Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1460 conjugacy classes of subgroups.
The subgroup lattice has 132 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C4 | C25×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#568 | C2 | direct (N × H) |
| 3. C2×C3×#131 | C2 | semidirect |
| 4. C22×C3×M16 | C2 | semidirect |
| 5. C2×#568 | C3 | direct (N × H) |
| 13. C3×#131 | C22 | semidirect |
| 15. C2×C3×M16 | C22 | semidirect |
| 16. C2×C3×M16 | C22 | semidirect |
| 17. C22×C3×C8 | C22 | semidirect |
| 19. #568 | C2×C3 | direct (N × H) |
| 20. C2×#131 | C2×C3 | semidirect |
| 21. C22×M16 | C2×C3 | semidirect |
| 37. C3×M16 | C23 | semidirect |
| 39. C2×C3×C8 | C23 | semidirect |
| 47. #131 | C22×C3 | semidirect |
| 49. C2×M16 | C22×C3 | semidirect |
| 50. C2×M16 | C22×C3 | semidirect |
| 51. C22×C8 | C22×C3 | semidirect |
| 83. M16 | C23×C3 | semidirect |
| 85. C2×C8 | C23×C3 | semidirect |
| 114. C2×C3 | #568 | direct (N × H) |
| 126. C3 | C2×#568 | direct (N × H) |
| 129. C2 | C3×#568 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#41 | 2 |
| C3×#568 | 2 |
| C2×C3×#131 | 2 |
| C22×C3×M16 | 2 |
| C2×#568 | 3 |