Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
236 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 135 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. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 4. C4×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 5. C7×#431 | C3 | semidirect |
| 15. C7×C4⋊C8 | C2×C3 | semidirect |
| 17. C4×C7×C8 | C2×C3 | semidirect |
| 22. C2×C4×ΓL(1,𝔽8) | C8 | semidirect |
| 52. #431 | ΓL(1,𝔽8) | direct (N × H) |
| 56. C2×C4×C7 | C3×C8 | semidirect |
| 77. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 79. C4×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 101. ΓL(1,𝔽8) | #431 | direct (N × H) |
| 127. C2×C4 | C8×ΓL(1,𝔽8) | semidirect |
| 135. C7 | C3×#431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C7×#431 | 3 |
| C3×#431 | 7 |