Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
3992 conjugacy classes of subgroups; only the 350 normal subgroups are shown.
This group has 3992 conjugacy classes of subgroups — too many to draw. Only its 350 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 ◁ C35 ◁ 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. C33×H(3,𝔽3) | C3 | semidirect |
| 3. C32×AΣL(1,𝔽27) | C3 | semidirect |
| 4. C3×#6891 | C3 | semidirect |
| 5. C3×C32⋊H(3,𝔽3) | C3 | semidirect |
| 6. #10624743 | C3 | direct (N × H) |
| 14. C32×3−1+2 | C32 | semidirect |
| 22. C32×H(3,𝔽3) | C32 | semidirect |
| 25. C3×AΣL(1,𝔽27) | C32 | semidirect |
| 27. #6891 | C32 | semidirect |
| 29. C32⋊H(3,𝔽3) | C32 | semidirect |
| 55. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 108. C33 | AΣL(1,𝔽27) | semidirect |
| 120. 3−1+2 | C3×H(3,𝔽3) | semidirect |
| 160. C32 | C3×AΣL(1,𝔽27) | semidirect |
| 172. C3 | #10624743 | 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 |
| C33×H(3,𝔽3) | 3 |
| C32×AΣL(1,𝔽27) | 3 |
| C3×#6891 | 3 |
| C3×C32⋊H(3,𝔽3) | 3 |
| #10624743 | 3 |