Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
412 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 192 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 |
|---|---|---|
| 4. #6453⋊C3 | C2 | semidirect |
| 5. #6451⋊C3 | C2 | semidirect |
| 6. #90191 | C3 | semidirect |
| 20. #6453 | C2×C3 | semidirect |
| 21. #6451 | C2×C3 | semidirect |
| 25. C2×C4×ΓL(1,𝔽8) | Q8 | semidirect |
| 57. C2×C4×C7 | C3×Q8 | semidirect |
| 98. ΓL(1,𝔽8) | #544 | semidirect |
| 126. C2×C4 | ΓL(1,𝔽8)⋊Q8 | semidirect |
| 129. C7 | C3×#544 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| #6489⋊C3 | 2 |
| #6487⋊C3 | 2 |
| #6453⋊C3 | 2 |
| #6451⋊C3 | 2 |
| #90191 | 3 |
| C3×#544 | 7 |