Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
146 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 32 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 |
|---|---|---|
| 1. 2−1+4⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. #131⋊ΓL(1,𝔽8) | C2 | semidirect |
| 3. C7×#578 | C3 | semidirect |
| 7. C7×2−1+4 | C2×C3 | semidirect |
| 11. C7×#131 | C2×C3 | semidirect |
| 31. #578 | ΓL(1,𝔽8) | semidirect |
| 48. 2−1+4 | C2×ΓL(1,𝔽8) | semidirect |
| 52. #131 | C2×ΓL(1,𝔽8) | semidirect |
| 89. C7 | #578⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊ΓL(1,𝔽8) | 2 |
| #131⋊ΓL(1,𝔽8) | 2 |
| C7×#578 | 3 |
| D8×ΓL(1,𝔽8) | 4 |
| #578⋊C3 | 7 |