Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
855 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 855 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #4175 |
| commutator, cosocle, hypocenter | C33 | C23×C32 |
| Frattini | C32 | C22×C32×D3 |
| Fitting | C22×C3×AΣL(1,𝔽27) | C2 |
| socle | C22×C32 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×AΣL(1,𝔽27) | C2 | semidirect |
| 2. C2×C3×#4175 | C2 | direct (N × H) |
| 3. C22×C3×C3⋊(C3⋊D3) | C3 | semidirect |
| 5. C22×#4175 | C3 | direct (N × H) |
| 6. C2×C3×AΣL(1,𝔽27) | C22 | semidirect |
| 7. C3×#4175 | C22 | direct (N × H) |
| 8. C22×C34 | C2×C3 | semidirect |
| 9. C2×C3×C3⋊(C3⋊D3) | C2×C3 | semidirect |
| 11. C22×C3×H(3,𝔽3) | D3 | semidirect |
| 14. C22×AΣL(1,𝔽27) | C2×C3 | semidirect |
| 15. C2×#4175 | C2×C3 | direct (N × H) |
| 16. C3×AΣL(1,𝔽27) | C23 | semidirect |
| 17. C22×C3⋊(C3⋊D3) | C32 | semidirect |
| 23. C2×C34 | C22×C3 | semidirect |
| 24. C3×C3⋊(C3⋊D3) | C22×C3 | semidirect |
| 26. C2×C3×H(3,𝔽3) | C2×D3 | semidirect |
| 29. C2×AΣL(1,𝔽27) | C22×C3 | semidirect |
| 30. #4175 | C22×C3 | direct (N × H) |
| 38. C22×C33 | C2×C32 | semidirect |
| 39. C2×C3⋊(C3⋊D3) | C2×C32 | semidirect |
| 47. C22×H(3,𝔽3) | C3×D3 | semidirect |
| 50. C34 | C23×C3 | semidirect |
| 52. C3×H(3,𝔽3) | C22×D3 | semidirect |
| 54. AΣL(1,𝔽27) | C23×C3 | semidirect |
| 70. C2×C33 | C22×C32 | semidirect |
| 71. C3⋊(C3⋊D3) | C22×C32 | semidirect |
| 79. C2×H(3,𝔽3) | C2×C3×D3 | semidirect |
| 114. C33 | C23×C32 | semidirect |
| 118. H(3,𝔽3) | C22×C3×D3 | semidirect |
| 158. C22×C3 | #4175 | direct (N × H) |
| 189. C2×C3 | C2×#4175 | direct (N × H) |
| 195. C22 | C3×#4175 | direct (N × H) |
| 202. C3 | C22×#4175 | direct (N × H) |
| 205. C2 | C2×C3×#4175 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×AΣL(1,𝔽27) | 2 |
| C2×C3×#4175 | 2 |
| C22×C3×C3⋊(C3⋊D3) | 3 |
| C22×C3×(C3⋊D3)⋊C3 | 3 |
| C22×#4175 | 3 |