| 185 | C3×C8 | — | 2 | 1 | no | — |
| 186 | C3×C8 | — | 2 | 1 | no | — |
| 187 | C3×C8 | — | 2 | 1 | no | — |
| 188 | C3×C8 | 2−1+4 | 1 | 2 | yes | — |
| 189 | C3⋊C8 | — | 2 | 2 | no | — |
| 190 | C3⋊C8 | — | 2 | 2 | no | — |
| 191 | C2×Q8 | — | 3 | 2 | no | — |
| 192 | C2×Q8 | — | 3 | 2 | no | — |
| 193 | C22×C4 | — | 3 | 2 | no | — |
| 194 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 195 | M16 | — | 4 | 1 | no | — |
| 196 | M16 | — | 12 | 1 | no | — |
| 197 | C2×C8 | — | 2 | 2 | no | — |
| 198 | C2×C8 | — | 3 | 2 | no | — |
| 199 | C2×C8 | — | 3 | 2 | no | — |
| 200 | C2×C8 | — | 12 | 1 | no | — |
| 201 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 202 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 203 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 204 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 205 | C4⋊C4 | — | 3 | 2 | no | — |
| 206 | C4⋊C4 | — | 3 | 2 | no | — |
| 207 | C4⋊C4 | — | 3 | 2 | no | — |
| 208 | C4⋊C4 | — | 3 | 2 | no | — |
| 209 | C4⋊C4 | — | 3 | 2 | no | — |
| 210 | C4⋊C4 | — | 3 | 2 | no | — |
| 211 | C4⋊C4 | — | 3 | 2 | no | — |
| 212 | C4⋊C4 | D3×C8 | 1 | 2 | yes | — |
| 213 | C22⋊C4 | — | 3 | 1 | no | — |
| 214 | C22⋊C4 | — | 3 | 1 | no | — |
| 215 | C22⋊C4 | D3×C8 | 1 | 2 | yes | — |
| 216 | C42 | — | 3 | 1 | no | — |
| 217 | C42 | — | 3 | 1 | no | — |
| 218 | C42 | — | 3 | 2 | no | — |
| 219 | C42 | — | 3 | 2 | no | — |
| 220 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 221 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 222 | C16 | — | 2 | 2 | no | — |
| 223 | C16 | — | 2 | 2 | no | — |
| 224 | C16 | — | 6 | 2 | no | — |
| 225 | C16 | — | 6 | 2 | no | — |
| 226 | C22×C3 | — | 2 | 2 | no | — |
| 227 | C22×C3 | — | 2 | 1 | no | — |
| 228 | C22×C3 | C23×C8 | 1 | 1 | yes | characteristic |
| 229 | C3×C4 | — | 2 | 2 | no | — |
| 230 | C3×C4 | — | 2 | 2 | no | — |
| 231 | C3×C4 | — | 2 | 1 | no | — |
| 232 | C3×C4 | #519 | 1 | 1 | yes | characteristic |
| 233 | C3×C4 | #568 | 1 | 1 | yes | characteristic |
| 234 | Dic3 | — | 2 | 2 | no | — |
| 235 | Dic3 | — | 2 | 2 | no | — |
| 236 | Dic3 | — | 2 | 2 | no | — |
| 237 | Dic3 | #504 | 1 | 4 | yes | — |
| 238 | C23 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 239 | Q8 | — | 6 | 4 | no | — |
| 240 | Q8 | — | 6 | 4 | no | — |
| 241 | C2×C4 | — | 2 | 1 | no | — |
| 242 | C2×C4 | — | 2 | 1 | no | — |
| 243 | C2×C4 | — | 3 | 2 | no | — |
| 244 | C2×C4 | — | 3 | 2 | no | — |
| 245 | C2×C4 | — | 3 | 2 | no | — |
| 246 | C2×C4 | — | 3 | 2 | no | — |
| 247 | C2×C4 | — | 6 | 4 | no | — |
| 248 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | C2×D3×C8 | 1 | 2 | yes | — |
| 250 | C2×C4 | C2×D3×C8 | 1 | 2 | yes | — |
| 251 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 252 | C8 | — | 2 | 1 | no | — |
| 253 | C8 | — | 2 | 1 | no | — |
| 254 | C8 | — | 2 | 1 | no | — |
| 255 | C8 | — | 6 | 2 | no | — |
| 256 | C8 | — | 6 | 2 | no | — |
| 257 | C8 | #1010 | 1 | 2 | yes | — |
| 258 | C2×C3 | — | 4 | 1 | no | — |
| 259 | C2×C3 | #2982 | 1 | 1 | yes | characteristic |
| 260 | C2×C3 | C2×#504 | 1 | 2 | yes | — |
| 261 | C22 | — | 2 | 2 | no | — |
| 262 | C22 | — | 2 | 1 | no | — |
| 263 | C22 | C22×D3×C8 | 1 | 1 | yes | characteristic |
| 264 | C4 | — | 2 | 2 | no | — |
| 265 | C4 | — | 2 | 2 | no | — |
| 266 | C4 | — | 2 | 1 | no | — |
| 267 | C4 | — | 3 | 4 | no | — |
| 268 | C4 | — | 6 | 2 | no | — |
| 269 | C4 | — | 6 | 2 | no | — |
| 270 | C4 | — | 6 | 2 | no | — |
| 271 | C4 | #5516 | 1 | 1 | yes | characteristic |
| 272 | C4 | #5745 | 1 | 1 | yes | characteristic |
| 273 | C3 | #33500 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 274 | C2 | — | 4 | 1 | no | — |
| 275 | C2 | #79263 | 1 | 1 | yes | characteristic |
| 276 | C2 | D3×#504 | 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