Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
516 conjugacy classes of subgroups; only the 357 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 357 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 |
|---|---|---|
| 2. ΓL(1,𝔽8)×#131 | C2 | semidirect |
| 3. C2×M16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C7×#568 | C3 | semidirect |
| 11. M16×ΓL(1,𝔽8) | C22 | semidirect |
| 12. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 14. C7×#131 | C2×C3 | semidirect |
| 15. C2×C7×M16 | C2×C3 | semidirect |
| 31. C7×M16 | C22×C3 | semidirect |
| 32. C2×C7×C8 | C22×C3 | semidirect |
| 42. #568 | ΓL(1,𝔽8) | direct (N × H) |
| 63. #131 | C2×ΓL(1,𝔽8) | semidirect |
| 64. C2×M16 | C2×ΓL(1,𝔽8) | semidirect |
| 79. ΓL(1,𝔽8) | #568 | direct (N × H) |
| 86. M16 | C22×ΓL(1,𝔽8) | semidirect |
| 87. C2×C8 | C22×ΓL(1,𝔽8) | semidirect |
| 105. C7 | C3×#568 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#131 | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| C7×#568 | 3 |
| C3×#568 | 7 |