Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
192 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 99 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. ΓL(1,𝔽8)×#97 | C2 | semidirect |
| 5. C4×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×#433 | C3 | semidirect |
| 15. C7×C4⋊C8 | C2×C3 | semidirect |
| 17. C7×#97 | C2×C3 | semidirect |
| 18. C4×C7×C8 | C2×C3 | semidirect |
| 47. #433 | ΓL(1,𝔽8) | direct (N × H) |
| 67. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 69. #97 | C2×ΓL(1,𝔽8) | semidirect |
| 70. C4×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 86. ΓL(1,𝔽8) | #433 | direct (N × H) |
| 112. C7 | C3×#433 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| ΓL(1,𝔽8)×#97 | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C7×#433 | 3 |
| C3×#433 | 7 |