Finite Group DatabasebetaFinite Group Databasebeta← C22×C3⋊((C22⋊C9)⋊C4) #13164508
Order 1728.
1554 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 1554 conjugacy classes of subgroups — too many to draw. Only its 267 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⋊((C22⋊C9)⋊C2) |
| commutator, hypocenter | C3×C22⋊C9 | C22×C4 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting, socle, prime radical | C25×C32 | D3 |
| cosocle | C2×C3×C22⋊C9 | C23 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊((C22⋊C9)⋊C4) | C2 | direct (N × H) |
| 5. C22×C3×C22⋊C9 | C4 | semidirect |
| 8. C3⋊((C22⋊C9)⋊C4) | C22 | direct (N × H) |
| 17. C2×C3×C22⋊C9 | C2×C4 | semidirect |
| 32. C22×C22⋊C9 | Dic3 | semidirect |
| 40. C3×C22⋊C9 | C22×C4 | semidirect |
| 70. C2×C22⋊C9 | C2×Dic3 | semidirect |
| 82. C24×C3 | Dic9 | semidirect |
| 116. C22⋊C9 | C22×Dic3 | semidirect |
| 142. C23×C3 | C2×Dic9 | semidirect |
| 168. C24 | C3⋊Dic9 | semidirect |
| 195. C22×C3 | (C22⋊C9)⋊C4 | semidirect |
| 196. C22×C3 | C22×Dic9 | semidirect |
| 216. C23 | C2×C3⋊Dic9 | semidirect |
| 230. C2×C3 | C2×(C22⋊C9)⋊C4 | semidirect |
| 243. C22 | C3⋊((C22⋊C9)⋊C4) | direct (N × H) |
| 244. C22 | C22×C3⋊Dic9 | semidirect |
| 248. C3 | C22×(C22⋊C9)⋊C4 | semidirect |
| 253. C2 | C2×C3⋊((C22⋊C9)⋊C4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C22⋊C9 | 2 |
| C2×C3⋊((C22⋊C9)⋊C4) | 2 |
| C22×#3757 | 3 |
| C22×(C22⋊C9)⋊C4 | 3 |
| C22×C3⋊Dic9 | 4 |