Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
171 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×GL(3,𝔽2) | C2 | 1 | 3 | yes | maximal |
| 2 | C2×GL(3,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 3 | Q8×S4 | — | 7 | 2 | no | maximal, Hall |
| 4 | GL(3,𝔽2) | Q8 | 1 | 1 | yes | characteristic, direct product factor |
| 5 | Q8×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 6 | Q8×A4 | — | 7 | 2 | no | — |
| 7 | C4×S4 | — | 7 | 6 | no | — |
| 8 | A4⋊Q8 | — | 7 | 6 | no | — |
| 9 | C4×ΓL(1,𝔽8) | — | 8 | 3 | no | — |
| 10 | D4×Q8 | — | 21 | 1 | no | Sylow |
| 11 | C7×Q8 | — | 8 | 1 | no | — |
| 12 | C2×S4 | — | 7 | 2 | no | — |
| 13 | D3×Q8 | — | 28 | 1 | no | — |
| 14 | C4×A4 | — | 7 | 6 | no | — |
| 15 | SL(2,ℤ/4) | — | 7 | 6 | no | — |
| 16 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 17 | C22×Q8 | — | 7 | 2 | no | — |
| 18 | C4⋊Q8 | — | 21 | 3 | no | — |
| 19 | C22⋊Q8 | — | 21 | 6 | no | — |
| 20 | C4×Q8 | — | 21 | 1 | no | — |
| 21 | C4×D4 | — | 21 | 3 | no | — |
| 22 | C4×C7 | — | 8 | 3 | no | — |
| 23 | C2×A4 | — | 7 | 2 | no | — |
| 24 | S4 | — | 7 | 2 | no | — |
| 25 | S4 | — | 7 | 2 | no | — |
| 26 | C3×Q8 | — | 28 | 1 | no | — |
| 27 | C4×D3 | — | 28 | 3 | no | — |
| 28 | Dic6 | — | 28 | 3 | no | — |
| 29 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 30 | C2×Q8 | — | 21 | 1 | no | — |
| 31 | C2×Q8 | — | 21 | 6 | no | — |
| 32 | C2×D4 | — | 21 | 1 | no | — |
| 33 | C22×C4 | — | 7 | 6 | no | — |
| 34 | C4⋊C4 | — | 21 | 3 | no | — |
| 35 | C4⋊C4 | — | 21 | 3 | no | — |
| 36 | C4⋊C4 | — | 21 | 6 | no | — |
| 37 | C22⋊C4 | — | 21 | 6 | no | — |
| 38 | C42 | — | 21 | 3 | no | — |
| 39 | C2×C7 | — | 8 | 1 | no | — |
| 40 | C2×D3 | — | 28 | 1 | no | — |
| 41 | A4 | — | 7 | 2 | no | — |
| 42 | C3×C4 | — | 28 | 3 | no | — |
| 43 | Dic3 | — | 28 | 3 | no | — |
| 44 | C23 | — | 7 | 2 | no | — |
| 45 | Q8 | — | 21 | 3 | no | — |
| 46 | Q8 | — | 42 | 2 | no | — |
| 47 | Q8 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 48 | D4 | — | 21 | 1 | no | — |
| 49 | D4 | — | 21 | 2 | no | — |
| 50 | D4 | — | 21 | 1 | no | — |
| 51 | C2×C4 | — | 21 | 1 | no | — |
| 52 | C2×C4 | — | 21 | 3 | no | — |
| 53 | C2×C4 | — | 21 | 6 | no | — |
| 54 | C2×C4 | — | 21 | 3 | no | — |
| 55 | C7 | — | 8 | 1 | no | Sylow |
| 56 | C2×C3 | — | 28 | 1 | no | — |
| 57 | D3 | — | 28 | 1 | no | — |
| 58 | D3 | — | 28 | 1 | no | — |
| 59 | C22 | — | 7 | 2 | no | — |
| 60 | C22 | — | 21 | 2 | no | — |
| 61 | C22 | — | 21 | 1 | no | — |
| 62 | C4 | — | 21 | 1 | no | — |
| 63 | C4 | — | 21 | 1 | no | — |
| 64 | C4 | — | 21 | 3 | no | — |
| 65 | C4 | — | 42 | 3 | no | — |
| 66 | C4 | C2×GL(3,𝔽2) | 1 | 3 | yes | — |
| 67 | C3 | — | 28 | 1 | no | Sylow |
| 68 | C2 | — | 21 | 1 | no | — |
| 69 | C2 | — | 21 | 1 | no | — |
| 70 | C2 | C22×GL(3,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2 | C22×GL(3,𝔽2) |
| commutator, socle, prime radical | C2×GL(3,𝔽2) | C22 |
| Fitting, hypercenter | Q8 | GL(3,𝔽2) |
| hypocenter | GL(3,𝔽2) | Q8 |
| LCS | G ▷ C2×GL(3,𝔽2) ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. GL(3,𝔽2) | Q8 | direct (N × H) |
| 47. Q8 | GL(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×GL(3,𝔽2) | 2 |
| Q8×S4 | 7 |
| Q8×ΓL(1,𝔽8) | 8 |