Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 92 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. #89956 | C3 | semidirect |
| 8. Q16×ΓL(1,𝔽8) | C4 | semidirect |
| 30. C7×Q16 | C3×C4 | semidirect |
| 86. ΓL(1,𝔽8) | #441 | semidirect |
| 88. Q16 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 111. C7 | C3×#441 | 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 |
| Q8×ΓL(1,𝔽8)⋊C4 | 2 |
| #6409⋊C3 | 2 |
| #6391⋊C3 | 2 |
| #6389⋊C3 | 2 |
| #89956 | 3 |
| C3×#441 | 7 |