Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
25 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #215 | — | 3 | 1 | no | maximal |
| 3 | C2×D9 | — | 4 | 1 | no | maximal |
| 4 | C3×Q8 | D3 | 1 | 1 | yes | characteristic |
| 5 | D12 | — | 3 | 1 | no | — |
| 6 | C3⋊C8 | — | 3 | 1 | no | — |
| 7 | C2×C9 | — | 4 | 1 | no | — |
| 8 | D9 | — | 4 | 2 | no | — |
| 9 | SD16 | — | 9 | 1 | no | Sylow |
| 10 | C2×D3 | — | 6 | 1 | no | — |
| 11 | C3×C4 | — | 3 | 1 | no | — |
| 12 | C9 | — | 4 | 1 | no | Sylow |
| 13 | Q8 | D9 | 1 | 1 | yes | characteristic, complemented |
| 14 | D4 | — | 9 | 1 | no | — |
| 15 | C8 | — | 9 | 1 | no | — |
| 16 | C2×C3 | S4 | 1 | 1 | yes | characteristic |
| 17 | D3 | — | 12 | 1 | no | — |
| 18 | C22 | — | 18 | 1 | no | — |
| 19 | C4 | — | 3 | 1 | no | — |
| 20 | C3 | GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 21 | C2 | — | 36 | 1 | no | — |
| 22 | 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 | C2×C3 | S4 |
| Fitting | C3×Q8 | D3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8⋊C9 | C2 | semidirect |
| 13. Q8 | D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8⋊C9 | 2 |
| #215 | 3 |
| C2×D9 | 4 |