Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
6209 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 6209 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 | C23×C4×A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C23×SL(2,ℤ/4) | C2 | 1 | 2 | yes | maximal |
| 3 | C22×A4⋊Q8 | C2 | 1 | 28 | yes | maximal, direct product factor |
| 4 | C23×C22⋊Q8 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C24×A4 | C22 | 1 | 1 | yes | characteristic |
| 6 | C23×Dic6 | — | 4 | 1 | no | maximal |
| 7 | C22×C4×A4 | C22 | 1 | 14 | yes | — |
| 8 | C22×SL(2,ℤ/4) | C22 | 1 | 28 | yes | — |
| 9 | C2×A4⋊Q8 | C22 | 1 | 112 | yes | direct product factor |
| 10 | C24×Q8 | — | 3 | 1 | no | — |
| 11 | C25×C4 | D3 | 1 | 1 | yes | characteristic |
| 12 | C22×C22⋊Q8 | — | 3 | 28 | no | — |
| 13 | C22×C22⋊Q8 | — | 3 | 28 | no | — |
| 14 | C23×C4⋊C4 | — | 3 | 1 | no | — |
| 15 | C23×C4⋊C4 | — | 3 | 2 | no | — |
| 16 | C23×C22⋊C4 | — | 3 | 2 | no | — |
| 17 | C23×A4 | Q8 | 1 | 8 | yes | complemented |
| 18 | C23×A4 | C23 | 1 | 7 | yes | — |
| 19 | C23×C3×C4 | — | 4 | 1 | no | — |
| 20 | C23×Dic3 | — | 4 | 2 | no | — |
| 21 | C22×Dic6 | — | 4 | 28 | no | — |
| 22 | C2×C4×A4 | C23 | 1 | 28 | yes | — |
| 23 | C2×SL(2,ℤ/4) | C23 | 1 | 56 | yes | — |
| 24 | A4⋊Q8 | C23 | 1 | 64 | yes | direct product factor |
| 25 | C26 | C2×D3 | 1 | 1 | yes | characteristic |
| 26 | C23×Q8 | — | 3 | 28 | no | — |
| 27 | C23×Q8 | — | 6 | 1 | no | — |
| 28 | C23×Q8 | — | 6 | 14 | no | — |
| 29 | C23×Q8 | — | 6 | 1 | no | — |
| 30 | C24×C4 | — | 3 | 1 | no | — |
| 31 | C24×C4 | — | 3 | 1 | no | — |
| 32 | C24×C4 | — | 3 | 14 | no | — |
| 33 | C24×C4 | — | 3 | 2 | no | — |
| 34 | C24×C4 | — | 3 | 2 | no | — |
| 35 | C24×C4 | C2×D3 | 1 | 14 | yes | — |
| 36 | C2×C22⋊Q8 | — | 3 | 336 | no | — |
| 37 | C2×C22⋊Q8 | — | 3 | 112 | no | — |
| 38 | C22×C4⋊C4 | — | 3 | 28 | no | — |
| 39 | C22×C4⋊C4 | — | 3 | 56 | no | — |
| 40 | C22×C22⋊C4 | — | 3 | 28 | no | — |
| 41 | C22×C22⋊C4 | — | 3 | 28 | no | — |
| 42 | C24×C3 | — | 4 | 1 | no | — |
| 43 | C22×A4 | C2×Q8 | 1 | 28 | yes | complemented |
| 44 | C22×A4 | C24 | 1 | 7 | yes | — |
| 45 | C22×C3×C4 | — | 4 | 14 | no | — |
| 46 | C22×Dic3 | — | 4 | 28 | no | — |
| 47 | C2×Dic6 | — | 4 | 112 | no | — |
| 48 | C4×A4 | C24 | 1 | 8 | yes | — |
| 49 | SL(2,ℤ/4) | C24 | 1 | 16 | yes | — |
| 50 | C25 | — | 3 | 1 | no | — |
| 51 | C25 | — | 3 | 7 | no | — |
| 52 | C25 | — | 3 | 8 | no | — |
| 53 | C25 | Dic6 | 1 | 8 | yes | complemented |
| 54 | C25 | C22×D3 | 1 | 7 | yes | — |
| 55 | C22×Q8 | — | 3 | 112 | no | — |
| 56 | C22×Q8 | — | 6 | 28 | no | — |
| 57 | C22×Q8 | — | 6 | 168 | no | — |
| 58 | C22×Q8 | — | 6 | 28 | no | — |
| 59 | C23×C4 | — | 3 | 1 | no | — |
| 60 | C23×C4 | — | 3 | 14 | no | — |
| 61 | C23×C4 | — | 3 | 14 | no | — |
| 62 | C23×C4 | — | 3 | 14 | no | — |
| 63 | C23×C4 | — | 3 | 84 | no | — |
| 64 | C23×C4 | — | 3 | 28 | no | — |
| 65 | C23×C4 | — | 3 | 28 | no | — |
| 66 | C23×C4 | — | 6 | 28 | no | — |
| 67 | C23×C4 | — | 6 | 7 | no | — |
| 68 | C23×C4 | — | 6 | 16 | no | — |
| 69 | C23×C4 | — | 6 | 14 | no | — |
| 70 | C23×C4 | — | 6 | 2 | no | — |
| 71 | C23×C4 | S4 | 1 | 1 | yes | characteristic |
| 72 | C23×C4 | C22×D3 | 1 | 28 | yes | — |
| 73 | C22⋊Q8 | — | 3 | 448 | no | — |
| 74 | C22⋊Q8 | — | 3 | 64 | no | — |
| 75 | C2×C4⋊C4 | — | 3 | 112 | no | — |
| 76 | C2×C4⋊C4 | — | 3 | 224 | no | — |
| 77 | C2×C22⋊C4 | — | 3 | 168 | no | — |
| 78 | C2×C22⋊C4 | — | 3 | 56 | no | — |
| 79 | C23×C3 | — | 4 | 8 | no | — |
| 80 | C23×C3 | — | 4 | 7 | no | — |
| 81 | C2×A4 | C22×Q8 | 1 | 14 | yes | complemented |
| 82 | C2×A4 | C25 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×C4 | — | 4 | 28 | no | — |
| 84 | C2×Dic3 | — | 4 | 56 | no | — |
| 85 | Dic6 | — | 4 | 64 | no | — |
| 86 | C24 | — | 3 | 8 | no | — |
| 87 | C24 | — | 3 | 8 | no | — |
| 88 | C24 | — | 3 | 7 | no | — |
| 89 | C24 | — | 3 | 7 | no | — |
| 90 | C24 | — | 3 | 21 | no | — |
| 91 | C24 | — | 3 | 56 | no | — |
| 92 | C24 | — | 3 | 28 | no | — |
| 93 | C24 | — | 6 | 7 | no | — |
| 94 | C24 | — | 6 | 28 | no | — |
| 95 | C24 | C2×Dic6 | 1 | 28 | yes | complemented |
| 96 | C24 | C2×S4 | 1 | 1 | yes | characteristic |
| 97 | C24 | C23×D3 | 1 | 7 | yes | — |
| 98 | C2×Q8 | — | 3 | 64 | no | — |
| 99 | C2×Q8 | — | 6 | 112 | no | — |
| 100 | C2×Q8 | — | 6 | 224 | no | — |
| 101 | C2×Q8 | — | 6 | 112 | no | — |
| 102 | C22×C4 | — | 3 | 14 | no | — |
| 103 | C22×C4 | — | 3 | 28 | no | — |
| 104 | C22×C4 | — | 3 | 84 | no | — |
| 105 | C22×C4 | — | 3 | 28 | no | — |
| 106 | C22×C4 | — | 3 | 56 | no | — |
| 107 | C22×C4 | — | 3 | 56 | no | — |
| 108 | C22×C4 | — | 3 | 56 | no | — |
| 109 | C22×C4 | — | 6 | 56 | no | — |
| 110 | C22×C4 | — | 6 | 42 | no | — |
| 111 | C22×C4 | — | 6 | 112 | no | — |
| 112 | C22×C4 | — | 6 | 84 | no | — |
| 113 | C22×C4 | — | 6 | 28 | no | — |
| 114 | C22×C4 | C2×S4 | 1 | 14 | yes | — |
| 115 | C22×C4 | C23×D3 | 1 | 8 | yes | — |
| 116 | C4⋊C4 | — | 3 | 64 | no | — |
| 117 | C4⋊C4 | — | 3 | 128 | no | — |
| 118 | C22⋊C4 | — | 3 | 112 | no | — |
| 119 | C22⋊C4 | — | 3 | 16 | no | — |
| 120 | C22×C3 | — | 4 | 28 | no | — |
| 121 | C22×C3 | — | 4 | 7 | no | — |
| 122 | A4 | C23×Q8 | 1 | 1 | yes | characteristic, complemented |
| 123 | C3×C4 | — | 4 | 8 | no | — |
| 124 | Dic3 | — | 4 | 16 | no | — |
| 125 | C23 | — | 3 | 28 | no | — |
| 126 | C23 | — | 3 | 28 | no | — |
| 127 | C23 | — | 3 | 56 | no | — |
| 128 | C23 | — | 3 | 7 | no | — |
| 129 | C23 | — | 3 | 21 | no | — |
| 130 | C23 | — | 3 | 7 | no | — |
| 131 | C23 | — | 3 | 84 | no | — |
| 132 | C23 | — | 3 | 14 | no | — |
| 133 | C23 | — | 6 | 7 | no | — |
| 134 | C23 | — | 6 | 42 | no | — |
| 135 | C23 | — | 6 | 56 | no | — |
| 136 | C23 | A4⋊Q8 | 1 | 8 | yes | direct product factor |
| 137 | C23 | C22×Dic6 | 1 | 14 | yes | complemented |
| 138 | C23 | C22×S4 | 1 | 7 | yes | — |
| 139 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 140 | Q8 | — | 6 | 64 | no | — |
| 141 | Q8 | — | 6 | 64 | no | — |
| 142 | C2×C4 | — | 3 | 28 | no | — |
| 143 | C2×C4 | — | 3 | 8 | no | — |
| 144 | C2×C4 | — | 3 | 56 | no | — |
| 145 | C2×C4 | — | 3 | 8 | no | — |
| 146 | C2×C4 | — | 3 | 16 | no | — |
| 147 | C2×C4 | — | 3 | 16 | no | — |
| 148 | C2×C4 | — | 6 | 28 | no | — |
| 149 | C2×C4 | — | 6 | 112 | no | — |
| 150 | C2×C4 | — | 6 | 56 | no | — |
| 151 | C2×C4 | — | 6 | 56 | no | — |
| 152 | C2×C4 | C22×S4 | 1 | 28 | yes | — |
| 153 | C2×C3 | — | 4 | 1 | no | — |
| 154 | C2×C3 | — | 4 | 14 | no | — |
| 155 | C22 | — | 3 | 14 | no | — |
| 156 | C22 | — | 3 | 1 | no | — |
| 157 | C22 | — | 3 | 1 | no | — |
| 158 | C22 | — | 3 | 7 | no | — |
| 159 | C22 | — | 3 | 14 | no | — |
| 160 | C22 | — | 3 | 14 | no | — |
| 161 | C22 | — | 3 | 84 | no | — |
| 162 | C22 | — | 6 | 7 | no | — |
| 163 | C22 | — | 6 | 28 | no | — |
| 164 | C22 | C2×A4⋊Q8 | 1 | 28 | yes | direct product factor |
| 165 | C22 | C23×Dic6 | 1 | 1 | yes | characteristic, complemented |
| 166 | C22 | C23×S4 | 1 | 7 | yes | — |
| 167 | C4 | — | 3 | 8 | no | — |
| 168 | C4 | — | 6 | 16 | no | — |
| 169 | C4 | — | 6 | 16 | no | — |
| 170 | C4 | C23×S4 | 1 | 8 | yes | — |
| 171 | C3 | — | 4 | 1 | no | Sylow |
| 172 | C2 | — | 3 | 1 | no | — |
| 173 | C2 | — | 3 | 1 | no | — |
| 174 | C2 | — | 3 | 14 | no | — |
| 175 | C2 | C22×A4⋊Q8 | 1 | 14 | yes | direct product factor |
| 176 | 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 |
| prime radical | C24×A4 | C22 |
| hypercenter | C23×C4 | S4 |
| hypocenter | A4 | C23×Q8 |
| 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 |
|---|---|---|
| 3. C22×A4⋊Q8 | C2 | direct (N × H) |
| 9. C2×A4⋊Q8 | C22 | direct (N × H) |
| 17. C23×A4 | Q8 | semidirect |
| 24. A4⋊Q8 | C23 | direct (N × H) |
| 43. C22×A4 | C2×Q8 | semidirect |
| 53. C25 | Dic6 | semidirect |
| 81. C2×A4 | C22×Q8 | semidirect |
| 95. C24 | C2×Dic6 | semidirect |
| 122. A4 | C23×Q8 | semidirect |
| 136. C23 | A4⋊Q8 | direct (N × H) |
| 137. C23 | C22×Dic6 | semidirect |
| 164. C22 | C2×A4⋊Q8 | direct (N × H) |
| 165. C22 | C23×Dic6 | semidirect |
| 175. C2 | C22×A4⋊Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×A4 | 2 |
| C23×SL(2,ℤ/4) | 2 |
| C22×A4⋊Q8 | 2 |
| C23×C22⋊Q8 | 3 |
| C23×Dic6 | 4 |