Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
958 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 958 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D131 |
| commutator, socle, cosocle | C22×C131 | C25 |
| Fitting | H(3,𝔽4)×C131 | C2 |
| hypercenter | H(3,𝔽4) | D131 |
| hypocenter | C131 | #3568 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22554062 | C2 | semidirect |
| 2. #22554031 | C2 | semidirect |
| 3. #22553993 | C2 | semidirect |
| 4. #22553926 | C2 | semidirect |
| 5. #22553897 | C2 | semidirect |
| 6. #22553841 | C2 | semidirect |
| 7. H(3,𝔽4)×C131 | C2 | semidirect |
| 8. #16862244 | C22 | semidirect |
| 9. #16862240 | C22 | semidirect |
| 10. #16862239 | C22 | semidirect |
| 11. #16862239 | C22 | semidirect |
| 12. #16862239 | C22 | semidirect |
| 13. #16862239 | C22 | semidirect |
| 14. #16862238 | C22 | semidirect |
| 16. #16862232 | C22 | semidirect |
| 17. #16862228 | C22 | semidirect |
| 20. #16862223 | C22 | semidirect |
| 21. #16862223 | C22 | semidirect |
| 22. #16862223 | C22 | semidirect |
| 23. #16862222 | C22 | semidirect |
| 26. #16862215 | C22 | semidirect |
| 27. Q8×Dic131 | C22 | semidirect |
| 28. C4×Dic262 | C22 | semidirect |
| 31. (C22⋊D131)⋊C4 | C22 | semidirect |
| 32. #16862202 | C22 | semidirect |
| 33. #16862202 | C22 | semidirect |
| 34. C2×C22⋊Dic131 | C22 | semidirect |
| 35. C42×D131 | C22 | semidirect |
| 100. C131 | #3568 | semidirect |
| 103. H(3,𝔽4) | D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22554062 | 2 |
| #22554031 | 2 |
| #22553993 | 2 |
| #22553926 | 2 |
| #22553897 | 2 |
| #22553841 | 2 |
| H(3,𝔽4)×C131 | 2 |
| #3568 | 131 |