Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
3000 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 725 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×#127 |
| 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×D524 | C2 | semidirect |
| 2. C23×C22⋊D131 | C2 | semidirect |
| 3. C2×#16862242 | C2 | direct (N × H) |
| 4. C22×C4×Dic131 | C2 | semidirect |
| 5. C23×D4×C131 | C2 | semidirect |
| 8. C22×D524 | C22 | semidirect |
| 10. C22×C22⋊D131 | C22 | semidirect |
| 12. #16862242 | C22 | direct (N × H) |
| 13. C2×C4×Dic131 | C22 | semidirect |
| 16. C22×D4×C131 | C22 | semidirect |
| 22. C2×D524 | C23 | semidirect |
| 24. C2×C22⋊D131 | C23 | semidirect |
| 25. C22×Dic131 | D4 | semidirect |
| 27. C4×Dic131 | C23 | semidirect |
| 33. C2×D4×C131 | C23 | semidirect |
| 34. C22×C4×C131 | D4 | semidirect |
| 42. C2×Dic131 | C2×D4 | semidirect |
| 52. C2×C4×C131 | C2×D4 | semidirect |
| 57. C22×C131 | #127 | semidirect |
| 64. C4×C131 | C22×D4 | semidirect |
| 65. Dic131 | C22×D4 | semidirect |
| 67. C2×C131 | C2×#127 | semidirect |
| 71. C131 | C22×#127 | semidirect |
| 75. C23×D4 | D131 | semidirect |
| 85. C22×D4 | C2×D131 | semidirect |
| 102. C2×D4 | C22×D131 | semidirect |
| 105. C22×C4 | C22⋊D131 | semidirect |
| 124. C2×C4 | C2×C22⋊D131 | semidirect |
| 132. C22 | #16862242 | direct (N × H) |
| 137. C4 | C22×C22⋊D131 | semidirect |
| 140. C2 | C2×#16862242 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D524 | 2 |
| C23×C22⋊D131 | 2 |
| C2×#16862242 | 2 |
| C22×C4×Dic131 | 2 |
| C23×D4×C131 | 2 |
| C22×#127 | 131 |