Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
584 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 86 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 | C34 | C33 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6894 | C3 | semidirect |
| 4. #10624548 | C3 | semidirect |
| 5. #10624452 | C3 | semidirect |
| 6. #10624450 | C3 | semidirect |
| 130. C9 | #6857 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6894 | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| C3×#6857 | 3 |
| #10624548 | 3 |
| #10624452 | 3 |
| #10624450 | 3 |
| #10624449 | 3 |