Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
660 conjugacy classes of subgroups; only the 336 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 336 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 |
|---|---|---|
| 3. #6489⋊C3 | C2 | direct (N × H) |
| 4. C2×C4⋊(ΓL(1,𝔽8)⋊C4) | C2 | semidirect |
| 5. C2×C4×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 6. C2×#6489 | C3 | semidirect |
| 8. C22×C4×ΓL(1,𝔽8) | C4 | semidirect |
| 14. C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | semidirect |
| 15. C4×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 18. #6489 | C2×C3 | semidirect |
| 19. C2×C4⋊Dic7 | C2×C3 | semidirect |
| 20. C2×C4×Dic7 | C2×C3 | semidirect |
| 28. C2×C4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 34. C22×C4×C7 | C3×C4 | semidirect |
| 40. C4⋊Dic7 | C22×C3 | semidirect |
| 41. C4×Dic7 | C22×C3 | semidirect |
| 63. C2×C4×C7 | C2×C3×C4 | semidirect |
| 78. C2×ΓL(1,𝔽8) | #117 | semidirect |
| 106. ΓL(1,𝔽8) | C2×#117 | semidirect |
| 109. C22×C4 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 117. C2×C7 | C3×#117 | semidirect |
| 137. C2×C4 | C2×ΓL(1,𝔽8)⋊C4 | semidirect |
| 140. C7 | C2×C3×#117 | semidirect |
| 156. C2 | #6489⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×ΓL(1,𝔽8) | 2 |
| C2×(C22⋊Dic7)⋊C3 | 2 |
| #6489⋊C3 | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| C2×C4×ΓL(1,𝔽8)⋊C4 | 2 |
| C2×#6489 | 3 |
| C2×C3×#117 | 7 |