Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
384 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 168 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)×#125 | C2 | semidirect |
| 4. #6453⋊C3 | C2 | semidirect |
| 5. #6450⋊C3 | C2 | semidirect |
| 7. #6432⋊C3 | C2 | semidirect |
| 8. #90388 | C3 | semidirect |
| 22. C7×#125 | C2×C3 | semidirect |
| 25. #6453 | C2×C3 | semidirect |
| 26. #6450 | C2×C3 | semidirect |
| 28. #6432 | C2×C3 | semidirect |
| 97. #125 | AGL(1,𝔽7) | semidirect |
| 119. ΓL(1,𝔽8) | #541 | semidirect |
| 154. C7 | C3×#541 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#125 | 2 |
| #6461⋊C3 | 2 |
| #6460⋊C3 | 2 |
| #6453⋊C3 | 2 |
| #6450⋊C3 | 2 |
| #6435⋊C3 | 2 |
| #6432⋊C3 | 2 |
| #90388 | 3 |
| C3×#541 | 7 |