Finite Group DatabasebetaFinite Group DatabasebetaOrder 1458.
584 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C3×AGL(1,ℤ/9) |
| commutator, cosocle | C3×C9 | C2×C33 |
| Frattini | C32 | C33×D3 |
| Fitting | C3×C9⋊H(3,𝔽3) | C2 |
| socle | C33 | C32×D3 |
| hypercenter | C33 | AGL(1,ℤ/9) |
| hypocenter | C9 | C2×C3×H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C9 | |
| UCS | C1 ◁ C32 ◁ C33 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C9⋊H(3,𝔽3) | C2 | semidirect |
| 2. C32×AGL(1,ℤ/9) | C3 | semidirect |
| 4. C33×D9 | C3 | semidirect |
| 5. D9⋊H(3,𝔽3) | C3 | direct (N × H) |
| 6. C32×3−1+2 | C2×C3 | semidirect |
| 8. C33×C9 | C2×C3 | semidirect |
| 10. C9⋊H(3,𝔽3) | C2×C3 | semidirect |
| 17. C3×AGL(1,ℤ/9) | C32 | semidirect |
| 21. C32×D9 | C32 | semidirect |
| 27. C3×3−1+2 | C2×C32 | semidirect |
| 38. C32×C9 | C2×C32 | semidirect |
| 51. C3×D9 | H(3,𝔽3) | semidirect |
| 61. C33 | AGL(1,ℤ/9) | semidirect |
| 79. C3×C9 | C2×H(3,𝔽3) | semidirect |
| 90. D9 | C3×H(3,𝔽3) | semidirect |
| 106. C32 | C3×AGL(1,ℤ/9) | semidirect |
| 112. C9 | C2×C3×H(3,𝔽3) | semidirect |
| 123. C3 | D9⋊H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9⋊H(3,𝔽3) | 2 |
| C32×AGL(1,ℤ/9) | 3 |
| C3×D3×H(3,𝔽3) | 3 |
| C33×D9 | 3 |
| D9⋊H(3,𝔽3) | 3 |