Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
767 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 767 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C33 | C32×C9 |
| Frattini, cosocle | C34 | C33 |
| prime radical | C32×#732 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×H(3,𝔽3) | C3 | semidirect |
| 3. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 4. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 5. C3×#6868 | C3 | semidirect |
| 6. #10624555 | C3 | semidirect |
| 7. C3×#6855 | C3 | semidirect |
| 16. C3×#732 | C9 | semidirect |
| 41. C3×H(3,𝔽3) | C3×C9 | semidirect |
| 64. C32×C9 | C3×C9 | semidirect |
| 109. H(3,𝔽3) | C32⋊C9 | semidirect |
| 136. C3×C9 | #732 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#732 | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| C3×#6868 | 3 |
| #10624555 | 3 |
| C3×#6855 | 3 |