Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1987 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 1987 conjugacy classes of subgroups — too many to draw. Only its 311 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 | C3×C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624943 | C3 | semidirect |
| 2. #10624929 | C3 | semidirect |
| 3. #10624929 | C3 | semidirect |
| 4. C3×#6913 | C3 | semidirect |
| 5. #10624882 | C3 | semidirect |
| 6. #10624753 | C3 | semidirect |
| 7. #10624752 | C3 | semidirect |
| 8. #10624751 | C3 | semidirect |
| 9. #10624750 | C3 | semidirect |
| 10. #10624749 | C3 | semidirect |
| 11. #10624748 | C3 | semidirect |
| 12. UT(4,𝔽3) | C3 | semidirect |
| 22. C3×#737 | C32 | semidirect |
| 23. C3×#737 | C32 | semidirect |
| 24. C32×3−1+2 | C32 | semidirect |
| 30. #6912 | C32 | semidirect |
| 31. #6912 | C32 | semidirect |
| 33. #6911 | C32 | semidirect |
| 35. #6910 | C32 | semidirect |
| 36. #6910 | C32 | semidirect |
| 37. #6910 | C32 | semidirect |
| 41. #6909 | C32 | semidirect |
| 44. C3×#733 | C32 | semidirect |
| 47. C3×#731 | C32 | semidirect |
| 49. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 51. #6893 | C32 | semidirect |
| 54. #6891 | C32 | semidirect |
| 80. #737 | H(3,𝔽3) | semidirect |
| 81. #737 | H(3,𝔽3) | semidirect |
| 95. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 112. C32×C9 | H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624943 | 3 |
| #10624929 | 3 |
| C3×#6913 | 3 |
| #10624882 | 3 |
| #10624753 | 3 |
| #10624752 | 3 |
| #10624751 | 3 |
| #10624750 | 3 |
| #10624749 | 3 |
| #10624748 | 3 |
| UT(4,𝔽3) | 3 |