← #10712839
Subgroups — #10712839
Order 768.
306 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×#366 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #70612 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | Dic3×#107 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #67806 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #67804 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67722 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67722 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #13109 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×#107 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×#107 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 16 | #4984 | — | 2 | 1 | no | — |
| 17 | Dic3×C4⋊C4 | — | 2 | 1 | no | — |
| 18 | C3×#366 | C4 | 1 | 4 | yes | complemented |
| 19 | #4529 | — | 2 | 1 | no | — |
| 20 | #4529 | — | 2 | 1 | no | — |
| 21 | C2×#366 | D3 | 1 | 1 | yes | characteristic |
| 22 | #1942 | — | 3 | 1 | no | — |
| 23 | C4×#107 | — | 3 | 1 | no | — |
| 24 | #1380 | — | 3 | 1 | no | — |
| 25 | #1380 | — | 3 | 1 | no | — |
| 26 | #1380 | — | 3 | 1 | no | — |
| 27 | #1374 | — | 3 | 1 | no | — |
| 28 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 30 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 31 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 32 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 33 | C2×C4⋊Dic3 | — | 4 | 1 | no | — |
| 34 | C2×C4×Dic3 | — | 4 | 1 | no | — |
| 35 | C2×C4×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×C3×C16 | — | 2 | 1 | no | — |
| 38 | C2×C3×C16 | — | 2 | 1 | no | — |
| 39 | C2×C3×C16 | C2×C4 | 1 | 2 | yes | — |
| 40 | C3×#107 | — | 2 | 1 | no | — |
| 41 | C3×#107 | — | 2 | 1 | no | — |
| 42 | C3×#107 | C2×C4 | 1 | 2 | yes | — |
| 43 | C3×#107 | C2×C4 | 1 | 2 | yes | — |
| 44 | C3⋊#95 | — | 4 | 1 | no | — |
| 45 | C3⋊#95 | — | 4 | 1 | no | — |
| 46 | #818 | — | 2 | 1 | no | — |
| 47 | #818 | — | 2 | 1 | no | — |
| 48 | #818 | — | 2 | 1 | no | — |
| 49 | #818 | Q8 | 1 | 2 | yes | — |
| 50 | C8×Dic3 | — | 2 | 1 | no | — |
| 51 | C8×Dic3 | — | 2 | 1 | no | — |
| 52 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 53 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 54 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 55 | C22×C16 | — | 3 | 1 | no | — |
| 56 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C2×#107 | — | 3 | 1 | no | — |
| 58 | C2×#107 | — | 3 | 1 | no | — |
| 59 | C2×#107 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | C2×#107 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C8 | — | 3 | 1 | no | — |
| 62 | #389 | — | 6 | 1 | no | — |
| 63 | C4×C4⋊C4 | — | 6 | 1 | no | — |
| 64 | #366 | Dic3 | 1 | 4 | yes | complemented |
| 65 | #340 | — | 6 | 1 | no | — |
| 66 | #340 | — | 6 | 1 | no | — |
| 67 | C22×C3×C4 | — | 4 | 1 | no | — |
| 68 | C22×C3×C4 | — | 4 | 1 | no | — |
| 69 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C22×Dic3 | — | 4 | 1 | no | — |
| 71 | C22×Dic3 | — | 4 | 1 | no | — |
| 72 | C22×Dic3 | D8 | 1 | 1 | yes | characteristic |
| 73 | C22×Dic3 | D8 | 1 | 1 | yes | characteristic |
| 74 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 76 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 78 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 79 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 80 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 81 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 82 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 83 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 84 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 85 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 86 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 87 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 88 | C4⋊Dic3 | — | 8 | 2 | no | — |
| 89 | C4×Dic3 | — | 2 | 1 | no | — |
| 90 | C4×Dic3 | — | 2 | 1 | no | — |
| 91 | C4×Dic3 | — | 8 | 2 | no | — |
| 92 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 93 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 94 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 95 | C3×C16 | — | 2 | 2 | no | — |
| 96 | C3⋊C16 | — | 4 | 1 | no | — |
| 97 | C22×C8 | — | 3 | 1 | no | — |
| 98 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 99 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 100 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 101 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 102 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 103 | C2×C4⋊C4 | — | 12 | 1 | no | — |
| 104 | C2×C42 | — | 3 | 1 | no | — |
| 105 | C2×C42 | — | 12 | 1 | no | — |
| 106 | C2×C16 | — | 2 | 1 | no | — |
| 107 | C2×C16 | — | 2 | 1 | no | — |
| 108 | C2×C16 | — | 6 | 1 | no | — |
| 109 | C2×C16 | — | 6 | 1 | no | — |
| 110 | C2×C16 | — | 6 | 1 | no | — |
| 111 | C2×C16 | C2×Dic3 | 1 | 2 | yes | — |
| 112 | #107 | — | 2 | 1 | no | — |
| 113 | #107 | — | 2 | 1 | no | — |
| 114 | #107 | — | 3 | 2 | no | — |
| 115 | #107 | — | 6 | 1 | no | — |
| 116 | #107 | — | 6 | 1 | no | — |
| 117 | #107 | — | 6 | 1 | no | — |
| 118 | #107 | C2×Dic3 | 1 | 2 | yes | — |
| 119 | #107 | C2×Dic3 | 1 | 2 | yes | — |
| 120 | C4×C8 | — | 6 | 1 | no | — |
| 121 | C4×C8 | — | 6 | 1 | no | — |
| 122 | #95 | — | 12 | 1 | no | — |
| 123 | #95 | — | 12 | 1 | no | — |
| 124 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 125 | C2×C3×C4 | — | 4 | 2 | no | — |
| 126 | C2×C3×C4 | — | 4 | 2 | no | — |
| 127 | C2×C3×C4 | — | 8 | 1 | no | — |
| 128 | C2×C3×C4 | — | 8 | 1 | no | — |
| 129 | C2×C3×C4 | — | 8 | 1 | no | — |
| 130 | C2×C3×C4 | — | 8 | 1 | no | — |
| 131 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 132 | C2×C3×C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 133 | C2×C3×C4 | #123 | 1 | 1 | yes | characteristic |
| 134 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 135 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 137 | C2×Dic3 | — | 2 | 1 | no | — |
| 138 | C2×Dic3 | — | 2 | 1 | no | — |
| 139 | C2×Dic3 | — | 2 | 1 | no | — |
| 140 | C2×Dic3 | — | 2 | 1 | no | — |
| 141 | C2×Dic3 | — | 2 | 1 | no | — |
| 142 | C2×Dic3 | — | 4 | 2 | no | — |
| 143 | C2×Dic3 | — | 8 | 1 | no | — |
| 144 | C2×Dic3 | — | 8 | 1 | no | — |
| 145 | C2×Dic3 | — | 8 | 1 | no | — |
| 146 | C2×Dic3 | — | 8 | 1 | no | — |
| 147 | C2×Dic3 | — | 8 | 1 | no | — |
| 148 | C2×Dic3 | D16 | 1 | 2 | yes | — |
| 149 | C3×C8 | #117 | 1 | 1 | yes | characteristic |
| 150 | C3×C8 | C4×Q8 | 1 | 1 | yes | characteristic |
| 151 | C3×C8 | #125 | 1 | 1 | yes | characteristic |
| 152 | C3×C8 | #126 | 1 | 1 | yes | characteristic |
| 153 | C3⋊C8 | — | 4 | 1 | no | — |
| 154 | C22×C4 | — | 3 | 1 | no | — |
| 155 | C22×C4 | — | 3 | 1 | no | — |
| 156 | C22×C4 | — | 4 | 1 | no | — |
| 157 | C22×C4 | — | 4 | 1 | no | — |
| 158 | C22×C4 | — | 12 | 1 | no | — |
| 159 | C22×C4 | — | 12 | 1 | no | — |
| 160 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 161 | C2×C8 | — | 6 | 1 | no | — |
| 162 | C2×C8 | — | 6 | 1 | no | — |
| 163 | C2×C8 | — | 6 | 1 | no | — |
| 164 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 165 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 166 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 167 | C2×C8 | D3×Q8 | 1 | 1 | yes | characteristic |
| 168 | C2×C8 | #239 | 1 | 1 | yes | characteristic |
| 169 | C2×C8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 170 | C4⋊C4 | — | 2 | 2 | no | — |
| 171 | C4⋊C4 | — | 2 | 2 | no | — |
| 172 | C4⋊C4 | — | 4 | 1 | no | — |
| 173 | C4⋊C4 | — | 4 | 1 | no | — |
| 174 | C4⋊C4 | — | 6 | 2 | no | — |
| 175 | C4⋊C4 | — | 12 | 1 | no | — |
| 176 | C4⋊C4 | — | 12 | 1 | no | — |
| 177 | C4⋊C4 | — | 12 | 1 | no | — |
| 178 | C4⋊C4 | — | 24 | 2 | no | — |
| 179 | C42 | — | 6 | 1 | no | — |
| 180 | C42 | — | 6 | 1 | no | — |
| 181 | C42 | — | 24 | 2 | no | — |
| 182 | C16 | — | 2 | 2 | no | — |
| 183 | C16 | — | 12 | 1 | no | — |
| 184 | C22×C3 | C4×D8 | 1 | 1 | yes | characteristic |
| 185 | C22×C3 | #474 | 1 | 1 | yes | characteristic |
| 186 | C22×C3 | #480 | 1 | 1 | yes | characteristic |
| 187 | C22×C3 | C2×D16 | 1 | 1 | yes | characteristic |
| 188 | C22×C3 | #508 | 1 | 1 | yes | characteristic |
| 189 | C22×C3 | #510 | 1 | 1 | yes | characteristic |
| 190 | C22×C3 | #510 | 1 | 1 | yes | characteristic |
| 191 | C3×C4 | — | 8 | 2 | no | — |
| 192 | C3×C4 | — | 8 | 2 | no | — |
| 193 | C3×C4 | #387 | 1 | 1 | yes | characteristic |
| 194 | C3×C4 | #441 | 1 | 1 | yes | characteristic |
| 195 | C3×C4 | #479 | 1 | 1 | yes | characteristic |
| 196 | C3×C4 | #485 | 1 | 1 | yes | characteristic |
| 197 | Dic3 | — | 2 | 2 | no | — |
| 198 | Dic3 | — | 4 | 1 | no | — |
| 199 | Dic3 | — | 8 | 2 | no | — |
| 200 | Dic3 | — | 16 | 1 | no | — |
| 201 | C23 | D3×D8 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | — | 3 | 2 | no | — |
| 203 | C2×C4 | — | 4 | 2 | no | — |
| 204 | C2×C4 | — | 4 | 2 | no | — |
| 205 | C2×C4 | — | 6 | 1 | no | — |
| 206 | C2×C4 | — | 6 | 1 | no | — |
| 207 | C2×C4 | — | 6 | 1 | no | — |
| 208 | C2×C4 | — | 6 | 1 | no | — |
| 209 | C2×C4 | — | 6 | 1 | no | — |
| 210 | C2×C4 | — | 8 | 1 | no | — |
| 211 | C2×C4 | — | 8 | 1 | no | — |
| 212 | C2×C4 | — | 8 | 1 | no | — |
| 213 | C2×C4 | — | 8 | 1 | no | — |
| 214 | C2×C4 | — | 12 | 2 | no | — |
| 215 | C2×C4 | — | 24 | 1 | no | — |
| 216 | C2×C4 | — | 24 | 1 | no | — |
| 217 | C2×C4 | — | 24 | 1 | no | — |
| 218 | C2×C4 | — | 24 | 1 | no | — |
| 219 | C2×C4 | — | 24 | 1 | no | — |
| 220 | C2×C4 | #894 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #896 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #912 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #918 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #918 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | D4×Dic3 | 1 | 1 | yes | characteristic |
| 226 | C8 | — | 12 | 1 | no | — |
| 227 | C8 | #889 | 1 | 1 | yes | characteristic |
| 228 | C8 | #898 | 1 | 1 | yes | characteristic |
| 229 | C8 | #926 | 1 | 1 | yes | characteristic |
| 230 | C8 | Q8×Dic3 | 1 | 1 | yes | characteristic |
| 231 | C2×C3 | #1918 | 1 | 1 | yes | characteristic |
| 232 | C2×C3 | C4×D16 | 1 | 1 | yes | characteristic |
| 233 | C2×C3 | #2176 | 1 | 1 | yes | characteristic |
| 234 | C2×C3 | #2224 | 1 | 1 | yes | characteristic |
| 235 | C2×C3 | #2229 | 1 | 1 | yes | characteristic |
| 236 | C2×C3 | #2232 | 1 | 1 | yes | characteristic |
| 237 | C2×C3 | #2237 | 1 | 1 | yes | characteristic |
| 238 | C22 | #4882 | 1 | 1 | yes | characteristic |
| 239 | C22 | #4890 | 1 | 1 | yes | characteristic |
| 240 | C22 | D3×D16 | 1 | 1 | yes | characteristic |
| 241 | C22 | #4912 | 1 | 1 | yes | characteristic |
| 242 | C22 | #4918 | 1 | 1 | yes | characteristic |
| 243 | C22 | #4918 | 1 | 1 | yes | characteristic |
| 244 | C22 | Dic3×D8 | 1 | 1 | yes | characteristic |
| 245 | C4 | — | 6 | 2 | no | — |
| 246 | C4 | — | 8 | 2 | no | — |
| 247 | C4 | — | 8 | 2 | no | — |
| 248 | C4 | — | 12 | 1 | no | — |
| 249 | C4 | — | 24 | 2 | no | — |
| 250 | C4 | — | 48 | 1 | no | — |
| 251 | C4 | #4877 | 1 | 1 | yes | characteristic |
| 252 | C4 | #4888 | 1 | 1 | yes | characteristic |
| 253 | C4 | #4980 | 1 | 1 | yes | characteristic |
| 254 | C4 | #5184 | 1 | 1 | yes | characteristic |
| 255 | C3 | #13109 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 256 | C2 | #69330 | 1 | 1 | yes | characteristic |
| 257 | C2 | #69332 | 1 | 1 | yes | characteristic |
| 258 | C2 | #69333 | 1 | 1 | yes | characteristic |
| 259 | C2 | #69336 | 1 | 1 | yes | characteristic |
| 260 | C2 | #70609 | 1 | 1 | yes | characteristic |
| 261 | C2 | Dic3×D16 | 1 | 1 | yes | characteristic |
| 262 | C2 | #71496 | 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