Finite Group DatabasebetaFinite Group DatabasebetaOrder 26944.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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 | C23×D421 |
| commutator, socle, cosocle | C22×C421 | C24 |
| Fitting | #120×C421 | C2 |
| hypercenter | #120 | D421 |
| hypocenter | C421 | #538 |
| LCS | G ▷ C22×C421 ▷ C421 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27212359 | C2 | semidirect |
| 2. #27212359 | C2 | semidirect |
| 3. #27212353 | C2 | semidirect |
| 4. #27212352 | C2 | semidirect |
| 5. #27212348 | C2 | semidirect |
| 6. #27212347 | C2 | semidirect |
| 7. #27212343 | C2 | semidirect |
| 8. #27212338 | C2 | semidirect |
| 9. #27212338 | C2 | semidirect |
| 10. #27212337 | C2 | semidirect |
| 11. #27212335 | C2 | semidirect |
| 12. D4×Dic421 | C2 | semidirect |
| 13. #27212326 | C2 | semidirect |
| 14. C2×C22⋊Dic421 | C2 | semidirect |
| 15. #120×C421 | C2 | semidirect |
| 16. C2×Dic842 | C22 | semidirect |
| 20. C2×C22⋊D421 | C22 | semidirect |
| 23. C22×Dic421 | C22 | semidirect |
| 24. C2×C4×D421 | C22 | semidirect |
| 25. C4⋊Dic421 | C22 | semidirect |
| 26. C4⋊Dic421 | C22 | semidirect |
| 27. #20269750 | C22 | semidirect |
| 28. #20269750 | C22 | semidirect |
| 29. #20269750 | C22 | semidirect |
| 30. #20269750 | C22 | semidirect |
| 38. C22⋊Dic421 | C22 | semidirect |
| 39. C22⋊Dic421 | C22 | semidirect |
| 40. #20269748 | C22 | semidirect |
| 41. #20269748 | C22 | semidirect |
| 42. C4×Dic421 | C22 | semidirect |
| 43. C4×Dic421 | C22 | semidirect |
| 116. C421 | #538 | semidirect |
| 129. #120 | D421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27212359 | 2 |
| #27212353 | 2 |
| #27212352 | 2 |
| #27212348 | 2 |
| #27212347 | 2 |
| #27212343 | 2 |
| #27212338 | 2 |
| #27212337 | 2 |
| #27212335 | 2 |
| D4×Dic421 | 2 |
| #27212326 | 2 |
| C2×C22⋊Dic421 | 2 |
| #120×C421 | 2 |
| #538 | 421 |