Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
251 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 12 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 | C22×GL(3,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C4×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×GL(3,𝔽2) | — | 2 | 2 | no | — |
| 4 | C2×GL(3,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 5 | D4×S4 | — | 7 | 2 | no | maximal, Hall |
| 6 | GL(3,𝔽2) | D4 | 1 | 1 | yes | characteristic, direct product factor |
| 7 | D4×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 8 | C22×S4 | — | 7 | 4 | no | — |
| 9 | D4×A4 | — | 7 | 2 | no | — |
| 10 | GL(2,ℤ/4) | — | 7 | 4 | no | — |
| 11 | C4⋊S4 | — | 7 | 2 | no | — |
| 12 | C4×S4 | — | 7 | 2 | no | — |
| 13 | C22×ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 14 | C4×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 15 | D42 | — | 21 | 1 | no | Sylow |
| 16 | C7×D4 | — | 8 | 1 | no | — |
| 17 | C22×A4 | — | 7 | 4 | no | — |
| 18 | C2×S4 | — | 7 | 2 | no | — |
| 19 | C2×S4 | — | 7 | 4 | no | — |
| 20 | C2×S4 | — | 14 | 4 | no | — |
| 21 | C2×S4 | — | 14 | 4 | no | — |
| 22 | D3×D4 | — | 28 | 1 | no | — |
| 23 | C4×A4 | — | 7 | 2 | no | — |
| 24 | SL(2,ℤ/4) | — | 7 | 2 | no | — |
| 25 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 26 | C2×ΓL(1,𝔽8) | — | 16 | 2 | no | — |
| 27 | C22×D4 | — | 7 | 2 | no | — |
| 28 | C22×D4 | — | 21 | 2 | no | — |
| 29 | #127 | — | 21 | 1 | no | — |
| 30 | #121 | — | 21 | 2 | no | — |
| 31 | #121 | — | 21 | 2 | no | — |
| 32 | #120 | — | 21 | 4 | no | — |
| 33 | C4×D4 | — | 21 | 1 | no | — |
| 34 | C4×D4 | — | 21 | 1 | no | — |
| 35 | C22×C7 | — | 8 | 2 | no | — |
| 36 | C4×C7 | — | 8 | 1 | no | — |
| 37 | C22×D3 | — | 28 | 2 | no | — |
| 38 | C2×A4 | — | 7 | 2 | no | — |
| 39 | C2×A4 | — | 14 | 4 | no | — |
| 40 | S4 | — | 7 | 2 | no | — |
| 41 | S4 | — | 7 | 2 | no | — |
| 42 | S4 | — | 14 | 4 | no | — |
| 43 | C3×D4 | — | 28 | 1 | no | — |
| 44 | #67 | — | 28 | 2 | no | — |
| 45 | D12 | — | 28 | 1 | no | — |
| 46 | C4×D3 | — | 28 | 1 | no | — |
| 47 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 48 | C24 | — | 7 | 4 | no | — |
| 49 | C2×D4 | — | 21 | 1 | no | — |
| 50 | C2×D4 | — | 21 | 1 | no | — |
| 51 | C2×D4 | — | 21 | 4 | no | — |
| 52 | C2×D4 | — | 21 | 2 | no | — |
| 53 | C2×D4 | — | 21 | 4 | no | — |
| 54 | C2×D4 | — | 21 | 2 | no | — |
| 55 | C2×D4 | — | 21 | 2 | no | — |
| 56 | C2×D4 | — | 42 | 2 | no | — |
| 57 | C2×D4 | — | 42 | 4 | no | — |
| 58 | C2×D4 | — | 42 | 2 | no | — |
| 59 | C22×C4 | — | 7 | 2 | no | — |
| 60 | C22×C4 | — | 21 | 2 | no | — |
| 61 | C4⋊C4 | — | 21 | 1 | no | — |
| 62 | C4⋊C4 | — | 21 | 1 | no | — |
| 63 | C22⋊C4 | — | 21 | 2 | no | — |
| 64 | C22⋊C4 | — | 21 | 2 | no | — |
| 65 | C22⋊C4 | — | 21 | 4 | no | — |
| 66 | C42 | — | 21 | 1 | no | — |
| 67 | C2×C7 | — | 8 | 1 | no | — |
| 68 | C2×C7 | — | 16 | 2 | no | — |
| 69 | C22×C3 | — | 28 | 2 | no | — |
| 70 | C2×D3 | — | 28 | 1 | no | — |
| 71 | C2×D3 | — | 28 | 2 | no | — |
| 72 | C2×D3 | — | 56 | 2 | no | — |
| 73 | C2×D3 | — | 56 | 2 | no | — |
| 74 | A4 | — | 7 | 2 | no | — |
| 75 | C3×C4 | — | 28 | 1 | no | — |
| 76 | Dic3 | — | 28 | 1 | no | — |
| 77 | C23 | — | 7 | 2 | no | — |
| 78 | C23 | — | 14 | 4 | no | — |
| 79 | C23 | — | 21 | 4 | no | — |
| 80 | C23 | — | 21 | 2 | no | — |
| 81 | C23 | — | 42 | 4 | no | — |
| 82 | D4 | — | 21 | 1 | no | — |
| 83 | D4 | — | 21 | 2 | no | — |
| 84 | D4 | — | 21 | 1 | no | — |
| 85 | D4 | — | 21 | 2 | no | — |
| 86 | D4 | — | 21 | 1 | no | — |
| 87 | D4 | — | 42 | 2 | no | — |
| 88 | D4 | — | 42 | 2 | no | — |
| 89 | D4 | — | 42 | 2 | no | — |
| 90 | D4 | — | 42 | 4 | no | — |
| 91 | D4 | — | 42 | 4 | no | — |
| 92 | D4 | — | 84 | 2 | no | — |
| 93 | D4 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 94 | C2×C4 | — | 21 | 1 | no | — |
| 95 | C2×C4 | — | 21 | 1 | no | — |
| 96 | C2×C4 | — | 21 | 2 | no | — |
| 97 | C2×C4 | — | 21 | 1 | no | — |
| 98 | C2×C4 | — | 21 | 2 | no | — |
| 99 | C2×C4 | — | 42 | 2 | no | — |
| 100 | C2×C4 | — | 42 | 2 | no | — |
| 101 | C7 | — | 8 | 1 | no | Sylow |
| 102 | C2×C3 | — | 28 | 1 | no | — |
| 103 | C2×C3 | — | 56 | 2 | no | — |
| 104 | D3 | — | 28 | 1 | no | — |
| 105 | D3 | — | 28 | 1 | no | — |
| 106 | D3 | — | 56 | 2 | no | — |
| 107 | C22 | — | 7 | 2 | no | — |
| 108 | C22 | — | 21 | 2 | no | — |
| 109 | C22 | — | 21 | 1 | no | — |
| 110 | C22 | — | 21 | 2 | no | — |
| 111 | C22 | — | 42 | 2 | no | — |
| 112 | C22 | — | 42 | 2 | no | — |
| 113 | C22 | — | 42 | 4 | no | — |
| 114 | C22 | — | 42 | 4 | no | — |
| 115 | C22 | C2×GL(3,𝔽2) | 1 | 2 | yes | complemented |
| 116 | C4 | — | 21 | 1 | no | — |
| 117 | C4 | — | 21 | 1 | no | — |
| 118 | C4 | — | 21 | 1 | no | — |
| 119 | C4 | — | 42 | 2 | no | — |
| 120 | C4 | — | 42 | 1 | no | — |
| 121 | C4 | C2×GL(3,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 122 | C3 | — | 28 | 1 | no | Sylow |
| 123 | C2 | — | 2 | 2 | no | — |
| 124 | C2 | — | 21 | 1 | no | — |
| 125 | C2 | — | 21 | 1 | no | — |
| 126 | C2 | — | 42 | 2 | no | — |
| 127 | C2 | C22×GL(3,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2 | C22×GL(3,𝔽2) |
| commutator, socle | C2×GL(3,𝔽2) | C22 |
| Fitting, hypercenter | D4 | GL(3,𝔽2) |
| hypocenter | GL(3,𝔽2) | D4 |
| LCS | G ▷ C2×GL(3,𝔽2) ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×GL(3,𝔽2) | C2 | semidirect |
| 2. C4×GL(3,𝔽2) | C2 | semidirect |
| 6. GL(3,𝔽2) | D4 | direct (N × H) |
| 93. D4 | GL(3,𝔽2) | direct (N × H) |
| 115. C22 | C2×GL(3,𝔽2) | semidirect |
| 121. C4 | C2×GL(3,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×GL(3,𝔽2) | 2 |
| C4×GL(3,𝔽2) | 2 |
| D4×S4 | 7 |
| D4×ΓL(1,𝔽8) | 8 |