Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
116 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 51 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 |
|---|---|---|
| 2. ΓL(1,𝔽8)×M32 | C2 | semidirect |
| 3. C7×#349 | C3 | semidirect |
| 10. C7×M32 | C2×C3 | semidirect |
| 32. #349 | ΓL(1,𝔽8) | direct (N × H) |
| 49. M32 | C2×ΓL(1,𝔽8) | semidirect |
| 63. ΓL(1,𝔽8) | #349 | direct (N × H) |
| 85. C7 | C3×#349 | 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)×M32 | 2 |
| C7×#349 | 3 |
| C3×#349 | 7 |