Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
300 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 5. C7×#419 | C3 | semidirect |
| 8. C2×Q8×ΓL(1,𝔽8) | C4 | semidirect |
| 20. C7×#103 | C2×C3 | semidirect |
| 36. C2×C7×Q8 | C3×C4 | semidirect |
| 57. #419 | ΓL(1,𝔽8) | direct (N × H) |
| 87. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 108. ΓL(1,𝔽8) | #419 | direct (N × H) |
| 111. C2×Q8 | C4×ΓL(1,𝔽8) | semidirect |
| 143. C7 | C3×#419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| C7×#419 | 3 |
| C3×#419 | 7 |