Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
19772 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C36 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C32×C32⋊C9 | C3 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#6866 | C3 | 1 | 36 | yes | maximal, direct product factor |
| 4 | C35 | — | 3 | 81 | no | — |
| 5 | C35 | — | 3 | 8 | no | — |
| 6 | C35 | — | 3 | 1 | no | — |
| 7 | C35 | — | 3 | 18 | no | — |
| 8 | C35 | C9 | 1 | 27 | yes | complemented |
| 9 | C35 | C32 | 1 | 12 | yes | — |
| 10 | C35 | C32 | 1 | 1 | yes | characteristic |
| 11 | C33×C9 | — | 3 | 3 | no | — |
| 12 | C3×C32⋊C9 | — | 3 | 24 | no | — |
| 13 | C3×C32⋊C9 | C32 | 1 | 36 | yes | complemented |
| 14 | #6866 | C32 | 1 | 81 | yes | direct product factor |
| 15 | C34 | — | 3 | 729 | no | — |
| 16 | C34 | — | 3 | 324 | no | — |
| 17 | C34 | — | 3 | 1458 | no | — |
| 18 | C34 | — | 3 | 648 | no | — |
| 19 | C34 | — | 3 | 216 | no | — |
| 20 | C34 | — | 3 | 27 | no | — |
| 21 | C34 | — | 3 | 81 | no | — |
| 22 | C34 | — | 3 | 27 | no | — |
| 23 | C34 | — | 3 | 24 | no | — |
| 24 | C34 | — | 3 | 12 | no | — |
| 25 | C34 | — | 3 | 72 | no | — |
| 26 | C34 | C3×C9 | 1 | 9 | yes | — |
| 27 | C34 | C3×C9 | 1 | 108 | yes | complemented |
| 28 | C34 | H(3,𝔽3) | 1 | 8 | yes | — |
| 29 | C34 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 30 | C34 | 3−1+2 | 1 | 18 | yes | — |
| 31 | C34 | C33 | 1 | 9 | yes | — |
| 32 | C34 | C33 | 1 | 4 | yes | — |
| 33 | C32×C9 | — | 3 | 36 | no | — |
| 34 | C32×C9 | — | 9 | 27 | no | — |
| 35 | C32⋊C9 | — | 3 | 72 | no | — |
| 36 | C32⋊C9 | C33 | 1 | 27 | yes | complemented |
| 37 | C33 | — | 3 | 972 | no | — |
| 38 | C33 | — | 3 | 81 | no | — |
| 39 | C33 | — | 3 | 648 | no | — |
| 40 | C33 | — | 3 | 5832 | no | — |
| 41 | C33 | — | 3 | 1944 | no | — |
| 42 | C33 | — | 3 | 9 | no | — |
| 43 | C33 | — | 3 | 18 | no | — |
| 44 | C33 | — | 3 | 729 | no | — |
| 45 | C33 | — | 3 | 81 | no | — |
| 46 | C33 | — | 3 | 162 | no | — |
| 47 | C33 | — | 3 | 216 | no | — |
| 48 | C33 | — | 3 | 108 | no | — |
| 49 | C33 | — | 3 | 324 | no | — |
| 50 | C33 | — | 3 | 72 | no | — |
| 51 | C33 | — | 3 | 36 | no | — |
| 52 | C33 | C32⋊C9 | 1 | 72 | yes | — |
| 53 | C33 | C32⋊C9 | 1 | 9 | yes | — |
| 54 | C33 | AΣL(1,𝔽27) | 1 | 27 | yes | — |
| 55 | C33 | C32×C9 | 1 | 12 | yes | — |
| 56 | C33 | C32×C9 | 1 | 27 | yes | complemented |
| 57 | C33 | C3×H(3,𝔽3) | 1 | 8 | yes | — |
| 58 | C33 | C3×H(3,𝔽3) | 1 | 4 | yes | — |
| 59 | C33 | C3×3−1+2 | 1 | 24 | yes | — |
| 60 | C33 | C34 | 1 | 1 | yes | characteristic |
| 61 | C3×C9 | — | 3 | 27 | no | — |
| 62 | C3×C9 | — | 9 | 108 | no | — |
| 63 | C32 | — | 3 | 81 | no | — |
| 64 | C32 | — | 3 | 162 | no | — |
| 65 | C32 | — | 3 | 1944 | no | — |
| 66 | C32 | — | 3 | 27 | no | — |
| 67 | C32 | — | 3 | 81 | no | — |
| 68 | C32 | — | 3 | 24 | no | — |
| 69 | C32 | — | 3 | 3 | no | — |
| 70 | C32 | — | 3 | 972 | no | — |
| 71 | C32 | — | 3 | 216 | no | — |
| 72 | C32 | — | 3 | 36 | no | — |
| 73 | C32 | — | 3 | 72 | no | — |
| 74 | C32 | #6866 | 1 | 81 | yes | direct product factor |
| 75 | C32 | C3×C32⋊C9 | 1 | 24 | yes | — |
| 76 | C32 | C3×C32⋊C9 | 1 | 12 | yes | — |
| 77 | C32 | C3×AΣL(1,𝔽27) | 1 | 36 | yes | — |
| 78 | C32 | C33×C9 | 1 | 1 | yes | characteristic |
| 79 | C32 | C32×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 80 | C32 | C32×3−1+2 | 1 | 2 | yes | — |
| 81 | C9 | — | 9 | 27 | no | — |
| 82 | C3 | — | 3 | 81 | no | — |
| 83 | C3 | — | 3 | 1 | no | — |
| 84 | C3 | — | 3 | 2 | no | — |
| 85 | C3 | — | 3 | 24 | no | — |
| 86 | C3 | C3×#6866 | 1 | 36 | yes | direct product factor |
| 87 | C3 | C32×C32⋊C9 | 1 | 1 | yes | characteristic |
| 88 | C3 | C32×AΣL(1,𝔽27) | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C33 | C34 |
| prime radical | C36 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ C35 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C32×C32⋊C9 | C3 | semidirect |
| 3. C3×#6866 | C3 | direct (N × H) |
| 8. C35 | C9 | semidirect |
| 13. C3×C32⋊C9 | C32 | semidirect |
| 14. #6866 | C32 | direct (N × H) |
| 27. C34 | C3×C9 | semidirect |
| 36. C32⋊C9 | C33 | semidirect |
| 56. C33 | C32×C9 | semidirect |
| 74. C32 | #6866 | direct (N × H) |
| 86. C3 | C3×#6866 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C36 | 3 |
| C32×C32⋊C9 | 3 |
| C3×#6866 | 3 |