Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
268 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 122 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. C22×ΓL(1,𝔽8)⋊C8 | C2 | semidirect |
| 3. (C22⋊(C7⋊C8))⋊C3 | C2 | semidirect |
| 4. C4⋊(ΓL(1,𝔽8)⋊C8) | C2 | semidirect |
| 5. C4×ΓL(1,𝔽8)⋊C8 | C2 | semidirect |
| 6. D4×C7⋊C8 | C3 | semidirect |
| 17. C22×C7⋊C8 | C2×C3 | semidirect |
| 18. C22⋊(C7⋊C8) | C2×C3 | semidirect |
| 19. C4⋊(C7⋊C8) | C2×C3 | semidirect |
| 20. C4×C7⋊C8 | C2×C3 | semidirect |
| 23. D4×ΓL(1,𝔽8) | C8 | semidirect |
| 32. ΓL(1,𝔽8)⋊C8 | D4 | direct (N × H) |
| 49. C22×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 53. C4×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 58. C7×D4 | C3×C8 | semidirect |
| 67. C7⋊C8 | C3×D4 | semidirect |
| 88. C22×C7 | C2×C3×C8 | semidirect |
| 92. C4×C7 | C2×C3×C8 | semidirect |
| 106. ΓL(1,𝔽8) | C8×D4 | semidirect |
| 130. D4 | ΓL(1,𝔽8)⋊C8 | direct (N × H) |
| 140. C7 | C3×C8×D4 | semidirect |
| 147. C22 | C2×ΓL(1,𝔽8)⋊C8 | semidirect |
| 151. C4 | C2×ΓL(1,𝔽8)⋊C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D4×ΓL(1,𝔽8) | 2 |
| C22×ΓL(1,𝔽8)⋊C8 | 2 |
| (C22⋊(C7⋊C8))⋊C3 | 2 |
| C4⋊(ΓL(1,𝔽8)⋊C8) | 2 |
| C4×ΓL(1,𝔽8)⋊C8 | 2 |
| D4×C7⋊C8 | 3 |
| C3×C8×D4 | 7 |