Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
118 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #73037 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #73035 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | A4×C32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13696 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #5402 | — | 2 | 1 | no | — |
| 6 | #5398 | — | 2 | 1 | no | — |
| 7 | A4×C16 | C22 | 1 | 1 | yes | characteristic |
| 8 | #4449 | — | 4 | 1 | no | maximal |
| 9 | C2×SD64 | — | 3 | 1 | no | — |
| 10 | C22×C32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #2216 | — | 3 | 1 | no | — |
| 12 | #2215 | — | 3 | 1 | no | — |
| 13 | #1424 | — | 3 | 1 | no | — |
| 14 | #1416 | — | 3 | 1 | no | — |
| 15 | #1415 | — | 3 | 1 | no | — |
| 16 | C4⋊S4 | — | 4 | 1 | no | — |
| 17 | A4⋊Q8 | — | 4 | 1 | no | — |
| 18 | C8×A4 | D4 | 1 | 1 | yes | characteristic |
| 19 | Dic24 | — | 4 | 1 | no | — |
| 20 | D48 | — | 4 | 1 | no | — |
| 21 | C3×C32 | — | 4 | 1 | no | — |
| 22 | C2×Q32 | — | 3 | 1 | no | — |
| 23 | C2×D16 | — | 3 | 1 | no | — |
| 24 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 25 | #467 | — | 6 | 1 | no | — |
| 26 | #466 | — | 6 | 1 | no | — |
| 27 | SD64 | — | 6 | 1 | no | — |
| 28 | SD64 | — | 6 | 1 | no | — |
| 29 | C2×C32 | — | 3 | 1 | no | — |
| 30 | C2×C32 | — | 3 | 1 | no | — |
| 31 | #366 | — | 3 | 1 | no | — |
| 32 | #366 | — | 3 | 1 | no | — |
| 33 | #358 | — | 6 | 1 | no | — |
| 34 | #357 | — | 6 | 1 | no | — |
| 35 | C2×S4 | — | 8 | 1 | no | — |
| 36 | C4×A4 | D8 | 1 | 1 | yes | characteristic |
| 37 | SL(2,ℤ/4) | — | 8 | 1 | no | — |
| 38 | Dic12 | — | 8 | 1 | no | — |
| 39 | D24 | — | 8 | 1 | no | — |
| 40 | C3×C16 | — | 4 | 1 | no | — |
| 41 | C2×Q16 | — | 6 | 1 | no | — |
| 42 | C2×D8 | — | 6 | 1 | no | — |
| 43 | C22×C8 | D12 | 1 | 1 | yes | characteristic |
| 44 | C22⋊Q8 | — | 12 | 1 | no | — |
| 45 | #121 | — | 12 | 1 | no | — |
| 46 | Q32 | — | 6 | 1 | no | — |
| 47 | Q32 | — | 6 | 1 | no | — |
| 48 | D16 | — | 6 | 1 | no | — |
| 49 | D16 | — | 6 | 1 | no | — |
| 50 | C2×C16 | — | 3 | 1 | no | — |
| 51 | C2×C16 | — | 3 | 1 | no | — |
| 52 | #107 | — | 6 | 1 | no | — |
| 53 | #107 | — | 6 | 1 | no | — |
| 54 | #103 | — | 12 | 1 | no | — |
| 55 | #102 | — | 12 | 1 | no | — |
| 56 | C32 | — | 3 | 1 | no | — |
| 57 | C32 | S4 | 1 | 1 | yes | characteristic, complemented |
| 58 | C2×A4 | D16 | 1 | 1 | yes | characteristic |
| 59 | S4 | — | 16 | 1 | no | — |
| 60 | D12 | — | 16 | 1 | no | — |
| 61 | Dic6 | — | 16 | 1 | no | — |
| 62 | C3×C8 | — | 4 | 1 | no | — |
| 63 | C2×Q8 | — | 12 | 1 | no | — |
| 64 | C2×D4 | — | 12 | 1 | no | — |
| 65 | C2×D4 | — | 24 | 1 | no | — |
| 66 | C22×C4 | D24 | 1 | 1 | yes | characteristic |
| 67 | Q16 | — | 12 | 1 | no | — |
| 68 | Q16 | — | 12 | 1 | no | — |
| 69 | D8 | — | 12 | 1 | no | — |
| 70 | D8 | — | 12 | 1 | no | — |
| 71 | C2×C8 | — | 3 | 1 | no | — |
| 72 | C2×C8 | — | 3 | 1 | no | — |
| 73 | C4⋊C4 | — | 12 | 1 | no | — |
| 74 | C4⋊C4 | — | 12 | 1 | no | — |
| 75 | C4⋊C4 | — | 24 | 1 | no | — |
| 76 | C22⋊C4 | — | 24 | 1 | no | — |
| 77 | C22⋊C4 | — | 24 | 1 | no | — |
| 78 | C16 | — | 3 | 1 | no | — |
| 79 | C16 | C2×S4 | 1 | 1 | yes | characteristic |
| 80 | C2×D3 | — | 32 | 1 | no | — |
| 81 | A4 | SD64 | 1 | 1 | yes | characteristic, complemented |
| 82 | C3×C4 | — | 4 | 1 | no | — |
| 83 | Dic3 | — | 32 | 1 | no | — |
| 84 | C23 | — | 24 | 1 | no | — |
| 85 | C23 | D48 | 1 | 1 | yes | characteristic |
| 86 | Q8 | — | 24 | 1 | no | — |
| 87 | Q8 | — | 24 | 1 | no | — |
| 88 | D4 | — | 24 | 1 | no | — |
| 89 | D4 | — | 24 | 1 | no | — |
| 90 | D4 | — | 48 | 1 | no | — |
| 91 | D4 | — | 48 | 1 | no | — |
| 92 | C2×C4 | — | 3 | 1 | no | — |
| 93 | C2×C4 | — | 3 | 1 | no | — |
| 94 | C2×C4 | — | 24 | 1 | no | — |
| 95 | C2×C4 | — | 24 | 1 | no | — |
| 96 | C2×C4 | — | 24 | 1 | no | — |
| 97 | C8 | — | 3 | 1 | no | — |
| 98 | C8 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 99 | C2×C3 | — | 4 | 1 | no | — |
| 100 | D3 | — | 64 | 1 | no | — |
| 101 | C22 | — | 3 | 1 | no | — |
| 102 | C22 | — | 3 | 1 | no | — |
| 103 | C22 | — | 48 | 1 | no | — |
| 104 | C22 | — | 48 | 1 | no | — |
| 105 | C22 | — | 48 | 1 | no | — |
| 106 | C22 | #4449 | 1 | 1 | yes | characteristic, complemented |
| 107 | C4 | — | 3 | 1 | no | — |
| 108 | C4 | — | 48 | 1 | no | — |
| 109 | C4 | — | 48 | 1 | no | — |
| 110 | C4 | — | 48 | 1 | no | — |
| 111 | C4 | #5402 | 1 | 1 | yes | characteristic |
| 112 | C3 | — | 4 | 1 | no | Sylow |
| 113 | C2 | — | 3 | 1 | no | — |
| 114 | C2 | — | 3 | 1 | no | — |
| 115 | C2 | — | 96 | 1 | no | — |
| 116 | C2 | #73035 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #73035 |
| commutator, cosocle | A4×C16 | C22 |
| Frattini | C16 | C2×S4 |
| Fitting | C22×C32 | D3 |
| socle | C23 | D48 |
| prime radical | #73035 | C2 |
| hypercenter | C32 | S4 |
| hypocenter | A4 | SD64 |
| LCS | G ▷ A4×C16 ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #73037 | C2 | semidirect |
| 3. A4×C32 | C2 | semidirect |
| 10. C22×C32 | D3 | semidirect |
| 57. C32 | S4 | semidirect |
| 81. A4 | SD64 | semidirect |
| 106. C22 | #4449 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #73037 | 2 |
| #73035 | 2 |
| A4×C32 | 2 |
| #13696 | 3 |
| #4449 | 4 |