Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
21 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #216 | — | 3 | 1 | no | maximal |
| 3 | Dic9 | — | 4 | 1 | no | maximal |
| 4 | C3×Q8 | D3 | 1 | 1 | yes | characteristic |
| 5 | Dic6 | — | 3 | 1 | no | — |
| 6 | C3⋊C8 | — | 3 | 1 | no | — |
| 7 | C2×C9 | — | 4 | 1 | no | — |
| 8 | Q16 | — | 9 | 1 | no | Sylow |
| 9 | C3×C4 | — | 3 | 1 | no | — |
| 10 | Dic3 | — | 6 | 1 | no | — |
| 11 | C9 | — | 4 | 1 | no | Sylow |
| 12 | Q8 | — | 9 | 1 | no | — |
| 13 | Q8 | D9 | 1 | 1 | yes | characteristic |
| 14 | C8 | — | 9 | 1 | no | — |
| 15 | C2×C3 | S4 | 1 | 1 | yes | characteristic |
| 16 | C4 | — | 3 | 1 | no | — |
| 17 | C4 | — | 18 | 1 | no | — |
| 18 | C3 | #226 | 1 | 1 | yes | characteristic |
| 19 | C2 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C2 |
| Frattini, socle, prime radical | C2×C3 | S4 |
| Fitting | C3×Q8 | D3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8⋊C9 | 2 |
| #216 | 3 |
| Dic9 | 4 |