Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
905 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 905 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⋊GL(2,𝔽3) | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C42×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33161 | — | 3 | 1 | no | maximal, Sylow |
| 4 | C22×GL(2,𝔽3) | C22 | 1 | 4 | yes | — |
| 5 | C2×#874 | — | 4 | 1 | no | maximal |
| 6 | C2×C4×SL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 7 | C4⋊GL(2,𝔽3) | — | 2 | 12 | no | — |
| 8 | C2×#460 | — | 3 | 6 | no | — |
| 9 | #2867 | — | 3 | 1 | no | — |
| 10 | C42×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #1982 | — | 3 | 6 | no | — |
| 12 | #1842 | — | 3 | 1 | no | — |
| 13 | C22×D12 | — | 4 | 6 | no | — |
| 14 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 15 | C2×GL(2,𝔽3) | — | 2 | 12 | no | — |
| 16 | C2×C3×C42 | — | 4 | 1 | no | — |
| 17 | #874 | — | 4 | 8 | no | — |
| 18 | C4×SL(2,𝔽3) | D4 | 1 | 6 | yes | complemented |
| 19 | C22×SD16 | — | 3 | 4 | no | — |
| 20 | C2×#127 | — | 3 | 6 | no | — |
| 21 | C2×#127 | — | 6 | 3 | no | — |
| 22 | C2×#127 | — | 6 | 1 | no | — |
| 23 | C2×C4×Q8 | C2×D3 | 1 | 3 | yes | — |
| 24 | #460 | — | 6 | 12 | no | — |
| 25 | #460 | — | 6 | 12 | no | — |
| 26 | C2×C4⋊C8 | — | 3 | 6 | no | — |
| 27 | C2×#102 | — | 3 | 12 | no | — |
| 28 | C4×C4⋊C4 | — | 3 | 3 | no | — |
| 29 | C43 | — | 3 | 1 | no | — |
| 30 | C23×D3 | — | 4 | 4 | no | — |
| 31 | C22×C3×C4 | — | 4 | 3 | no | — |
| 32 | C2×D12 | — | 4 | 24 | no | — |
| 33 | C2×D12 | — | 8 | 12 | no | — |
| 34 | C2×D12 | — | 8 | 12 | no | — |
| 35 | C2×SL(2,𝔽3) | C2×D4 | 1 | 3 | yes | — |
| 36 | GL(2,𝔽3) | — | 4 | 4 | no | — |
| 37 | C3×C42 | — | 4 | 4 | no | — |
| 38 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 39 | C22×D4 | — | 3 | 4 | no | — |
| 40 | C22×D4 | — | 6 | 6 | no | — |
| 41 | C22×D4 | — | 6 | 6 | no | — |
| 42 | C2×SD16 | — | 6 | 12 | no | — |
| 43 | C2×SD16 | — | 6 | 12 | no | — |
| 44 | C22×C8 | — | 3 | 4 | no | — |
| 45 | #127 | — | 6 | 12 | no | — |
| 46 | #127 | — | 6 | 12 | no | — |
| 47 | #127 | — | 12 | 12 | no | — |
| 48 | #127 | — | 12 | 4 | no | — |
| 49 | C4×Q8 | — | 3 | 6 | no | — |
| 50 | C4×Q8 | D12 | 1 | 6 | yes | complemented |
| 51 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 52 | C2×C42 | — | 3 | 3 | no | — |
| 53 | C2×C42 | — | 3 | 3 | no | — |
| 54 | C2×C42 | S4 | 1 | 1 | yes | characteristic |
| 55 | C4⋊C8 | — | 6 | 12 | no | — |
| 56 | #102 | — | 6 | 24 | no | — |
| 57 | #95 | — | 6 | 1 | no | — |
| 58 | C23×C3 | — | 4 | 1 | no | — |
| 59 | C22×D3 | — | 4 | 8 | no | — |
| 60 | C22×D3 | — | 8 | 12 | no | — |
| 61 | C22×D3 | — | 8 | 12 | no | — |
| 62 | C2×C3×C4 | — | 4 | 6 | no | — |
| 63 | C2×C3×C4 | — | 4 | 6 | no | — |
| 64 | C2×C3×C4 | — | 4 | 6 | no | — |
| 65 | D12 | — | 8 | 48 | no | — |
| 66 | SL(2,𝔽3) | #127 | 1 | 1 | yes | characteristic, complemented |
| 67 | C24 | — | 6 | 4 | no | — |
| 68 | C2×Q8 | — | 3 | 3 | no | — |
| 69 | C2×Q8 | C2×D12 | 1 | 3 | yes | — |
| 70 | C2×D4 | — | 6 | 12 | no | — |
| 71 | C2×D4 | — | 6 | 12 | no | — |
| 72 | C2×D4 | — | 12 | 12 | no | — |
| 73 | C2×D4 | — | 12 | 24 | no | — |
| 74 | C2×D4 | — | 12 | 12 | no | — |
| 75 | C2×D4 | — | 12 | 12 | no | — |
| 76 | C2×D4 | — | 12 | 12 | no | — |
| 77 | C22×C4 | — | 3 | 1 | no | — |
| 78 | C22×C4 | — | 3 | 3 | no | — |
| 79 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 80 | SD16 | — | 12 | 12 | no | — |
| 81 | SD16 | — | 12 | 4 | no | — |
| 82 | C2×C8 | — | 6 | 12 | no | — |
| 83 | C4⋊C4 | — | 3 | 6 | no | — |
| 84 | C4⋊C4 | — | 6 | 3 | no | — |
| 85 | C42 | — | 3 | 6 | no | — |
| 86 | C42 | — | 3 | 6 | no | — |
| 87 | C42 | — | 6 | 6 | no | — |
| 88 | C42 | GL(2,𝔽3) | 1 | 4 | yes | complemented |
| 89 | C22×C3 | — | 4 | 3 | no | — |
| 90 | C22×C3 | — | 4 | 1 | no | — |
| 91 | C22×C3 | — | 4 | 3 | no | — |
| 92 | C2×D3 | — | 8 | 24 | no | — |
| 93 | C2×D3 | — | 16 | 12 | no | — |
| 94 | C2×D3 | — | 16 | 4 | no | — |
| 95 | C3×C4 | — | 4 | 12 | no | — |
| 96 | C23 | — | 12 | 12 | no | — |
| 97 | C23 | — | 12 | 4 | no | — |
| 98 | C23 | — | 12 | 12 | no | — |
| 99 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 100 | Q8 | — | 3 | 3 | no | — |
| 101 | Q8 | — | 6 | 1 | no | — |
| 102 | Q8 | #874 | 1 | 1 | yes | characteristic, complemented |
| 103 | D4 | — | 12 | 4 | no | — |
| 104 | D4 | — | 12 | 12 | no | — |
| 105 | D4 | — | 24 | 24 | no | — |
| 106 | D4 | — | 24 | 24 | no | — |
| 107 | C2×C4 | — | 3 | 3 | no | — |
| 108 | C2×C4 | — | 3 | 3 | no | — |
| 109 | C2×C4 | — | 3 | 6 | no | — |
| 110 | C2×C4 | — | 6 | 6 | no | — |
| 111 | C2×C4 | C4⋊S4 | 1 | 6 | yes | — |
| 112 | C2×C4 | C2×GL(2,𝔽3) | 1 | 6 | yes | — |
| 113 | C2×C4 | #986 | 1 | 6 | yes | — |
| 114 | C8 | — | 12 | 4 | no | — |
| 115 | C2×C3 | — | 4 | 1 | no | — |
| 116 | C2×C3 | — | 4 | 3 | no | — |
| 117 | C2×C3 | — | 4 | 3 | no | — |
| 118 | D3 | — | 16 | 8 | no | — |
| 119 | C22 | — | 24 | 4 | no | — |
| 120 | C22 | — | 24 | 12 | no | — |
| 121 | C22 | — | 24 | 12 | no | — |
| 122 | C22 | C2×C4⋊S4 | 1 | 3 | yes | — |
| 123 | C22 | C22×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 124 | C22 | C2×#986 | 1 | 3 | yes | — |
| 125 | C4 | — | 3 | 1 | no | — |
| 126 | C4 | — | 3 | 3 | no | — |
| 127 | C4 | — | 6 | 6 | no | — |
| 128 | C4 | C4⋊GL(2,𝔽3) | 1 | 12 | yes | complemented |
| 129 | C3 | — | 4 | 1 | no | Sylow |
| 130 | C2 | — | 48 | 4 | no | — |
| 131 | C2 | #85289 | 1 | 1 | yes | characteristic |
| 132 | C2 | C2×C4⋊GL(2,𝔽3) | 1 | 3 | yes | — |
| 133 | C2 | #85346 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C22×S4 |
| commutator, 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 |
|---|---|---|
| 1. C2×C4⋊GL(2,𝔽3) | C2 | semidirect |
| 2. C42×SL(2,𝔽3) | C2 | semidirect |
| 10. C42×Q8 | D3 | semidirect |
| 18. C4×SL(2,𝔽3) | D4 | semidirect |
| 50. C4×Q8 | D12 | semidirect |
| 66. SL(2,𝔽3) | #127 | semidirect |
| 88. C42 | GL(2,𝔽3) | semidirect |
| 102. Q8 | #874 | semidirect |
| 128. C4 | C4⋊GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊GL(2,𝔽3) | 2 |
| C42×SL(2,𝔽3) | 2 |
| #33161 | 3 |
| C2×#874 | 4 |