Finite Group DatabasebetaFinite Group DatabasebetaOrder 6144.
At least 9349 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has at least 9349 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72217073 |
| commutator, cosocle, hypocenter | #11710730 | C23 |
| Frattini | C24 | C22×#1020 |
| Fitting | #72216292 | D3 |
| hypercenter | C22 | #72215837 |
| LCS | G ▷ #11710730 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72216913 | C2 | semidirect |
| 2. #72216887 | C2 | semidirect |
| 3. #72216292 | D3 | semidirect |
| 4. #72210418 | C2 | semidirect |
| 5. #72209845 | C2 | semidirect |
| 6. #72209832 | C2 | semidirect |
| 7. #72209831 | C22 | semidirect |
| 8. #72209772 | C22 | semidirect |
| 9. #72209717 | C22 | semidirect |
| 10. #72209645 | C22 | semidirect |
| 11. #72209641 | C22 | semidirect |
| 12. #11710730 | C23 | semidirect |
| 13. #6499874 | C2×D3 | semidirect |
| 14. #6499867 | C2×D3 | semidirect |
| 15. #62817 | C22×D3 | semidirect |
| 16. #60399 | S4 | semidirect |
| 17. C2×#2846 | S4 | semidirect |
| 18. C27 | C2×S4 | semidirect |
| 19. H(7,𝔽2) | C2×S4 | semidirect |
| 20. #3026 | C2×S4 | semidirect |
| 21. C2×B(4,𝔽2) | C2×S4 | semidirect |
| 22. #2846 | C2×S4 | semidirect |
| 26. C25 | #5934 | semidirect |
| 28. C22×D4 | #5934 | semidirect |
| 30. C23×C4 | #5934 | semidirect |