Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1235 conjugacy classes of subgroups; only the 359 normal subgroups are shown.
This group has 1235 conjugacy classes of subgroups — too many to draw. Only its 359 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C33 | C34 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C33 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6896 | C3 | semidirect |
| 3. C32×C92 | C3 | semidirect |
| 4. #10624560 | C3 | semidirect |
| 5. #10624557 | C3 | semidirect |
| 6. #10624556 | C3 | semidirect |
| 8. #10624548 | C3 | semidirect |
| 40. C32×C9 | 3−1+2 | semidirect |
| 56. C92 | H(3,𝔽3) | semidirect |
| 100. C32 | #6896 | semidirect |
| 111. C9 | C9⋊H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6896 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| C32×C92 | 3 |
| #10624560 | 3 |
| #10624557 | 3 |
| #10624556 | 3 |
| #10624555 | 3 |
| #10624548 | 3 |