Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
4058 conjugacy classes of subgroups; only the 764 normal subgroups are shown.
This group has 4058 conjugacy classes of subgroups — too many to draw. Only its 764 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C33 | C34 |
| prime radical | C33×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ C35 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C34×C9 | C3 | semidirect |
| 3. C32×C32⋊C9 | C3 | semidirect |
| 4. C3×H(3,𝔽3)⋊C9 | C3 | direct (N × H) |
| 9. C32×H(3,𝔽3) | C9 | semidirect |
| 15. C33×C9 | C32 | semidirect |
| 17. C3×C32⋊C9 | C32 | semidirect |
| 18. H(3,𝔽3)⋊C9 | C32 | direct (N × H) |
| 35. C3×H(3,𝔽3) | C3×C9 | semidirect |
| 44. C32×C9 | C33 | semidirect |
| 46. C32⋊C9 | C33 | semidirect |
| 73. H(3,𝔽3) | C32×C9 | semidirect |
| 94. C32 | H(3,𝔽3)⋊C9 | direct (N × H) |
| 109. C3 | C3×H(3,𝔽3)⋊C9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×H(3,𝔽3) | 3 |
| C34×C9 | 3 |
| C32×C32⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |