Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
504 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 504 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×#127 | C2×C3 |
| hypercenter | #127 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#560 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×#127 | C2 | semidirect |
| 2. #6502⋊C3 | C2 | semidirect |
| 3. #6500⋊C3 | C2 | semidirect |
| 4. #6498⋊C3 | C2 | semidirect |
| 5. #10620712 | C2 | semidirect |
| 6. #90411 | C3 | semidirect |
| 9. C2×D4×ΓL(1,𝔽8) | C22 | semidirect |
| 11. C22×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 12. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 14. (C22⋊Dic7)⋊C3 | C22 | semidirect |
| 16. (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | semidirect |
| 17. C7×#127 | C2×C3 | semidirect |
| 18. #6502 | C2×C3 | semidirect |
| 19. #6500 | C2×C3 | semidirect |
| 20. #6498 | C2×C3 | semidirect |
| 21. #6439 | C2×C3 | semidirect |
| 35. C2×C7×D4 | C22×C3 | semidirect |
| 37. C22×Dic7 | C22×C3 | semidirect |
| 38. C42×C7 | C22×C3 | semidirect |
| 40. C22⋊Dic7 | C22×C3 | semidirect |
| 42. Dic7⋊C4 | C22×C3 | semidirect |
| 81. #127 | AGL(1,𝔽7) | semidirect |
| 104. ΓL(1,𝔽8) | #560 | semidirect |
| 108. C2×D4 | C2×AGL(1,𝔽7) | semidirect |
| 114. C42 | C2×AGL(1,𝔽7) | semidirect |
| 137. C7 | C3×#560 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#127 | 2 |
| #6502⋊C3 | 2 |
| #6500⋊C3 | 2 |
| #6498⋊C3 | 2 |
| #10620712 | 2 |
| #90411 | 3 |
| C3×#560 | 7 |