Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
592 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C22×C3×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×Q16 | C2 | semidirect |
| 2. C22×C3×SD16 | C2 | semidirect |
| 4. C2×C3×#124 | C2 | semidirect |
| 5. C3×#495 | C2 | direct (N × H) |
| 6. C2×C3×C4×C8 | C2 | semidirect |
| 7. C2×#495 | C3 | direct (N × H) |
| 11. C2×C3×Q16 | C22 | semidirect |
| 13. C2×C3×SD16 | C22 | semidirect |
| 19. C3×#124 | C22 | semidirect |
| 22. C3×C4×C8 | C22 | semidirect |
| 23. C22×Q16 | C2×C3 | semidirect |
| 24. C22×SD16 | C2×C3 | semidirect |
| 26. C2×#124 | C2×C3 | semidirect |
| 27. #495 | C2×C3 | direct (N × H) |
| 28. C2×C4×C8 | C2×C3 | semidirect |
| 54. C2×Q16 | C22×C3 | semidirect |
| 56. C2×SD16 | C22×C3 | semidirect |
| 62. #124 | C22×C3 | semidirect |
| 65. C4×C8 | C22×C3 | semidirect |
| 140. C2×C3 | #495 | direct (N × H) |
| 154. C3 | C2×#495 | direct (N × H) |
| 156. C2 | C3×#495 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q16 | 2 |
| C22×C3×SD16 | 2 |
| C22×C3×D8 | 2 |
| C2×C3×#124 | 2 |
| C3×#495 | 2 |
| C2×C3×C4×C8 | 2 |
| C2×#495 | 3 |