Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
221 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C23 |
| Frattini | C2×C3 | C2×D32 |
| Fitting | C22×H(3,𝔽3) | C22 |
| socle | C22×C3 | D32 |
| hypocenter | H(3,𝔽3) | C2×D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#272 | C2 | semidirect |
| 2. C22×(C3⋊D3)⋊C3 | C2 | semidirect |
| 3. C2×(C3⋊Dic3)⋊C3 | C2 | semidirect |
| 4. #6200 | C2 | direct (N × H) |
| 9. C2×#272 | C22 | semidirect |
| 11. C2×(C3⋊D3)⋊C3 | C22 | semidirect |
| 12. (C3⋊Dic3)⋊C3 | C22 | semidirect |
| 13. C22×C3⋊D3 | D3 | semidirect |
| 18. C2×C3⋊Dic3 | D3 | semidirect |
| 22. C2×H(3,𝔽3) | D4 | semidirect |
| 33. C2×C3⋊D3 | C2×D3 | semidirect |
| 42. C3⋊Dic3 | C2×D3 | semidirect |
| 44. H(3,𝔽3) | C2×D4 | semidirect |
| 60. C2×C32 | D12 | semidirect |
| 61. C2×C32 | #67 | semidirect |
| 92. C32 | C2×D12 | semidirect |
| 93. C32 | C2×#67 | semidirect |
| 125. C2 | #6200 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#272 | 2 |
| C22×(C3⋊D3)⋊C3 | 2 |
| C2×(C3⋊Dic3)⋊C3 | 2 |
| #6200 | 2 |
| C2×#626 | 3 |
| C2×#625 | 3 |