Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
344 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | ΓL(1,𝔽8)×#125 | C2 | 1 | 15 | yes | maximal |
| 2 | C7×#564 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C4⋊C4×ΓL(1,𝔽8) | C22 | 1 | 30 | yes | — |
| 4 | C42×ΓL(1,𝔽8) | C22 | 1 | 5 | yes | — |
| 5 | C7×#125 | C2×C3 | 1 | 15 | yes | — |
| 6 | C3×#564 | — | 7 | 1 | no | maximal, Hall |
| 7 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 15 | yes | — |
| 8 | C7×C4⋊C4 | C22×C3 | 1 | 30 | yes | — |
| 9 | C42×C7 | C22×C3 | 1 | 5 | yes | — |
| 10 | C3×#125 | — | 7 | 15 | no | — |
| 11 | C22×ΓL(1,𝔽8) | C24 | 1 | 1 | yes | characteristic |
| 12 | C4×ΓL(1,𝔽8) | — | 2 | 15 | no | — |
| 13 | #564 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C2×C4×C7 | C23×C3 | 1 | 15 | yes | — |
| 15 | C3×C4⋊C4 | — | 7 | 30 | no | — |
| 16 | C3×C42 | — | 7 | 5 | no | — |
| 17 | C2×ΓL(1,𝔽8) | 2−1+4 | 1 | 3 | yes | — |
| 18 | #125 | C2×ΓL(1,𝔽8) | 1 | 15 | yes | — |
| 19 | C22×C7 | C24×C3 | 1 | 1 | yes | characteristic |
| 20 | C4×C7 | — | 2 | 15 | no | — |
| 21 | C2×C3×C4 | — | 7 | 15 | no | — |
| 22 | ΓL(1,𝔽8) | #564 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 23 | C4⋊C4 | C22×ΓL(1,𝔽8) | 1 | 30 | yes | — |
| 24 | C42 | C22×ΓL(1,𝔽8) | 1 | 5 | yes | — |
| 25 | C2×C7 | C3×2−1+4 | 1 | 3 | yes | — |
| 26 | C22×C3 | — | 7 | 1 | no | — |
| 27 | C3×C4 | — | 14 | 15 | no | — |
| 28 | C2×C4 | C23×ΓL(1,𝔽8) | 1 | 15 | yes | — |
| 29 | C7 | C3×#564 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 30 | C2×C3 | — | 7 | 3 | no | — |
| 31 | C22 | C24×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 32 | C4 | — | 2 | 15 | no | — |
| 33 | C3 | — | 7 | 1 | no | Sylow |
| 34 | C2 | ΓL(1,𝔽8)×2−1+4 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#564 | C3 | semidirect |
| 13. #564 | ΓL(1,𝔽8) | direct (N × H) |
| 22. ΓL(1,𝔽8) | #564 | direct (N × H) |
| 29. C7 | C3×#564 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#125 | 2 |
| C7×#564 | 3 |
| C3×#564 | 7 |