Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| socle | C2×C3 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#135 | C2 | semidirect |
| 2. C3×#510 | C2 | semidirect |
| 3. C3×#509 | C2 | semidirect |
| 4. C3×#508 | C2 | semidirect |
| 5. C2×C3×M32 | C2 | semidirect |
| 6. #3414 | C3 | direct (N × H) |
| 11. C3×#135 | C22 | semidirect |
| 12. C2×C3×Q16 | C22 | semidirect |
| 14. C2×C3×D8 | C22 | semidirect |
| 16. C3×Q32 | C22 | semidirect |
| 17. C3×SD32 | C22 | semidirect |
| 18. C3×D16 | C22 | semidirect |
| 19. C3×M32 | C22 | semidirect |
| 20. C2×C3×C16 | C22 | semidirect |
| 21. C2×#135 | C2×C3 | semidirect |
| 22. #510 | C2×C3 | semidirect |
| 23. #509 | C2×C3 | semidirect |
| 24. #508 | C2×C3 | semidirect |
| 25. C2×M32 | C2×C3 | semidirect |
| 50. #135 | C22×C3 | semidirect |
| 51. C2×Q16 | C22×C3 | semidirect |
| 53. C2×D8 | C22×C3 | semidirect |
| 55. Q32 | C22×C3 | semidirect |
| 56. SD32 | C22×C3 | semidirect |
| 57. D16 | C22×C3 | semidirect |
| 58. M32 | C22×C3 | semidirect |
| 59. C2×C16 | C22×C3 | semidirect |
| 138. C3 | #3414 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#135 | 2 |
| C3×#510 | 2 |
| C3×#509 | 2 |
| C3×#508 | 2 |
| C2×C3×M32 | 2 |
| #3414 | 3 |