Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1646 conjugacy classes of subgroups; only the 584 normal subgroups are shown.
This group has 1646 conjugacy classes of subgroups — too many to draw. Only its 584 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C3×H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C33 | C34 |
| socle | C32 | C32×H(3,𝔽3) |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×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. #10624692 | C3 | semidirect |
| 4. #10624691 | C3 | semidirect |
| 5. #10624690 | C3 | semidirect |
| 7. #10624688 | C3 | semidirect |
| 11. C3×#737 | C9 | semidirect |
| 40. C3×3−1+2 | C3×C9 | semidirect |
| 49. C32×C9 | C3×C9 | semidirect |
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 |
| #10624692 | 3 |
| #10624691 | 3 |
| #10624690 | 3 |
| #10624689 | 3 |
| #10624688 | 3 |