Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
934 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 934 conjugacy classes of subgroups — too many to draw. Only its 169 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×#4695 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #69772 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#1273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10304 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C22×D24 | — | 2 | 6 | no | — |
| 6 | C2×#874 | C22 | 1 | 3 | yes | — |
| 7 | C2×C3×C4×C8 | C22 | 1 | 3 | yes | — |
| 8 | C2×#821 | — | 2 | 6 | no | — |
| 9 | #4935 | C22 | 1 | 1 | yes | characteristic |
| 10 | #4695 | — | 2 | 12 | no | — |
| 11 | C2×#493 | — | 3 | 3 | no | — |
| 12 | #1957 | — | 3 | 3 | no | — |
| 13 | #1273 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×D12 | — | 2 | 6 | no | — |
| 15 | C22×D12 | — | 2 | 3 | no | — |
| 16 | C22×C3×C8 | D4 | 1 | 6 | yes | — |
| 17 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 18 | C2×C4⋊Dic3 | — | 2 | 3 | no | — |
| 19 | C2×D24 | — | 2 | 24 | no | — |
| 20 | C2×D24 | — | 4 | 12 | no | — |
| 21 | C2×D24 | — | 4 | 12 | no | — |
| 22 | #874 | — | 2 | 6 | no | — |
| 23 | #874 | — | 2 | 3 | no | — |
| 24 | #874 | D4 | 1 | 6 | yes | — |
| 25 | C3×C4×C8 | — | 2 | 6 | no | — |
| 26 | #821 | — | 4 | 12 | no | — |
| 27 | C22×D8 | — | 6 | 6 | no | — |
| 28 | C2×#127 | — | 3 | 3 | no | — |
| 29 | #493 | — | 6 | 12 | no | — |
| 30 | C2×#102 | — | 6 | 6 | no | — |
| 31 | C2×C4×C8 | C2×D3 | 1 | 3 | yes | — |
| 32 | #384 | — | 3 | 1 | no | — |
| 33 | C23×D3 | — | 4 | 3 | no | — |
| 34 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 35 | C22×Dic3 | — | 4 | 1 | no | — |
| 36 | C2×D12 | — | 2 | 12 | no | — |
| 37 | C2×D12 | — | 2 | 6 | no | — |
| 38 | C2×D12 | — | 4 | 6 | no | — |
| 39 | C2×D12 | — | 4 | 12 | no | — |
| 40 | C2×D12 | — | 4 | 12 | no | — |
| 41 | C2×D12 | — | 4 | 6 | no | — |
| 42 | C2×D12 | — | 4 | 3 | no | — |
| 43 | C2×D12 | — | 4 | 6 | no | — |
| 44 | C2×C3×C8 | — | 2 | 6 | no | — |
| 45 | C2×C3×C8 | — | 2 | 6 | no | — |
| 46 | C2×C3×C8 | D8 | 1 | 12 | yes | complemented |
| 47 | C3×C42 | C2×D4 | 1 | 3 | yes | — |
| 48 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 49 | C4⋊Dic3 | — | 4 | 6 | no | — |
| 50 | D24 | — | 4 | 48 | no | — |
| 51 | C22×D4 | — | 6 | 6 | no | — |
| 52 | C22×D4 | — | 6 | 3 | no | — |
| 53 | C2×D8 | — | 6 | 24 | no | — |
| 54 | C2×D8 | — | 12 | 12 | no | — |
| 55 | C2×D8 | — | 12 | 12 | no | — |
| 56 | C22×C8 | D12 | 1 | 6 | yes | — |
| 57 | #127 | — | 3 | 6 | no | — |
| 58 | #127 | — | 6 | 6 | no | — |
| 59 | #127 | — | 6 | 3 | no | — |
| 60 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 61 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 62 | #102 | — | 12 | 12 | no | — |
| 63 | C4×C8 | — | 2 | 6 | no | — |
| 64 | C23×C3 | #127 | 1 | 1 | yes | characteristic |
| 65 | C22×D3 | — | 4 | 6 | no | — |
| 66 | C22×D3 | — | 8 | 3 | no | — |
| 67 | C22×D3 | — | 8 | 6 | no | — |
| 68 | C22×D3 | — | 8 | 6 | no | — |
| 69 | C22×D3 | — | 8 | 3 | no | — |
| 70 | C2×C3×C4 | #120 | 1 | 3 | yes | — |
| 71 | C2×C3×C4 | #121 | 1 | 3 | yes | — |
| 72 | C2×C3×C4 | C2×D8 | 1 | 6 | yes | — |
| 73 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 74 | C2×Dic3 | — | 8 | 3 | no | — |
| 75 | D12 | — | 4 | 24 | no | — |
| 76 | D12 | — | 4 | 12 | no | — |
| 77 | D12 | — | 8 | 6 | no | — |
| 78 | D12 | — | 8 | 12 | no | — |
| 79 | C3×C8 | — | 2 | 12 | no | — |
| 80 | C24 | — | 12 | 3 | no | — |
| 81 | C2×D4 | — | 6 | 12 | no | — |
| 82 | C2×D4 | — | 6 | 6 | no | — |
| 83 | C2×D4 | — | 12 | 3 | no | — |
| 84 | C2×D4 | — | 12 | 6 | no | — |
| 85 | C2×D4 | — | 12 | 6 | no | — |
| 86 | C2×D4 | — | 12 | 12 | no | — |
| 87 | C2×D4 | — | 12 | 12 | no | — |
| 88 | C2×D4 | — | 12 | 6 | no | — |
| 89 | C22×C4 | — | 12 | 1 | no | — |
| 90 | C22×C4 | C2×D12 | 1 | 3 | yes | — |
| 91 | D8 | — | 12 | 48 | no | — |
| 92 | C2×C8 | — | 2 | 6 | no | — |
| 93 | C2×C8 | — | 2 | 6 | no | — |
| 94 | C2×C8 | D24 | 1 | 12 | yes | complemented |
| 95 | C4⋊C4 | — | 12 | 6 | no | — |
| 96 | C42 | D3×D4 | 1 | 3 | yes | — |
| 97 | C42 | #239 | 1 | 1 | yes | characteristic |
| 98 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 99 | C22×C3 | #493 | 1 | 3 | yes | — |
| 100 | C22×C3 | #496 | 1 | 3 | yes | — |
| 101 | C2×D3 | — | 8 | 12 | no | — |
| 102 | C2×D3 | — | 8 | 6 | no | — |
| 103 | C2×D3 | — | 16 | 3 | no | — |
| 104 | C2×D3 | — | 16 | 3 | no | — |
| 105 | C2×D3 | — | 16 | 6 | no | — |
| 106 | C3×C4 | #447 | 1 | 6 | yes | — |
| 107 | C3×C4 | #459 | 1 | 6 | yes | — |
| 108 | Dic3 | — | 16 | 1 | no | — |
| 109 | C23 | — | 12 | 6 | no | — |
| 110 | C23 | — | 24 | 3 | no | — |
| 111 | C23 | — | 24 | 6 | no | — |
| 112 | C23 | — | 24 | 6 | no | — |
| 113 | C23 | — | 24 | 3 | no | — |
| 114 | C23 | #874 | 1 | 1 | yes | characteristic |
| 115 | D4 | — | 12 | 24 | no | — |
| 116 | D4 | — | 12 | 12 | no | — |
| 117 | D4 | — | 24 | 6 | no | — |
| 118 | D4 | — | 24 | 12 | no | — |
| 119 | C2×C4 | — | 24 | 3 | no | — |
| 120 | C2×C4 | #882 | 1 | 3 | yes | — |
| 121 | C2×C4 | #895 | 1 | 3 | yes | — |
| 122 | C2×C4 | C2×D24 | 1 | 6 | yes | — |
| 123 | C2×C4 | #908 | 1 | 6 | yes | — |
| 124 | C8 | — | 2 | 12 | no | — |
| 125 | C2×C3 | #1712 | 1 | 1 | yes | characteristic |
| 126 | C2×C3 | #1721 | 1 | 3 | yes | — |
| 127 | C2×C3 | #2011 | 1 | 3 | yes | — |
| 128 | D3 | — | 16 | 6 | no | — |
| 129 | C22 | — | 24 | 12 | no | — |
| 130 | C22 | — | 24 | 6 | no | — |
| 131 | C22 | — | 48 | 3 | no | — |
| 132 | C22 | — | 48 | 3 | no | — |
| 133 | C22 | — | 48 | 6 | no | — |
| 134 | C22 | #4671 | 1 | 1 | yes | characteristic |
| 135 | C22 | #4695 | 1 | 3 | yes | — |
| 136 | C22 | #4712 | 1 | 3 | yes | — |
| 137 | C4 | — | 48 | 1 | no | — |
| 138 | C4 | #4732 | 1 | 6 | yes | — |
| 139 | C4 | #4843 | 1 | 6 | yes | — |
| 140 | C3 | #10304 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 141 | C2 | — | 48 | 6 | no | — |
| 142 | C2 | #68062 | 1 | 1 | yes | characteristic |
| 143 | C2 | #68094 | 1 | 3 | yes | — |
| 144 | C2 | #68674 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4695 | C2 | semidirect |
| 2. #69772 | C2 | semidirect |
| 3. C3×#1273 | C2 | semidirect |
| 13. #1273 | D3 | semidirect |
| 46. C2×C3×C8 | D8 | semidirect |
| 94. C2×C8 | D24 | semidirect |
| 140. C3 | #10304 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4695 | 2 |
| #69772 | 2 |
| C3×#1273 | 2 |
| #10304 | 3 |