Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D3 |
| hypocenter | C3 | #3529 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×H(3,𝔽4) | C2 | semidirect |
| 2. #5684 | C2 | semidirect |
| 3. #5680 | C2 | semidirect |
| 4. #5672 | C2 | semidirect |
| 5. #5668 | C2 | semidirect |
| 6. #5595 | C2 | semidirect |
| 7. #5586 | C2 | semidirect |
| 9. C23×Dic3 | C22 | semidirect |
| 10. C2×#239 | C22 | semidirect |
| 15. #953 | C22 | semidirect |
| 16. #946 | C22 | semidirect |
| 17. Q8×Dic3 | C22 | semidirect |
| 18. #944 | C22 | semidirect |
| 24. #886 | C22 | semidirect |
| 25. #884 | C22 | semidirect |
| 26. #883 | C22 | semidirect |
| 27. #67⋊C4 | C22 | semidirect |
| 28. #879 | C22 | semidirect |
| 29. #878 | C22 | semidirect |
| 30. #877 | C22 | semidirect |
| 31. #875 | C22 | semidirect |
| 32. C42×D3 | C22 | semidirect |
| 33. #870 | C22 | semidirect |
| 36. H(3,𝔽4) | D3 | semidirect |
| 190. C3 | #3529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×H(3,𝔽4) | 2 |
| #5684 | 2 |
| #5680 | 2 |
| #5672 | 2 |
| #5668 | 2 |
| #5595 | 2 |
| #5586 | 2 |
| #3529 | 3 |