Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
593 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 593 conjugacy classes of subgroups — too many to draw. Only its 95 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×C4⋊#226 | C2 | 1 | 6 | yes | maximal |
| 2 | C42×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33163 | — | 3 | 1 | no | maximal, Sylow |
| 4 | C22×#226 | C22 | 1 | 4 | yes | — |
| 5 | C2×C4×SL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 6 | C4⋊#226 | — | 2 | 12 | no | — |
| 7 | #4935 | — | 4 | 1 | no | maximal |
| 8 | C2×#462 | — | 3 | 6 | no | — |
| 9 | #2868 | — | 3 | 1 | no | — |
| 10 | C42×Q8 | D3 | 1 | 1 | yes | characteristic |
| 11 | #1981 | — | 3 | 6 | no | — |
| 12 | #1842 | — | 3 | 1 | no | — |
| 13 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 14 | C2×#226 | — | 2 | 12 | no | — |
| 15 | C2×C3×C42 | — | 4 | 1 | no | — |
| 16 | C2×C4⋊Dic3 | — | 4 | 6 | no | — |
| 17 | C4×SL(2,𝔽3) | D4 | 1 | 6 | yes | — |
| 18 | C22×Q16 | — | 3 | 4 | no | — |
| 19 | C2×C4⋊Q8 | — | 3 | 6 | no | — |
| 20 | C2×C4×Q8 | C2×D3 | 1 | 3 | yes | — |
| 21 | #462 | — | 6 | 12 | no | — |
| 22 | #462 | — | 6 | 12 | no | — |
| 23 | C2×C4⋊C8 | — | 3 | 6 | no | — |
| 24 | C2×#103 | — | 3 | 12 | no | — |
| 25 | #384 | — | 6 | 3 | no | — |
| 26 | #384 | — | 6 | 1 | no | — |
| 27 | C4×C4⋊C4 | — | 3 | 3 | no | — |
| 28 | C43 | — | 3 | 1 | no | — |
| 29 | C22×C3×C4 | — | 4 | 3 | no | — |
| 30 | C22×Dic3 | — | 4 | 4 | no | — |
| 31 | C2×SL(2,𝔽3) | C2×D4 | 1 | 3 | yes | — |
| 32 | #226 | — | 4 | 4 | no | — |
| 33 | C3×C42 | — | 4 | 4 | no | — |
| 34 | C4⋊Dic3 | — | 8 | 12 | no | — |
| 35 | C22×Q8 | — | 3 | 4 | no | — |
| 36 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 37 | C2×Q16 | — | 6 | 12 | no | — |
| 38 | C2×Q16 | — | 6 | 12 | no | — |
| 39 | C22×C8 | — | 3 | 4 | no | — |
| 40 | C4⋊Q8 | — | 6 | 12 | no | — |
| 41 | C4⋊Q8 | — | 6 | 12 | no | — |
| 42 | C4×Q8 | — | 3 | 6 | no | — |
| 43 | C4×Q8 | D12 | 1 | 6 | yes | — |
| 44 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 45 | C2×C4⋊C4 | — | 6 | 6 | no | — |
| 46 | C2×C4⋊C4 | — | 6 | 6 | no | — |
| 47 | C2×C42 | — | 3 | 3 | no | — |
| 48 | C2×C42 | — | 3 | 3 | no | — |
| 49 | C2×C42 | S4 | 1 | 1 | yes | characteristic |
| 50 | C4⋊C8 | — | 6 | 12 | no | — |
| 51 | #103 | — | 6 | 24 | no | — |
| 52 | #95 | — | 6 | 1 | no | — |
| 53 | C23×C3 | — | 4 | 1 | no | — |
| 54 | C2×C3×C4 | — | 4 | 6 | no | — |
| 55 | C2×C3×C4 | — | 4 | 6 | no | — |
| 56 | C2×C3×C4 | — | 4 | 6 | no | — |
| 57 | C2×Dic3 | — | 8 | 12 | no | — |
| 58 | SL(2,𝔽3) | #127 | 1 | 1 | yes | characteristic |
| 59 | C2×Q8 | — | 3 | 3 | no | — |
| 60 | C2×Q8 | — | 6 | 12 | no | — |
| 61 | C2×Q8 | — | 6 | 12 | no | — |
| 62 | C2×Q8 | C2×D12 | 1 | 3 | yes | — |
| 63 | C22×C4 | — | 3 | 1 | no | — |
| 64 | C22×C4 | — | 3 | 3 | no | — |
| 65 | C22×C4 | — | 6 | 4 | no | — |
| 66 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 67 | Q16 | — | 12 | 12 | no | — |
| 68 | Q16 | — | 12 | 4 | no | — |
| 69 | C2×C8 | — | 6 | 12 | no | — |
| 70 | C4⋊C4 | — | 3 | 6 | no | — |
| 71 | C4⋊C4 | — | 6 | 3 | no | — |
| 72 | C4⋊C4 | — | 12 | 12 | no | — |
| 73 | C4⋊C4 | — | 12 | 12 | no | — |
| 74 | C42 | — | 3 | 6 | no | — |
| 75 | C42 | — | 3 | 6 | no | — |
| 76 | C42 | — | 6 | 6 | no | — |
| 77 | C42 | #226 | 1 | 4 | yes | complemented |
| 78 | C22×C3 | — | 4 | 3 | no | — |
| 79 | C22×C3 | — | 4 | 1 | no | — |
| 80 | C22×C3 | — | 4 | 3 | no | — |
| 81 | C3×C4 | — | 4 | 12 | no | — |
| 82 | Dic3 | — | 16 | 4 | no | — |
| 83 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 84 | Q8 | — | 3 | 3 | no | — |
| 85 | Q8 | — | 6 | 1 | no | — |
| 86 | Q8 | — | 12 | 4 | no | — |
| 87 | Q8 | — | 12 | 12 | no | — |
| 88 | Q8 | #874 | 1 | 1 | yes | characteristic |
| 89 | C2×C4 | — | 3 | 3 | no | — |
| 90 | C2×C4 | — | 3 | 3 | no | — |
| 91 | C2×C4 | — | 3 | 6 | no | — |
| 92 | C2×C4 | — | 6 | 6 | no | — |
| 93 | C2×C4 | — | 12 | 12 | no | — |
| 94 | C2×C4 | C4⋊S4 | 1 | 6 | yes | — |
| 95 | C2×C4 | C2×#226 | 1 | 6 | yes | — |
| 96 | C2×C4 | #984 | 1 | 6 | yes | — |
| 97 | C8 | — | 12 | 4 | no | — |
| 98 | C2×C3 | — | 4 | 1 | no | — |
| 99 | C2×C3 | — | 4 | 3 | no | — |
| 100 | C2×C3 | — | 4 | 3 | no | — |
| 101 | C22 | C2×C4⋊S4 | 1 | 3 | yes | — |
| 102 | C22 | C22×#226 | 1 | 1 | yes | characteristic |
| 103 | C22 | C2×#984 | 1 | 3 | yes | — |
| 104 | C4 | — | 3 | 1 | no | — |
| 105 | C4 | — | 3 | 3 | no | — |
| 106 | C4 | — | 6 | 6 | no | — |
| 107 | C4 | — | 24 | 4 | no | — |
| 108 | C4 | C4⋊#226 | 1 | 12 | yes | complemented |
| 109 | C3 | — | 4 | 1 | no | Sylow |
| 110 | C2 | #85289 | 1 | 1 | yes | characteristic |
| 111 | C2 | C2×C4⋊#226 | 1 | 3 | yes | — |
| 112 | C2 | #85336 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C22×S4 |
| commutator, prime radical, cosocle | C22×SL(2,𝔽3) | C23 |
| Fitting | C42×Q8 | D3 |
| hypercenter | C2×C42 | S4 |
| hypocenter | SL(2,𝔽3) | #127 |
| LCS | G ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 77. C42 | #226 | semidirect |
| 108. C4 | C4⋊#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊#226 | 2 |
| C42×SL(2,𝔽3) | 2 |
| #33163 | 3 |
| #4935 | 4 |