Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
248 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 68 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)×#124 | C2 | semidirect |
| 2. C2×#1143⋊C3 | C2 | semidirect |
| 4. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 5. #6409⋊C3 | C2 | semidirect |
| 6. #6406⋊C3 | C2 | semidirect |
| 7. C4⋊(ΓL(1,𝔽8)⋊C8) | C2 | semidirect |
| 8. #89829 | C3 | semidirect |
| 22. C7×#124 | C2×C3 | semidirect |
| 23. C2×#1143 | C2×C3 | semidirect |
| 25. C4×D28 | C2×C3 | semidirect |
| 26. #6409 | C2×C3 | semidirect |
| 27. #6406 | C2×C3 | semidirect |
| 28. C4⋊(C7⋊C8) | C2×C3 | semidirect |
| 116. #124 | AGL(1,𝔽7) | semidirect |
| 150. ΓL(1,𝔽8) | #463 | semidirect |
| 198. C7 | C3×#463 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#124 | 2 |
| C2×#1143⋊C3 | 2 |
| C2×#1141⋊C3 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #6409⋊C3 | 2 |
| #6406⋊C3 | 2 |
| C4⋊(ΓL(1,𝔽8)⋊C8) | 2 |
| #89829 | 3 |
| C3×#463 | 7 |