Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
9681 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 9681 conjugacy classes of subgroups — too many to draw. Only its 661 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 | C24×S4 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C23×C4×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C22×C4⋊S4 | C2 | 1 | 28 | yes | maximal, direct product factor |
| 4 | C23×#121 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C24×A4 | C22 | 1 | 1 | yes | characteristic |
| 6 | C23×S4 | — | 2 | 16 | no | — |
| 7 | C23×S4 | C22 | 1 | 28 | yes | complemented |
| 8 | C23×D12 | — | 4 | 1 | no | maximal |
| 9 | C22×C4×A4 | C22 | 1 | 14 | yes | complemented |
| 10 | C2×C4⋊S4 | C22 | 1 | 112 | yes | direct product factor |
| 11 | C24×D4 | — | 3 | 1 | no | — |
| 12 | C24×D4 | — | 3 | 2 | no | — |
| 13 | C25×C4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×#121 | — | 3 | 28 | no | — |
| 15 | C22×#121 | — | 3 | 28 | no | — |
| 16 | C23×C4⋊C4 | — | 3 | 1 | no | — |
| 17 | C23×C22⋊C4 | — | 3 | 2 | no | — |
| 18 | C24×D3 | — | 4 | 2 | no | — |
| 19 | C23×A4 | D4 | 1 | 8 | yes | complemented |
| 20 | C23×A4 | C23 | 1 | 7 | yes | — |
| 21 | C22×S4 | — | 2 | 112 | no | — |
| 22 | C22×S4 | C23 | 1 | 56 | yes | complemented |
| 23 | C23×C3×C4 | — | 4 | 1 | no | — |
| 24 | C22×D12 | — | 4 | 28 | no | — |
| 25 | C2×C4×A4 | C23 | 1 | 28 | yes | complemented |
| 26 | C4⋊S4 | C23 | 1 | 64 | yes | direct product factor |
| 27 | C26 | — | 3 | 2 | no | — |
| 28 | C26 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C23×D4 | — | 3 | 28 | no | — |
| 30 | C23×D4 | — | 3 | 28 | no | — |
| 31 | C23×D4 | — | 3 | 28 | no | — |
| 32 | C23×D4 | — | 6 | 1 | no | — |
| 33 | C23×D4 | — | 6 | 14 | no | — |
| 34 | C23×D4 | — | 6 | 16 | no | — |
| 35 | C23×D4 | — | 6 | 16 | no | — |
| 36 | C23×D4 | — | 6 | 1 | no | — |
| 37 | C24×C4 | — | 3 | 1 | no | — |
| 38 | C24×C4 | — | 3 | 1 | no | — |
| 39 | C24×C4 | — | 3 | 14 | no | — |
| 40 | C24×C4 | — | 3 | 2 | no | — |
| 41 | C24×C4 | C2×D3 | 1 | 14 | yes | complemented |
| 42 | C2×#121 | — | 3 | 336 | no | — |
| 43 | C2×#121 | — | 3 | 112 | no | — |
| 44 | C22×C4⋊C4 | — | 3 | 28 | no | — |
| 45 | C22×C22⋊C4 | — | 3 | 56 | no | — |
| 46 | C24×C3 | — | 4 | 1 | no | — |
| 47 | C23×D3 | — | 4 | 28 | no | — |
| 48 | C23×D3 | — | 8 | 16 | no | — |
| 49 | C22×A4 | C2×D4 | 1 | 28 | yes | complemented |
| 50 | C22×A4 | C24 | 1 | 7 | yes | — |
| 51 | C2×S4 | — | 2 | 112 | no | — |
| 52 | C2×S4 | C24 | 1 | 16 | yes | complemented |
| 53 | C22×C3×C4 | — | 4 | 14 | no | — |
| 54 | C2×D12 | — | 4 | 112 | no | — |
| 55 | C4×A4 | C24 | 1 | 8 | yes | complemented |
| 56 | C25 | — | 3 | 1 | no | — |
| 57 | C25 | — | 3 | 7 | no | — |
| 58 | C25 | — | 3 | 8 | no | — |
| 59 | C25 | — | 3 | 28 | no | — |
| 60 | C25 | — | 6 | 16 | no | — |
| 61 | C25 | — | 6 | 14 | no | — |
| 62 | C25 | — | 6 | 16 | no | — |
| 63 | C25 | — | 6 | 2 | no | — |
| 64 | C25 | D12 | 1 | 8 | yes | complemented |
| 65 | C25 | C22×D3 | 1 | 7 | yes | — |
| 66 | C22×D4 | — | 3 | 112 | no | — |
| 67 | C22×D4 | — | 3 | 168 | no | — |
| 68 | C22×D4 | — | 3 | 56 | no | — |
| 69 | C22×D4 | — | 6 | 28 | no | — |
| 70 | C22×D4 | — | 6 | 168 | no | — |
| 71 | C22×D4 | — | 6 | 224 | no | — |
| 72 | C22×D4 | — | 6 | 112 | no | — |
| 73 | C22×D4 | — | 6 | 112 | no | — |
| 74 | C22×D4 | — | 6 | 28 | no | — |
| 75 | C23×C4 | — | 3 | 1 | no | — |
| 76 | C23×C4 | — | 3 | 14 | no | — |
| 77 | C23×C4 | — | 3 | 14 | no | — |
| 78 | C23×C4 | — | 3 | 14 | no | — |
| 79 | C23×C4 | — | 3 | 84 | no | — |
| 80 | C23×C4 | — | 3 | 28 | no | — |
| 81 | C23×C4 | — | 6 | 28 | no | — |
| 82 | C23×C4 | — | 6 | 7 | no | — |
| 83 | C23×C4 | — | 6 | 16 | no | — |
| 84 | C23×C4 | S4 | 1 | 1 | yes | characteristic, complemented |
| 85 | C23×C4 | C22×D3 | 1 | 28 | yes | complemented |
| 86 | #121 | — | 3 | 448 | no | — |
| 87 | #121 | — | 3 | 64 | no | — |
| 88 | C2×C4⋊C4 | — | 3 | 112 | no | — |
| 89 | C2×C22⋊C4 | — | 3 | 224 | no | — |
| 90 | C23×C3 | — | 4 | 8 | no | — |
| 91 | C23×C3 | — | 4 | 7 | no | — |
| 92 | C22×D3 | — | 4 | 56 | no | — |
| 93 | C22×D3 | — | 8 | 112 | no | — |
| 94 | C2×A4 | C22×D4 | 1 | 14 | yes | complemented |
| 95 | C2×A4 | C25 | 1 | 1 | yes | characteristic |
| 96 | S4 | — | 2 | 16 | no | — |
| 97 | C2×C3×C4 | — | 4 | 28 | no | — |
| 98 | D12 | — | 4 | 64 | no | — |
| 99 | C24 | — | 3 | 8 | no | — |
| 100 | C24 | — | 3 | 8 | no | — |
| 101 | C24 | — | 3 | 7 | no | — |
| 102 | C24 | — | 3 | 7 | no | — |
| 103 | C24 | — | 3 | 21 | no | — |
| 104 | C24 | — | 3 | 56 | no | — |
| 105 | C24 | — | 3 | 28 | no | — |
| 106 | C24 | — | 3 | 56 | no | — |
| 107 | C24 | — | 6 | 7 | no | — |
| 108 | C24 | — | 6 | 28 | no | — |
| 109 | C24 | — | 6 | 112 | no | — |
| 110 | C24 | — | 6 | 84 | no | — |
| 111 | C24 | — | 6 | 112 | no | — |
| 112 | C24 | — | 6 | 28 | no | — |
| 113 | C24 | — | 12 | 112 | no | — |
| 114 | C24 | — | 12 | 16 | no | — |
| 115 | C24 | C2×D12 | 1 | 28 | yes | complemented |
| 116 | C24 | C2×S4 | 1 | 1 | yes | characteristic |
| 117 | C24 | C23×D3 | 1 | 7 | yes | — |
| 118 | C2×D4 | — | 3 | 64 | no | — |
| 119 | C2×D4 | — | 3 | 112 | no | — |
| 120 | C2×D4 | — | 3 | 16 | no | — |
| 121 | C2×D4 | — | 6 | 112 | no | — |
| 122 | C2×D4 | — | 6 | 224 | no | — |
| 123 | C2×D4 | — | 6 | 672 | no | — |
| 124 | C2×D4 | — | 6 | 112 | no | — |
| 125 | C2×D4 | — | 6 | 112 | no | — |
| 126 | C2×D4 | — | 6 | 112 | no | — |
| 127 | C22×C4 | — | 3 | 14 | no | — |
| 128 | C22×C4 | — | 3 | 28 | no | — |
| 129 | C22×C4 | — | 3 | 84 | no | — |
| 130 | C22×C4 | — | 3 | 28 | no | — |
| 131 | C22×C4 | — | 3 | 56 | no | — |
| 132 | C22×C4 | — | 3 | 56 | no | — |
| 133 | C22×C4 | — | 6 | 56 | no | — |
| 134 | C22×C4 | — | 6 | 42 | no | — |
| 135 | C22×C4 | — | 6 | 112 | no | — |
| 136 | C22×C4 | C2×S4 | 1 | 14 | yes | complemented |
| 137 | C22×C4 | C23×D3 | 1 | 8 | yes | complemented |
| 138 | C4⋊C4 | — | 3 | 64 | no | — |
| 139 | C22⋊C4 | — | 3 | 128 | no | — |
| 140 | C22×C3 | — | 4 | 28 | no | — |
| 141 | C22×C3 | — | 4 | 7 | no | — |
| 142 | C2×D3 | — | 4 | 16 | no | — |
| 143 | C2×D3 | — | 8 | 112 | no | — |
| 144 | A4 | C23×D4 | 1 | 1 | yes | characteristic, complemented |
| 145 | C3×C4 | — | 4 | 8 | no | — |
| 146 | C23 | — | 3 | 28 | no | — |
| 147 | C23 | — | 3 | 28 | no | — |
| 148 | C23 | — | 3 | 56 | no | — |
| 149 | C23 | — | 3 | 7 | no | — |
| 150 | C23 | — | 3 | 21 | no | — |
| 151 | C23 | — | 3 | 7 | no | — |
| 152 | C23 | — | 3 | 84 | no | — |
| 153 | C23 | — | 3 | 14 | no | — |
| 154 | C23 | — | 3 | 16 | no | — |
| 155 | C23 | — | 6 | 7 | no | — |
| 156 | C23 | — | 6 | 42 | no | — |
| 157 | C23 | — | 6 | 56 | no | — |
| 158 | C23 | — | 6 | 56 | no | — |
| 159 | C23 | — | 6 | 112 | no | — |
| 160 | C23 | — | 6 | 112 | no | — |
| 161 | C23 | — | 6 | 56 | no | — |
| 162 | C23 | — | 12 | 336 | no | — |
| 163 | C23 | — | 12 | 112 | no | — |
| 164 | C23 | C4⋊S4 | 1 | 8 | yes | direct product factor |
| 165 | C23 | C22×D12 | 1 | 14 | yes | complemented |
| 166 | C23 | C22×S4 | 1 | 7 | yes | — |
| 167 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 168 | D4 | — | 6 | 64 | no | — |
| 169 | D4 | — | 6 | 224 | no | — |
| 170 | D4 | — | 6 | 16 | no | — |
| 171 | D4 | — | 6 | 16 | no | — |
| 172 | D4 | — | 6 | 64 | no | — |
| 173 | C2×C4 | — | 3 | 28 | no | — |
| 174 | C2×C4 | — | 3 | 8 | no | — |
| 175 | C2×C4 | — | 3 | 56 | no | — |
| 176 | C2×C4 | — | 3 | 8 | no | — |
| 177 | C2×C4 | — | 3 | 16 | no | — |
| 178 | C2×C4 | — | 6 | 28 | no | — |
| 179 | C2×C4 | — | 6 | 112 | no | — |
| 180 | C2×C4 | C22×S4 | 1 | 28 | yes | complemented |
| 181 | C2×C3 | — | 4 | 1 | no | — |
| 182 | C2×C3 | — | 4 | 14 | no | — |
| 183 | D3 | — | 8 | 16 | no | — |
| 184 | C22 | — | 3 | 14 | no | — |
| 185 | C22 | — | 3 | 1 | no | — |
| 186 | C22 | — | 3 | 1 | no | — |
| 187 | C22 | — | 3 | 7 | no | — |
| 188 | C22 | — | 3 | 14 | no | — |
| 189 | C22 | — | 3 | 14 | no | — |
| 190 | C22 | — | 3 | 84 | no | — |
| 191 | C22 | — | 6 | 7 | no | — |
| 192 | C22 | — | 6 | 28 | no | — |
| 193 | C22 | — | 6 | 16 | no | — |
| 194 | C22 | — | 6 | 16 | no | — |
| 195 | C22 | — | 6 | 16 | no | — |
| 196 | C22 | — | 12 | 112 | no | — |
| 197 | C22 | — | 12 | 112 | no | — |
| 198 | C22 | C2×C4⋊S4 | 1 | 28 | yes | direct product factor |
| 199 | C22 | C23×D12 | 1 | 1 | yes | characteristic, complemented |
| 200 | C22 | C23×S4 | 1 | 7 | yes | — |
| 201 | C4 | — | 3 | 8 | no | — |
| 202 | C4 | — | 6 | 16 | no | — |
| 203 | C4 | C23×S4 | 1 | 8 | yes | complemented |
| 204 | C3 | — | 4 | 1 | no | Sylow |
| 205 | C2 | — | 3 | 1 | no | — |
| 206 | C2 | — | 3 | 1 | no | — |
| 207 | C2 | — | 3 | 14 | no | — |
| 208 | C2 | — | 12 | 16 | no | — |
| 209 | C2 | C22×C4⋊S4 | 1 | 14 | yes | direct product factor |
| 210 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C2×S4 |
| commutator, cosocle | C2×A4 | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C25×C4 | D3 |
| socle | C26 | C2×D3 |
| hypercenter | C23×C4 | S4 |
| hypocenter | A4 | C23×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C24 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×S4 | C2 | semidirect |
| 2. C23×C4×A4 | C2 | semidirect |
| 3. C22×C4⋊S4 | C2 | direct (N × H) |
| 7. C23×S4 | C22 | semidirect |
| 9. C22×C4×A4 | C22 | semidirect |
| 10. C2×C4⋊S4 | C22 | direct (N × H) |
| 13. C25×C4 | D3 | semidirect |
| 19. C23×A4 | D4 | semidirect |
| 22. C22×S4 | C23 | semidirect |
| 25. C2×C4×A4 | C23 | semidirect |
| 26. C4⋊S4 | C23 | direct (N × H) |
| 41. C24×C4 | C2×D3 | semidirect |
| 49. C22×A4 | C2×D4 | semidirect |
| 52. C2×S4 | C24 | semidirect |
| 55. C4×A4 | C24 | semidirect |
| 64. C25 | D12 | semidirect |
| 84. C23×C4 | S4 | semidirect |
| 85. C23×C4 | C22×D3 | semidirect |
| 94. C2×A4 | C22×D4 | semidirect |
| 115. C24 | C2×D12 | semidirect |
| 136. C22×C4 | C2×S4 | semidirect |
| 137. C22×C4 | C23×D3 | semidirect |
| 144. A4 | C23×D4 | semidirect |
| 164. C23 | C4⋊S4 | direct (N × H) |
| 165. C23 | C22×D12 | semidirect |
| 180. C2×C4 | C22×S4 | semidirect |
| 198. C22 | C2×C4⋊S4 | direct (N × H) |
| 199. C22 | C23×D12 | semidirect |
| 203. C4 | C23×S4 | semidirect |
| 209. C2 | C22×C4⋊S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×S4 | 2 |
| C23×C4×A4 | 2 |
| C22×C4⋊S4 | 2 |
| C23×#121 | 3 |
| C23×D12 | 4 |