Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#123 | C2 |
| prime radical | C2×#937 | C2 |
| hypercenter | C2×#123 | D3 |
| hypocenter | C3 | #2412 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#123 | C2 | semidirect |
| 3. C22×(C2×D3)⋊C4 | C2 | semidirect |
| 4. C2×D3×C22⋊C4 | C2 | semidirect |
| 5. #5220 | C2 | semidirect |
| 6. #5211 | C2 | semidirect |
| 7. #4992 | C2 | semidirect |
| 8. #4987 | C2 | semidirect |
| 9. #4961 | C2 | semidirect |
| 10. #4956 | C2 | semidirect |
| 11. #4950 | C2 | semidirect |
| 21. C2×C3×C4⋊C4 | C22 | semidirect |
| 23. C2×C3×C22⋊C4 | C22 | semidirect |
| 26. C2×#217 | C22 | semidirect |
| 34. C2×C4⋊Dic3 | C22 | semidirect |
| 36. C3⋊#95 | C22 | semidirect |
| 37. C3⋊#95 | C22 | semidirect |
| 38. C3⋊#95 | C22 | semidirect |
| 39. C3⋊#95 | C22 | semidirect |
| 40. C2×#123 | D3 | semidirect |
| 84. C3×C22⋊C4 | D4 | semidirect |
| 104. C2×C4⋊C4 | C2×D3 | semidirect |
| 116. C2×C22⋊C4 | C2×D3 | semidirect |
| 214. C22⋊C4 | #67 | semidirect |
| 240. C2×D3 | #123 | semidirect |
| 341. C3 | #2412 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#123 | 2 |
| C2×#937 | 2 |
| C22×(C2×D3)⋊C4 | 2 |
| C2×D3×C22⋊C4 | 2 |
| #5220 | 2 |
| #5211 | 2 |
| #4992 | 2 |
| #4987 | 2 |
| #4961 | 2 |
| #4956 | 2 |
| #4950 | 2 |
| #2412 | 3 |