Finite Group DatabasebetaFinite Group DatabasebetaOrder 4374.
4462 conjugacy classes of subgroups; only the 665 normal subgroups are shown.
This group has 4462 conjugacy classes of subgroups — too many to draw. Only its 665 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C33×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C34 |
| Frattini | C32 | C34×D3 |
| Fitting | #14555436 | C2 |
| hypercenter | C3×H(3,𝔽3) | (C3⋊D3)⋊C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14555436 | C2 | semidirect |
| 3. C3×#92392 | C3 | semidirect |
| 4. #13091645 | C3 | semidirect |
| 5. C3×#92281 | C3 | semidirect |
| 7. C3×H(5,𝔽3) | D3 | semidirect |
| 8. C3×H(5,𝔽3) | C2×C3 | semidirect |
| 10. #10624930 | C2×C3 | semidirect |
| 12. C3×#6909 | C2×C3 | semidirect |
| 14. #92392 | C32 | semidirect |
| 15. C32×#272 | C32 | semidirect |
| 20. #92281 | C32 | semidirect |
| 21. #92281 | C32 | semidirect |
| 22. C3×#4174 | C32 | semidirect |
| 32. H(5,𝔽3) | C2×C32 | semidirect |
| 39. C32×H(3,𝔽3) | C3×D3 | semidirect |
| 40. C32×H(3,𝔽3) | C3×D3 | semidirect |
| 41. C32×H(3,𝔽3) | C2×C32 | semidirect |
| 45. #6909 | C2×C32 | semidirect |
| 46. #6909 | C2×C32 | semidirect |
| 49. C3×AΣL(1,𝔽27) | C2×C32 | semidirect |
| 62. #4174 | C33 | semidirect |
| 66. C34 | C32×D3 | semidirect |
| 100. C3×H(3,𝔽3) | C32×D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14555436 | 2 |
| D3×H(5,𝔽3) | 3 |
| C3×#92392 | 3 |
| #13091645 | 3 |
| C3×#92281 | 3 |