Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
922 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 922 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #81799 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #81782 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #81557 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C3×#1709 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #71298 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71276 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #71180 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #68060 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | D3×C82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #19603 | — | 3 | 1 | no | maximal, Sylow |
| 11 | C2×#919 | — | 2 | 2 | no | — |
| 12 | C2×#912 | — | 2 | 2 | no | — |
| 13 | #5725 | C22 | 1 | 2 | yes | — |
| 14 | #5716 | C22 | 1 | 2 | yes | — |
| 15 | C3×#494 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#493 | C22 | 1 | 1 | yes | characteristic |
| 17 | C3×#486 | C22 | 1 | 4 | yes | — |
| 18 | #5191 | — | 2 | 2 | no | — |
| 19 | #5155 | — | 2 | 2 | no | — |
| 20 | #5085 | C22 | 1 | 2 | yes | — |
| 21 | #5078 | C22 | 1 | 4 | yes | — |
| 22 | #5076 | C22 | 1 | 2 | yes | — |
| 23 | #5073 | C22 | 1 | 4 | yes | — |
| 24 | #4775 | — | 2 | 8 | no | — |
| 25 | #4756 | — | 2 | 8 | no | — |
| 26 | #4699 | C22 | 1 | 1 | yes | characteristic |
| 27 | #4695 | C22 | 1 | 1 | yes | characteristic |
| 28 | #4694 | C22 | 1 | 4 | yes | — |
| 29 | C4×D3×C8 | C22 | 1 | 1 | yes | characteristic |
| 30 | C4×D3×C8 | C22 | 1 | 2 | yes | — |
| 31 | C3×C82 | C22 | 1 | 1 | yes | characteristic |
| 32 | C8×C3⋊C8 | C22 | 1 | 2 | yes | — |
| 33 | C8×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 34 | #3169 | — | 3 | 4 | no | — |
| 35 | #3147 | — | 3 | 1 | no | — |
| 36 | #3147 | — | 3 | 1 | no | — |
| 37 | #1713 | — | 3 | 1 | no | — |
| 38 | #1712 | — | 3 | 1 | no | — |
| 39 | #1709 | — | 3 | 1 | no | — |
| 40 | #1709 | D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | #1706 | — | 3 | 4 | no | — |
| 42 | C2×C82 | — | 3 | 1 | no | — |
| 43 | C2×#239 | — | 2 | 2 | no | — |
| 44 | C2×#237 | — | 2 | 2 | no | — |
| 45 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 46 | C2×C3×D8 | — | 2 | 2 | no | — |
| 47 | C3×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 48 | C3×#127 | C23 | 1 | 2 | yes | — |
| 49 | #947 | — | 2 | 2 | no | — |
| 50 | C2×#215 | — | 2 | 4 | no | — |
| 51 | #938 | — | 2 | 4 | no | — |
| 52 | #936 | — | 2 | 2 | no | — |
| 53 | D4×Dic3 | — | 2 | 4 | no | — |
| 54 | C2×#214 | — | 2 | 4 | no | — |
| 55 | C2×#213 | — | 2 | 8 | no | — |
| 56 | #919 | — | 4 | 8 | no | — |
| 57 | #912 | — | 4 | 8 | no | — |
| 58 | C2×Dic12 | — | 2 | 2 | no | — |
| 59 | C2×D24 | — | 2 | 2 | no | — |
| 60 | C2×D3×C8 | D4 | 1 | 4 | yes | — |
| 61 | C2×D3×C8 | D4 | 1 | 2 | yes | — |
| 62 | #895 | — | 2 | 4 | no | — |
| 63 | #893 | — | 2 | 4 | no | — |
| 64 | #874 | C23 | 1 | 2 | yes | — |
| 65 | C42×D3 | C23 | 1 | 1 | yes | characteristic |
| 66 | #869 | C23 | 1 | 2 | yes | — |
| 67 | C3×#102 | — | 2 | 8 | no | — |
| 68 | C3×C4×C8 | C23 | 1 | 2 | yes | — |
| 69 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 70 | #821 | — | 2 | 8 | no | — |
| 71 | C8×Dic3 | D4 | 1 | 2 | yes | — |
| 72 | C8×Dic3 | D4 | 1 | 4 | yes | — |
| 73 | #807 | — | 2 | 8 | no | — |
| 74 | C4×C3⋊C8 | C23 | 1 | 2 | yes | — |
| 75 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×#135 | — | 6 | 2 | no | — |
| 77 | C2×#135 | — | 6 | 2 | no | — |
| 78 | #532 | — | 3 | 2 | no | — |
| 79 | #532 | — | 3 | 2 | no | — |
| 80 | #500 | — | 3 | 4 | no | — |
| 81 | #495 | — | 6 | 2 | no | — |
| 82 | #495 | — | 6 | 2 | no | — |
| 83 | #494 | — | 3 | 1 | no | — |
| 84 | #494 | C2×D3 | 1 | 1 | yes | characteristic |
| 85 | #493 | — | 3 | 4 | no | — |
| 86 | #493 | — | 3 | 1 | no | — |
| 87 | #493 | C2×D3 | 1 | 1 | yes | characteristic |
| 88 | #492 | — | 3 | 2 | no | — |
| 89 | #492 | — | 3 | 2 | no | — |
| 90 | #486 | — | 3 | 4 | no | — |
| 91 | #486 | C2×D3 | 1 | 4 | yes | — |
| 92 | #466 | — | 6 | 8 | no | — |
| 93 | C4×D8 | — | 6 | 8 | no | — |
| 94 | C2×C4×C8 | — | 3 | 1 | no | — |
| 95 | C2×C4×C8 | — | 3 | 2 | no | — |
| 96 | C82 | — | 3 | 2 | no | — |
| 97 | C82 | — | 3 | 1 | no | — |
| 98 | C82 | C2×D3 | 1 | 1 | yes | characteristic |
| 99 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 100 | C2×C3×D4 | — | 2 | 4 | no | — |
| 101 | C2×C3×D4 | — | 2 | 2 | no | — |
| 102 | C2×#67 | — | 4 | 2 | no | — |
| 103 | C22×Dic3 | — | 4 | 2 | no | — |
| 104 | #239 | — | 4 | 8 | no | — |
| 105 | #237 | — | 4 | 8 | no | — |
| 106 | C2×D12 | — | 2 | 4 | no | — |
| 107 | C2×D12 | — | 2 | 2 | no | — |
| 108 | C2×D12 | — | 4 | 2 | no | — |
| 109 | C2×C4×D3 | — | 4 | 2 | no | — |
| 110 | C2×C4×D3 | C2×D4 | 1 | 2 | yes | — |
| 111 | C2×C4×D3 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×Dic6 | — | 2 | 2 | no | — |
| 113 | C3×Q16 | — | 4 | 4 | no | — |
| 114 | C3×D8 | — | 4 | 4 | no | — |
| 115 | C2×C3×C8 | C2×D4 | 1 | 4 | yes | — |
| 116 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 117 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 118 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 119 | #217 | — | 4 | 4 | no | — |
| 120 | #215 | — | 4 | 8 | no | — |
| 121 | #214 | — | 4 | 8 | no | — |
| 122 | #213 | — | 4 | 16 | no | — |
| 123 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 124 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 125 | C4×Dic3 | C2×D4 | 1 | 2 | yes | — |
| 126 | C4×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 128 | C2×C3⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 129 | Dic12 | — | 4 | 4 | no | — |
| 130 | D24 | — | 4 | 4 | no | — |
| 131 | D3×C8 | — | 2 | 8 | no | — |
| 132 | D3×C8 | D8 | 1 | 8 | yes | complemented |
| 133 | C2×#41 | — | 6 | 2 | no | — |
| 134 | C2×#41 | — | 6 | 2 | no | — |
| 135 | #135 | — | 12 | 8 | no | — |
| 136 | #135 | — | 12 | 8 | no | — |
| 137 | C2×Q16 | — | 2 | 2 | no | — |
| 138 | C2×Q16 | — | 6 | 2 | no | — |
| 139 | C2×SD16 | — | 6 | 4 | no | — |
| 140 | C2×SD16 | — | 6 | 4 | no | — |
| 141 | C2×D8 | — | 2 | 2 | no | — |
| 142 | C2×D8 | — | 6 | 8 | no | — |
| 143 | C2×D8 | — | 6 | 2 | no | — |
| 144 | C22×C8 | — | 3 | 2 | no | — |
| 145 | C22×C8 | — | 3 | 4 | no | — |
| 146 | C4⋊Q8 | — | 3 | 2 | no | — |
| 147 | C4⋊Q8 | C22×D3 | 1 | 2 | yes | — |
| 148 | #127 | — | 3 | 2 | no | — |
| 149 | #127 | C22×D3 | 1 | 2 | yes | — |
| 150 | #124 | — | 6 | 2 | no | — |
| 151 | #124 | — | 6 | 2 | no | — |
| 152 | #121 | — | 6 | 4 | no | — |
| 153 | #121 | — | 6 | 4 | no | — |
| 154 | C4×D4 | — | 6 | 4 | no | — |
| 155 | C4×D4 | — | 6 | 4 | no | — |
| 156 | C2×C42 | — | 3 | 1 | no | — |
| 157 | #107 | — | 6 | 8 | no | — |
| 158 | #102 | — | 2 | 8 | no | — |
| 159 | #102 | — | 6 | 8 | no | — |
| 160 | C4×C8 | — | 3 | 4 | no | — |
| 161 | C4×C8 | — | 3 | 2 | no | — |
| 162 | C4×C8 | — | 3 | 1 | no | — |
| 163 | C4×C8 | — | 3 | 2 | no | — |
| 164 | C4×C8 | C22×D3 | 1 | 2 | yes | — |
| 165 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 166 | C23×C3 | — | 4 | 2 | no | — |
| 167 | C22×D3 | — | 4 | 2 | no | — |
| 168 | C22×D3 | #127 | 1 | 1 | yes | characteristic |
| 169 | C3×Q8 | — | 4 | 4 | no | — |
| 170 | C3×D4 | — | 4 | 4 | no | — |
| 171 | C3×D4 | — | 4 | 8 | no | — |
| 172 | C2×C3×C4 | — | 4 | 2 | no | — |
| 173 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 174 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 175 | #67 | — | 8 | 4 | no | — |
| 176 | C2×Dic3 | — | 4 | 2 | no | — |
| 177 | C2×Dic3 | — | 8 | 4 | no | — |
| 178 | C2×Dic3 | #127 | 1 | 2 | yes | — |
| 179 | C2×Dic3 | #127 | 1 | 1 | yes | characteristic |
| 180 | D12 | — | 4 | 4 | no | — |
| 181 | D12 | — | 4 | 8 | no | — |
| 182 | D12 | — | 8 | 4 | no | — |
| 183 | C4×D3 | — | 2 | 4 | no | — |
| 184 | C4×D3 | — | 8 | 4 | no | — |
| 185 | C4×D3 | C2×D8 | 1 | 4 | yes | — |
| 186 | Dic6 | — | 4 | 4 | no | — |
| 187 | C3×C8 | — | 2 | 4 | no | — |
| 188 | C3×C8 | C2×D8 | 1 | 4 | yes | — |
| 189 | C3⋊C8 | C2×D8 | 1 | 4 | yes | — |
| 190 | C3⋊C8 | #135 | 1 | 8 | yes | complemented |
| 191 | #41 | — | 12 | 8 | no | — |
| 192 | #41 | — | 12 | 8 | no | — |
| 193 | C2×Q8 | — | 2 | 2 | no | — |
| 194 | C2×Q8 | — | 6 | 2 | no | — |
| 195 | C2×D4 | — | 2 | 4 | no | — |
| 196 | C2×D4 | — | 2 | 2 | no | — |
| 197 | C2×D4 | — | 6 | 4 | no | — |
| 198 | C2×D4 | — | 6 | 2 | no | — |
| 199 | C2×D4 | — | 12 | 2 | no | — |
| 200 | C2×D4 | — | 12 | 2 | no | — |
| 201 | C22×C4 | — | 3 | 1 | no | — |
| 202 | C22×C4 | — | 3 | 2 | no | — |
| 203 | C22×C4 | — | 12 | 2 | no | — |
| 204 | C22×C4 | — | 12 | 2 | no | — |
| 205 | Q16 | — | 4 | 4 | no | — |
| 206 | Q16 | — | 12 | 4 | no | — |
| 207 | SD16 | — | 12 | 8 | no | — |
| 208 | SD16 | — | 12 | 8 | no | — |
| 209 | D8 | — | 4 | 4 | no | — |
| 210 | D8 | — | 12 | 16 | no | — |
| 211 | D8 | — | 12 | 4 | no | — |
| 212 | C2×C8 | — | 3 | 4 | no | — |
| 213 | C2×C8 | — | 3 | 2 | no | — |
| 214 | C2×C8 | — | 3 | 8 | no | — |
| 215 | C2×C8 | — | 6 | 8 | no | — |
| 216 | C2×C8 | D3×D4 | 1 | 4 | yes | — |
| 217 | C2×C8 | D3×D4 | 1 | 2 | yes | — |
| 218 | C4⋊C4 | — | 2 | 4 | no | — |
| 219 | C4⋊C4 | — | 6 | 4 | no | — |
| 220 | C22⋊C4 | — | 12 | 4 | no | — |
| 221 | C22⋊C4 | — | 12 | 4 | no | — |
| 222 | C42 | — | 3 | 2 | no | — |
| 223 | C42 | — | 3 | 1 | no | — |
| 224 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 225 | C22×C3 | — | 8 | 2 | no | — |
| 226 | C22×C3 | — | 8 | 4 | no | — |
| 227 | C22×C3 | C2×#127 | 1 | 1 | yes | characteristic |
| 228 | C2×D3 | — | 2 | 2 | no | — |
| 229 | C2×D3 | — | 8 | 2 | no | — |
| 230 | C2×D3 | — | 8 | 4 | no | — |
| 231 | C2×D3 | #493 | 1 | 2 | yes | — |
| 232 | C3×C4 | — | 8 | 2 | no | — |
| 233 | C3×C4 | C22×D8 | 1 | 2 | yes | — |
| 234 | C3×C4 | C2×#135 | 1 | 4 | yes | — |
| 235 | Dic3 | — | 2 | 2 | no | — |
| 236 | Dic3 | — | 8 | 2 | no | — |
| 237 | Dic3 | #493 | 1 | 2 | yes | — |
| 238 | C23 | — | 3 | 1 | no | — |
| 239 | C23 | — | 4 | 2 | no | — |
| 240 | C23 | — | 12 | 2 | no | — |
| 241 | Q8 | — | 4 | 4 | no | — |
| 242 | Q8 | — | 12 | 4 | no | — |
| 243 | D4 | — | 4 | 4 | no | — |
| 244 | D4 | — | 4 | 8 | no | — |
| 245 | D4 | — | 12 | 4 | no | — |
| 246 | D4 | — | 12 | 8 | no | — |
| 247 | D4 | — | 24 | 4 | no | — |
| 248 | D4 | — | 24 | 4 | no | — |
| 249 | C2×C4 | — | 3 | 1 | no | — |
| 250 | C2×C4 | — | 3 | 4 | no | — |
| 251 | C2×C4 | — | 3 | 2 | no | — |
| 252 | C2×C4 | — | 4 | 2 | no | — |
| 253 | C2×C4 | — | 6 | 4 | no | — |
| 254 | C2×C4 | — | 12 | 2 | no | — |
| 255 | C2×C4 | — | 24 | 4 | no | — |
| 256 | C2×C4 | — | 24 | 4 | no | — |
| 257 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 259 | C8 | — | 2 | 4 | no | — |
| 260 | C8 | — | 3 | 4 | no | — |
| 261 | C8 | — | 3 | 8 | no | — |
| 262 | C8 | D3×D8 | 1 | 4 | yes | — |
| 263 | C2×C3 | — | 16 | 2 | no | — |
| 264 | C2×C3 | C2×#493 | 1 | 1 | yes | characteristic |
| 265 | C2×C3 | #3147 | 1 | 2 | yes | — |
| 266 | D3 | — | 2 | 2 | no | — |
| 267 | D3 | — | 16 | 2 | no | — |
| 268 | C22 | — | 3 | 2 | no | — |
| 269 | C22 | — | 6 | 2 | no | — |
| 270 | C22 | — | 8 | 2 | no | — |
| 271 | C22 | — | 8 | 4 | no | — |
| 272 | C22 | — | 24 | 2 | no | — |
| 273 | C22 | — | 24 | 4 | no | — |
| 274 | C22 | D3×#127 | 1 | 1 | yes | characteristic |
| 275 | C4 | — | 3 | 2 | no | — |
| 276 | C4 | — | 6 | 2 | no | — |
| 277 | C4 | — | 8 | 2 | no | — |
| 278 | C4 | — | 24 | 2 | no | — |
| 279 | C4 | C2×D3×D8 | 1 | 2 | yes | — |
| 280 | C4 | D3×#135 | 1 | 4 | yes | — |
| 281 | C3 | #19603 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 282 | C2 | — | 6 | 2 | no | — |
| 283 | C2 | — | 16 | 2 | no | — |
| 284 | C2 | — | 48 | 2 | no | — |
| 285 | C2 | D3×#493 | 1 | 1 | yes | characteristic |
| 286 | C2 | #81820 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#127 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#1709 | C2 |
| socle | C22×C3 | C2×#127 |
| hypercenter | #1709 | D3 |
| hypocenter | C3 | #19603 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1709 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #81799 | C2 | semidirect |
| 2. #81782 | C2 | semidirect |
| 3. #81557 | C2 | semidirect |
| 4. C3×#1709 | C2 | semidirect |
| 5. #71298 | C2 | semidirect |
| 6. #71276 | C2 | semidirect |
| 7. #71180 | C2 | semidirect |
| 8. #68060 | C2 | semidirect |
| 9. D3×C82 | C2 | semidirect |
| 40. #1709 | D3 | semidirect |
| 132. D3×C8 | D8 | semidirect |
| 190. C3⋊C8 | #135 | semidirect |
| 281. C3 | #19603 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #81799 | 2 |
| #81782 | 2 |
| #81557 | 2 |
| C3×#1709 | 2 |
| #71298 | 2 |
| #71276 | 2 |
| #71180 | 2 |
| #68060 | 2 |
| D3×C82 | 2 |
| #19603 | 3 |