Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
660 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 296 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 |
|---|---|---|
| 4. C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | C2 | semidirect |
| 5. C2×C4×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 6. #6440⋊C3 | C2 | direct (N × H) |
| 7. C2×#6440 | C3 | semidirect |
| 10. C22×ΓL(1,𝔽8)⋊C4 | C4 | semidirect |
| 17. (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | semidirect |
| 18. C4×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 22. C2×Dic7⋊C4 | C2×C3 | semidirect |
| 23. C2×C4×Dic7 | C2×C3 | semidirect |
| 24. #6440 | C2×C3 | semidirect |
| 35. C2×ΓL(1,𝔽8)⋊C4 | C2×C4 | semidirect |
| 41. C22×Dic7 | C3×C4 | semidirect |
| 48. Dic7⋊C4 | C22×C3 | semidirect |
| 49. C4×Dic7 | C22×C3 | semidirect |
| 76. C2×Dic7 | C2×C3×C4 | semidirect |
| 92. C2×ΓL(1,𝔽8) | #117 | semidirect |
| 124. ΓL(1,𝔽8) | C2×#117 | semidirect |
| 137. C2×C7 | C3×#117 | semidirect |
| 163. C7 | C2×C3×#117 | semidirect |
| 180. C2 | #6440⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×ΓL(1,𝔽8)⋊C4 | 2 |
| C2×C22⋊C4×ΓL(1,𝔽8) | 2 |
| C2×(C22⋊Dic7)⋊C3 | 2 |
| C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | 2 |
| C2×C4×ΓL(1,𝔽8)⋊C4 | 2 |
| #6440⋊C3 | 2 |
| C2×#6440 | 3 |
| C2×C3×#117 | 7 |