Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
192 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C22×C32 | C22×C3 |
| Frattini | C3 | C2×D3×A4 |
| Fitting | C23×C32 | C2×C3 |
| socle | C23×C3 | C3×D3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C22⋊H(3,𝔽3) | C2 | semidirect |
| 2. (C3⋊D3)⋊A4 | C2 | direct (N × H) |
| 3. C23×C3⋊D3 | C3 | semidirect |
| 6. C22⋊H(3,𝔽3) | C22 | semidirect |
| 7. C23×C32 | C2×C3 | semidirect |
| 11. C22×C3⋊D3 | C2×C3 | semidirect |
| 13. C2×C3×A4 | D3 | semidirect |
| 22. C22×C32 | C22×C3 | semidirect |
| 28. C2×C3⋊D3 | A4 | semidirect |
| 31. C3×A4 | C2×D3 | semidirect |
| 50. C2×C32 | C2×A4 | semidirect |
| 52. C3⋊D3 | C2×A4 | semidirect |
| 78. C32 | C22×A4 | semidirect |
| 83. C23 | (C3⋊D3)⋊C3 | semidirect |
| 105. C22 | C2×(C3⋊D3)⋊C3 | semidirect |
| 114. C2 | (C3⋊D3)⋊A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊H(3,𝔽3) | 2 |
| (C3⋊D3)⋊A4 | 2 |
| C23×C3⋊D3 | 3 |
| C2×D3×A4 | 3 |
| C2×(C3⋊D3)⋊C3 | 4 |