Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
196 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10620198 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊D3×#157 | C2 | 1 | 2 | yes | maximal |
| 3 | C3⋊D32 | C22 | 1 | 1 | yes | characteristic |
| 4 | #66472 | C4 | 1 | 2 | yes | complemented |
| 5 | C3⋊(C3⋊#157) | C4 | 1 | 2 | yes | complemented |
| 6 | C32×#157 | C4 | 1 | 2 | yes | complemented |
| 7 | D3×#157 | — | 6 | 4 | no | — |
| 8 | C3⋊(C3⋊(C3⋊D3)) | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | C32×C3⋊D3 | C2×C4 | 1 | 2 | yes | — |
| 10 | C4×#157 | — | 9 | 2 | no | maximal |
| 11 | D3×C3⋊D3 | — | 6 | 4 | no | — |
| 12 | C3⋊#157 | — | 6 | 4 | no | — |
| 13 | C3×#157 | — | 2 | 4 | no | — |
| 14 | C34 | C42 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C2×#157 | — | 9 | 2 | no | — |
| 16 | C2×#157 | — | 9 | 2 | no | — |
| 17 | C4×C3⋊D3 | — | 9 | 2 | no | — |
| 18 | C3⋊(C3⋊D3) | — | 6 | 4 | no | — |
| 19 | C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 20 | C3×C3⋊D3 | — | 2 | 4 | no | — |
| 21 | C32×D3 | — | 6 | 4 | no | — |
| 22 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 23 | D32 | — | 36 | 4 | no | — |
| 24 | #157 | — | 9 | 4 | no | — |
| 25 | #157 | — | 9 | 2 | no | — |
| 26 | #157 | — | 36 | 4 | no | — |
| 27 | #157 | #157 | 1 | 2 | yes | direct product factor |
| 28 | C32×C4 | — | 9 | 2 | no | — |
| 29 | C3⋊Dic3 | — | 9 | 2 | no | — |
| 30 | C33 | — | 2 | 4 | no | — |
| 31 | C33 | — | 8 | 4 | no | — |
| 32 | C4×D3 | — | 54 | 4 | no | — |
| 33 | C2×C32 | — | 9 | 2 | no | — |
| 34 | C3⋊D3 | — | 9 | 2 | no | — |
| 35 | C3⋊D3 | — | 36 | 4 | no | — |
| 36 | C3⋊D3 | — | 36 | 4 | no | — |
| 37 | C3⋊D3 | — | 72 | 8 | no | — |
| 38 | C3⋊D3 | — | 72 | 4 | no | — |
| 39 | C3⋊D3 | C2×#157 | 1 | 2 | yes | — |
| 40 | C3×D3 | — | 12 | 8 | no | — |
| 41 | C42 | — | 81 | 1 | no | Sylow |
| 42 | C2×D3 | — | 54 | 4 | no | — |
| 43 | C3×C4 | — | 18 | 4 | no | — |
| 44 | Dic3 | — | 54 | 4 | no | — |
| 45 | C32 | — | 4 | 4 | no | — |
| 46 | C32 | — | 4 | 4 | no | — |
| 47 | C32 | — | 8 | 8 | no | — |
| 48 | C32 | — | 8 | 4 | no | — |
| 49 | C32 | C4×#157 | 1 | 2 | yes | complemented |
| 50 | C2×C4 | — | 81 | 2 | no | — |
| 51 | C2×C4 | — | 81 | 1 | no | — |
| 52 | C2×C3 | — | 18 | 4 | no | — |
| 53 | D3 | — | 6 | 4 | no | — |
| 54 | D3 | — | 54 | 4 | no | — |
| 55 | D3 | — | 216 | 4 | no | — |
| 56 | C22 | — | 81 | 1 | no | — |
| 57 | C4 | — | 9 | 2 | no | — |
| 58 | C4 | — | 81 | 2 | no | — |
| 59 | C4 | — | 81 | 2 | no | — |
| 60 | C3 | — | 2 | 4 | no | — |
| 61 | C3 | — | 8 | 4 | no | — |
| 62 | C2 | — | 9 | 2 | no | — |
| 63 | C2 | — | 81 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C34 | C42 |
| prime radical, cosocle | C3⋊D32 | C22 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #66472 | C4 | semidirect |
| 5. C3⋊(C3⋊#157) | C4 | semidirect |
| 6. C32×#157 | C4 | semidirect |
| 14. C34 | C42 | semidirect |
| 27. #157 | #157 | direct (N × H) |
| 49. C32 | C4×#157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10620198 | 2 |
| C3⋊D3×#157 | 2 |
| C4×#157 | 9 |