Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
220 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 96 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. Q16×Dic7 | C3 | semidirect |
| 8. Q16×ΓL(1,𝔽8) | C4 | semidirect |
| 31. C7×Q16 | C3×C4 | semidirect |
| 49. ΓL(1,𝔽8)⋊C4 | Q16 | direct (N × H) |
| 82. Dic7 | C3×Q16 | semidirect |
| 92. ΓL(1,𝔽8) | C4×Q16 | semidirect |
| 94. Q16 | ΓL(1,𝔽8)⋊C4 | direct (N × H) |
| 120. C7 | C3×C4×Q16 | 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 |
| #6392⋊C3 | 2 |
| C8×ΓL(1,𝔽8)⋊C4 | 2 |
| Q16×Dic7 | 3 |
| C3×C4×Q16 | 7 |