Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 76 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. #6432⋊C3 | C2 | semidirect |
| 4. #6381⋊C3 | C2 | semidirect |
| 5. C4⋊(ΓL(1,𝔽8)⋊C8) | C2 | semidirect |
| 6. #89779 | C3 | semidirect |
| 17. #6432 | C2×C3 | semidirect |
| 19. #6381 | C2×C3 | semidirect |
| 20. C4⋊(C7⋊C8) | C2×C3 | semidirect |
| 24. D4×ΓL(1,𝔽8) | Q8 | semidirect |
| 60. C7×D4 | C3×Q8 | semidirect |
| 108. ΓL(1,𝔽8) | #474 | semidirect |
| 135. D4 | ΓL(1,𝔽8)⋊Q8 | semidirect |
| 143. C7 | C3×#474 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D4×ΓL(1,𝔽8) | 2 |
| #6432⋊C3 | 2 |
| #6406⋊C3 | 2 |
| #6381⋊C3 | 2 |
| C4⋊(ΓL(1,𝔽8)⋊C8) | 2 |
| #89779 | 3 |
| C3×#474 | 7 |