Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1826 conjugacy classes of subgroups; only the 602 normal subgroups are shown.
This group has 1826 conjugacy classes of subgroups — too many to draw. Only its 602 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C33 | C34 |
| socle | C33 | C3×H(3,𝔽3) |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#6887 | C3 | semidirect |
| 3. C32×C92 | C3 | semidirect |
| 4. #10624697 | C3 | direct (N × H) |
| 5. C3×3−1+2⋊C9 | C3 | semidirect |
| 9. C3×#737 | C9 | semidirect |
| 18. #6887 | C32 | semidirect |
| 25. C3×C92 | C32 | semidirect |
| 26. 3−1+2⋊C9 | C32 | semidirect |
| 32. #737 | C3×C9 | semidirect |
| 117. C3 | #10624697 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| C3×#6887 | 3 |
| C32×C92 | 3 |
| #10624697 | 3 |
| C3×3−1+2⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |