Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
205 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×D3 |
| Fitting | D4×H(3,𝔽3) | C2 |
| hypercenter | D4 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C2×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×(C3⋊D3)⋊C3 | C2 | semidirect |
| 2. D4×H(3,𝔽3) | C2 | semidirect |
| 3. #638⋊C3 | C2 | semidirect |
| 4. C4⋊((C3⋊D3)⋊C3) | C2 | semidirect |
| 5. C4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 6. D4×C3⋊D3 | C3 | semidirect |
| 8. C22×H(3,𝔽3) | C22 | semidirect |
| 11. C2×(C3⋊D3)⋊C3 | C22 | semidirect |
| 12. C4×H(3,𝔽3) | C22 | semidirect |
| 13. (C3⋊Dic3)⋊C3 | C22 | semidirect |
| 14. C22×C3⋊D3 | C2×C3 | semidirect |
| 17. C32×D4 | D3 | semidirect |
| 18. C32×D4 | C2×C3 | semidirect |
| 19. #638 | C2×C3 | semidirect |
| 20. C4⋊(C3⋊D3) | C2×C3 | semidirect |
| 21. C4×C3⋊D3 | C2×C3 | semidirect |
| 28. (C3⋊D3)⋊C3 | D4 | direct (N × H) |
| 33. C22×C32 | C2×D3 | semidirect |
| 34. C22×C32 | C22×C3 | semidirect |
| 37. C2×C3⋊D3 | C22×C3 | semidirect |
| 42. C32×C4 | C2×D3 | semidirect |
| 43. C32×C4 | C22×C3 | semidirect |
| 44. C3⋊Dic3 | C22×C3 | semidirect |
| 46. H(3,𝔽3) | C2×D4 | semidirect |
| 70. C3⋊D3 | C3×D4 | semidirect |
| 95. C32 | D3×D4 | semidirect |
| 96. C32 | C2×C3×D4 | semidirect |
| 100. D4 | (C3⋊D3)⋊C3 | direct (N × H) |
| 119. C22 | C2×(C3⋊D3)⋊C3 | semidirect |
| 121. C4 | C2×(C3⋊D3)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×(C3⋊D3)⋊C3 | 2 |
| D4×H(3,𝔽3) | 2 |
| #638⋊C3 | 2 |
| C4⋊((C3⋊D3)⋊C3) | 2 |
| C4×(C3⋊D3)⋊C3 | 2 |
| D4×C3⋊D3 | 3 |
| C3×D3×D4 | 3 |