Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
112 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 112 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. C2×C16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C7×#350 | C3 | semidirect |
| 13. C7×M32 | C2×C3 | semidirect |
| 14. C2×C7×C16 | C2×C3 | semidirect |
| 39. #350 | ΓL(1,𝔽8) | direct (N × H) |
| 59. M32 | C2×ΓL(1,𝔽8) | semidirect |
| 60. C2×C16 | C2×ΓL(1,𝔽8) | semidirect |
| 74. ΓL(1,𝔽8) | #350 | direct (N × H) |
| 98. C7 | C3×#350 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#131 | 2 |
| ΓL(1,𝔽8)×M32 | 2 |
| C2×C16×ΓL(1,𝔽8) | 2 |
| C7×#350 | 3 |
| C3×#350 | 7 |