Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1047 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 219 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 | C9⋊S4 |
| commutator, hypocenter | C9×A4 | C22×C4 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C25×C9 | D3 |
| socle | C25×C3 | C3⋊D3 |
| prime radical | C23×C3×A4 | D3 |
| cosocle | C2×C9×A4 | C23 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C9⋊SL(2,ℤ/4) | C2 | direct (N × H) |
| 6. C22×C9×A4 | C4 | semidirect |
| 9. C9⋊SL(2,ℤ/4) | C22 | direct (N × H) |
| 20. C2×C9×A4 | C2×C4 | semidirect |
| 32. C24×C9 | Dic3 | semidirect |
| 34. C22×C22⋊C9 | Dic3 | semidirect |
| 47. C9×A4 | C22×C4 | semidirect |
| 70. C23×C9 | C2×Dic3 | semidirect |
| 73. C2×C22⋊C9 | C2×Dic3 | semidirect |
| 89. C22×A4 | Dic9 | semidirect |
| 115. C22×C9 | SL(2,ℤ/4) | semidirect |
| 116. C22×C9 | C22×Dic3 | semidirect |
| 118. C22⋊C9 | C22×Dic3 | semidirect |
| 141. C2×A4 | C2×Dic9 | semidirect |
| 156. C2×C9 | C2×SL(2,ℤ/4) | semidirect |
| 161. C24 | C3⋊Dic9 | semidirect |
| 184. A4 | C22×Dic9 | semidirect |
| 190. C9 | C22×SL(2,ℤ/4) | semidirect |
| 200. C23 | C2×C3⋊Dic9 | semidirect |
| 224. C22 | C9⋊SL(2,ℤ/4) | direct (N × H) |
| 225. C22 | C22×C3⋊Dic9 | semidirect |
| 234. C2 | 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 |
|---|---|
| C23×C9×A4 | 2 |
| C2×C9⋊SL(2,ℤ/4) | 2 |
| C22×C3⋊SL(2,ℤ/4) | 3 |
| C22×(C22⋊C9)⋊C4 | 3 |
| C22×#3676 | 3 |
| C22×C3⋊Dic9 | 4 |