Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
168 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 58 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. ΓL(1,𝔽8)×#135 | C2 | semidirect |
| 2. #6467⋊C3 | C2 | semidirect |
| 3. #6403⋊C3 | C2 | semidirect |
| 4. #6402⋊C3 | C2 | semidirect |
| 5. #6401⋊C3 | C2 | semidirect |
| 6. #6400⋊C3 | C2 | semidirect |
| 7. C2×ΓL(1,𝔽8)⋊C16 | C2 | semidirect |
| 8. #89966 | C3 | semidirect |
| 20. C7×#135 | C2×C3 | semidirect |
| 21. #6467 | C2×C3 | semidirect |
| 22. #6403 | C2×C3 | semidirect |
| 23. #6402 | C2×C3 | semidirect |
| 24. #6401 | C2×C3 | semidirect |
| 25. #6400 | C2×C3 | semidirect |
| 26. C2×C7⋊C16 | C2×C3 | semidirect |
| 92. #135 | AGL(1,𝔽7) | semidirect |
| 116. ΓL(1,𝔽8) | #508 | semidirect |
| 150. C7 | C3×#508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#135 | 2 |
| #6467⋊C3 | 2 |
| #6403⋊C3 | 2 |
| #6402⋊C3 | 2 |
| #6401⋊C3 | 2 |
| #6400⋊C3 | 2 |
| C2×ΓL(1,𝔽8)⋊C16 | 2 |
| #89966 | 3 |
| C3×#508 | 7 |