Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
66 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | A4⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A42 | C4 | 1 | 1 | yes | characteristic, complemented |
| 3 | #1020 | — | 6 | 1 | no | — |
| 4 | GL(2,ℤ/4) | — | 6 | 1 | no | — |
| 5 | C3⋊S4 | — | 8 | 1 | no | — |
| 6 | AΣL(1,𝔽16) | — | 9 | 1 | no | maximal, Sylow |
| 7 | C22⋊A4 | — | 2 | 1 | no | — |
| 8 | C22×A4 | — | 2 | 1 | no | — |
| 9 | C2×S4 | — | 6 | 1 | no | — |
| 10 | SL(2,ℤ/4) | — | 6 | 1 | no | — |
| 11 | C3×A4 | — | 8 | 1 | no | — |
| 12 | #157 | — | 16 | 1 | no | maximal |
| 13 | #120 | — | 9 | 1 | no | — |
| 14 | #100 | — | 9 | 1 | no | — |
| 15 | #99 | — | 9 | 1 | no | — |
| 16 | C2×A4 | — | 6 | 1 | no | — |
| 17 | S4 | — | 12 | 1 | no | — |
| 18 | S4 | — | 24 | 2 | no | — |
| 19 | S4 | — | 24 | 1 | no | — |
| 20 | #67 | — | 24 | 1 | no | — |
| 21 | C3⋊D3 | — | 16 | 1 | no | — |
| 22 | C24 | #157 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×D4 | — | 9 | 1 | no | — |
| 24 | C2×D4 | — | 18 | 1 | no | — |
| 25 | M16 | — | 18 | 1 | no | — |
| 26 | C22⋊C4 | — | 9 | 1 | no | — |
| 27 | C22⋊C4 | — | 18 | 1 | no | — |
| 28 | C22⋊C4 | — | 18 | 1 | no | — |
| 29 | C22×C3 | — | 8 | 1 | no | — |
| 30 | C2×D3 | — | 24 | 1 | no | — |
| 31 | A4 | — | 2 | 1 | no | — |
| 32 | A4 | — | 8 | 2 | no | — |
| 33 | A4 | — | 24 | 1 | no | — |
| 34 | Dic3 | — | 24 | 1 | no | — |
| 35 | C32 | — | 16 | 1 | no | Sylow |
| 36 | C23 | — | 6 | 1 | no | — |
| 37 | C23 | — | 9 | 1 | no | — |
| 38 | C23 | — | 9 | 1 | no | — |
| 39 | D4 | — | 36 | 1 | no | — |
| 40 | D4 | — | 36 | 1 | no | — |
| 41 | D4 | — | 36 | 1 | no | — |
| 42 | C2×C4 | — | 9 | 1 | no | — |
| 43 | C2×C4 | — | 18 | 1 | no | — |
| 44 | C2×C4 | — | 36 | 1 | no | — |
| 45 | C8 | — | 36 | 1 | no | — |
| 46 | C2×C3 | — | 24 | 1 | no | — |
| 47 | D3 | — | 48 | 1 | no | — |
| 48 | D3 | — | 96 | 1 | no | — |
| 49 | C22 | — | 2 | 1 | no | — |
| 50 | C22 | — | 6 | 1 | no | — |
| 51 | C22 | — | 9 | 1 | no | — |
| 52 | C22 | — | 18 | 1 | no | — |
| 53 | C22 | — | 18 | 1 | no | — |
| 54 | C22 | — | 36 | 1 | no | — |
| 55 | C4 | — | 18 | 1 | no | — |
| 56 | C4 | — | 36 | 1 | no | — |
| 57 | C4 | — | 72 | 1 | no | — |
| 58 | C3 | — | 8 | 1 | no | — |
| 59 | C3 | — | 32 | 1 | no | — |
| 60 | C2 | — | 6 | 1 | no | — |
| 61 | C2 | — | 9 | 1 | no | — |
| 62 | C2 | — | 36 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | A42 | C4 |
| Fitting, socle | C24 | #157 |
| prime radical, cosocle | A4⋊S4 | C2 |
| LCS | G ▷ A42 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A42 | C4 | semidirect |
| 22. C24 | #157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4⋊S4 | 2 |
| AΣL(1,𝔽16) | 9 |
| #157 | 16 |