Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
516 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×#125 | C2 | 1 | 15 | yes | maximal |
| 2 | C3×#564 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 3 | C32×C4⋊C4 | C22 | 1 | 30 | yes | — |
| 4 | C32×C42 | C22 | 1 | 5 | yes | — |
| 5 | C3×#125 | C2×C3 | 1 | 60 | yes | — |
| 6 | C2×C32×C4 | C23 | 1 | 15 | yes | — |
| 7 | #564 | C32 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C3×C4⋊C4 | C22×C3 | 1 | 120 | yes | — |
| 9 | C3×C42 | C22×C3 | 1 | 20 | yes | — |
| 10 | C22×C32 | C24 | 1 | 1 | yes | characteristic |
| 11 | C32×C4 | — | 2 | 15 | no | — |
| 12 | #125 | C2×C32 | 1 | 15 | yes | — |
| 13 | C2×C3×C4 | C23×C3 | 1 | 60 | yes | — |
| 14 | C2×C32 | 2−1+4 | 1 | 3 | yes | — |
| 15 | C4⋊C4 | C22×C32 | 1 | 30 | yes | — |
| 16 | C42 | C22×C32 | 1 | 5 | yes | — |
| 17 | C22×C3 | C24×C3 | 1 | 4 | yes | — |
| 18 | C3×C4 | — | 2 | 60 | no | — |
| 19 | C32 | #564 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 20 | C2×C4 | C23×C32 | 1 | 15 | yes | — |
| 21 | C2×C3 | C3×2−1+4 | 1 | 12 | yes | — |
| 22 | C22 | C24×C32 | 1 | 1 | yes | characteristic |
| 23 | C4 | — | 2 | 15 | no | — |
| 24 | C3 | C3×#564 | 1 | 4 | yes | direct product factor |
| 25 | C2 | C32×2−1+4 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#564 | C3 | direct (N × H) |
| 7. #564 | C32 | direct (N × H) |
| 19. C32 | #564 | direct (N × H) |
| 24. C3 | C3×#564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#125 | 2 |
| C3×#564 | 3 |