Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2899 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 2899 conjugacy classes of subgroups — too many to draw. Only its 157 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 | C2×D4×S4 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | A4×#127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #85496 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #85289 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C42×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | D4×#127 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C23×S4 | C22 | 1 | 4 | yes | — |
| 8 | C2×D4×A4 | C22 | 1 | 6 | yes | complemented |
| 9 | C2×GL(2,ℤ/4) | C22 | 1 | 12 | yes | complemented |
| 10 | D4×S4 | — | 2 | 24 | no | — |
| 11 | C2×C4⋊S4 | C22 | 1 | 6 | yes | complemented |
| 12 | C2×C4×S4 | C22 | 1 | 3 | yes | — |
| 13 | D3×#127 | — | 4 | 1 | no | maximal |
| 14 | C42×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C4×SL(2,ℤ/4) | C22 | 1 | 3 | yes | complemented |
| 16 | C2×D42 | — | 3 | 6 | no | — |
| 17 | C22×#127 | — | 3 | 1 | no | — |
| 18 | C22×#127 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | #2867 | — | 3 | 1 | no | — |
| 20 | #2856 | — | 3 | 1 | no | — |
| 21 | #2856 | — | 3 | 1 | no | — |
| 22 | #2541 | — | 3 | 6 | no | — |
| 23 | #2433 | — | 3 | 6 | no | — |
| 24 | #2433 | — | 3 | 6 | no | — |
| 25 | C4×#127 | — | 3 | 1 | no | — |
| 26 | C42×D4 | — | 3 | 1 | no | — |
| 27 | C23×A4 | C23 | 1 | 4 | yes | — |
| 28 | C22×S4 | — | 2 | 24 | no | — |
| 29 | C22×S4 | C23 | 1 | 4 | yes | — |
| 30 | C22×S4 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×D3×D4 | — | 4 | 6 | no | — |
| 32 | D4×A4 | — | 2 | 12 | no | — |
| 33 | C2×C4×A4 | C23 | 1 | 3 | yes | — |
| 34 | GL(2,ℤ/4) | — | 2 | 24 | no | — |
| 35 | C2×SL(2,ℤ/4) | C23 | 1 | 3 | yes | — |
| 36 | C4⋊S4 | — | 2 | 12 | no | — |
| 37 | C4×S4 | D4 | 1 | 12 | yes | complemented |
| 38 | C3×#127 | — | 4 | 1 | no | — |
| 39 | #940 | — | 4 | 6 | no | — |
| 40 | #874 | — | 4 | 1 | no | — |
| 41 | C42×D3 | — | 4 | 1 | no | — |
| 42 | C23×D4 | — | 3 | 4 | no | — |
| 43 | C23×D4 | — | 3 | 6 | no | — |
| 44 | C23×D4 | C2×D3 | 1 | 6 | yes | complemented |
| 45 | D42 | — | 6 | 24 | no | — |
| 46 | D42 | — | 6 | 24 | no | — |
| 47 | C2×#127 | — | 3 | 1 | no | — |
| 48 | C2×#127 | — | 3 | 1 | no | — |
| 49 | C2×#127 | — | 3 | 6 | no | — |
| 50 | C2×#127 | — | 3 | 1 | no | — |
| 51 | C2×#127 | — | 3 | 6 | no | — |
| 52 | C2×#127 | — | 3 | 6 | no | — |
| 53 | C2×#127 | — | 3 | 6 | no | — |
| 54 | C2×#127 | — | 6 | 6 | no | — |
| 55 | C2×#127 | — | 6 | 1 | no | — |
| 56 | C2×#127 | — | 6 | 1 | no | — |
| 57 | C2×#121 | — | 3 | 6 | no | — |
| 58 | C2×#121 | — | 3 | 12 | no | — |
| 59 | C2×#121 | — | 3 | 6 | no | — |
| 60 | C2×#120 | — | 3 | 12 | no | — |
| 61 | C2×#120 | — | 3 | 12 | no | — |
| 62 | C2×C4×D4 | — | 3 | 3 | no | — |
| 63 | C2×C4×D4 | — | 3 | 6 | no | — |
| 64 | C22×C42 | — | 3 | 1 | no | — |
| 65 | C22×C42 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 66 | #392 | — | 3 | 24 | no | — |
| 67 | #390 | — | 3 | 6 | no | — |
| 68 | #390 | — | 3 | 6 | no | — |
| 69 | #390 | — | 3 | 6 | no | — |
| 70 | #384 | — | 3 | 1 | no | — |
| 71 | C4×C4⋊C4 | — | 3 | 3 | no | — |
| 72 | C4×C22⋊C4 | — | 3 | 3 | no | — |
| 73 | C4×C22⋊C4 | — | 3 | 3 | no | — |
| 74 | C43 | — | 3 | 1 | no | — |
| 75 | C23×D3 | — | 4 | 4 | no | — |
| 76 | C22×A4 | — | 2 | 12 | no | — |
| 77 | C22×A4 | C24 | 1 | 1 | yes | characteristic |
| 78 | C2×S4 | — | 2 | 12 | no | — |
| 79 | C2×S4 | — | 4 | 16 | no | — |
| 80 | C2×S4 | C2×D4 | 1 | 6 | yes | — |
| 81 | C2×C3×D4 | — | 4 | 6 | no | — |
| 82 | C2×#67 | — | 4 | 12 | no | — |
| 83 | D3×D4 | — | 8 | 24 | no | — |
| 84 | C2×D12 | — | 4 | 6 | no | — |
| 85 | C2×C4×D3 | — | 4 | 3 | no | — |
| 86 | C4×A4 | C2×D4 | 1 | 6 | yes | complemented |
| 87 | SL(2,ℤ/4) | C2×D4 | 1 | 6 | yes | complemented |
| 88 | C3×C42 | — | 4 | 1 | no | — |
| 89 | C4×Dic3 | — | 4 | 3 | no | — |
| 90 | C25 | — | 3 | 4 | no | — |
| 91 | C25 | C22×D3 | 1 | 4 | yes | — |
| 92 | C22×D4 | — | 2 | 12 | no | — |
| 93 | C22×D4 | — | 3 | 1 | no | — |
| 94 | C22×D4 | — | 3 | 6 | no | — |
| 95 | C22×D4 | — | 3 | 6 | no | — |
| 96 | C22×D4 | — | 3 | 12 | no | — |
| 97 | C22×D4 | — | 3 | 4 | no | — |
| 98 | C22×D4 | — | 3 | 4 | no | — |
| 99 | C22×D4 | — | 3 | 6 | no | — |
| 100 | C22×D4 | — | 3 | 4 | no | — |
| 101 | C22×D4 | — | 3 | 12 | no | — |
| 102 | C22×D4 | — | 3 | 12 | no | — |
| 103 | C22×D4 | — | 6 | 12 | no | — |
| 104 | C22×D4 | — | 6 | 24 | no | — |
| 105 | C22×D4 | — | 6 | 24 | no | — |
| 106 | C22×D4 | — | 6 | 12 | no | — |
| 107 | C22×D4 | — | 6 | 6 | no | — |
| 108 | C22×D4 | — | 6 | 24 | no | — |
| 109 | C22×D4 | — | 6 | 6 | no | — |
| 110 | C22×D4 | — | 12 | 12 | no | — |
| 111 | C23×C4 | — | 3 | 3 | no | — |
| 112 | C23×C4 | — | 3 | 4 | no | — |
| 113 | C23×C4 | C22×D3 | 1 | 3 | yes | — |
| 114 | #127 | — | 3 | 6 | no | — |
| 115 | #127 | — | 3 | 1 | no | — |
| 116 | #127 | — | 6 | 3 | no | — |
| 117 | #127 | — | 6 | 3 | no | — |
| 118 | #127 | — | 6 | 24 | no | — |
| 119 | #127 | — | 6 | 4 | no | — |
| 120 | #127 | — | 6 | 12 | no | — |
| 121 | #127 | — | 6 | 6 | no | — |
| 122 | #127 | — | 6 | 6 | no | — |
| 123 | #127 | — | 6 | 1 | no | — |
| 124 | #127 | — | 12 | 12 | no | — |
| 125 | #127 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 126 | #121 | — | 6 | 48 | no | — |
| 127 | #121 | — | 6 | 12 | no | — |
| 128 | #121 | — | 6 | 24 | no | — |
| 129 | #121 | — | 6 | 12 | no | — |
| 130 | #120 | — | 6 | 24 | no | — |
| 131 | #120 | — | 6 | 48 | no | — |
| 132 | #120 | — | 6 | 24 | no | — |
| 133 | C4×D4 | — | 3 | 12 | no | — |
| 134 | C4×D4 | — | 3 | 12 | no | — |
| 135 | C4×D4 | — | 6 | 24 | no | — |
| 136 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 137 | C2×C4⋊C4 | — | 3 | 6 | no | — |
| 138 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 139 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 140 | C2×C22⋊C4 | — | 3 | 12 | no | — |
| 141 | C2×C22⋊C4 | — | 3 | 12 | no | — |
| 142 | C2×C22⋊C4 | — | 3 | 12 | no | — |
| 143 | C2×C42 | — | 3 | 3 | no | — |
| 144 | C2×C42 | — | 3 | 1 | no | — |
| 145 | C2×C42 | — | 3 | 3 | no | — |
| 146 | C2×C42 | — | 3 | 3 | no | — |
| 147 | C2×C42 | — | 3 | 3 | no | — |
| 148 | C2×C42 | — | 6 | 3 | no | — |
| 149 | C2×C42 | — | 6 | 1 | no | — |
| 150 | #95 | — | 3 | 6 | no | — |
| 151 | C23×C3 | — | 4 | 4 | no | — |
| 152 | C22×D3 | — | 4 | 1 | no | — |
| 153 | C22×D3 | — | 4 | 4 | no | — |
| 154 | C22×D3 | — | 8 | 24 | no | — |
| 155 | C2×A4 | — | 4 | 4 | no | — |
| 156 | C2×A4 | C22×D4 | 1 | 3 | yes | — |
| 157 | S4 | — | 4 | 4 | no | — |
| 158 | S4 | #127 | 1 | 4 | yes | direct product factor |
| 159 | C3×D4 | — | 8 | 12 | no | — |
| 160 | C2×C3×C4 | — | 4 | 3 | no | — |
| 161 | #67 | — | 8 | 24 | no | — |
| 162 | C2×Dic3 | — | 4 | 3 | no | — |
| 163 | D12 | — | 8 | 12 | no | — |
| 164 | C4×D3 | — | 4 | 12 | no | — |
| 165 | C24 | — | 2 | 12 | no | — |
| 166 | C24 | — | 3 | 1 | no | — |
| 167 | C24 | — | 3 | 4 | no | — |
| 168 | C24 | — | 3 | 4 | no | — |
| 169 | C24 | — | 3 | 4 | no | — |
| 170 | C24 | — | 6 | 12 | no | — |
| 171 | C24 | — | 6 | 4 | no | — |
| 172 | C24 | — | 6 | 24 | no | — |
| 173 | C24 | — | 6 | 4 | no | — |
| 174 | C24 | — | 12 | 12 | no | — |
| 175 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 176 | C2×D4 | — | 3 | 6 | no | — |
| 177 | C2×D4 | — | 3 | 6 | no | — |
| 178 | C2×D4 | — | 3 | 12 | no | — |
| 179 | C2×D4 | — | 3 | 6 | no | — |
| 180 | C2×D4 | — | 6 | 12 | no | — |
| 181 | C2×D4 | — | 6 | 12 | no | — |
| 182 | C2×D4 | — | 6 | 24 | no | — |
| 183 | C2×D4 | — | 6 | 12 | no | — |
| 184 | C2×D4 | — | 6 | 12 | no | — |
| 185 | C2×D4 | — | 6 | 6 | no | — |
| 186 | C2×D4 | — | 6 | 24 | no | — |
| 187 | C2×D4 | — | 6 | 24 | no | — |
| 188 | C2×D4 | — | 6 | 12 | no | — |
| 189 | C2×D4 | — | 6 | 6 | no | — |
| 190 | C2×D4 | — | 6 | 12 | no | — |
| 191 | C2×D4 | — | 6 | 24 | no | — |
| 192 | C2×D4 | — | 6 | 12 | no | — |
| 193 | C2×D4 | — | 6 | 24 | no | — |
| 194 | C2×D4 | — | 6 | 12 | no | — |
| 195 | C2×D4 | — | 6 | 12 | no | — |
| 196 | C2×D4 | — | 6 | 12 | no | — |
| 197 | C2×D4 | — | 6 | 6 | no | — |
| 198 | C2×D4 | — | 12 | 24 | no | — |
| 199 | C2×D4 | — | 12 | 12 | no | — |
| 200 | C2×D4 | — | 12 | 48 | no | — |
| 201 | C2×D4 | — | 12 | 24 | no | — |
| 202 | C2×D4 | — | 12 | 12 | no | — |
| 203 | C2×D4 | — | 12 | 24 | no | — |
| 204 | C2×D4 | — | 12 | 16 | no | — |
| 205 | C2×D4 | — | 12 | 48 | no | — |
| 206 | C2×D4 | C2×S4 | 1 | 6 | yes | complemented |
| 207 | C22×C4 | — | 3 | 1 | no | — |
| 208 | C22×C4 | — | 3 | 3 | no | — |
| 209 | C22×C4 | — | 3 | 3 | no | — |
| 210 | C22×C4 | — | 3 | 6 | no | — |
| 211 | C22×C4 | — | 3 | 3 | no | — |
| 212 | C22×C4 | — | 3 | 3 | no | — |
| 213 | C22×C4 | — | 3 | 12 | no | — |
| 214 | C22×C4 | — | 3 | 4 | no | — |
| 215 | C22×C4 | — | 6 | 3 | no | — |
| 216 | C22×C4 | — | 6 | 6 | no | — |
| 217 | C22×C4 | — | 6 | 24 | no | — |
| 218 | C22×C4 | — | 6 | 3 | no | — |
| 219 | C22×C4 | D3×D4 | 1 | 6 | yes | complemented |
| 220 | C4⋊C4 | — | 3 | 12 | no | — |
| 221 | C4⋊C4 | — | 6 | 12 | no | — |
| 222 | C22⋊C4 | — | 3 | 12 | no | — |
| 223 | C22⋊C4 | — | 3 | 6 | no | — |
| 224 | C22⋊C4 | — | 3 | 6 | no | — |
| 225 | C22⋊C4 | — | 6 | 24 | no | — |
| 226 | C22⋊C4 | — | 6 | 24 | no | — |
| 227 | C22⋊C4 | — | 6 | 24 | no | — |
| 228 | C42 | — | 3 | 12 | no | — |
| 229 | C42 | — | 3 | 3 | no | — |
| 230 | C42 | — | 6 | 1 | no | — |
| 231 | C42 | — | 6 | 3 | no | — |
| 232 | C42 | — | 6 | 6 | no | — |
| 233 | C42 | — | 6 | 3 | no | — |
| 234 | C42 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 235 | C22×C3 | — | 4 | 1 | no | — |
| 236 | C22×C3 | — | 8 | 12 | no | — |
| 237 | C2×D3 | — | 4 | 6 | no | — |
| 238 | C2×D3 | — | 8 | 12 | no | — |
| 239 | C2×D3 | — | 16 | 16 | no | — |
| 240 | A4 | C2×#127 | 1 | 1 | yes | characteristic, complemented |
| 241 | C3×C4 | — | 4 | 6 | no | — |
| 242 | Dic3 | — | 4 | 6 | no | — |
| 243 | C23 | — | 3 | 1 | no | — |
| 244 | C23 | — | 3 | 3 | no | — |
| 245 | C23 | — | 3 | 6 | no | — |
| 246 | C23 | — | 3 | 4 | no | — |
| 247 | C23 | — | 4 | 4 | no | — |
| 248 | C23 | — | 6 | 12 | no | — |
| 249 | C23 | — | 6 | 12 | no | — |
| 250 | C23 | — | 6 | 3 | no | — |
| 251 | C23 | — | 6 | 12 | no | — |
| 252 | C23 | — | 6 | 12 | no | — |
| 253 | C23 | — | 6 | 12 | no | — |
| 254 | C23 | — | 6 | 12 | no | — |
| 255 | C23 | — | 6 | 1 | no | — |
| 256 | C23 | — | 6 | 4 | no | — |
| 257 | C23 | — | 12 | 24 | no | — |
| 258 | C23 | — | 12 | 12 | no | — |
| 259 | C23 | — | 12 | 16 | no | — |
| 260 | C23 | — | 12 | 24 | no | — |
| 261 | C23 | — | 24 | 4 | no | — |
| 262 | C23 | — | 24 | 24 | no | — |
| 263 | C23 | C2×D3×D4 | 1 | 3 | yes | — |
| 264 | C23 | C22×S4 | 1 | 4 | yes | — |
| 265 | D4 | — | 2 | 12 | no | — |
| 266 | D4 | — | 3 | 12 | no | — |
| 267 | D4 | — | 3 | 4 | no | — |
| 268 | D4 | — | 6 | 24 | no | — |
| 269 | D4 | — | 6 | 12 | no | — |
| 270 | D4 | — | 12 | 12 | no | — |
| 271 | D4 | — | 12 | 16 | no | — |
| 272 | D4 | — | 12 | 12 | no | — |
| 273 | D4 | — | 12 | 24 | no | — |
| 274 | D4 | — | 12 | 24 | no | — |
| 275 | D4 | — | 12 | 12 | no | — |
| 276 | D4 | — | 12 | 16 | no | — |
| 277 | D4 | — | 12 | 4 | no | — |
| 278 | D4 | — | 12 | 12 | no | — |
| 279 | D4 | — | 24 | 24 | no | — |
| 280 | C2×C4 | — | 3 | 6 | no | — |
| 281 | C2×C4 | — | 3 | 3 | no | — |
| 282 | C2×C4 | — | 3 | 6 | no | — |
| 283 | C2×C4 | — | 3 | 6 | no | — |
| 284 | C2×C4 | — | 3 | 6 | no | — |
| 285 | C2×C4 | — | 3 | 6 | no | — |
| 286 | C2×C4 | — | 3 | 6 | no | — |
| 287 | C2×C4 | — | 6 | 3 | no | — |
| 288 | C2×C4 | — | 6 | 12 | no | — |
| 289 | C2×C4 | — | 6 | 6 | no | — |
| 290 | C2×C4 | — | 6 | 3 | no | — |
| 291 | C2×C4 | — | 6 | 3 | no | — |
| 292 | C2×C4 | — | 6 | 12 | no | — |
| 293 | C2×C4 | — | 6 | 12 | no | — |
| 294 | C2×C4 | — | 12 | 16 | no | — |
| 295 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 296 | C2×C3 | — | 4 | 3 | no | — |
| 297 | C2×C3 | — | 16 | 4 | no | — |
| 298 | D3 | — | 4 | 4 | no | — |
| 299 | D3 | — | 16 | 4 | no | — |
| 300 | C22 | — | 2 | 12 | no | — |
| 301 | C22 | — | 3 | 3 | no | — |
| 302 | C22 | — | 3 | 3 | no | — |
| 303 | C22 | — | 3 | 3 | no | — |
| 304 | C22 | — | 3 | 4 | no | — |
| 305 | C22 | — | 6 | 1 | no | — |
| 306 | C22 | — | 6 | 6 | no | — |
| 307 | C22 | — | 6 | 6 | no | — |
| 308 | C22 | — | 6 | 12 | no | — |
| 309 | C22 | — | 12 | 4 | no | — |
| 310 | C22 | — | 12 | 12 | no | — |
| 311 | C22 | — | 12 | 4 | no | — |
| 312 | C22 | — | 12 | 12 | no | — |
| 313 | C22 | — | 12 | 4 | no | — |
| 314 | C22 | — | 12 | 12 | no | — |
| 315 | C22 | — | 12 | 12 | no | — |
| 316 | C22 | — | 24 | 16 | no | — |
| 317 | C22 | — | 24 | 16 | no | — |
| 318 | C22 | D3×#127 | 1 | 1 | yes | characteristic, complemented |
| 319 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 3 | 4 | no | — |
| 321 | C4 | — | 3 | 6 | no | — |
| 322 | C4 | — | 6 | 6 | no | — |
| 323 | C4 | — | 6 | 6 | no | — |
| 324 | C4 | — | 12 | 4 | no | — |
| 325 | C4 | D4×S4 | 1 | 6 | yes | complemented |
| 326 | C3 | — | 4 | 1 | no | Sylow |
| 327 | C2 | — | 3 | 1 | no | — |
| 328 | C2 | — | 3 | 3 | no | — |
| 329 | C2 | — | 4 | 4 | no | — |
| 330 | C2 | — | 6 | 4 | no | — |
| 331 | C2 | — | 12 | 4 | no | — |
| 332 | C2 | — | 24 | 4 | no | — |
| 333 | C2 | C2×D4×S4 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Fitting | C22×#127 | D3 |
| socle | C24 | C23×D3 |
| hypercenter | #127 | S4 |
| hypocenter | A4 | C2×#127 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×S4 | C2 | semidirect |
| 2. A4×#127 | C2 | semidirect |
| 3. #85496 | C2 | semidirect |
| 4. #85289 | C2 | semidirect |
| 5. C42×S4 | C2 | semidirect |
| 8. C2×D4×A4 | C22 | semidirect |
| 9. C2×GL(2,ℤ/4) | C22 | semidirect |
| 11. C2×C4⋊S4 | C22 | semidirect |
| 14. C42×A4 | C22 | semidirect |
| 15. C4×SL(2,ℤ/4) | C22 | semidirect |
| 18. C22×#127 | D3 | semidirect |
| 37. C4×S4 | D4 | semidirect |
| 44. C23×D4 | C2×D3 | semidirect |
| 65. C22×C42 | C2×D3 | semidirect |
| 86. C4×A4 | C2×D4 | semidirect |
| 87. SL(2,ℤ/4) | C2×D4 | semidirect |
| 125. #127 | S4 | direct (N × H) |
| 158. S4 | #127 | direct (N × H) |
| 206. C2×D4 | C2×S4 | semidirect |
| 219. C22×C4 | D3×D4 | semidirect |
| 234. C42 | C2×S4 | semidirect |
| 240. A4 | C2×#127 | semidirect |
| 318. C22 | D3×#127 | semidirect |
| 325. C4 | D4×S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×S4 | 2 |
| A4×#127 | 2 |
| #85496 | 2 |
| #85289 | 2 |
| C42×S4 | 2 |
| D4×#127 | 3 |
| D3×#127 | 4 |