Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
209 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | S4 |
| commutator, hypocenter | A4 | C2×C4×C9 |
| Frattini | C2×C3 | C2×C3×S4 |
| Fitting | C24×C9 | D3 |
| socle | C24×C3 | C3×D3 |
| prime radical | C22×C3×A4 | C2×C3 |
| cosocle | C2×C3×A4 | C22×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C9×SL(2,ℤ/4) | C2 | direct (N × H) |
| 5. C2×C9×A4 | C4 | semidirect |
| 17. C9×A4 | C2×C4 | semidirect |
| 19. C2×SL(2,ℤ/4) | C9 | direct (N × H) |
| 27. C23×C9 | Dic3 | semidirect |
| 43. SL(2,ℤ/4) | C2×C9 | direct (N × H) |
| 57. C22×C9 | C2×Dic3 | semidirect |
| 71. C2×A4 | C4×C9 | semidirect |
| 86. C2×C9 | SL(2,ℤ/4) | direct (N × H) |
| 105. A4 | C2×C4×C9 | semidirect |
| 111. C9 | C2×SL(2,ℤ/4) | direct (N × H) |
| 115. C23 | C9×Dic3 | semidirect |
| 138. C22 | C2×C9×Dic3 | semidirect |
| 148. C2 | C9×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 |
|---|---|
| C22×C9×A4 | 2 |
| C9×SL(2,ℤ/4) | 2 |
| C2×C3×SL(2,ℤ/4) | 3 |
| C2×C9×C22⋊C4 | 3 |
| C2×C9×Dic3 | 4 |