Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
196 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 80 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 |
|---|---|---|
| 6. #89627 | C3 | semidirect |
| 25. C8×ΓL(1,𝔽8) | Q8 | semidirect |
| 26. ΓL(1,𝔽8)⋊C8 | Q8 | semidirect |
| 51. C7×C8 | C3×Q8 | semidirect |
| 52. C7⋊C8 | C3×Q8 | semidirect |
| 80. ΓL(1,𝔽8) | #498 | semidirect |
| 102. C8 | ΓL(1,𝔽8)⋊Q8 | semidirect |
| 103. C7 | C3×#498 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6451⋊C3 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| #6391⋊C3 | 2 |
| C8×ΓL(1,𝔽8)⋊C4 | 2 |
| #6382⋊C3 | 2 |
| #89627 | 3 |
| C3×#498 | 7 |