Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
520 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#124 | C2×C3 |
| hypercenter | #124 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×H(3,𝔽4) |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×#124 | C2 | semidirect |
| 2. #6510⋊C3 | C2 | semidirect |
| 3. #6500⋊C3 | C2 | semidirect |
| 4. #6448⋊C3 | C2 | semidirect |
| 5. #6445⋊C3 | C2 | semidirect |
| 6. #6438⋊C3 | C2 | semidirect |
| 7. #90374 | C3 | semidirect |
| 8. C23×AGL(1,𝔽7) | C22 | semidirect |
| 9. C2×Q8×ΓL(1,𝔽8) | C22 | semidirect |
| 14. C2×ΓL(1,𝔽8)⋊Q8 | C22 | semidirect |
| 15. C22⋊C4×ΓL(1,𝔽8) | C22 | semidirect |
| 16. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 17. (C22⋊Dic7)⋊C3 | C22 | semidirect |
| 18. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 19. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 20. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 21. C4×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 22. C7×#124 | C2×C3 | semidirect |
| 23. #6510 | C2×C3 | semidirect |
| 24. #6500 | C2×C3 | semidirect |
| 25. #6448 | C2×C3 | semidirect |
| 26. #6445 | C2×C3 | semidirect |
| 27. #6438 | C2×C3 | semidirect |
| 43. C23×D7 | C22×C3 | semidirect |
| 44. C2×C7×Q8 | C22×C3 | semidirect |
| 49. C2×Dic14 | C22×C3 | semidirect |
| 50. C7×C22⋊C4 | C22×C3 | semidirect |
| 51. C42×C7 | C22×C3 | semidirect |
| 52. C22⋊Dic7 | C22×C3 | semidirect |
| 53. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 54. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 55. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 56. C4×Dic7 | C22×C3 | semidirect |
| 110. #124 | AGL(1,𝔽7) | semidirect |
| 138. ΓL(1,𝔽8) | H(3,𝔽4) | semidirect |
| 141. C2×Q8 | C2×AGL(1,𝔽7) | semidirect |
| 150. C22⋊C4 | C2×AGL(1,𝔽7) | semidirect |
| 152. C42 | C2×AGL(1,𝔽7) | semidirect |
| 182. C7 | C3×H(3,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#124 | 2 |
| #6510⋊C3 | 2 |
| #6500⋊C3 | 2 |
| #6448⋊C3 | 2 |
| #6445⋊C3 | 2 |
| #6438⋊C3 | 2 |
| #90374 | 3 |
| C3×H(3,𝔽4) | 7 |