Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
92 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,ℤ/9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #88937 | — | 3 | 1 | no | maximal |
| 3 | C2×#4174 | — | 4 | 1 | no | maximal |
| 4 | #6296 | D3 | 1 | 1 | yes | characteristic, complemented |
| 5 | C3×#625 | — | 3 | 1 | no | — |
| 6 | C3⋊#622 | — | 3 | 1 | no | — |
| 7 | C2×AΣL(1,𝔽27) | — | 4 | 1 | no | — |
| 8 | #4174 | — | 4 | 2 | no | — |
| 9 | #3775 | — | 9 | 1 | no | — |
| 10 | C2×C32×D3 | — | 6 | 1 | no | — |
| 11 | C3×C3⋊Dic3 | — | 3 | 1 | no | — |
| 12 | C2×#272 | — | 4 | 1 | no | — |
| 13 | AΣL(1,𝔽27) | — | 4 | 1 | no | Sylow |
| 14 | C3×#67 | — | 18 | 1 | no | — |
| 15 | #627 | — | 9 | 1 | no | — |
| 16 | #625 | — | 3 | 1 | no | — |
| 17 | #622 | — | 9 | 1 | no | — |
| 18 | C2×C33 | S4 | 1 | 1 | yes | characteristic |
| 19 | C32×D3 | — | 12 | 1 | no | — |
| 20 | C2×3−1+2 | — | 8 | 1 | no | — |
| 21 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 22 | #272 | — | 4 | 2 | no | — |
| 23 | GL(2,𝔽3) | — | 27 | 1 | no | maximal |
| 24 | #214 | — | 27 | 1 | no | — |
| 25 | C22×C32 | — | 18 | 1 | no | — |
| 26 | C2×C3×D3 | — | 6 | 1 | no | — |
| 27 | C2×C3×D3 | — | 6 | 1 | no | — |
| 28 | C2×C3×D3 | — | 12 | 1 | no | — |
| 29 | C2×C3×D3 | — | 36 | 1 | no | — |
| 30 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 31 | C3×Dic3 | — | 18 | 1 | no | — |
| 32 | C3×Dic3 | — | 18 | 1 | no | — |
| 33 | C33 | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 34 | 3−1+2 | — | 8 | 1 | no | — |
| 35 | H(3,𝔽3) | — | 4 | 1 | no | — |
| 36 | C3×D4 | — | 27 | 1 | no | — |
| 37 | #67 | — | 18 | 1 | no | — |
| 38 | Dic6 | — | 27 | 1 | no | — |
| 39 | SL(2,𝔽3) | — | 27 | 1 | no | — |
| 40 | C3⋊C8 | — | 27 | 1 | no | — |
| 41 | C2×C32 | — | 3 | 1 | no | — |
| 42 | C2×C32 | — | 4 | 1 | no | — |
| 43 | C2×C32 | — | 6 | 1 | no | — |
| 44 | C2×C32 | — | 12 | 1 | no | — |
| 45 | C2×C32 | — | 36 | 1 | no | — |
| 46 | C3×D3 | — | 12 | 1 | no | — |
| 47 | C3×D3 | — | 12 | 2 | no | — |
| 48 | C3×D3 | — | 12 | 1 | no | — |
| 49 | C3×D3 | — | 36 | 2 | no | — |
| 50 | C2×C9 | — | 24 | 1 | no | — |
| 51 | SD16 | — | 81 | 1 | no | Sylow |
| 52 | C22×C3 | — | 18 | 1 | no | — |
| 53 | C22×C3 | — | 18 | 1 | no | — |
| 54 | C22×C3 | — | 36 | 1 | no | — |
| 55 | C2×D3 | — | 6 | 1 | no | — |
| 56 | C2×D3 | — | 36 | 1 | no | — |
| 57 | C3×C4 | — | 27 | 1 | no | — |
| 58 | Dic3 | — | 18 | 1 | no | — |
| 59 | Dic3 | — | 18 | 1 | no | — |
| 60 | C32 | — | 3 | 1 | no | — |
| 61 | C32 | — | 4 | 1 | no | — |
| 62 | C32 | — | 6 | 1 | no | — |
| 63 | C32 | — | 12 | 1 | no | — |
| 64 | C9 | — | 24 | 1 | no | — |
| 65 | Q8 | — | 27 | 1 | no | — |
| 66 | D4 | — | 27 | 1 | no | — |
| 67 | C8 | — | 81 | 1 | no | — |
| 68 | C2×C3 | — | 3 | 1 | no | — |
| 69 | C2×C3 | — | 4 | 1 | no | — |
| 70 | C2×C3 | — | 6 | 1 | no | — |
| 71 | C2×C3 | — | 36 | 1 | no | — |
| 72 | C2×C3 | — | 36 | 2 | no | — |
| 73 | C2×C3 | — | 36 | 1 | no | — |
| 74 | C2×C3 | — | 36 | 1 | no | — |
| 75 | D3 | — | 12 | 1 | no | — |
| 76 | D3 | — | 36 | 2 | no | — |
| 77 | C22 | — | 18 | 1 | no | — |
| 78 | C4 | — | 27 | 1 | no | — |
| 79 | C3 | — | 3 | 1 | no | — |
| 80 | C3 | — | 4 | 1 | no | — |
| 81 | C3 | — | 6 | 1 | no | — |
| 82 | C3 | — | 36 | 1 | no | — |
| 83 | C2 | — | 36 | 1 | no | — |
| 84 | C2 | #10620185 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620185 |
| commutator, cosocle, hypocenter | SL(2,ℤ/9) | C2 |
| Fitting, socle | C2×C33 | S4 |
| LCS | G ▷ SL(2,ℤ/9) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,ℤ/9) | C2 | semidirect |
| 4. #6296 | D3 | semidirect |
| 33. C33 | GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,ℤ/9) | 2 |
| #88937 | 3 |
| C2×#4174 | 4 |
| GL(2,𝔽3) | 27 |