Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
372 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 159 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 |
|---|---|---|
| 5. C7×#391 | C3 | semidirect |
| 8. C2×Q8×ΓL(1,𝔽8) | C4 | semidirect |
| 32. C2×C4×ΓL(1,𝔽8) | Q8 | semidirect |
| 35. C2×C7×Q8 | C3×C4 | semidirect |
| 55. #391 | ΓL(1,𝔽8) | direct (N × H) |
| 68. C2×C4×C7 | C3×Q8 | semidirect |
| 109. ΓL(1,𝔽8) | #391 | direct (N × H) |
| 112. C2×Q8 | C4×ΓL(1,𝔽8) | semidirect |
| 145. C2×C4 | Q8×ΓL(1,𝔽8) | semidirect |
| 146. C7 | C3×#391 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| C2×C42×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#95 | 2 |
| C7×#391 | 3 |
| C3×#391 | 7 |