Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
220 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 81 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. C2×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 6. C7×#470 | C3 | semidirect |
| 16. C2×C7×Q16 | C2×C3 | semidirect |
| 19. C7×#106 | C2×C3 | semidirect |
| 20. C7×#103 | C2×C3 | semidirect |
| 55. #470 | ΓL(1,𝔽8) | direct (N × H) |
| 79. C2×Q16 | C2×ΓL(1,𝔽8) | semidirect |
| 82. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 83. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 103. ΓL(1,𝔽8) | #470 | direct (N × H) |
| 135. C7 | C3×#470 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×ΓL(1,𝔽8) | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| C7×#470 | 3 |
| C3×#470 | 7 |