Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
90 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×Q32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #63628 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3⋊(C3⋊C32) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #4522 | — | 3 | 4 | no | maximal |
| 5 | C32×Q16 | — | 2 | 1 | no | — |
| 6 | #3746 | — | 2 | 1 | no | — |
| 7 | C32×C16 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×Q32 | D3 | 1 | 4 | yes | — |
| 9 | Dic24 | — | 3 | 4 | no | — |
| 10 | C3⋊C32 | — | 3 | 4 | no | — |
| 11 | C32×Q8 | — | 4 | 1 | no | — |
| 12 | #634 | — | 4 | 1 | no | — |
| 13 | C32×C8 | D4 | 1 | 1 | yes | characteristic |
| 14 | Q64 | — | 9 | 1 | no | Sylow |
| 15 | C3×Q16 | — | 2 | 4 | no | — |
| 16 | Dic12 | — | 6 | 4 | no | — |
| 17 | C3×C16 | C2×D3 | 1 | 4 | yes | — |
| 18 | C32×C4 | — | 8 | 1 | no | — |
| 19 | C32×C4 | D8 | 1 | 1 | yes | characteristic |
| 20 | C3⋊Dic3 | — | 8 | 1 | no | — |
| 21 | Q32 | — | 9 | 1 | no | — |
| 22 | Q32 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 23 | C32 | — | 9 | 1 | no | — |
| 24 | C3×Q8 | — | 4 | 4 | no | — |
| 25 | Dic6 | — | 12 | 4 | no | — |
| 26 | C3×C8 | #67 | 1 | 4 | yes | — |
| 27 | C2×C32 | D16 | 1 | 1 | yes | characteristic |
| 28 | Q16 | — | 2 | 1 | no | — |
| 29 | Q16 | — | 18 | 1 | no | — |
| 30 | C16 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 31 | C3×C4 | — | 8 | 4 | no | — |
| 32 | C3×C4 | #213 | 1 | 4 | yes | — |
| 33 | Dic3 | — | 24 | 4 | no | — |
| 34 | C32 | Q64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 35 | Q8 | — | 4 | 1 | no | — |
| 36 | Q8 | — | 36 | 1 | no | — |
| 37 | C8 | #638 | 1 | 1 | yes | characteristic |
| 38 | C2×C3 | #826 | 1 | 4 | yes | — |
| 39 | C4 | — | 8 | 1 | no | — |
| 40 | C4 | — | 72 | 1 | no | — |
| 41 | C4 | #3753 | 1 | 1 | yes | characteristic |
| 42 | C3 | #4522 | 1 | 4 | yes | complemented |
| 43 | C2 | #63653 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63653 |
| commutator, cosocle | C32×C16 | C22 |
| Frattini | C16 | C2×C3⋊D3 |
| Fitting | C32×Q32 | C2 |
| socle, prime radical | C2×C32 | D16 |
| hypercenter | Q32 | C3⋊D3 |
| hypocenter | C32 | Q64 |
| LCS | G ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. C32 | Q64 | semidirect |
| 42. C3 | #4522 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q32 | 2 |
| #63628 | 2 |
| C3⋊(C3⋊C32) | 2 |
| #4522 | 3 |