Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
2536 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 2536 conjugacy classes of subgroups — too many to draw. Only its 581 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D131 |
| commutator, cosocle | C22×C131 | C25 |
| Frattini | C22 | C24×D131 |
| Fitting | C2×#127×C131 | C2 |
| socle | C23×C131 | C24 |
| hypercenter | C2×#127 | D131 |
| hypocenter | C131 | C2×D42 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D4×D131 | C2 | semidirect |
| 2. #22553947 | C2 | direct (N × H) |
| 3. C2×#16862217 | C2 | semidirect |
| 4. C2×#16862214 | C2 | semidirect |
| 5. C2×C4×D524 | C2 | semidirect |
| 6. C2×#127×C131 | C2 | semidirect |
| 7. C24×D131 | C22 | semidirect |
| 9. C22×C22⋊D131 | C22 | semidirect |
| 11. C2×D4×D131 | C22 | semidirect |
| 12. C22×C4×D131 | C22 | semidirect |
| 13. #16862217 | C22 | semidirect |
| 14. #16862214 | C22 | semidirect |
| 15. C4×D524 | C22 | semidirect |
| 16. C2×C4⋊Dic131 | C22 | semidirect |
| 17. C2×C22⋊Dic131 | C22 | semidirect |
| 18. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 19. C22×D4×C131 | C22 | semidirect |
| 20. C22×D4×C131 | C22 | semidirect |
| 21. #127×C131 | C22 | semidirect |
| 22. C2×C42×C131 | C22 | semidirect |
| 26. C23×D131 | C23 | semidirect |
| 27. C2×D524 | D4 | semidirect |
| 30. C2×C22⋊D131 | C23 | semidirect |
| 34. C2×C4×D131 | C23 | semidirect |
| 35. C4⋊Dic131 | C23 | semidirect |
| 36. C22⋊Dic131 | C23 | semidirect |
| 37. (C2×D131)⋊C4 | C23 | semidirect |
| 41. C2×D4×C131 | C23 | semidirect |
| 42. C2×D4×C131 | C23 | semidirect |
| 45. C42×C131 | C23 | semidirect |
| 51. C22×D131 | C2×D4 | semidirect |
| 53. D524 | C2×D4 | semidirect |
| 65. C2×C4×C131 | C2×D4 | semidirect |
| 78. C2×D131 | C22×D4 | semidirect |
| 80. C4×C131 | C22×D4 | semidirect |
| 83. C2×C131 | D42 | semidirect |
| 87. C131 | C2×D42 | semidirect |
| 91. C2×#127 | D131 | semidirect |
| 100. C22×D4 | C2×D131 | semidirect |
| 101. C22×D4 | C2×D131 | semidirect |
| 103. #127 | C2×D131 | semidirect |
| 110. C2×C42 | C2×D131 | semidirect |
| 123. C2×D4 | C22×D131 | semidirect |
| 124. C2×D4 | C22×D131 | semidirect |
| 133. C42 | C22×D131 | semidirect |
| 153. C2×C4 | D4×D131 | semidirect |
| 169. C4 | C2×D4×D131 | semidirect |
| 172. C2 | #22553947 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×D131 | 2 |
| #22553947 | 2 |
| C2×#16862217 | 2 |
| C2×#16862214 | 2 |
| C2×C4×D524 | 2 |
| C2×#127×C131 | 2 |
| C2×D42 | 131 |