Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2117 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 2117 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 ◁ 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×#6910 | C3 | semidirect |
| 4. C3×#6910 | C3 | semidirect |
| 5. C3×#6909 | C3 | semidirect |
| 6. C3×#6909 | C3 | semidirect |
| 7. C3×#6909 | C3 | semidirect |
| 8. #10624881 | C3 | semidirect |
| 9. #10624880 | C3 | semidirect |
| 10. #10624749 | C3 | semidirect |
| 16. 3−1+4 | C32 | semidirect |
| 30. #6911 | C32 | semidirect |
| 34. #6910 | C32 | semidirect |
| 35. #6910 | C32 | semidirect |
| 43. #6909 | C32 | semidirect |
| 44. #6909 | C32 | semidirect |
| 50. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 53. #6891 | C32 | semidirect |
| 57. C32⋊H(3,𝔽3) | C32 | semidirect |
| 109. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 110. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 111. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 112. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 166. 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 |
|---|---|
| C3×3−1+4 | 3 |
| C3×#6911 | 3 |
| C3×#6910 | 3 |
| C3×#6909 | 3 |
| #10624881 | 3 |
| #10624880 | 3 |
| #10624749 | 3 |