Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
156 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×D3 |
| Fitting | C2×C4×H(3,𝔽3) | C2 |
| hypercenter | C2×C4 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×H(3,𝔽3) | C2 | semidirect |
| 2. #638⋊C3 | C2 | semidirect |
| 3. C4⋊((C3⋊D3)⋊C3) | C2 | semidirect |
| 4. C4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 5. #634⋊C3 | C2 | semidirect |
| 6. #3828 | C3 | semidirect |
| 14. C2×C32×C4 | D3 | semidirect |
| 15. C2×C32×C4 | C2×C3 | semidirect |
| 16. #638 | C2×C3 | semidirect |
| 17. C4⋊(C3⋊D3) | C2×C3 | semidirect |
| 18. C4×C3⋊D3 | C2×C3 | semidirect |
| 19. #634 | C2×C3 | semidirect |
| 43. H(3,𝔽3) | #41 | semidirect |
| 88. C32 | #235 | semidirect |
| 89. C32 | C3×#41 | semidirect |
| 94. C2×C4 | (C3⋊D3)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×H(3,𝔽3) | 2 |
| #638⋊C3 | 2 |
| C4⋊((C3⋊D3)⋊C3) | 2 |
| C4×(C3⋊D3)⋊C3 | 2 |
| #634⋊C3 | 2 |
| #3828 | 3 |
| C3×#235 | 3 |