Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
551 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 410 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C3×C9 | C34 |
| commutator, socle | C32 | C33×C9 |
| prime radical | C3×H(3,𝔽3) | C3×C9 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624729 | C3 | semidirect |
| 3. #10624729 | C3 | semidirect |
| 4. #10624727 | C3 | semidirect |
| 5. C27×3−1+2 | C3 | semidirect |
| 6. #10624713 | C3 | semidirect |
| 7. #10624711 | C3 | semidirect |
| 8. #10624710 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624879 | 3 |
| #10624729 | 3 |
| #10624727 | 3 |
| C27×3−1+2 | 3 |
| #10624713 | 3 |
| #10624711 | 3 |
| #10624710 | 3 |
| #10624684 | 3 |