Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5969 conjugacy classes of subgroups; only the 541 normal subgroups are shown.
This group has 5969 conjugacy classes of subgroups — too many to draw. Only its 541 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 | C22×Q8×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#5919 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 3 | C22×C4⋊S4 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C22×C4×S4 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C22×#548 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C23×S4 | C22 | 1 | 3 | yes | — |
| 7 | C22×#239 | — | 4 | 1 | no | maximal |
| 8 | C2×Q8×A4 | C22 | 1 | 12 | yes | complemented |
| 9 | C22×C4×A4 | C22 | 1 | 3 | yes | — |
| 10 | C22×SL(2,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 11 | #5919 | C22 | 1 | 64 | yes | direct product factor |
| 12 | C2×C4⋊S4 | C22 | 1 | 36 | yes | complemented |
| 13 | C2×C4×S4 | C22 | 1 | 36 | yes | complemented |
| 14 | C23×#41 | — | 3 | 1 | no | — |
| 15 | C24×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#548 | — | 3 | 24 | no | — |
| 17 | C2×#548 | — | 3 | 24 | no | — |
| 18 | C22×#124 | — | 3 | 3 | no | — |
| 19 | C22×C22⋊Q8 | — | 3 | 3 | no | — |
| 20 | C22×#121 | — | 3 | 3 | no | — |
| 21 | C22×C4×Q8 | — | 3 | 1 | no | — |
| 22 | C22×C4×D4 | — | 3 | 3 | no | — |
| 23 | C23×A4 | C23 | 1 | 1 | yes | characteristic |
| 24 | C22×S4 | — | 2 | 12 | no | — |
| 25 | C22×S4 | C23 | 1 | 18 | yes | — |
| 26 | C22×C3×Q8 | — | 4 | 1 | no | — |
| 27 | C2×#239 | — | 4 | 24 | no | — |
| 28 | C22×D12 | — | 4 | 3 | no | — |
| 29 | C22×C4×D3 | — | 4 | 3 | no | — |
| 30 | Q8×A4 | C23 | 1 | 16 | yes | complemented |
| 31 | C2×C4×A4 | C23 | 1 | 18 | yes | — |
| 32 | C2×SL(2,ℤ/4) | C23 | 1 | 6 | yes | — |
| 33 | C4⋊S4 | C23 | 1 | 48 | yes | complemented |
| 34 | C4×S4 | C23 | 1 | 48 | yes | complemented |
| 35 | C22×#41 | — | 3 | 24 | no | — |
| 36 | C22×#41 | — | 6 | 12 | no | — |
| 37 | C22×#41 | — | 6 | 3 | no | — |
| 38 | C22×#41 | — | 6 | 1 | no | — |
| 39 | C23×Q8 | — | 3 | 1 | no | — |
| 40 | C23×Q8 | — | 3 | 3 | no | — |
| 41 | C23×Q8 | — | 3 | 12 | no | — |
| 42 | C23×Q8 | C2×D3 | 1 | 12 | yes | complemented |
| 43 | C23×D4 | — | 3 | 3 | no | — |
| 44 | C23×D4 | — | 3 | 3 | no | — |
| 45 | C24×C4 | — | 3 | 3 | no | — |
| 46 | C24×C4 | C2×D3 | 1 | 3 | yes | — |
| 47 | #548 | — | 3 | 192 | no | — |
| 48 | #548 | — | 3 | 64 | no | — |
| 49 | C2×#124 | — | 3 | 72 | no | — |
| 50 | C2×C22⋊Q8 | — | 3 | 72 | no | — |
| 51 | C2×#121 | — | 3 | 36 | no | — |
| 52 | C2×#121 | — | 3 | 36 | no | — |
| 53 | C2×C4×Q8 | — | 3 | 24 | no | — |
| 54 | C2×C4×D4 | — | 3 | 36 | no | — |
| 55 | C2×C4×D4 | — | 3 | 36 | no | — |
| 56 | C22×C4⋊C4 | — | 3 | 3 | no | — |
| 57 | C22×C4⋊C4 | — | 3 | 3 | no | — |
| 58 | C22×C22⋊C4 | — | 3 | 3 | no | — |
| 59 | C22×C22⋊C4 | — | 3 | 6 | no | — |
| 60 | C22×C22⋊C4 | — | 3 | 1 | no | — |
| 61 | C22×C42 | — | 3 | 3 | no | — |
| 62 | C23×D3 | — | 4 | 3 | no | — |
| 63 | C22×A4 | #41 | 1 | 4 | yes | complemented |
| 64 | C22×A4 | C24 | 1 | 3 | yes | — |
| 65 | C2×S4 | — | 2 | 36 | no | — |
| 66 | C2×S4 | C24 | 1 | 12 | yes | — |
| 67 | C2×C3×Q8 | — | 4 | 12 | no | — |
| 68 | C22×C3×C4 | — | 4 | 3 | no | — |
| 69 | C22×Dic3 | — | 4 | 1 | no | — |
| 70 | #239 | — | 4 | 64 | no | — |
| 71 | C2×D12 | — | 4 | 36 | no | — |
| 72 | C2×C4×D3 | — | 4 | 36 | no | — |
| 73 | C4×A4 | C24 | 1 | 12 | yes | — |
| 74 | SL(2,ℤ/4) | C24 | 1 | 4 | yes | — |
| 75 | C25 | — | 3 | 3 | no | — |
| 76 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 77 | C2×#41 | — | 3 | 64 | no | — |
| 78 | C2×#41 | — | 6 | 96 | no | — |
| 79 | C2×#41 | — | 6 | 72 | no | — |
| 80 | C2×#41 | — | 6 | 24 | no | — |
| 81 | C22×Q8 | — | 3 | 12 | no | — |
| 82 | C22×Q8 | — | 3 | 12 | no | — |
| 83 | C22×Q8 | — | 3 | 36 | no | — |
| 84 | C22×Q8 | — | 3 | 48 | no | — |
| 85 | C22×Q8 | — | 3 | 3 | no | — |
| 86 | C22×Q8 | — | 6 | 1 | no | — |
| 87 | C22×Q8 | — | 6 | 18 | no | — |
| 88 | C22×Q8 | — | 6 | 16 | no | — |
| 89 | C22×Q8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 90 | C22×Q8 | C22×D3 | 1 | 16 | yes | complemented |
| 91 | C22×D4 | — | 3 | 18 | no | — |
| 92 | C22×D4 | — | 3 | 36 | no | — |
| 93 | C22×D4 | — | 3 | 18 | no | — |
| 94 | C22×D4 | — | 6 | 3 | no | — |
| 95 | C22×D4 | — | 6 | 18 | no | — |
| 96 | C22×D4 | — | 6 | 12 | no | — |
| 97 | C22×D4 | — | 6 | 12 | no | — |
| 98 | C22×D4 | — | 6 | 3 | no | — |
| 99 | C23×C4 | — | 3 | 3 | no | — |
| 100 | C23×C4 | — | 3 | 3 | no | — |
| 101 | C23×C4 | — | 3 | 18 | no | — |
| 102 | C23×C4 | — | 3 | 1 | no | — |
| 103 | C23×C4 | — | 3 | 3 | no | — |
| 104 | C23×C4 | — | 3 | 1 | no | — |
| 105 | C23×C4 | — | 3 | 36 | no | — |
| 106 | C23×C4 | — | 6 | 3 | no | — |
| 107 | C23×C4 | — | 6 | 18 | no | — |
| 108 | C23×C4 | — | 6 | 3 | no | — |
| 109 | C23×C4 | C22×D3 | 1 | 18 | yes | — |
| 110 | #124 | — | 3 | 192 | no | — |
| 111 | C22⋊Q8 | — | 3 | 192 | no | — |
| 112 | #121 | — | 3 | 144 | no | — |
| 113 | #121 | — | 3 | 48 | no | — |
| 114 | C4×Q8 | — | 3 | 64 | no | — |
| 115 | C4×D4 | — | 3 | 144 | no | — |
| 116 | C4×D4 | — | 3 | 48 | no | — |
| 117 | C2×C4⋊C4 | — | 3 | 36 | no | — |
| 118 | C2×C4⋊C4 | — | 3 | 36 | no | — |
| 119 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 120 | C2×C22⋊C4 | — | 3 | 36 | no | — |
| 121 | C2×C22⋊C4 | — | 3 | 72 | no | — |
| 122 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 123 | C2×C42 | — | 3 | 36 | no | — |
| 124 | C23×C3 | — | 4 | 1 | no | — |
| 125 | C22×D3 | — | 4 | 18 | no | — |
| 126 | C22×D3 | — | 8 | 12 | no | — |
| 127 | C2×A4 | C2×#41 | 1 | 6 | yes | complemented |
| 128 | C2×A4 | C25 | 1 | 1 | yes | characteristic |
| 129 | S4 | — | 2 | 12 | no | — |
| 130 | C3×Q8 | — | 4 | 16 | no | — |
| 131 | C2×C3×C4 | — | 4 | 18 | no | — |
| 132 | C2×Dic3 | — | 4 | 6 | no | — |
| 133 | D12 | — | 4 | 48 | no | — |
| 134 | C4×D3 | — | 4 | 48 | no | — |
| 135 | C24 | — | 3 | 1 | no | — |
| 136 | C24 | — | 3 | 3 | no | — |
| 137 | C24 | — | 3 | 4 | no | — |
| 138 | C24 | — | 3 | 18 | no | — |
| 139 | C24 | — | 6 | 9 | no | — |
| 140 | C24 | — | 6 | 12 | no | — |
| 141 | C24 | — | 6 | 12 | no | — |
| 142 | C24 | — | 6 | 3 | no | — |
| 143 | C24 | #239 | 1 | 4 | yes | complemented |
| 144 | C24 | C23×D3 | 1 | 3 | yes | — |
| 145 | #41 | — | 6 | 192 | no | — |
| 146 | #41 | — | 6 | 64 | no | — |
| 147 | C2×Q8 | — | 3 | 16 | no | — |
| 148 | C2×Q8 | — | 3 | 48 | no | — |
| 149 | C2×Q8 | — | 3 | 48 | no | — |
| 150 | C2×Q8 | — | 3 | 36 | no | — |
| 151 | C2×Q8 | — | 6 | 12 | no | — |
| 152 | C2×Q8 | — | 6 | 72 | no | — |
| 153 | C2×Q8 | C2×S4 | 1 | 12 | yes | complemented |
| 154 | C2×D4 | — | 3 | 36 | no | — |
| 155 | C2×D4 | — | 3 | 48 | no | — |
| 156 | C2×D4 | — | 3 | 12 | no | — |
| 157 | C2×D4 | — | 6 | 36 | no | — |
| 158 | C2×D4 | — | 6 | 72 | no | — |
| 159 | C2×D4 | — | 6 | 72 | no | — |
| 160 | C2×D4 | — | 6 | 36 | no | — |
| 161 | C2×D4 | — | 6 | 36 | no | — |
| 162 | C2×D4 | — | 6 | 36 | no | — |
| 163 | C22×C4 | — | 3 | 3 | no | — |
| 164 | C22×C4 | — | 3 | 18 | no | — |
| 165 | C22×C4 | — | 3 | 18 | no | — |
| 166 | C22×C4 | — | 3 | 18 | no | — |
| 167 | C22×C4 | — | 3 | 36 | no | — |
| 168 | C22×C4 | — | 3 | 6 | no | — |
| 169 | C22×C4 | — | 3 | 18 | no | — |
| 170 | C22×C4 | — | 3 | 6 | no | — |
| 171 | C22×C4 | — | 3 | 48 | no | — |
| 172 | C22×C4 | — | 6 | 12 | no | — |
| 173 | C22×C4 | — | 6 | 9 | no | — |
| 174 | C22×C4 | — | 6 | 36 | no | — |
| 175 | C22×C4 | — | 6 | 72 | no | — |
| 176 | C22×C4 | — | 6 | 4 | no | — |
| 177 | C22×C4 | — | 6 | 3 | no | — |
| 178 | C22×C4 | — | 6 | 12 | no | — |
| 179 | C22×C4 | — | 6 | 1 | no | — |
| 180 | C22×C4 | — | 6 | 36 | no | — |
| 181 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 182 | C22×C4 | C23×D3 | 1 | 12 | yes | — |
| 183 | C4⋊C4 | — | 3 | 48 | no | — |
| 184 | C4⋊C4 | — | 3 | 48 | no | — |
| 185 | C22⋊C4 | — | 3 | 12 | no | — |
| 186 | C22⋊C4 | — | 3 | 48 | no | — |
| 187 | C22⋊C4 | — | 3 | 96 | no | — |
| 188 | C22⋊C4 | — | 3 | 4 | no | — |
| 189 | C42 | — | 3 | 48 | no | — |
| 190 | C22×C3 | — | 4 | 4 | no | — |
| 191 | C22×C3 | — | 4 | 3 | no | — |
| 192 | C2×D3 | — | 4 | 12 | no | — |
| 193 | C2×D3 | — | 8 | 36 | no | — |
| 194 | A4 | C22×#41 | 1 | 1 | yes | characteristic, complemented |
| 195 | C3×C4 | — | 4 | 12 | no | — |
| 196 | Dic3 | — | 4 | 4 | no | — |
| 197 | C23 | — | 3 | 4 | no | — |
| 198 | C23 | — | 3 | 4 | no | — |
| 199 | C23 | — | 3 | 3 | no | — |
| 200 | C23 | — | 3 | 3 | no | — |
| 201 | C23 | — | 3 | 3 | no | — |
| 202 | C23 | — | 3 | 12 | no | — |
| 203 | C23 | — | 3 | 6 | no | — |
| 204 | C23 | — | 3 | 12 | no | — |
| 205 | C23 | — | 6 | 1 | no | — |
| 206 | C23 | — | 6 | 6 | no | — |
| 207 | C23 | — | 6 | 18 | no | — |
| 208 | C23 | — | 6 | 36 | no | — |
| 209 | C23 | — | 6 | 36 | no | — |
| 210 | C23 | — | 6 | 18 | no | — |
| 211 | C23 | — | 12 | 36 | no | — |
| 212 | C23 | — | 12 | 12 | no | — |
| 213 | C23 | C2×#239 | 1 | 6 | yes | complemented |
| 214 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 215 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 216 | Q8 | — | 3 | 48 | no | — |
| 217 | Q8 | — | 6 | 16 | no | — |
| 218 | Q8 | C22×S4 | 1 | 16 | yes | complemented |
| 219 | D4 | — | 6 | 48 | no | — |
| 220 | D4 | — | 6 | 72 | no | — |
| 221 | D4 | — | 6 | 12 | no | — |
| 222 | D4 | — | 6 | 12 | no | — |
| 223 | D4 | — | 6 | 48 | no | — |
| 224 | C2×C4 | — | 3 | 18 | no | — |
| 225 | C2×C4 | — | 3 | 12 | no | — |
| 226 | C2×C4 | — | 3 | 36 | no | — |
| 227 | C2×C4 | — | 3 | 12 | no | — |
| 228 | C2×C4 | — | 3 | 4 | no | — |
| 229 | C2×C4 | — | 3 | 12 | no | — |
| 230 | C2×C4 | — | 3 | 4 | no | — |
| 231 | C2×C4 | — | 6 | 18 | no | — |
| 232 | C2×C4 | — | 6 | 48 | no | — |
| 233 | C2×C4 | — | 6 | 12 | no | — |
| 234 | C2×C4 | — | 6 | 6 | no | — |
| 235 | C2×C4 | — | 6 | 36 | no | — |
| 236 | C2×C4 | — | 6 | 6 | no | — |
| 237 | C2×C4 | — | 6 | 48 | no | — |
| 238 | C2×C4 | C22×S4 | 1 | 18 | yes | — |
| 239 | C2×C3 | — | 4 | 1 | no | — |
| 240 | C2×C3 | — | 4 | 6 | no | — |
| 241 | D3 | — | 8 | 12 | no | — |
| 242 | C22 | — | 3 | 6 | no | — |
| 243 | C22 | — | 3 | 1 | no | — |
| 244 | C22 | — | 3 | 1 | no | — |
| 245 | C22 | — | 3 | 3 | no | — |
| 246 | C22 | — | 3 | 6 | no | — |
| 247 | C22 | — | 3 | 6 | no | — |
| 248 | C22 | — | 3 | 12 | no | — |
| 249 | C22 | — | 6 | 3 | no | — |
| 250 | C22 | — | 6 | 4 | no | — |
| 251 | C22 | — | 6 | 12 | no | — |
| 252 | C22 | — | 6 | 12 | no | — |
| 253 | C22 | — | 6 | 12 | no | — |
| 254 | C22 | — | 12 | 36 | no | — |
| 255 | C22 | — | 12 | 36 | no | — |
| 256 | C22 | #5919 | 1 | 4 | yes | direct product factor |
| 257 | C22 | C22×#239 | 1 | 1 | yes | characteristic, complemented |
| 258 | C22 | C23×S4 | 1 | 3 | yes | — |
| 259 | C4 | — | 3 | 12 | no | — |
| 260 | C4 | — | 6 | 4 | no | — |
| 261 | C4 | — | 6 | 12 | no | — |
| 262 | C4 | — | 6 | 4 | no | — |
| 263 | C4 | C23×S4 | 1 | 12 | yes | — |
| 264 | C3 | — | 4 | 1 | no | Sylow |
| 265 | C2 | — | 3 | 1 | no | — |
| 266 | C2 | — | 3 | 1 | no | — |
| 267 | C2 | — | 3 | 6 | no | — |
| 268 | C2 | — | 12 | 12 | no | — |
| 269 | C2 | C2×#5919 | 1 | 6 | yes | direct product factor |
| 270 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C24×Q8 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | C22×Q8 | S4 |
| hypocenter | A4 | C22×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×A4 | C2 | semidirect |
| 2. C2×#5919 | C2 | direct (N × H) |
| 3. C22×C4⋊S4 | C2 | semidirect |
| 4. C22×C4×S4 | C2 | semidirect |
| 8. C2×Q8×A4 | C22 | semidirect |
| 11. #5919 | C22 | direct (N × H) |
| 12. C2×C4⋊S4 | C22 | semidirect |
| 13. C2×C4×S4 | C22 | semidirect |
| 15. C24×Q8 | D3 | semidirect |
| 30. Q8×A4 | C23 | semidirect |
| 33. C4⋊S4 | C23 | semidirect |
| 34. C4×S4 | C23 | semidirect |
| 42. C23×Q8 | C2×D3 | semidirect |
| 63. C22×A4 | #41 | semidirect |
| 89. C22×Q8 | S4 | semidirect |
| 90. C22×Q8 | C22×D3 | semidirect |
| 127. C2×A4 | C2×#41 | semidirect |
| 143. C24 | #239 | semidirect |
| 153. C2×Q8 | C2×S4 | semidirect |
| 194. A4 | C22×#41 | semidirect |
| 213. C23 | C2×#239 | semidirect |
| 218. Q8 | C22×S4 | semidirect |
| 256. C22 | #5919 | direct (N × H) |
| 257. C22 | C22×#239 | semidirect |
| 269. C2 | C2×#5919 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×A4 | 2 |
| C2×#5919 | 2 |
| C22×C4⋊S4 | 2 |
| C22×C4×S4 | 2 |
| C22×#548 | 3 |
| C22×#239 | 4 |