Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D3 |
| commutator, socle, cosocle | C2×C3 | C26 |
| Fitting | C3×#585 | C2 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | 2−1+6 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#585 | C2 | semidirect |
| 2. D3×2−1+4 | C2 | semidirect |
| 3. #5966 | C2 | semidirect |
| 4. #5964 | C2 | semidirect |
| 5. C2×#1010 | C2 | semidirect |
| 7. C3×2−1+4 | C22 | semidirect |
| 8. C3×H(5,𝔽2) | C22 | semidirect |
| 9. C2×C3×#41 | C22 | semidirect |
| 10. #1010 | C22 | semidirect |
| 11. #1010 | C22 | semidirect |
| 12. #1009 | C22 | semidirect |
| 13. D3×#41 | C22 | semidirect |
| 14. #1007 | C22 | semidirect |
| 15. C2×D3×Q8 | C22 | semidirect |
| 16. #1004 | C22 | semidirect |
| 17. C2×#237 | C22 | semidirect |
| 18. C2×#235 | C22 | semidirect |
| 19. C22×Dic6 | C22 | semidirect |
| 22. #585 | D3 | semidirect |
| 45. 2−1+4 | C2×D3 | semidirect |
| 48. H(5,𝔽2) | C2×D3 | semidirect |
| 52. C2×#41 | C2×D3 | semidirect |
| 115. C3 | 2−1+6 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| D3×2−1+4 | 2 |
| #5966 | 2 |
| #5964 | 2 |
| C2×#1010 | 2 |
| 2−1+6 | 3 |