Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
7334 conjugacy classes of subgroups; only the 2924 normal subgroups are shown.
This group has 7334 conjugacy classes of subgroups — too many to draw. Only its 2924 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | C33 |
| commutator, Frattini, cosocle | C32 | C35 |
| socle | C33 | C34 |
| prime radical | C3×C32⋊H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#737 | C3 | semidirect |
| 2. C34×C9 | C3 | semidirect |
| 3. #10624869 | C3 | direct (N × H) |
| 4. C3×#6893 | C3 | semidirect |
| 5. C3×C9⋊H(3,𝔽3) | C3 | semidirect |
| 6. C3×#6891 | C3 | semidirect |
| 8. C3×C9×H(3,𝔽3) | C3 | semidirect |
| 11. C3×#737 | C32 | semidirect |
| 13. C32×3−1+2 | C32 | semidirect |
| 21. C33×C9 | C32 | semidirect |
| 22. C33×C9 | C32 | semidirect |
| 23. #6893 | C32 | semidirect |
| 24. C9⋊H(3,𝔽3) | C32 | semidirect |
| 25. #6891 | C32 | semidirect |
| 27. C9×H(3,𝔽3) | C32 | semidirect |
| 29. C3×C32⋊C9 | C32 | semidirect |
| 40. C3×3−1+2 | C33 | semidirect |
| 50. C32×C9 | H(3,𝔽3) | semidirect |
| 59. C32×C9 | C33 | semidirect |
| 61. C32⋊C9 | C33 | semidirect |
| 71. C33 | #737 | semidirect |
| 88. C3×C9 | C3×H(3,𝔽3) | semidirect |
| 105. C32 | C3×#737 | semidirect |
| 114. C3 | #10624869 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#737 | 3 |
| C34×C9 | 3 |
| #10624869 | 3 |
| C3×#6893 | 3 |
| C3×C9⋊H(3,𝔽3) | 3 |
| C3×#6891 | 3 |
| C3×C32⋊H(3,𝔽3) | 3 |
| C3×C9×H(3,𝔽3) | 3 |