← #10950885
Subgroups — #10950885
Order 768.
504 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #80741 | C2 | 1 | 2 | yes | maximal |
| 2 | C3×#2219 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #71499 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #71496 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #71447 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | Dic3×M32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #71083 | C2 | 1 | 2 | yes | maximal |
| 8 | #69340 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #69338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #69215 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #69209 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #33749 | — | 3 | 1 | no | maximal, Sylow |
| 13 | #5622 | — | 2 | 2 | no | — |
| 14 | C2×C3×Q32 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#467 | C22 | 1 | 2 | yes | — |
| 17 | #5183 | C22 | 1 | 2 | yes | — |
| 18 | Dic3×Q16 | C22 | 1 | 2 | yes | — |
| 19 | C2×#829 | C22 | 1 | 2 | yes | — |
| 20 | #5116 | C22 | 1 | 2 | yes | — |
| 21 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 23 | #5051 | — | 2 | 2 | no | — |
| 24 | C2×Dic24 | C22 | 1 | 1 | yes | characteristic |
| 25 | #4874 | C22 | 1 | 2 | yes | — |
| 26 | #4873 | C22 | 1 | 2 | yes | — |
| 27 | #4792 | — | 2 | 2 | no | — |
| 28 | #4790 | — | 2 | 2 | no | — |
| 29 | C3×#367 | C22 | 1 | 1 | yes | characteristic |
| 30 | C3×#358 | C22 | 1 | 2 | yes | — |
| 31 | #4564 | C22 | 1 | 2 | yes | — |
| 32 | #4505 | C22 | 1 | 1 | yes | characteristic |
| 33 | #4503 | C22 | 1 | 2 | yes | — |
| 34 | #4502 | C22 | 1 | 1 | yes | characteristic |
| 35 | #4502 | C22 | 1 | 1 | yes | characteristic |
| 36 | Dic3×C16 | C22 | 1 | 1 | yes | characteristic |
| 37 | Dic3×C16 | C22 | 1 | 1 | yes | characteristic |
| 38 | #4492 | C22 | 1 | 2 | yes | — |
| 39 | #4490 | C22 | 1 | 2 | yes | — |
| 40 | #3170 | — | 3 | 2 | no | — |
| 41 | #2254 | — | 3 | 1 | no | — |
| 42 | #2247 | — | 3 | 1 | no | — |
| 43 | #2241 | — | 3 | 2 | no | — |
| 44 | #2219 | — | 3 | 1 | no | — |
| 45 | #2219 | — | 3 | 2 | no | — |
| 46 | #2219 | D3 | 1 | 1 | yes | characteristic |
| 47 | #2176 | — | 3 | 1 | no | — |
| 48 | #2176 | — | 3 | 1 | no | — |
| 49 | #2176 | — | 3 | 2 | no | — |
| 50 | C4×M32 | — | 3 | 1 | no | — |
| 51 | C2×C3×Q16 | C23 | 1 | 2 | yes | — |
| 52 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 53 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 54 | Q8×Dic3 | — | 2 | 2 | no | — |
| 55 | #944 | — | 2 | 2 | no | — |
| 56 | C2×#216 | — | 2 | 2 | no | — |
| 57 | #924 | — | 2 | 2 | no | — |
| 58 | C2×C4×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 59 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 60 | C2×Dic12 | C23 | 1 | 2 | yes | — |
| 61 | #889 | — | 4 | 2 | no | — |
| 62 | #888 | — | 2 | 2 | no | — |
| 63 | #887 | — | 2 | 2 | no | — |
| 64 | #887 | — | 4 | 2 | no | — |
| 65 | #878 | — | 4 | 2 | no | — |
| 66 | #877 | — | 4 | 2 | no | — |
| 67 | C3×Q32 | — | 2 | 2 | no | — |
| 68 | C3×M32 | — | 2 | 2 | no | — |
| 69 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 70 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 71 | C3×#107 | C23 | 1 | 2 | yes | — |
| 72 | C3×#103 | — | 2 | 2 | no | — |
| 73 | #835 | — | 2 | 2 | no | — |
| 74 | #829 | — | 2 | 4 | no | — |
| 75 | #818 | C23 | 1 | 2 | yes | — |
| 76 | #816 | — | 2 | 2 | no | — |
| 77 | C8×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 78 | C8×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 79 | C8×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 80 | C8×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 81 | #812 | — | 2 | 2 | no | — |
| 82 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 83 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 84 | #810 | — | 2 | 2 | no | — |
| 85 | #807 | — | 2 | 2 | no | — |
| 86 | Dic24 | — | 2 | 2 | no | — |
| 87 | #533 | — | 6 | 2 | no | — |
| 88 | C2×Q32 | — | 3 | 2 | no | — |
| 89 | C2×Q32 | — | 3 | 1 | no | — |
| 90 | C2×Q32 | C2×D3 | 1 | 1 | yes | characteristic |
| 91 | C2×M32 | — | 3 | 1 | no | — |
| 92 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 93 | #500 | — | 3 | 2 | no | — |
| 94 | #494 | — | 3 | 2 | no | — |
| 95 | #487 | — | 6 | 2 | no | — |
| 96 | #467 | — | 3 | 2 | no | — |
| 97 | #467 | — | 6 | 2 | no | — |
| 98 | #467 | C2×D3 | 1 | 2 | yes | — |
| 99 | C4×Q16 | — | 3 | 2 | no | — |
| 100 | C4×Q16 | — | 3 | 2 | no | — |
| 101 | C4×Q16 | — | 6 | 2 | no | — |
| 102 | C2×C4×C8 | — | 3 | 1 | no | — |
| 103 | #367 | — | 3 | 1 | no | — |
| 104 | #367 | — | 3 | 2 | no | — |
| 105 | #367 | C2×D3 | 1 | 1 | yes | characteristic |
| 106 | #358 | — | 3 | 2 | no | — |
| 107 | #358 | — | 3 | 2 | no | — |
| 108 | #358 | — | 3 | 2 | no | — |
| 109 | #358 | C2×D3 | 1 | 2 | yes | — |
| 110 | #346 | — | 3 | 1 | no | — |
| 111 | #346 | — | 3 | 1 | no | — |
| 112 | C4×C16 | — | 3 | 1 | no | — |
| 113 | C4×C16 | — | 3 | 1 | no | — |
| 114 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 115 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C22×Dic3 | D8 | 1 | 2 | yes | — |
| 117 | C2×Dic6 | — | 2 | 2 | no | — |
| 118 | C2×Dic6 | — | 4 | 2 | no | — |
| 119 | C3×Q16 | — | 2 | 2 | no | — |
| 120 | C3×Q16 | — | 2 | 2 | no | — |
| 121 | C2×C3×C8 | — | 2 | 1 | no | — |
| 122 | C2×C3×C8 | — | 2 | 1 | no | — |
| 123 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 125 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 126 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 127 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 128 | #217 | — | 4 | 2 | no | — |
| 129 | #216 | — | 4 | 4 | no | — |
| 130 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 131 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 132 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 133 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 134 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 135 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 136 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 137 | C4×Dic3 | — | 4 | 2 | no | — |
| 138 | C4×Dic3 | — | 4 | 2 | no | — |
| 139 | C4×Dic3 | D8 | 1 | 2 | yes | — |
| 140 | C4×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 141 | C4×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 142 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 143 | C2×C3⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 144 | C2×C3⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 145 | Dic12 | — | 2 | 2 | no | — |
| 146 | Dic12 | — | 2 | 2 | no | — |
| 147 | C3×C16 | — | 2 | 1 | no | — |
| 148 | C3×C16 | #41 | 1 | 2 | yes | — |
| 149 | C3⋊C16 | — | 2 | 1 | no | — |
| 150 | C3⋊C16 | #41 | 1 | 2 | yes | — |
| 151 | C2×Q16 | — | 3 | 2 | no | — |
| 152 | C2×Q16 | — | 6 | 2 | no | — |
| 153 | C2×Q16 | C22×D3 | 1 | 2 | yes | — |
| 154 | C22×C8 | — | 3 | 1 | no | — |
| 155 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 156 | C4⋊Q8 | — | 6 | 2 | no | — |
| 157 | C4⋊Q8 | — | 6 | 2 | no | — |
| 158 | #125 | — | 12 | 2 | no | — |
| 159 | C22⋊Q8 | — | 2 | 2 | no | — |
| 160 | C22⋊Q8 | — | 6 | 2 | no | — |
| 161 | C22⋊Q8 | — | 12 | 2 | no | — |
| 162 | C4×Q8 | — | 6 | 2 | no | — |
| 163 | C4×Q8 | — | 6 | 2 | no | — |
| 164 | C4×Q8 | — | 12 | 2 | no | — |
| 165 | #117 | — | 12 | 2 | no | — |
| 166 | C2×C42 | — | 3 | 1 | no | — |
| 167 | Q32 | — | 2 | 2 | no | — |
| 168 | Q32 | — | 6 | 4 | no | — |
| 169 | Q32 | — | 6 | 2 | no | — |
| 170 | M32 | — | 2 | 2 | no | — |
| 171 | M32 | — | 6 | 2 | no | — |
| 172 | C2×C16 | — | 3 | 1 | no | — |
| 173 | C2×C16 | — | 3 | 1 | no | — |
| 174 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 175 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 176 | #107 | — | 3 | 2 | no | — |
| 177 | #107 | — | 6 | 2 | no | — |
| 178 | #107 | C22×D3 | 1 | 2 | yes | — |
| 179 | #103 | — | 2 | 2 | no | — |
| 180 | #103 | — | 6 | 2 | no | — |
| 181 | #103 | — | 6 | 2 | no | — |
| 182 | #103 | — | 6 | 2 | no | — |
| 183 | C4×C8 | — | 3 | 1 | no | — |
| 184 | C4×C8 | — | 3 | 1 | no | — |
| 185 | C4×C8 | — | 3 | 1 | no | — |
| 186 | C4×C8 | — | 3 | 1 | no | — |
| 187 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 188 | C3×Q8 | — | 4 | 2 | no | — |
| 189 | C3×Q8 | — | 4 | 2 | no | — |
| 190 | C2×C3×C4 | — | 2 | 1 | no | — |
| 191 | C2×C3×C4 | — | 2 | 1 | no | — |
| 192 | C2×C3×C4 | — | 4 | 2 | no | — |
| 193 | C2×C3×C4 | — | 4 | 2 | no | — |
| 194 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 195 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 196 | C2×Dic3 | — | 2 | 4 | no | — |
| 197 | C2×Dic3 | — | 4 | 2 | no | — |
| 198 | C2×Dic3 | — | 4 | 2 | no | — |
| 199 | C2×Dic3 | C2×D8 | 1 | 2 | yes | — |
| 200 | C2×Dic3 | C2×D8 | 1 | 2 | yes | — |
| 201 | Dic6 | — | 4 | 2 | no | — |
| 202 | Dic6 | — | 4 | 2 | no | — |
| 203 | Dic6 | — | 8 | 4 | no | — |
| 204 | C3×C8 | — | 2 | 1 | no | — |
| 205 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 206 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 207 | C3⋊C8 | — | 2 | 1 | no | — |
| 208 | C3⋊C8 | #135 | 1 | 2 | yes | — |
| 209 | C2×Q8 | — | 2 | 2 | no | — |
| 210 | C2×Q8 | — | 6 | 2 | no | — |
| 211 | C2×Q8 | — | 12 | 2 | no | — |
| 212 | C22×C4 | — | 3 | 2 | no | — |
| 213 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 214 | Q16 | — | 2 | 2 | no | — |
| 215 | Q16 | — | 2 | 2 | no | — |
| 216 | Q16 | — | 6 | 2 | no | — |
| 217 | Q16 | — | 6 | 2 | no | — |
| 218 | Q16 | — | 12 | 4 | no | — |
| 219 | C2×C8 | — | 2 | 1 | no | — |
| 220 | C2×C8 | — | 2 | 1 | no | — |
| 221 | C2×C8 | — | 3 | 1 | no | — |
| 222 | C2×C8 | — | 3 | 1 | no | — |
| 223 | C2×C8 | — | 6 | 2 | no | — |
| 224 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 226 | C4⋊C4 | — | 2 | 2 | no | — |
| 227 | C4⋊C4 | — | 4 | 2 | no | — |
| 228 | C4⋊C4 | — | 6 | 2 | no | — |
| 229 | C4⋊C4 | — | 12 | 2 | no | — |
| 230 | C4⋊C4 | — | 12 | 2 | no | — |
| 231 | C4⋊C4 | — | 12 | 2 | no | — |
| 232 | C4⋊C4 | — | 12 | 2 | no | — |
| 233 | C4⋊C4 | — | 12 | 2 | no | — |
| 234 | C4⋊C4 | — | 12 | 2 | no | — |
| 235 | C22⋊C4 | — | 4 | 2 | no | — |
| 236 | C22⋊C4 | — | 12 | 2 | no | — |
| 237 | C42 | — | 3 | 2 | no | — |
| 238 | C42 | — | 3 | 1 | no | — |
| 239 | C42 | — | 3 | 1 | no | — |
| 240 | C42 | — | 12 | 2 | no | — |
| 241 | C42 | — | 12 | 2 | no | — |
| 242 | C16 | — | 2 | 1 | no | — |
| 243 | C16 | — | 3 | 2 | no | — |
| 244 | C16 | — | 6 | 1 | no | — |
| 245 | C16 | #237 | 1 | 2 | yes | — |
| 246 | C22×C3 | — | 2 | 2 | no | — |
| 247 | C22×C3 | — | 2 | 1 | no | — |
| 248 | C22×C3 | C22×D8 | 1 | 1 | yes | characteristic |
| 249 | C3×C4 | — | 2 | 1 | no | — |
| 250 | C3×C4 | — | 8 | 2 | no | — |
| 251 | C3×C4 | — | 8 | 2 | no | — |
| 252 | C3×C4 | #532 | 1 | 1 | yes | characteristic |
| 253 | C3×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 254 | Dic3 | — | 2 | 2 | no | — |
| 255 | Dic3 | — | 8 | 2 | no | — |
| 256 | Dic3 | — | 8 | 2 | no | — |
| 257 | Dic3 | #510 | 1 | 4 | yes | — |
| 258 | C23 | D3×D8 | 1 | 1 | yes | characteristic |
| 259 | Q8 | — | 4 | 2 | no | — |
| 260 | Q8 | — | 4 | 2 | no | — |
| 261 | Q8 | — | 12 | 2 | no | — |
| 262 | Q8 | — | 12 | 2 | no | — |
| 263 | Q8 | — | 24 | 4 | no | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 3 | 2 | no | — |
| 267 | C2×C4 | — | 3 | 2 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 4 | 2 | no | — |
| 270 | C2×C4 | — | 6 | 4 | no | — |
| 271 | C2×C4 | — | 12 | 2 | no | — |
| 272 | C2×C4 | — | 12 | 2 | no | — |
| 273 | C2×C4 | D3×D8 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 3 | 2 | no | — |
| 277 | C8 | — | 6 | 1 | no | — |
| 278 | C8 | C2×#237 | 1 | 1 | yes | characteristic |
| 279 | C8 | D3×#41 | 1 | 1 | yes | characteristic |
| 280 | C2×C3 | — | 4 | 1 | no | — |
| 281 | C2×C3 | #3169 | 1 | 1 | yes | characteristic |
| 282 | C2×C3 | C2×#510 | 1 | 2 | yes | — |
| 283 | C22 | — | 2 | 1 | no | — |
| 284 | C22 | — | 2 | 2 | no | — |
| 285 | C22 | C2×D3×D8 | 1 | 1 | yes | characteristic |
| 286 | C4 | — | 2 | 1 | no | — |
| 287 | C4 | — | 3 | 4 | no | — |
| 288 | C4 | — | 6 | 2 | no | — |
| 289 | C4 | — | 8 | 2 | no | — |
| 290 | C4 | — | 8 | 2 | no | — |
| 291 | C4 | — | 24 | 2 | no | — |
| 292 | C4 | — | 24 | 2 | no | — |
| 293 | C4 | #5596 | 1 | 1 | yes | characteristic |
| 294 | C4 | D3×#135 | 1 | 1 | yes | characteristic |
| 295 | C3 | #33749 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 296 | C2 | — | 4 | 1 | no | — |
| 297 | C2 | #80697 | 1 | 1 | yes | characteristic |
| 298 | C2 | D3×#510 | 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