Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.
This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D131 |
| commutator, cosocle | C22×C131 | C25 |
| Frattini | C22 | C24×D131 |
| Fitting | C23×D4×C131 | C2 |
| socle | C24×C131 | C23 |
| hypercenter | C23×D4 | D131 |
| hypocenter | C131 | C22×#121 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊D131 | C2 | semidirect |
| 2. C23×C22⋊D131 | C2 | semidirect |
| 3. C24×Dic131 | C2 | semidirect |
| 4. C2×#16862218 | C2 | direct (N × H) |
| 5. C22×Dic131⋊C4 | C2 | semidirect |
| 6. C22×C22⋊Dic131 | C2 | semidirect |
| 7. C22×(C2×D131)⋊C4 | C2 | semidirect |
| 8. C23×D4×C131 | C2 | semidirect |
| 12. C22×C22⋊D131 | C22 | semidirect |
| 13. C22×C22⋊D131 | C22 | semidirect |
| 18. C23×Dic131 | C22 | semidirect |
| 19. #16862218 | C22 | direct (N × H) |
| 20. C2×Dic131⋊C4 | C22 | semidirect |
| 21. C2×C22⋊Dic131 | C22 | semidirect |
| 22. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 26. C22×D4×C131 | C22 | semidirect |
| 34. C2×C22⋊D131 | C23 | semidirect |
| 35. C2×C22⋊D131 | C23 | semidirect |
| 39. C22×Dic131 | D4 | semidirect |
| 43. C22×Dic131 | C23 | semidirect |
| 44. Dic131⋊C4 | C23 | semidirect |
| 45. C22⋊Dic131 | C23 | semidirect |
| 46. (C2×D131)⋊C4 | C23 | semidirect |
| 52. C24×C131 | D4 | semidirect |
| 57. C2×D4×C131 | C23 | semidirect |
| 70. C2×Dic131 | C2×D4 | semidirect |
| 83. C23×C131 | C2×D4 | semidirect |
| 98. C22×C131 | #121 | semidirect |
| 101. C22×C131 | C22×D4 | semidirect |
| 111. Dic131 | C22×D4 | semidirect |
| 114. C2×C131 | C2×#121 | semidirect |
| 119. C131 | C22×#121 | semidirect |
| 123. C23×D4 | D131 | semidirect |
| 137. C22×D4 | C2×D131 | semidirect |
| 157. C24 | C22⋊D131 | semidirect |
| 166. C2×D4 | C22×D131 | semidirect |
| 194. C23 | C2×C22⋊D131 | semidirect |
| 222. C22 | #16862218 | direct (N × H) |
| 225. C22 | C22×C22⋊D131 | semidirect |
| 235. C2 | C2×#16862218 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊D131 | 2 |
| C24×Dic131 | 2 |
| C2×#16862218 | 2 |
| C22×Dic131⋊C4 | 2 |
| C22×C22⋊Dic131 | 2 |
| C22×(C2×D131)⋊C4 | 2 |
| C23×D4×C131 | 2 |
| C22×#121 | 131 |