Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 81 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. C7×#344 | C3 | semidirect |
| 9. M16×ΓL(1,𝔽8) | C4 | semidirect |
| 11. C2×C8×ΓL(1,𝔽8) | C4 | semidirect |
| 33. C7×M16 | C3×C4 | semidirect |
| 35. C2×C7×C8 | C3×C4 | semidirect |
| 51. #344 | ΓL(1,𝔽8) | direct (N × H) |
| 98. ΓL(1,𝔽8) | #344 | direct (N × H) |
| 103. M16 | C4×ΓL(1,𝔽8) | semidirect |
| 105. C2×C8 | C4×ΓL(1,𝔽8) | semidirect |
| 132. C7 | C3×#344 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| C7×#344 | 3 |
| C3×#344 | 7 |