Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 228 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)×C4⋊C8 | C2 | semidirect |
| 5. C7×#424 | C3 | semidirect |
| 17. C7×C4⋊C8 | C2×C3 | semidirect |
| 46. #424 | ΓL(1,𝔽8) | direct (N × H) |
| 69. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 85. ΓL(1,𝔽8) | #424 | direct (N × H) |
| 112. C7 | C3×#424 | 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 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| C7×#424 | 3 |
| C3×#424 | 7 |