Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
6520 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.
This group has 6520 conjugacy classes of subgroups — too many to draw. Only its 3301 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D131 |
| commutator, cosocle | C2×C131 | C26 |
| Frattini | C2 | C25×D131 |
| Fitting | C24×C4×C131 | C2 |
| socle | C24×C131 | C23 |
| hypercenter | C24×C4 | D131 |
| hypocenter | C131 | C23×#41 |
| LCS | G ▷ C2×C131 ▷ C131 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14523513 | C2 | direct (N × H) |
| 2. C23×Dic262 | C2 | semidirect |
| 3. C23×D524 | C2 | semidirect |
| 4. C23×C22⋊D131 | C2 | semidirect |
| 5. C23×C4×D131 | C2 | semidirect |
| 6. C24×C4×C131 | C2 | semidirect |
| 8. C2×#14523513 | C22 | direct (N × H) |
| 9. C22×Dic262 | C22 | semidirect |
| 10. C22×D524 | C22 | semidirect |
| 11. C22×C22⋊D131 | C22 | semidirect |
| 13. C22×C4×D131 | C22 | semidirect |
| 17. C23×C4×C131 | C22 | semidirect |
| 20. #14523513 | C23 | direct (N × H) |
| 21. C2×Dic262 | C23 | semidirect |
| 22. C2×D524 | C23 | semidirect |
| 23. C2×C22⋊D131 | C23 | semidirect |
| 25. C2×C4×D131 | C23 | semidirect |
| 31. C22×C4×C131 | C23 | semidirect |
| 34. Dic262 | C24 | semidirect |
| 35. D524 | C24 | semidirect |
| 36. C22⋊D131 | C24 | semidirect |
| 38. C4×D131 | C24 | semidirect |
| 40. C23×C131 | #41 | semidirect |
| 45. C2×C4×C131 | C24 | semidirect |
| 47. C22×C131 | C2×#41 | semidirect |
| 56. C2×C131 | C22×#41 | semidirect |
| 59. C131 | C23×#41 | semidirect |
| 66. C24×C4 | D131 | semidirect |
| 77. C23×C4 | C2×D131 | semidirect |
| 91. C22×C4 | C22×D131 | semidirect |
| 95. C23 | #14523513 | direct (N × H) |
| 105. C2×C4 | C23×D131 | semidirect |
| 109. C22 | C2×#14523513 | direct (N × H) |
| 117. C2 | C22×#14523513 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14523513 | 2 |
| C23×Dic262 | 2 |
| C23×D524 | 2 |
| C23×C22⋊D131 | 2 |
| C23×C4×D131 | 2 |
| C24×C4×C131 | 2 |
| C23×#41 | 131 |