Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 112 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. #6386⋊C3 | C2 | direct (N × H) |
| 3. C2×ΓL(1,𝔽8)⋊C16 | C2 | semidirect |
| 4. C2×#6386 | C3 | semidirect |
| 10. ΓL(1,𝔽8)⋊C16 | C22 | semidirect |
| 12. #6386 | C2×C3 | semidirect |
| 13. C2×C7⋊C16 | C2×C3 | semidirect |
| 28. C7⋊C16 | C22×C3 | semidirect |
| 54. C2×ΓL(1,𝔽8) | M32 | semidirect |
| 73. ΓL(1,𝔽8) | C2×M32 | semidirect |
| 81. C2×C7 | C3×M32 | semidirect |
| 97. C7 | C2×C3×M32 | semidirect |
| 109. C2 | #6386⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C8×ΓL(1,𝔽8) | 2 |
| #6386⋊C3 | 2 |
| C2×ΓL(1,𝔽8)⋊C16 | 2 |
| C2×#6386 | 3 |
| C2×C3×M32 | 7 |