Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
292 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 168 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. #6389⋊C3 | C2 | direct (N × H) |
| 5. C2×#6389 | C3 | semidirect |
| 8. C2×C8×ΓL(1,𝔽8) | C4 | semidirect |
| 12. C2×ΓL(1,𝔽8)⋊C8 | C4 | semidirect |
| 17. #6389 | C2×C3 | semidirect |
| 26. C8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 27. ΓL(1,𝔽8)⋊C8 | C2×C4 | semidirect |
| 30. C2×C7×C8 | C3×C4 | semidirect |
| 34. C2×C7⋊C8 | C3×C4 | semidirect |
| 55. C7×C8 | C2×C3×C4 | semidirect |
| 56. C7⋊C8 | C2×C3×C4 | semidirect |
| 65. C2×ΓL(1,𝔽8) | #97 | semidirect |
| 88. ΓL(1,𝔽8) | C2×#97 | semidirect |
| 93. C2×C8 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 97. C2×C7 | C3×#97 | semidirect |
| 115. C8 | C2×ΓL(1,𝔽8)⋊C4 | semidirect |
| 116. C7 | C2×C3×#97 | semidirect |
| 128. C2 | #6389⋊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 |
| C2×C4×ΓL(1,𝔽8)⋊C4 | 2 |
| C22×ΓL(1,𝔽8)⋊C8 | 2 |
| #6389⋊C3 | 2 |
| C2×#6389 | 3 |
| C2×C3×#97 | 7 |