Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
400 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C22×D4 |
| prime radical | C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#533 | C2 | semidirect |
| 3. C3×#488 | C2 | semidirect |
| 4. C3×#487 | C2 | semidirect |
| 5. C3×#486 | C2 | semidirect |
| 6. C3×#416 | C2 | semidirect |
| 7. C2×C3×C4×C8 | C2 | semidirect |
| 8. #3131 | C3 | direct (N × H) |
| 26. #533 | C2×C3 | semidirect |
| 28. #488 | C2×C3 | semidirect |
| 29. #487 | C2×C3 | semidirect |
| 30. #486 | C2×C3 | semidirect |
| 31. #416 | C2×C3 | semidirect |
| 32. C2×C4×C8 | C2×C3 | semidirect |
| 152. C3 | #3131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#533 | 2 |
| C3×#532 | 2 |
| C3×#488 | 2 |
| C3×#487 | 2 |
| C3×#486 | 2 |
| C3×#416 | 2 |
| C2×C3×C4×C8 | 2 |
| #3131 | 3 |