Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
312 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C33 | C2×C32 |
| Frattini | C32 | C3×D3×A4 |
| Fitting | C22×AΣL(1,𝔽27) | C2×C3 |
| socle | C22×C3 | C3×(C3⋊D3)⋊C3 |
| LCS | G ▷ C22×C33 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×AΣL(1,𝔽27) | C2 | semidirect |
| 2. A4×C3⋊(C3⋊D3) | C3 | semidirect |
| 4. C22×#4175 | C3 | semidirect |
| 5. #10619775 | C3 | semidirect |
| 7. C33×A4 | C2×C3 | semidirect |
| 9. A4×H(3,𝔽3) | D3 | semidirect |
| 10. C22×AΣL(1,𝔽27) | C2×C3 | semidirect |
| 13. #66368 | C2×C3 | semidirect |
| 17. C22×C3⋊(C3⋊D3) | C32 | semidirect |
| 29. #4175 | A4 | direct (N × H) |
| 30. C22×C33 | C2×C32 | semidirect |
| 41. C22×H(3,𝔽3) | C3×D3 | semidirect |
| 45. C22⋊H(3,𝔽3) | C3×D3 | semidirect |
| 55. AΣL(1,𝔽27) | C2×A4 | semidirect |
| 67. C3⋊(C3⋊D3) | C3×A4 | semidirect |
| 111. C33 | C2×C3×A4 | semidirect |
| 117. H(3,𝔽3) | D3×A4 | semidirect |
| 159. A4 | #4175 | direct (N × H) |
| 196. C22 | C3×#4175 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×AΣL(1,𝔽27) | 2 |
| A4×C3⋊(C3⋊D3) | 3 |
| A4×(C3⋊D3)⋊C3 | 3 |
| C22×#4175 | 3 |
| #10619775 | 3 |
| C3×#4175 | 4 |