Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
120 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C32×D4 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C32×C4 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C3×#41 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 5 | C22×C32 | C22 | 1 | 3 | yes | — |
| 6 | C32×C4 | C22 | 1 | 3 | yes | — |
| 7 | C32×C4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×Q8 | C2×C3 | 1 | 4 | yes | complemented |
| 9 | C3×D4 | C2×C3 | 1 | 12 | yes | complemented |
| 10 | C2×C3×C4 | C2×C3 | 1 | 12 | yes | complemented |
| 11 | C2×C32 | — | 2 | 3 | no | — |
| 12 | C2×C32 | C23 | 1 | 1 | yes | characteristic |
| 13 | #41 | C32 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C22×C3 | C22×C3 | 1 | 12 | yes | — |
| 15 | C3×C4 | C22×C3 | 1 | 4 | yes | — |
| 16 | C3×C4 | C22×C3 | 1 | 12 | yes | — |
| 17 | C32 | #41 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 18 | Q8 | C2×C32 | 1 | 1 | yes | characteristic, complemented |
| 19 | D4 | C2×C32 | 1 | 3 | yes | complemented |
| 20 | C2×C4 | C2×C32 | 1 | 3 | yes | complemented |
| 21 | C2×C3 | — | 2 | 12 | no | — |
| 22 | C2×C3 | C23×C3 | 1 | 4 | yes | — |
| 23 | C22 | C22×C32 | 1 | 3 | yes | — |
| 24 | C4 | C22×C32 | 1 | 1 | yes | characteristic |
| 25 | C4 | C22×C32 | 1 | 3 | yes | — |
| 26 | C3 | C3×#41 | 1 | 4 | yes | direct product factor |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C23×C32 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C32 |
| socle | C2×C32 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×Q8 | C2 | semidirect |
| 2. C32×D4 | C2 | semidirect |
| 3. C2×C32×C4 | C2 | semidirect |
| 4. C3×#41 | C3 | direct (N × H) |
| 8. C3×Q8 | C2×C3 | semidirect |
| 9. C3×D4 | C2×C3 | semidirect |
| 10. C2×C3×C4 | C2×C3 | semidirect |
| 13. #41 | C32 | direct (N × H) |
| 17. C32 | #41 | direct (N × H) |
| 18. Q8 | C2×C32 | semidirect |
| 19. D4 | C2×C32 | semidirect |
| 20. C2×C4 | C2×C32 | semidirect |
| 26. C3 | C3×#41 | 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×D4 | 2 |
| C2×C32×C4 | 2 |
| C3×#41 | 3 |