Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
76 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#564 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C42⋊C9 | — | 4 | 5 | no | maximal |
| 3 | C3×#125 | — | 3 | 5 | no | — |
| 4 | #564 | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 5 | C3×C4⋊C4 | — | 3 | 10 | no | — |
| 6 | C3×C42 | A4 | 1 | 5 | yes | — |
| 7 | C22⋊C9 | — | 16 | 1 | no | — |
| 8 | #125 | — | 3 | 5 | no | — |
| 9 | C2×C3×C4 | — | 3 | 5 | no | — |
| 10 | C4⋊C4 | — | 3 | 10 | no | — |
| 11 | C42 | C22⋊C9 | 1 | 5 | yes | — |
| 12 | C22×C3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 13 | C3×C4 | — | 6 | 5 | no | — |
| 14 | C9 | — | 64 | 1 | no | Sylow |
| 15 | C2×C4 | — | 3 | 5 | no | — |
| 16 | C2×C3 | — | 3 | 1 | no | — |
| 17 | C22 | C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 18 | C4 | — | 6 | 5 | no | — |
| 19 | C3 | #564⋊C3 | 1 | 1 | yes | characteristic |
| 20 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #564⋊C3 |
| commutator, hypocenter | #564 | C9 |
| Frattini, socle, prime radical | C22×C3 | C22⋊A4 |
| Fitting, cosocle | C3×#564 | C3 |
| LCS | G ▷ #564 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #564 | C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#564 | 3 |
| C42⋊C9 | 4 |