Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
110 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×SL(2,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10621462 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | SL(2,𝔽7) | C22 | 1 | 1 | yes | characteristic, complemented |
| 4 | C22×AGL(1,𝔽7) | — | 8 | 1 | no | maximal |
| 5 | C2×#226 | — | 14 | 1 | no | — |
| 6 | C22×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 7 | C2×AGL(1,𝔽7) | — | 8 | 2 | no | — |
| 8 | C2×AGL(1,𝔽7) | — | 8 | 4 | no | — |
| 9 | C2×SD32 | — | 21 | 1 | no | maximal, Sylow |
| 10 | C22×D7 | — | 8 | 1 | no | — |
| 11 | C2×#67 | — | 28 | 1 | no | maximal |
| 12 | C2×SL(2,𝔽3) | — | 14 | 1 | no | — |
| 13 | #226 | — | 14 | 1 | no | — |
| 14 | #226 | — | 14 | 1 | no | — |
| 15 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 16 | C2×ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 17 | AGL(1,𝔽7) | — | 8 | 4 | no | — |
| 18 | C2×Q16 | — | 21 | 1 | no | — |
| 19 | C2×D8 | — | 21 | 1 | no | — |
| 20 | SD32 | — | 21 | 2 | no | — |
| 21 | SD32 | — | 21 | 2 | no | — |
| 22 | C2×C16 | — | 21 | 1 | no | — |
| 23 | C22×C7 | — | 8 | 1 | no | — |
| 24 | C2×D7 | — | 8 | 2 | no | — |
| 25 | C2×D7 | — | 8 | 4 | no | — |
| 26 | C23×C3 | — | 28 | 1 | no | — |
| 27 | C22×D3 | — | 28 | 1 | no | — |
| 28 | #67 | — | 28 | 2 | no | — |
| 29 | #67 | — | 28 | 2 | no | — |
| 30 | C2×Dic3 | — | 28 | 1 | no | — |
| 31 | SL(2,𝔽3) | — | 14 | 1 | no | — |
| 32 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 33 | C2×Q8 | — | 14 | 1 | no | — |
| 34 | C2×D4 | — | 42 | 1 | no | — |
| 35 | Q16 | — | 21 | 1 | no | — |
| 36 | Q16 | — | 21 | 1 | no | — |
| 37 | Q16 | — | 42 | 1 | no | — |
| 38 | D8 | — | 21 | 2 | no | — |
| 39 | D8 | — | 42 | 1 | no | — |
| 40 | C2×C8 | — | 21 | 1 | no | — |
| 41 | C16 | — | 21 | 2 | no | — |
| 42 | C2×C7 | — | 8 | 2 | no | — |
| 43 | C2×C7 | — | 8 | 1 | no | — |
| 44 | D7 | — | 8 | 4 | no | — |
| 45 | C22×C3 | — | 28 | 2 | no | — |
| 46 | C22×C3 | — | 28 | 1 | no | — |
| 47 | C22×C3 | — | 56 | 2 | no | — |
| 48 | C2×D3 | — | 28 | 2 | no | — |
| 49 | C2×D3 | — | 56 | 2 | no | — |
| 50 | Dic3 | — | 28 | 1 | no | — |
| 51 | Dic3 | — | 28 | 1 | no | — |
| 52 | C23 | — | 28 | 1 | no | — |
| 53 | Q8 | — | 14 | 1 | no | — |
| 54 | Q8 | — | 42 | 1 | no | — |
| 55 | D4 | — | 42 | 2 | no | — |
| 56 | D4 | — | 84 | 1 | no | — |
| 57 | C2×C4 | — | 21 | 1 | no | — |
| 58 | C8 | — | 21 | 1 | no | — |
| 59 | C8 | — | 21 | 1 | no | — |
| 60 | C7 | — | 8 | 1 | no | Sylow |
| 61 | C2×C3 | — | 28 | 2 | no | — |
| 62 | C2×C3 | — | 28 | 1 | no | — |
| 63 | C2×C3 | — | 56 | 2 | no | — |
| 64 | D3 | — | 56 | 2 | no | — |
| 65 | C22 | — | 28 | 2 | no | — |
| 66 | C22 | — | 56 | 2 | no | — |
| 67 | C22 | SO(3,𝔽7) | 1 | 1 | yes | characteristic |
| 68 | C4 | — | 21 | 1 | no | — |
| 69 | C4 | — | 21 | 1 | no | — |
| 70 | C3 | — | 28 | 1 | no | Sylow |
| 71 | C2 | — | 56 | 2 | no | — |
| 72 | C2 | #10621462 | 1 | 2 | yes | direct product factor |
| 73 | C2 | C2×SO(3,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C22 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | SL(2,𝔽7) | C22 |
| Frattini | C2 | C2×SO(3,𝔽7) |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SL(2,𝔽7) | C2 | semidirect |
| 2. #10621462 | C2 | direct (N × H) |
| 3. SL(2,𝔽7) | C22 | semidirect |
| 72. C2 | #10621462 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SL(2,𝔽7) | 2 |
| #10621462 | 2 |
| C22×AGL(1,𝔽7) | 8 |
| C2×SD32 | 21 |
| C2×#67 | 28 |