Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
96 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | GU(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #6249⋊C2 | — | 3 | 1 | no | maximal |
| 3 | #4174⋊C2 | — | 4 | 1 | no | maximal |
| 4 | SU(3,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #1088⋊C2 | — | 3 | 1 | no | — |
| 6 | #6249 | — | 3 | 1 | no | — |
| 7 | #4183 | — | 4 | 1 | no | — |
| 8 | #4175 | — | 4 | 1 | no | — |
| 9 | #4174 | — | 4 | 1 | no | — |
| 10 | C3⋊GL(2,𝔽3) | — | 9 | 1 | no | maximal |
| 11 | D3×C3⋊D3 | — | 12 | 1 | no | — |
| 12 | ((C3⋊D3)⋊C3)⋊C2 | — | 6 | 1 | no | — |
| 13 | #1088 | — | 3 | 1 | no | — |
| 14 | AΣL(1,𝔽27) | — | 4 | 1 | no | Sylow |
| 15 | C3×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 16 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 17 | C3×C3⋊D3 | — | 12 | 1 | no | — |
| 18 | C32×D3 | — | 12 | 1 | no | — |
| 19 | #272 | S4 | 1 | 1 | yes | characteristic |
| 20 | AGL(1,ℤ/9) | — | 24 | 1 | no | — |
| 21 | (C3⋊D3)⋊C3 | — | 12 | 1 | no | — |
| 22 | GL(2,𝔽3) | — | 27 | 3 | no | — |
| 23 | #215 | — | 27 | 1 | no | — |
| 24 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 25 | D32 | — | 36 | 1 | no | — |
| 26 | D32 | — | 36 | 3 | no | — |
| 27 | C33 | — | 4 | 1 | no | — |
| 28 | 3−1+2 | — | 8 | 1 | no | — |
| 29 | H(3,𝔽3) | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 30 | C3×Q8 | — | 9 | 1 | no | — |
| 31 | D12 | — | 27 | 1 | no | — |
| 32 | SL(2,𝔽3) | — | 9 | 3 | no | — |
| 33 | C3⋊C8 | — | 27 | 1 | no | — |
| 34 | C2×C32 | — | 36 | 1 | no | — |
| 35 | C3⋊D3 | — | 12 | 1 | no | — |
| 36 | C3⋊D3 | — | 12 | 1 | no | — |
| 37 | C3⋊D3 | — | 36 | 1 | no | — |
| 38 | C3⋊D3 | — | 36 | 3 | no | — |
| 39 | C3×D3 | — | 12 | 1 | no | — |
| 40 | C3×D3 | — | 12 | 3 | no | — |
| 41 | C3×D3 | — | 36 | 1 | no | — |
| 42 | C3×D3 | — | 36 | 3 | no | — |
| 43 | D9 | — | 24 | 1 | no | — |
| 44 | SD16 | — | 81 | 1 | no | Sylow |
| 45 | C2×D3 | — | 54 | 1 | no | — |
| 46 | C2×D3 | — | 108 | 1 | no | — |
| 47 | C2×D3 | — | 108 | 3 | no | — |
| 48 | C3×C4 | — | 27 | 1 | no | — |
| 49 | C32 | — | 4 | 1 | no | — |
| 50 | C32 | — | 12 | 3 | no | — |
| 51 | C32 | — | 12 | 1 | no | — |
| 52 | C9 | — | 24 | 1 | no | — |
| 53 | Q8 | — | 9 | 1 | no | — |
| 54 | D4 | — | 81 | 1 | no | — |
| 55 | C8 | — | 81 | 1 | no | — |
| 56 | C2×C3 | — | 9 | 1 | no | — |
| 57 | C2×C3 | — | 36 | 3 | no | — |
| 58 | C2×C3 | — | 108 | 1 | no | — |
| 59 | D3 | — | 12 | 1 | no | — |
| 60 | D3 | — | 36 | 3 | no | — |
| 61 | D3 | — | 36 | 1 | no | — |
| 62 | D3 | — | 108 | 1 | no | — |
| 63 | D3 | — | 108 | 3 | no | — |
| 64 | C22 | — | 162 | 1 | no | — |
| 65 | C4 | — | 27 | 1 | no | — |
| 66 | C3 | — | 12 | 1 | no | — |
| 67 | C3 | — | 12 | 3 | no | — |
| 68 | C3 | AGL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 69 | C2 | — | 9 | 1 | no | — |
| 70 | C2 | — | 108 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | GU(3,𝔽2) | C2 |
| Frattini, socle | C3 | AGL(2,𝔽3) |
| Fitting | H(3,𝔽3) | GL(2,𝔽3) |
| LCS | G ▷ GU(3,𝔽2) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. GU(3,𝔽2) | C2 | semidirect |
| 4. SU(3,𝔽2) | D3 | semidirect |
| 29. H(3,𝔽3) | GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| GU(3,𝔽2) | 2 |
| #6249⋊C2 | 3 |
| #4174⋊C2 | 4 |
| C3⋊GL(2,𝔽3) | 9 |