Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
224 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 68 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 |
|---|---|---|
| 2. C2×#1144⋊C3 | C2 | semidirect |
| 4. C4×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 5. #6409⋊C3 | C2 | semidirect |
| 7. C4⋊(ΓL(1,𝔽8)⋊C8) | C2 | semidirect |
| 8. #89826 | C3 | semidirect |
| 23. C2×#1144 | C2×C3 | semidirect |
| 25. C4×Dic14 | C2×C3 | semidirect |
| 26. #6409 | C2×C3 | semidirect |
| 28. C4⋊(C7⋊C8) | C2×C3 | semidirect |
| 142. ΓL(1,𝔽8) | #464 | semidirect |
| 186. C7 | C3×#464 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#124 | 2 |
| C2×#1144⋊C3 | 2 |
| C2×#1142⋊C3 | 2 |
| C4×ΓL(1,𝔽8)⋊Q8 | 2 |
| #6409⋊C3 | 2 |
| #6406⋊C3 | 2 |
| C4⋊(ΓL(1,𝔽8)⋊C8) | 2 |
| #89826 | 3 |
| C3×#464 | 7 |