Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 72 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 |
|---|---|---|
| 8. #89628 | C3 | semidirect |
| 36. C8×ΓL(1,𝔽8) | Q8 | semidirect |
| 37. ΓL(1,𝔽8)⋊C8 | Q8 | semidirect |
| 74. C7×C8 | C3×Q8 | semidirect |
| 75. C7⋊C8 | C3×Q8 | semidirect |
| 116. ΓL(1,𝔽8) | #501 | semidirect |
| 149. C8 | ΓL(1,𝔽8)⋊Q8 | semidirect |
| 150. C7 | C3×#501 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6453⋊C3 | 2 |
| #6451⋊C3 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| #6392⋊C3 | 2 |
| #6389⋊C3 | 2 |
| #6382⋊C3 | 2 |
| #6381⋊C3 | 2 |
| #89628 | 3 |
| C3×#501 | 7 |