Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
149 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 13 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 | 1 | yes | maximal, characteristic |
| 2 | GL(3,𝔽2)⋊C4 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×GL(3,𝔽2) | C4 | 1 | 2 | yes | complemented |
| 4 | C2×GL(3,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 5 | GL(3,𝔽2) | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×ΓL(1,𝔽8)⋊C4 | — | 8 | 1 | no | maximal |
| 7 | C22×S4 | — | 14 | 1 | no | — |
| 8 | C22×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 9 | ΓL(1,𝔽8)⋊C4 | — | 8 | 2 | no | — |
| 10 | C2×#102 | — | 21 | 1 | no | maximal, Sylow |
| 11 | C2×Dic7 | — | 8 | 1 | no | — |
| 12 | C22×A4 | — | 14 | 1 | no | — |
| 13 | C2×S4 | — | 14 | 2 | no | — |
| 14 | C2×S4 | — | 14 | 1 | no | — |
| 15 | C2×S4 | — | 14 | 1 | no | — |
| 16 | C2×S4 | — | 14 | 2 | no | — |
| 17 | C2×C4×D3 | — | 28 | 1 | no | maximal |
| 18 | C2×ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 19 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 20 | C22×D4 | — | 21 | 1 | no | — |
| 21 | C22×C8 | — | 21 | 1 | no | — |
| 22 | C2×C4⋊C4 | — | 21 | 1 | no | — |
| 23 | #102 | — | 21 | 2 | no | — |
| 24 | #102 | — | 21 | 2 | no | — |
| 25 | C22×C7 | — | 8 | 1 | no | — |
| 26 | Dic7 | — | 8 | 2 | no | — |
| 27 | C22×D3 | — | 28 | 1 | no | — |
| 28 | C2×A4 | — | 14 | 2 | no | — |
| 29 | C2×A4 | — | 14 | 1 | no | — |
| 30 | S4 | — | 14 | 1 | no | — |
| 31 | S4 | — | 14 | 2 | no | — |
| 32 | S4 | — | 14 | 1 | no | — |
| 33 | C2×C3×C4 | — | 28 | 1 | no | — |
| 34 | C2×Dic3 | — | 28 | 1 | no | — |
| 35 | C4×D3 | — | 28 | 2 | no | — |
| 36 | C4×D3 | — | 28 | 2 | no | — |
| 37 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 38 | C24 | — | 14 | 1 | no | — |
| 39 | C2×D4 | — | 21 | 1 | no | — |
| 40 | C2×D4 | — | 21 | 2 | no | — |
| 41 | C2×D4 | — | 21 | 1 | no | — |
| 42 | C2×D4 | — | 21 | 2 | no | — |
| 43 | C2×D4 | — | 42 | 1 | no | — |
| 44 | C2×D4 | — | 42 | 2 | no | — |
| 45 | C22×C4 | — | 21 | 1 | no | — |
| 46 | C22×C4 | — | 42 | 1 | no | — |
| 47 | C2×C8 | — | 21 | 2 | no | — |
| 48 | C2×C8 | — | 42 | 2 | no | — |
| 49 | C4⋊C4 | — | 21 | 2 | no | — |
| 50 | C4⋊C4 | — | 42 | 1 | no | — |
| 51 | C2×C7 | — | 8 | 2 | no | — |
| 52 | C2×C7 | — | 8 | 1 | no | — |
| 53 | C22×C3 | — | 28 | 1 | no | — |
| 54 | C2×D3 | — | 28 | 1 | no | — |
| 55 | C2×D3 | — | 28 | 1 | no | — |
| 56 | C2×D3 | — | 28 | 2 | no | — |
| 57 | C2×D3 | — | 28 | 2 | no | — |
| 58 | A4 | — | 14 | 1 | no | — |
| 59 | C3×C4 | — | 28 | 2 | no | — |
| 60 | Dic3 | — | 28 | 2 | no | — |
| 61 | C23 | — | 14 | 2 | no | — |
| 62 | C23 | — | 14 | 1 | no | — |
| 63 | C23 | — | 21 | 1 | no | — |
| 64 | C23 | — | 42 | 2 | no | — |
| 65 | C23 | — | 42 | 1 | no | — |
| 66 | D4 | — | 21 | 1 | no | — |
| 67 | D4 | — | 21 | 1 | no | — |
| 68 | D4 | — | 21 | 2 | no | — |
| 69 | D4 | — | 42 | 1 | no | — |
| 70 | D4 | — | 42 | 2 | no | — |
| 71 | D4 | — | 42 | 1 | no | — |
| 72 | D4 | — | 42 | 2 | no | — |
| 73 | C2×C4 | — | 21 | 1 | no | — |
| 74 | C2×C4 | — | 21 | 1 | no | — |
| 75 | C2×C4 | — | 21 | 2 | no | — |
| 76 | C2×C4 | — | 21 | 2 | no | — |
| 77 | C2×C4 | — | 28 | 1 | no | — |
| 78 | C2×C4 | — | 42 | 2 | no | — |
| 79 | C2×C4 | — | 84 | 1 | no | — |
| 80 | C8 | — | 42 | 2 | no | — |
| 81 | C7 | — | 8 | 1 | no | Sylow |
| 82 | C2×C3 | — | 28 | 2 | no | — |
| 83 | C2×C3 | — | 28 | 1 | no | — |
| 84 | D3 | — | 28 | 1 | no | — |
| 85 | D3 | — | 28 | 2 | no | — |
| 86 | D3 | — | 28 | 1 | no | — |
| 87 | C22 | — | 14 | 1 | no | — |
| 88 | C22 | — | 21 | 2 | no | — |
| 89 | C22 | — | 21 | 2 | no | — |
| 90 | C22 | — | 21 | 1 | no | — |
| 91 | C22 | — | 21 | 1 | no | — |
| 92 | C22 | — | 42 | 2 | no | — |
| 93 | C22 | — | 42 | 1 | no | — |
| 94 | C22 | — | 84 | 1 | no | — |
| 95 | C22 | SO(3,𝔽7) | 1 | 1 | yes | characteristic |
| 96 | C4 | — | 21 | 1 | no | — |
| 97 | C4 | — | 21 | 1 | no | — |
| 98 | C4 | — | 21 | 2 | no | — |
| 99 | C4 | — | 28 | 2 | no | — |
| 100 | C3 | — | 28 | 1 | no | Sylow |
| 101 | C2 | — | 21 | 1 | no | — |
| 102 | C2 | — | 21 | 2 | no | — |
| 103 | C2 | — | 21 | 1 | no | — |
| 104 | C2 | GL(3,𝔽2)⋊C4 | 1 | 2 | yes | direct product factor |
| 105 | C2 | C2×SO(3,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C22 | SO(3,𝔽7) |
| commutator, hypocenter | GL(3,𝔽2) | C2×C4 |
| Frattini | C2 | C2×SO(3,𝔽7) |
| socle, prime radical | C22×GL(3,𝔽2) | C2 |
| cosocle | C2×GL(3,𝔽2) | C22 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. GL(3,𝔽2)⋊C4 | C2 | direct (N × H) |
| 3. C2×GL(3,𝔽2) | C4 | semidirect |
| 5. GL(3,𝔽2) | C2×C4 | semidirect |
| 104. C2 | GL(3,𝔽2)⋊C4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×GL(3,𝔽2) | 2 |
| GL(3,𝔽2)⋊C4 | 2 |
| C2×ΓL(1,𝔽8)⋊C4 | 8 |
| C2×#102 | 21 |
| C2×C4×D3 | 28 |