Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
57194 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10624943 | C3 | 1 | 243 | yes | maximal, complemented |
| 2 | C3×3−1+4 | C3 | 1 | 120 | yes | maximal, complemented |
| 3 | C3×H(5,𝔽3) | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | 3−1+4 | C32 | 1 | 810 | yes | complemented |
| 5 | 3−1+4 | C32 | 1 | 6480 | yes | complemented |
| 6 | H(5,𝔽3) | C32 | 1 | 81 | yes | — |
| 7 | C3×#737 | C32 | 1 | 3240 | yes | complemented |
| 8 | C32×3−1+2 | C32 | 1 | 360 | yes | complemented |
| 9 | C32×H(3,𝔽3) | C32 | 1 | 40 | yes | — |
| 10 | C34 | C33 | 1 | 40 | yes | — |
| 11 | #737 | C33 | 1 | 21870 | yes | complemented |
| 12 | C3×3−1+2 | C33 | 1 | 1080 | yes | — |
| 13 | C3×3−1+2 | C33 | 1 | 8640 | yes | complemented |
| 14 | C3×H(3,𝔽3) | C33 | 1 | 90 | yes | — |
| 15 | C3×H(3,𝔽3) | C33 | 1 | 1080 | yes | — |
| 16 | C32×C9 | C33 | 1 | 1080 | yes | complemented |
| 17 | C33 | — | 27 | 40 | no | — |
| 18 | C33 | C34 | 1 | 360 | yes | — |
| 19 | C33 | C34 | 1 | 40 | yes | — |
| 20 | 3−1+2 | C34 | 1 | 81 | yes | — |
| 21 | 3−1+2 | C34 | 1 | 6480 | yes | — |
| 22 | H(3,𝔽3) | C34 | 1 | 810 | yes | — |
| 23 | C3×C9 | C34 | 1 | 3240 | yes | — |
| 24 | C32 | — | 9 | 40 | no | — |
| 25 | C32 | — | 9 | 360 | no | — |
| 26 | C32 | C35 | 1 | 120 | yes | — |
| 27 | C32 | C35 | 1 | 1 | yes | characteristic |
| 28 | C9 | C35 | 1 | 243 | yes | — |
| 29 | C3 | — | 3 | 1 | no | — |
| 30 | C3 | — | 3 | 120 | no | — |
| 31 | C3 | C36 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C3 | C36 |
| prime radical | C3×H(5,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624943 | C3 | semidirect |
| 2. C3×3−1+4 | C3 | semidirect |
| 4. 3−1+4 | C32 | semidirect |
| 5. 3−1+4 | C32 | semidirect |
| 7. C3×#737 | C32 | semidirect |
| 8. C32×3−1+2 | C32 | semidirect |
| 11. #737 | C33 | semidirect |
| 13. C3×3−1+2 | C33 | semidirect |
| 16. C32×C9 | C33 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624943 | 3 |
| C3×3−1+4 | 3 |
| C3×H(5,𝔽3) | 3 |