Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
127 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Ω+(4,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5935 | — | 3 | 2 | no | maximal |
| 3 | #5429 | — | 3 | 2 | no | maximal |
| 4 | C3⋊GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 5 | #997 | D3 | 1 | 2 | yes | complemented |
| 6 | #994 | D3 | 1 | 2 | yes | complemented |
| 7 | #986 | — | 3 | 2 | no | — |
| 8 | GU(2,𝔽3) | — | 3 | 2 | no | — |
| 9 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 10 | #453 | — | 9 | 1 | no | Sylow |
| 11 | C2×S4 | — | 12 | 2 | no | — |
| 12 | #41⋊C3 | — | 3 | 2 | no | — |
| 13 | GL(2,𝔽3) | — | 6 | 2 | no | — |
| 14 | GL(2,𝔽3) | — | 12 | 4 | no | — |
| 15 | #215 | — | 12 | 2 | no | — |
| 16 | C2×C3⋊D3 | — | 16 | 1 | no | — |
| 17 | H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 18 | AGL(1,ℤ/8) | — | 9 | 2 | no | — |
| 19 | #127 | — | 9 | 1 | no | — |
| 20 | #104 | — | 9 | 2 | no | — |
| 21 | #100 | — | 9 | 1 | no | — |
| 22 | C2×A4 | — | 12 | 2 | no | — |
| 23 | S4 | — | 12 | 4 | no | — |
| 24 | C3×Q8 | — | 4 | 2 | no | — |
| 25 | D12 | — | 12 | 2 | no | — |
| 26 | SL(2,𝔽3) | — | 4 | 4 | no | — |
| 27 | SL(2,𝔽3) | S4 | 1 | 2 | yes | complemented |
| 28 | C3⋊C8 | — | 12 | 2 | no | — |
| 29 | C2×C32 | — | 16 | 1 | no | — |
| 30 | C3⋊D3 | — | 16 | 2 | no | — |
| 31 | #41 | — | 3 | 2 | no | — |
| 32 | C2×D4 | — | 9 | 1 | no | — |
| 33 | C2×D4 | — | 9 | 1 | no | — |
| 34 | C2×D4 | — | 18 | 2 | no | — |
| 35 | SD16 | — | 18 | 2 | no | — |
| 36 | D8 | — | 18 | 2 | no | — |
| 37 | M16 | — | 9 | 2 | no | — |
| 38 | C42 | — | 9 | 1 | no | — |
| 39 | C2×D3 | — | 24 | 2 | no | — |
| 40 | C2×D3 | — | 48 | 2 | no | — |
| 41 | A4 | — | 12 | 2 | no | — |
| 42 | C3×C4 | — | 12 | 2 | no | — |
| 43 | C32 | — | 16 | 1 | no | Sylow |
| 44 | C23 | — | 3 | 2 | no | — |
| 45 | C23 | — | 18 | 1 | no | — |
| 46 | Q8 | C3⋊S4 | 1 | 2 | yes | — |
| 47 | D4 | — | 9 | 2 | no | — |
| 48 | D4 | — | 18 | 2 | no | — |
| 49 | D4 | — | 36 | 4 | no | — |
| 50 | C2×C4 | — | 9 | 1 | no | — |
| 51 | C2×C4 | — | 18 | 1 | no | — |
| 52 | C8 | — | 18 | 2 | no | — |
| 53 | C2×C3 | — | 4 | 2 | no | — |
| 54 | C2×C3 | — | 16 | 2 | no | — |
| 55 | D3 | — | 48 | 2 | no | — |
| 56 | D3 | — | 48 | 4 | no | — |
| 57 | C22 | — | 9 | 1 | no | — |
| 58 | C22 | — | 12 | 2 | no | — |
| 59 | C22 | — | 36 | 2 | no | — |
| 60 | C22 | — | 36 | 1 | no | — |
| 61 | C4 | — | 3 | 2 | no | — |
| 62 | C4 | — | 18 | 2 | no | — |
| 63 | C3 | — | 4 | 2 | no | — |
| 64 | C3 | — | 16 | 2 | no | — |
| 65 | C2 | — | 18 | 1 | no | — |
| 66 | C2 | — | 72 | 1 | no | — |
| 67 | C2 | A4⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C2 |
| Fitting | H(5,𝔽2) | C3⋊D3 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Ω+(4,𝔽3) | C2 | semidirect |
| 5. #997 | D3 | semidirect |
| 6. #994 | D3 | semidirect |
| 17. H(5,𝔽2) | C3⋊D3 | semidirect |
| 27. SL(2,𝔽3) | S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Ω+(4,𝔽3) | 2 |
| #5935 | 3 |
| #5429 | 3 |
| C3⋊GL(2,𝔽3) | 4 |