Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D131 |
| commutator, socle, cosocle | C23×C131 | C24 |
| Fitting | C22×C22⋊C4×C131 | C2 |
| hypercenter | C22×C22⋊C4 | D131 |
| hypocenter | C131 | #2404 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14523513 | C2 | semidirect |
| 2. C2×#16862215 | C2 | semidirect |
| 3. #22553224 | C2 | semidirect |
| 4. #22553221 | C2 | semidirect |
| 5. #22553191 | C2 | semidirect |
| 6. #22553188 | C2 | semidirect |
| 7. #22553175 | C2 | semidirect |
| 8. C22×C22⋊C4×C131 | C2 | semidirect |
| 11. C22×Dic262 | C22 | semidirect |
| 12. C22×D524 | C22 | semidirect |
| 15. C22×C4×D131 | C22 | semidirect |
| 19. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 20. C131⋊#95 | C22 | semidirect |
| 21. C131⋊#95 | C22 | semidirect |
| 22. C131⋊#95 | C22 | semidirect |
| 40. C2×C22⋊D131 | D4 | semidirect |
| 58. C22×C4×C131 | D4 | semidirect |
| 92. C23×C131 | #41 | semidirect |
| 133. C131 | #2404 | semidirect |
| 137. C22×C22⋊C4 | D131 | semidirect |
| 189. C22×C4 | C22⋊D131 | semidirect |
| 215. C23 | #14523513 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14523513 | 2 |
| C2×#16862215 | 2 |
| #22553224 | 2 |
| #22553221 | 2 |
| #22553191 | 2 |
| #22553188 | 2 |
| #22553175 | 2 |
| C22×C22⋊C4×C131 | 2 |
| #2404 | 131 |