Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2096 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 2096 conjugacy classes of subgroups — too many to draw. Only its 314 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×3−1+4 | C3 | semidirect |
| 2. C3×#6911 | C3 | semidirect |
| 3. C3×#6911 | C3 | semidirect |
| 4. C3×#6910 | C3 | semidirect |
| 5. C3×#6909 | C3 | semidirect |
| 6. C3×#6909 | C3 | semidirect |
| 7. C32×#732 | C3 | semidirect |
| 8. C32×#731 | C3 | semidirect |
| 9. C3×#6893 | C3 | semidirect |
| 10. #10624753 | C3 | direct (N × H) |
| 14. 3−1+4 | C32 | semidirect |
| 29. #6911 | C32 | semidirect |
| 30. #6911 | C32 | semidirect |
| 32. #6910 | C32 | semidirect |
| 35. #6909 | C32 | semidirect |
| 36. #6909 | C32 | semidirect |
| 41. C3×#732 | C32 | semidirect |
| 45. C3×#731 | C32 | semidirect |
| 49. #6893 | C32 | semidirect |
| 269. C3 | #10624753 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×3−1+4 | 3 |
| C3×#6911 | 3 |
| C3×#6910 | 3 |
| C3×#6909 | 3 |
| C32×#732 | 3 |
| C32×#731 | 3 |
| C3×#6893 | 3 |
| #10624753 | 3 |