Finite Group DatabasebetaFinite Group Databasebeta← C2×D3×C22⋊(C22⋊C9) #13180098
Order 1728.
4078 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 4078 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×C22⋊A4 |
| commutator, hypocenter | C24×C3 | C22×C9 |
| Frattini | C3 | C2×D3×C22⋊A4 |
| Fitting, socle | C25×C32 | C2×C3 |
| prime radical | C25×C3×D3 | C3 |
| cosocle | C24×C32 | C22×C3 |
| LCS | G ▷ C24×C3 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C22⋊(C22⋊C9) | C2 | semidirect |
| 2. D3×C22⋊(C22⋊C9) | C2 | direct (N × H) |
| 5. C3×C22⋊(C22⋊C9) | C22 | semidirect |
| 14. C2×C22⋊(C22⋊C9) | D3 | direct (N × H) |
| 18. C25×D3 | C9 | semidirect |
| 33. C22⋊(C22⋊C9) | C2×D3 | direct (N × H) |
| 42. C25×C3 | C2×C9 | semidirect |
| 47. C24×D3 | C2×C9 | semidirect |
| 88. C24×C3 | C22×C9 | semidirect |
| 99. C23×D3 | C22⋊C9 | semidirect |
| 120. C25 | D3×C9 | semidirect |
| 147. C23×C3 | C2×C22⋊C9 | semidirect |
| 160. C22×D3 | C2×C22⋊C9 | semidirect |
| 181. C24 | C2×D3×C9 | semidirect |
| 203. C22×C3 | C22×C22⋊C9 | semidirect |
| 212. C2×D3 | C22⋊(C22⋊C9) | direct (N × H) |
| 232. C23 | D3×C22⋊C9 | semidirect |
| 242. C2×C3 | C2×C22⋊(C22⋊C9) | semidirect |
| 245. D3 | C2×C22⋊(C22⋊C9) | direct (N × H) |
| 258. C22 | C2×D3×C22⋊C9 | semidirect |
| 260. C3 | C22×C22⋊(C22⋊C9) | semidirect |
| 266. C2 | D3×C22⋊(C22⋊C9) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C22⋊(C22⋊C9) | 2 |
| D3×C22⋊(C22⋊C9) | 2 |
| C25×C3×D3 | 3 |
| C22×C22⋊(C22⋊C9) | 3 |
| C2×D3×C22⋊C9 | 4 |