Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1998 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#3504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85827 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #83471 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #83337 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #83335 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #83012 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#5069 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #82570 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #82511 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #60627 | — | 3 | 1 | no | maximal, Sylow |
| 11 | C2×#1010 | C22 | 1 | 2 | yes | — |
| 12 | C3×#557 | — | 2 | 2 | no | — |
| 13 | C3×#554 | — | 2 | 6 | no | — |
| 14 | C3×#544 | C22 | 1 | 6 | yes | — |
| 15 | C3×#533 | C22 | 1 | 3 | yes | — |
| 16 | C2×C3×#125 | C22 | 1 | 3 | yes | — |
| 17 | C3×#520 | C22 | 1 | 2 | yes | — |
| 18 | C3×C4×#41 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#948 | C22 | 1 | 2 | yes | — |
| 20 | Q8×D12 | — | 2 | 2 | no | — |
| 21 | #5555 | — | 2 | 6 | no | — |
| 22 | #5526 | C22 | 1 | 6 | yes | — |
| 23 | #5477 | C22 | 1 | 3 | yes | — |
| 24 | C2×#869 | C22 | 1 | 3 | yes | — |
| 25 | #5093 | C22 | 1 | 12 | yes | — |
| 26 | #5089 | C22 | 1 | 12 | yes | — |
| 27 | #5069 | C22 | 1 | 12 | yes | — |
| 28 | #5069 | C22 | 1 | 3 | yes | — |
| 29 | #5069 | C22 | 1 | 6 | yes | — |
| 30 | #5069 | C22 | 1 | 3 | yes | — |
| 31 | #5068 | C22 | 1 | 12 | yes | — |
| 32 | #5067 | C22 | 1 | 3 | yes | — |
| 33 | #5067 | C22 | 1 | 1 | yes | characteristic |
| 34 | #5004 | C22 | 1 | 12 | yes | — |
| 35 | #5002 | C22 | 1 | 12 | yes | — |
| 36 | #4997 | C22 | 1 | 6 | yes | — |
| 37 | C2×#810 | C22 | 1 | 12 | yes | — |
| 38 | #4966 | C22 | 1 | 12 | yes | — |
| 39 | C4×#208 | C22 | 1 | 3 | yes | — |
| 40 | C2×#803 | C22 | 1 | 3 | yes | — |
| 41 | #3504 | D3 | 1 | 1 | yes | characteristic, complemented |
| 42 | #3499 | — | 3 | 1 | no | — |
| 43 | #3142 | — | 3 | 3 | no | — |
| 44 | #3139 | — | 3 | 3 | no | — |
| 45 | #3137 | — | 3 | 12 | no | — |
| 46 | #3135 | — | 3 | 3 | no | — |
| 47 | C2×#491 | — | 3 | 3 | no | — |
| 48 | #2898 | — | 3 | 4 | no | — |
| 49 | #2875 | — | 3 | 1 | no | — |
| 50 | C2×C3×#41 | C23 | 1 | 1 | yes | characteristic |
| 51 | #1010 | — | 2 | 16 | no | — |
| 52 | C2×D3×Q8 | — | 2 | 2 | no | — |
| 53 | C2×#237 | — | 2 | 6 | no | — |
| 54 | C2×#235 | C23 | 1 | 6 | yes | — |
| 55 | C22×Dic6 | C23 | 1 | 6 | yes | — |
| 56 | C3×C4⋊Q8 | — | 2 | 6 | no | — |
| 57 | C3×C4⋊Q8 | C23 | 1 | 6 | yes | — |
| 58 | C3×#125 | — | 2 | 6 | no | — |
| 59 | C3×#125 | — | 2 | 6 | no | — |
| 60 | C3×#125 | C23 | 1 | 3 | yes | — |
| 61 | C3×#125 | C23 | 1 | 1 | yes | characteristic |
| 62 | C3×#125 | C23 | 1 | 6 | yes | — |
| 63 | C3×C22⋊Q8 | — | 2 | 6 | no | — |
| 64 | C3×C22⋊Q8 | — | 2 | 12 | no | — |
| 65 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 66 | C3×C4×Q8 | — | 2 | 6 | no | — |
| 67 | C3×C4×Q8 | D4 | 1 | 2 | yes | — |
| 68 | C3×C4×D4 | — | 2 | 6 | no | — |
| 69 | C3×C4×D4 | D4 | 1 | 6 | yes | — |
| 70 | C3×#117 | C23 | 1 | 6 | yes | — |
| 71 | C3×#117 | C23 | 1 | 3 | yes | — |
| 72 | C2×C3×C4⋊C4 | — | 2 | 6 | no | — |
| 73 | C2×C3×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 74 | C2×C3×C42 | C23 | 1 | 3 | yes | — |
| 75 | #948 | — | 2 | 8 | no | — |
| 76 | #930 | — | 2 | 6 | no | — |
| 77 | C2×C4⋊Dic3 | — | 2 | 6 | no | — |
| 78 | C2×#208 | C23 | 1 | 6 | yes | — |
| 79 | C22×C3⋊C8 | C23 | 1 | 6 | yes | — |
| 80 | #897 | — | 2 | 12 | no | — |
| 81 | #886 | — | 2 | 12 | no | — |
| 82 | #875 | C23 | 1 | 6 | yes | — |
| 83 | #874 | C23 | 1 | 1 | yes | characteristic |
| 84 | C4×D12 | — | 2 | 6 | no | — |
| 85 | #869 | — | 2 | 6 | no | — |
| 86 | #869 | C23 | 1 | 3 | yes | — |
| 87 | #869 | C23 | 1 | 6 | yes | — |
| 88 | #810 | C23 | 1 | 24 | yes | — |
| 89 | #810 | C23 | 1 | 12 | yes | — |
| 90 | #810 | C23 | 1 | 12 | yes | — |
| 91 | #809 | C23 | 1 | 12 | yes | — |
| 92 | #809 | C23 | 1 | 4 | yes | — |
| 93 | #803 | C23 | 1 | 1 | yes | characteristic |
| 94 | #803 | C23 | 1 | 3 | yes | — |
| 95 | #803 | C23 | 1 | 6 | yes | — |
| 96 | C4×C3⋊C8 | C23 | 1 | 6 | yes | — |
| 97 | C2×2−1+4 | — | 3 | 2 | no | — |
| 98 | C2×#131 | — | 3 | 2 | no | — |
| 99 | #557 | — | 2 | 2 | no | — |
| 100 | #554 | — | 2 | 6 | no | — |
| 101 | D4×Q8 | — | 6 | 2 | no | — |
| 102 | #547 | — | 6 | 6 | no | — |
| 103 | #544 | C2×D3 | 1 | 6 | yes | — |
| 104 | #536 | — | 3 | 6 | no | — |
| 105 | #533 | C2×D3 | 1 | 3 | yes | — |
| 106 | #532 | — | 3 | 3 | no | — |
| 107 | C2×C4⋊Q8 | — | 3 | 3 | no | — |
| 108 | C2×#125 | C2×D3 | 1 | 3 | yes | — |
| 109 | #520 | C2×D3 | 1 | 2 | yes | — |
| 110 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 111 | #491 | — | 3 | 3 | no | — |
| 112 | #491 | — | 3 | 12 | no | — |
| 113 | #491 | — | 3 | 6 | no | — |
| 114 | #491 | — | 3 | 3 | no | — |
| 115 | #490 | — | 3 | 1 | no | — |
| 116 | #490 | — | 3 | 3 | no | — |
| 117 | #489 | — | 3 | 12 | no | — |
| 118 | #488 | — | 3 | 12 | no | — |
| 119 | #487 | — | 3 | 12 | no | — |
| 120 | #419 | — | 3 | 12 | no | — |
| 121 | #417 | — | 3 | 12 | no | — |
| 122 | #416 | — | 3 | 12 | no | — |
| 123 | C2×#103 | — | 3 | 12 | no | — |
| 124 | #405 | — | 3 | 6 | no | — |
| 125 | C4×M16 | — | 3 | 3 | no | — |
| 126 | C2×#97 | — | 3 | 3 | no | — |
| 127 | C3×#41 | #41 | 1 | 8 | yes | — |
| 128 | C2×C3×Q8 | — | 2 | 6 | no | — |
| 129 | C2×C3×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C2×C3×D4 | C2×D4 | 1 | 3 | yes | — |
| 131 | C22×C3×C4 | — | 2 | 6 | no | — |
| 132 | C22×C3×C4 | C2×D4 | 1 | 6 | yes | — |
| 133 | C22×C3×C4 | C24 | 1 | 3 | yes | — |
| 134 | C2×#67 | — | 2 | 6 | no | — |
| 135 | C22×Dic3 | — | 2 | 6 | no | — |
| 136 | D3×Q8 | — | 4 | 8 | no | — |
| 137 | #237 | — | 4 | 24 | no | — |
| 138 | #235 | — | 2 | 24 | no | — |
| 139 | C2×D12 | — | 2 | 2 | no | — |
| 140 | C2×D12 | C24 | 1 | 2 | yes | — |
| 141 | C2×C4×D3 | — | 2 | 6 | no | — |
| 142 | C2×Dic6 | — | 2 | 24 | no | — |
| 143 | C2×Dic6 | — | 2 | 6 | no | — |
| 144 | C2×Dic6 | C24 | 1 | 6 | yes | — |
| 145 | C3×C4⋊C4 | — | 2 | 6 | no | — |
| 146 | C3×C4⋊C4 | — | 2 | 12 | no | — |
| 147 | C3×C4⋊C4 | — | 2 | 6 | no | — |
| 148 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 149 | C3×C4⋊C4 | — | 2 | 6 | no | — |
| 150 | C3×C4⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 151 | C3×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 152 | C3×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 153 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 154 | C3×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 155 | C3×C42 | — | 2 | 6 | no | — |
| 156 | C3×C42 | C2×D4 | 1 | 6 | yes | — |
| 157 | C3×C42 | C24 | 1 | 3 | yes | — |
| 158 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 159 | (C2×D3)⋊C4 | — | 2 | 12 | no | — |
| 160 | C4⋊Dic3 | — | 2 | 6 | no | — |
| 161 | C4⋊Dic3 | — | 2 | 12 | no | — |
| 162 | #208 | — | 2 | 12 | no | — |
| 163 | C2×C3⋊C8 | — | 2 | 12 | no | — |
| 164 | C2×C3⋊C8 | C24 | 1 | 6 | yes | — |
| 165 | C2×C3⋊C8 | C24 | 1 | 2 | yes | — |
| 166 | 2−1+4 | — | 6 | 16 | no | — |
| 167 | C2×#41 | — | 3 | 6 | no | — |
| 168 | C2×#41 | — | 6 | 6 | no | — |
| 169 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 170 | C22×Q8 | — | 3 | 6 | no | — |
| 171 | C22×Q8 | — | 6 | 2 | no | — |
| 172 | #131 | — | 6 | 8 | no | — |
| 173 | C2×M16 | — | 3 | 6 | no | — |
| 174 | C22×C8 | — | 3 | 6 | no | — |
| 175 | C4⋊Q8 | — | 2 | 6 | no | — |
| 176 | C4⋊Q8 | — | 3 | 3 | no | — |
| 177 | C4⋊Q8 | — | 3 | 6 | no | — |
| 178 | C4⋊Q8 | — | 6 | 6 | no | — |
| 179 | C4⋊Q8 | C22×D3 | 1 | 6 | yes | — |
| 180 | #127 | — | 3 | 1 | no | — |
| 181 | #125 | — | 2 | 6 | no | — |
| 182 | #125 | — | 2 | 6 | no | — |
| 183 | #125 | C22×D3 | 1 | 3 | yes | — |
| 184 | #125 | C22×D3 | 1 | 6 | yes | — |
| 185 | #125 | C22×D3 | 1 | 1 | yes | characteristic |
| 186 | #124 | — | 3 | 6 | no | — |
| 187 | #123 | — | 6 | 12 | no | — |
| 188 | C22⋊Q8 | — | 2 | 6 | no | — |
| 189 | C22⋊Q8 | — | 2 | 12 | no | — |
| 190 | C22⋊Q8 | — | 6 | 12 | no | — |
| 191 | #121 | — | 6 | 6 | no | — |
| 192 | C4×Q8 | — | 2 | 2 | no | — |
| 193 | C4×Q8 | — | 2 | 6 | no | — |
| 194 | C4×Q8 | #67 | 1 | 2 | yes | — |
| 195 | C4×D4 | — | 2 | 6 | no | — |
| 196 | C4×D4 | — | 6 | 6 | no | — |
| 197 | C4×D4 | #67 | 1 | 6 | yes | — |
| 198 | #117 | C22×D3 | 1 | 6 | yes | — |
| 199 | #117 | C22×D3 | 1 | 3 | yes | — |
| 200 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 201 | C2×C4⋊C4 | — | 6 | 6 | no | — |
| 202 | C2×C4⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 203 | C2×C42 | C22×D3 | 1 | 3 | yes | — |
| 204 | #103 | — | 3 | 12 | no | — |
| 205 | #103 | — | 3 | 24 | no | — |
| 206 | #103 | — | 3 | 12 | no | — |
| 207 | #102 | — | 3 | 4 | no | — |
| 208 | #102 | — | 3 | 12 | no | — |
| 209 | #97 | — | 3 | 1 | no | — |
| 210 | #97 | — | 3 | 3 | no | — |
| 211 | #97 | — | 3 | 6 | no | — |
| 212 | C4×C8 | — | 3 | 6 | no | — |
| 213 | C23×C3 | C22×D4 | 1 | 3 | yes | — |
| 214 | C22×D3 | — | 2 | 2 | no | — |
| 215 | C3×Q8 | — | 8 | 6 | no | — |
| 216 | C3×Q8 | #124 | 1 | 4 | yes | — |
| 217 | C3×D4 | #124 | 1 | 12 | yes | — |
| 218 | C2×C3×C4 | — | 2 | 2 | no | — |
| 219 | C2×C3×C4 | — | 2 | 6 | no | — |
| 220 | C2×C3×C4 | — | 4 | 6 | no | — |
| 221 | C2×C3×C4 | — | 8 | 6 | no | — |
| 222 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 223 | C2×C3×C4 | C22×D4 | 1 | 6 | yes | — |
| 224 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 225 | C2×C3×C4 | C2×#41 | 1 | 12 | yes | — |
| 226 | C2×C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 227 | #67 | — | 4 | 12 | no | — |
| 228 | C2×Dic3 | — | 2 | 6 | no | — |
| 229 | C2×Dic3 | — | 4 | 12 | no | — |
| 230 | D12 | — | 2 | 4 | no | — |
| 231 | D12 | — | 4 | 4 | no | — |
| 232 | C4×D3 | — | 4 | 12 | no | — |
| 233 | Dic6 | — | 2 | 12 | no | — |
| 234 | Dic6 | — | 4 | 12 | no | — |
| 235 | C3⋊C8 | — | 2 | 2 | no | — |
| 236 | C3⋊C8 | — | 2 | 6 | no | — |
| 237 | #41 | — | 6 | 24 | no | — |
| 238 | #41 | — | 12 | 24 | no | — |
| 239 | #41 | #239 | 1 | 8 | yes | — |
| 240 | C2×Q8 | — | 2 | 6 | no | — |
| 241 | C2×Q8 | — | 3 | 6 | no | — |
| 242 | C2×Q8 | — | 6 | 6 | no | — |
| 243 | C2×Q8 | — | 6 | 24 | no | — |
| 244 | C2×Q8 | — | 12 | 8 | no | — |
| 245 | C2×Q8 | C2×#67 | 1 | 1 | yes | characteristic |
| 246 | C2×D4 | — | 3 | 2 | no | — |
| 247 | C2×D4 | — | 6 | 6 | no | — |
| 248 | C2×D4 | — | 6 | 2 | no | — |
| 249 | C2×D4 | C2×#67 | 1 | 3 | yes | — |
| 250 | C22×C4 | — | 2 | 6 | no | — |
| 251 | C22×C4 | — | 6 | 6 | no | — |
| 252 | C22×C4 | — | 6 | 6 | no | — |
| 253 | C22×C4 | C2×#67 | 1 | 6 | yes | — |
| 254 | C22×C4 | C23×D3 | 1 | 3 | yes | — |
| 255 | M16 | — | 6 | 12 | no | — |
| 256 | C2×C8 | — | 3 | 6 | no | — |
| 257 | C2×C8 | — | 3 | 2 | no | — |
| 258 | C2×C8 | — | 6 | 12 | no | — |
| 259 | C4⋊C4 | — | 2 | 6 | no | — |
| 260 | C4⋊C4 | — | 2 | 12 | no | — |
| 261 | C4⋊C4 | — | 2 | 6 | no | — |
| 262 | C4⋊C4 | — | 2 | 2 | no | — |
| 263 | C4⋊C4 | — | 2 | 6 | no | — |
| 264 | C4⋊C4 | — | 6 | 6 | no | — |
| 265 | C4⋊C4 | — | 6 | 12 | no | — |
| 266 | C4⋊C4 | C2×#67 | 1 | 6 | yes | — |
| 267 | C4⋊C4 | C23×D3 | 1 | 6 | yes | — |
| 268 | C4⋊C4 | C23×D3 | 1 | 2 | yes | — |
| 269 | C22⋊C4 | — | 2 | 6 | no | — |
| 270 | C22⋊C4 | — | 6 | 12 | no | — |
| 271 | C22⋊C4 | C2×#67 | 1 | 6 | yes | — |
| 272 | C42 | — | 2 | 6 | no | — |
| 273 | C42 | C2×#67 | 1 | 6 | yes | — |
| 274 | C42 | C23×D3 | 1 | 3 | yes | — |
| 275 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 276 | C22×C3 | — | 2 | 6 | no | — |
| 277 | C22×C3 | C2×#124 | 1 | 6 | yes | — |
| 278 | C22×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 279 | C2×D3 | — | 4 | 4 | no | — |
| 280 | C2×D3 | — | 4 | 2 | no | — |
| 281 | C3×C4 | — | 2 | 2 | no | — |
| 282 | C3×C4 | — | 2 | 6 | no | — |
| 283 | C3×C4 | — | 4 | 6 | no | — |
| 284 | C3×C4 | — | 4 | 2 | no | — |
| 285 | C3×C4 | C2×#124 | 1 | 6 | yes | — |
| 286 | C3×C4 | C22×#41 | 1 | 2 | yes | — |
| 287 | Dic3 | — | 4 | 6 | no | — |
| 288 | C23 | — | 6 | 2 | no | — |
| 289 | C23 | C22×#67 | 1 | 3 | yes | — |
| 290 | Q8 | — | 6 | 12 | no | — |
| 291 | Q8 | — | 8 | 6 | no | — |
| 292 | Q8 | — | 12 | 12 | no | — |
| 293 | Q8 | #947 | 1 | 4 | yes | — |
| 294 | D4 | — | 6 | 4 | no | — |
| 295 | D4 | — | 12 | 4 | no | — |
| 296 | D4 | — | 12 | 12 | no | — |
| 297 | D4 | #947 | 1 | 12 | yes | — |
| 298 | C2×C4 | — | 2 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 6 | no | — |
| 300 | C2×C4 | — | 4 | 6 | no | — |
| 301 | C2×C4 | — | 6 | 6 | no | — |
| 302 | C2×C4 | — | 8 | 6 | no | — |
| 303 | C2×C4 | — | 12 | 12 | no | — |
| 304 | C2×C4 | — | 12 | 12 | no | — |
| 305 | C2×C4 | C2×#239 | 1 | 12 | yes | — |
| 306 | C2×C4 | C22×#67 | 1 | 2 | yes | — |
| 307 | C2×C4 | C22×#67 | 1 | 6 | yes | — |
| 308 | C2×C4 | C22×#67 | 1 | 3 | yes | — |
| 309 | C2×C4 | C24×D3 | 1 | 1 | yes | characteristic |
| 310 | C8 | — | 6 | 2 | no | — |
| 311 | C8 | — | 6 | 6 | no | — |
| 312 | C2×C3 | — | 2 | 6 | no | — |
| 313 | C2×C3 | C22×#124 | 1 | 1 | yes | characteristic |
| 314 | C2×C3 | #3586 | 1 | 2 | yes | — |
| 315 | D3 | — | 8 | 2 | no | — |
| 316 | C22 | — | 2 | 6 | no | — |
| 317 | C22 | — | 12 | 4 | no | — |
| 318 | C22 | — | 12 | 2 | no | — |
| 319 | C22 | C2×#947 | 1 | 6 | yes | — |
| 320 | C22 | C23×#67 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 2 | 2 | no | — |
| 322 | C4 | — | 2 | 6 | no | — |
| 323 | C4 | — | 4 | 6 | no | — |
| 324 | C4 | — | 4 | 2 | no | — |
| 325 | C4 | — | 12 | 6 | no | — |
| 326 | C4 | C2×#947 | 1 | 6 | yes | — |
| 327 | C4 | C22×#239 | 1 | 2 | yes | — |
| 328 | C3 | #60627 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 329 | C2 | — | 2 | 6 | no | — |
| 330 | C2 | — | 24 | 2 | no | — |
| 331 | C2 | C22×#947 | 1 | 1 | yes | characteristic |
| 332 | C2 | #87299 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3504 | C2 | semidirect |
| 3. #83471 | C2 | semidirect |
| 4. #83337 | C2 | semidirect |
| 5. #83335 | C2 | semidirect |
| 6. #83012 | C2 | semidirect |
| 7. C2×#5069 | C2 | semidirect |
| 8. #82570 | C2 | semidirect |
| 9. #82511 | C2 | semidirect |
| 41. #3504 | D3 | semidirect |
| 328. C3 | #60627 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3504 | 2 |
| #85827 | 2 |
| #83471 | 2 |
| #83337 | 2 |
| #83335 | 2 |
| #83012 | 2 |
| C2×#5069 | 2 |
| #82570 | 2 |
| #82511 | 2 |
| #60627 | 3 |