Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 87 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 |
|---|---|---|
| 6. C7×#462 | C3 | semidirect |
| 54. C4×ΓL(1,𝔽8) | Q16 | semidirect |
| 57. #462 | ΓL(1,𝔽8) | direct (N × H) |
| 92. C4×C7 | C3×Q16 | semidirect |
| 106. ΓL(1,𝔽8) | #462 | direct (N × H) |
| 139. C7 | C3×#462 | semidirect |
| 148. C4 | Q16×ΓL(1,𝔽8) | 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 |
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| C7×#462 | 3 |
| C3×#462 | 7 |