Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
662 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊S4 |
| commutator, hypocenter | C3×A4 | C22×C4 |
| Frattini | C2 | C22×C3⋊S4 |
| Fitting, socle | C25×C3 | D3 |
| prime radical | C23×C3×A4 | C2 |
| cosocle | C2×C3×A4 | C23 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊SL(2,ℤ/4) | C2 | direct (N × H) |
| 5. C22×C3×A4 | C4 | semidirect |
| 8. C3⋊SL(2,ℤ/4) | C22 | direct (N × H) |
| 17. C2×C3×A4 | C2×C4 | semidirect |
| 24. C24×C3 | Dic3 | semidirect |
| 26. C22×A4 | Dic3 | semidirect |
| 39. C3×A4 | C22×C4 | semidirect |
| 56. C23×C3 | C2×Dic3 | semidirect |
| 59. C2×A4 | C2×Dic3 | semidirect |
| 72. C24 | C3⋊Dic3 | semidirect |
| 92. C22×C3 | SL(2,ℤ/4) | semidirect |
| 93. C22×C3 | C22×Dic3 | semidirect |
| 95. A4 | C22×Dic3 | semidirect |
| 109. C23 | C2×C3⋊Dic3 | semidirect |
| 122. C2×C3 | C2×SL(2,ℤ/4) | semidirect |
| 133. C22 | C3⋊SL(2,ℤ/4) | direct (N × H) |
| 134. C22 | C22×C3⋊Dic3 | semidirect |
| 139. C3 | C22×SL(2,ℤ/4) | semidirect |
| 143. C2 | C2×C3⋊SL(2,ℤ/4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×A4 | 2 |
| C2×C3⋊SL(2,ℤ/4) | 2 |
| C22×SL(2,ℤ/4) | 3 |
| C22×#217 | 3 |
| C22×C3⋊Dic3 | 4 |