Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
770 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624880 | C3 | semidirect |
| 2. #10624773 | C3 | semidirect |
| 3. #10624742 | C3 | semidirect |
| 4. #10624589 | C3 | semidirect |
| 5. #10624562 | C3 | semidirect |
| 21. C34 | H(3,𝔽3) | semidirect |
| 39. C3×H(3,𝔽3) | H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624880 | 3 |
| #10624773 | 3 |
| #10624742 | 3 |
| #10624589 | 3 |
| #10624562 | 3 |