Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
54 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×Q8 | C2 | 1 | 2 | yes | maximal |
| 2 | C32×C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×Q16 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C32×C4 | — | 2 | 2 | no | — |
| 5 | C32×C4 | C22 | 1 | 1 | yes | characteristic |
| 6 | C3×Q8 | C2×C3 | 1 | 8 | yes | — |
| 7 | C3×C8 | C2×C3 | 1 | 4 | yes | — |
| 8 | C2×C32 | D4 | 1 | 1 | yes | characteristic |
| 9 | Q16 | C32 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C3×C4 | — | 2 | 8 | no | — |
| 11 | C3×C4 | C22×C3 | 1 | 4 | yes | — |
| 12 | C32 | Q16 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | Q8 | C2×C32 | 1 | 2 | yes | — |
| 14 | C8 | C2×C32 | 1 | 1 | yes | characteristic |
| 15 | C2×C3 | C3×D4 | 1 | 4 | yes | — |
| 16 | C4 | — | 2 | 2 | no | — |
| 17 | C4 | C22×C32 | 1 | 1 | yes | characteristic |
| 18 | C3 | C3×Q16 | 1 | 4 | yes | direct product factor |
| 19 | C2 | C32×D4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C32 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×Q16 | C3 | direct (N × H) |
| 9. Q16 | C32 | direct (N × H) |
| 12. C32 | Q16 | direct (N × H) |
| 18. C3 | C3×Q16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q8 | 2 |
| C32×C8 | 2 |
| C3×Q16 | 3 |