Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #226 | — | 3 | 3 | no | maximal |
| 3 | #216 | — | 3 | 1 | no | maximal |
| 4 | C3⋊Dic3 | — | 4 | 1 | no | maximal |
| 5 | C3×Q8 | D3 | 1 | 1 | yes | characteristic |
| 6 | Dic6 | — | 3 | 1 | no | — |
| 7 | SL(2,𝔽3) | D3 | 1 | 3 | yes | — |
| 8 | C3⋊C8 | — | 3 | 1 | no | — |
| 9 | C2×C32 | — | 4 | 1 | no | — |
| 10 | Q16 | — | 9 | 1 | no | Sylow |
| 11 | C3×C4 | — | 3 | 1 | no | — |
| 12 | Dic3 | — | 6 | 1 | no | — |
| 13 | Dic3 | — | 12 | 3 | no | — |
| 14 | C32 | — | 4 | 1 | no | Sylow |
| 15 | Q8 | — | 9 | 1 | no | — |
| 16 | Q8 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 17 | C8 | — | 9 | 1 | no | — |
| 18 | C2×C3 | — | 4 | 3 | no | — |
| 19 | C2×C3 | S4 | 1 | 1 | yes | characteristic |
| 20 | C4 | — | 3 | 1 | no | — |
| 21 | C4 | — | 18 | 1 | no | — |
| 22 | C3 | — | 4 | 3 | no | — |
| 23 | C3 | #226 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2 | C3⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊S4 |
| commutator, prime radical, cosocle, hypocenter | C3×SL(2,𝔽3) | C2 |
| Fitting | C3×Q8 | D3 |
| socle | C2×C3 | S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 23. C3 | #226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×SL(2,𝔽3) | 2 |
| #226 | 3 |
| #216 | 3 |
| C3⋊Dic3 | 4 |