Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
57 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Ω+(4,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #5932 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #458⋊C3 | — | 3 | 1 | no | maximal |
| 4 | #997 | D3 | 1 | 1 | yes | characteristic |
| 5 | #997 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | #994 | — | 2 | 1 | no | — |
| 7 | C3×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 8 | #458 | — | 3 | 1 | no | Sylow |
| 9 | #41⋊C3 | — | 6 | 1 | no | — |
| 10 | C4×A4 | — | 12 | 1 | no | — |
| 11 | SL(2,ℤ/4) | — | 12 | 1 | no | — |
| 12 | C3×Dic3 | — | 16 | 1 | no | maximal |
| 13 | H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 14 | #123 | — | 9 | 1 | no | — |
| 15 | #99 | — | 9 | 1 | no | — |
| 16 | C2×A4 | — | 12 | 1 | no | — |
| 17 | C2×A4 | — | 12 | 1 | no | — |
| 18 | C3×Q8 | — | 8 | 1 | no | — |
| 19 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 20 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 21 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 22 | C2×C32 | — | 16 | 1 | no | — |
| 23 | #41 | — | 6 | 1 | no | — |
| 24 | C2×D4 | — | 9 | 1 | no | — |
| 25 | C22×C4 | — | 3 | 1 | no | — |
| 26 | C4⋊C4 | — | 18 | 1 | no | — |
| 27 | C22⋊C4 | — | 9 | 1 | no | — |
| 28 | C22⋊C4 | — | 18 | 1 | no | — |
| 29 | A4 | — | 12 | 1 | no | — |
| 30 | A4 | — | 12 | 1 | no | — |
| 31 | C3×C4 | — | 24 | 1 | no | — |
| 32 | C3×C4 | — | 48 | 1 | no | — |
| 33 | Dic3 | — | 16 | 1 | no | — |
| 34 | C32 | — | 16 | 1 | no | Sylow |
| 35 | C23 | — | 3 | 1 | no | — |
| 36 | C23 | — | 3 | 1 | no | — |
| 37 | Q8 | — | 2 | 1 | no | — |
| 38 | D4 | — | 18 | 1 | no | — |
| 39 | C2×C4 | — | 9 | 1 | no | — |
| 40 | C2×C4 | — | 18 | 1 | no | — |
| 41 | C2×C4 | — | 18 | 1 | no | — |
| 42 | C2×C3 | — | 8 | 1 | no | — |
| 43 | C2×C3 | — | 16 | 1 | no | — |
| 44 | C2×C3 | — | 16 | 1 | no | — |
| 45 | C22 | — | 9 | 1 | no | — |
| 46 | C22 | — | 12 | 1 | no | — |
| 47 | C22 | — | 12 | 1 | no | — |
| 48 | C4 | — | 6 | 1 | no | — |
| 49 | C4 | — | 12 | 1 | no | — |
| 50 | C4 | — | 36 | 1 | no | — |
| 51 | C3 | — | 8 | 1 | no | — |
| 52 | C3 | — | 16 | 1 | no | — |
| 53 | C3 | — | 16 | 1 | no | — |
| 54 | C2 | — | 18 | 1 | no | — |
| 55 | C2 | #1020⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | #1020⋊C3 |
| commutator, cosocle, hypocenter | #997 | C2×C3 |
| Fitting | H(5,𝔽2) | C3×D3 |
| prime radical | Ω+(4,𝔽3) | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5932 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Ω+(4,𝔽3) | 2 |
| #5932 | 3 |
| #458⋊C3 | 3 |
| C3×Dic3 | 16 |