Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×#67 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C3×#585 | C2 |
| socle | C2×C3 | C23×D4 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | #3586 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#585 | C2 | semidirect |
| 2. C2×#1010 | C2 | semidirect |
| 3. #5838 | C2 | semidirect |
| 4. #5836 | C2 | semidirect |
| 5. C2×#951 | C2 | semidirect |
| 6. #5822 | C2 | semidirect |
| 7. #5819 | C2 | semidirect |
| 10. C3×2−1+4 | C22 | semidirect |
| 12. C3×H(5,𝔽2) | C22 | semidirect |
| 17. #1010 | C22 | semidirect |
| 20. C2×#235 | C22 | semidirect |
| 21. C22×Dic6 | C22 | semidirect |
| 22. #951 | C22 | semidirect |
| 23. #951 | C22 | semidirect |
| 24. #950 | C22 | semidirect |
| 25. #949 | C22 | semidirect |
| 26. #948 | C22 | semidirect |
| 27. C2×#216 | C22 | semidirect |
| 28. #942 | C22 | semidirect |
| 29. C2×#214 | C22 | semidirect |
| 30. #932 | C22 | semidirect |
| 31. C2×#208 | C22 | semidirect |
| 32. #585 | D3 | semidirect |
| 78. 2−1+4 | C2×D3 | semidirect |
| 80. H(5,𝔽2) | C2×D3 | semidirect |
| 219. C3 | #3586 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| C2×#1010 | 2 |
| #5838 | 2 |
| #5836 | 2 |
| C2×#951 | 2 |
| #5822 | 2 |
| #5819 | 2 |
| #3586 | 3 |