Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
268 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 106 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 |
|---|---|---|
| 1. C2×C8×AGL(1,𝔽7) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 4. (C2×AGL(1,𝔽7))⋊C8 | C2 | semidirect |
| 5. C4×ΓL(1,𝔽8)⋊C8 | C2 | semidirect |
| 6. #89606 | C3 | semidirect |
| 16. C2×C8×D7 | C2×C3 | semidirect |
| 18. C7×C4⋊C8 | C2×C3 | semidirect |
| 19. (C2×D7)⋊C8 | C2×C3 | semidirect |
| 20. C4×C7⋊C8 | C2×C3 | semidirect |
| 27. C4⋊AGL(1,𝔽7) | C8 | semidirect |
| 32. ΓL(1,𝔽8)⋊C8 | D4 | semidirect |
| 50. C2×AGL(1,𝔽7) | C2×C8 | semidirect |
| 52. C4×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 62. D28 | C3×C8 | semidirect |
| 67. C7⋊C8 | C3×D4 | semidirect |
| 83. C4⋊C8 | AGL(1,𝔽7) | semidirect |
| 89. C2×D7 | C2×C3×C8 | semidirect |
| 91. C4×C7 | C2×C3×C8 | semidirect |
| 106. ΓL(1,𝔽8) | C8×D4 | semidirect |
| 140. C7 | C3×C8×D4 | semidirect |
| 151. C4 | C8×AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×AGL(1,𝔽7) | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| (C2×AGL(1,𝔽7))⋊C8 | 2 |
| C4×ΓL(1,𝔽8)⋊C8 | 2 |
| #89606 | 3 |
| C3×C8×D4 | 7 |