Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
364 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 129 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. C7×#395 | C3 | semidirect |
| 21. C2×C4×ΓL(1,𝔽8) | Q8 | semidirect |
| 36. #395 | ΓL(1,𝔽8) | direct (N × H) |
| 44. C2×C4×C7 | C3×Q8 | semidirect |
| 70. ΓL(1,𝔽8) | #395 | direct (N × H) |
| 92. C2×C4 | Q8×ΓL(1,𝔽8) | semidirect |
| 93. C7 | C3×#395 | 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 |
| ΓL(1,𝔽8)×#95 | 2 |
| C7×#395 | 3 |
| C3×#395 | 7 |