Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#125 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | C2×#540 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#125 | C2 | semidirect |
| 2. #5697 | C2 | direct (N × H) |
| 3. C2×#895 | C2 | semidirect |
| 4. C2×#895 | C2 | semidirect |
| 5. C2×#894 | C2 | semidirect |
| 6. C2×#893 | C2 | semidirect |
| 7. C2×#874 | C2 | semidirect |
| 8. C2×#872 | C2 | semidirect |
| 15. C3×#125 | C22 | semidirect |
| 24. #895 | C22 | semidirect |
| 25. #895 | C22 | semidirect |
| 26. #894 | C22 | semidirect |
| 27. #893 | C22 | semidirect |
| 28. #874 | C22 | semidirect |
| 29. #872 | C22 | semidirect |
| 32. C2×#125 | D3 | semidirect |
| 69. #125 | C2×D3 | semidirect |
| 169. C2×C3 | #540 | semidirect |
| 189. C3 | C2×#540 | semidirect |
| 192. C2 | #5697 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#125 | 2 |
| #5697 | 2 |
| C2×#895 | 2 |
| C2×#894 | 2 |
| C2×#893 | 2 |
| C2×#874 | 2 |
| C2×#872 | 2 |
| C2×#540 | 3 |