Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
2330 conjugacy classes of subgroups; only the 332 normal subgroups are shown.
This group has 2330 conjugacy classes of subgroups — too many to draw. Only its 332 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×3−1+2 | C3 | semidirect |
| 2. C32×#733 | C3 | semidirect |
| 3. C32×#732 | C3 | semidirect |
| 4. C32×#731 | C3 | semidirect |
| 5. C32×AΣL(1,𝔽27) | C3 | semidirect |
| 6. C3×#6893 | C3 | semidirect |
| 7. #10624739 | C3 | direct (N × H) |
| 15. C32×3−1+2 | C32 | semidirect |
| 21. C3×#733 | C32 | semidirect |
| 23. C3×#732 | C32 | semidirect |
| 25. C3×#731 | C32 | semidirect |
| 27. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 29. #6893 | C32 | semidirect |
| 163. C3 | #10624739 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×3−1+2 | 3 |
| C32×#733 | 3 |
| C32×#732 | 3 |
| C32×#731 | 3 |
| C32×AΣL(1,𝔽27) | 3 |
| C3×#6893 | 3 |
| #10624739 | 3 |