Finite Group DatabasebetaFinite Group Databasebeta← C23×((C22⋊C9)⋊C2)⋊C3 #13180080
Order 1728.
2682 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 2682 conjugacy classes of subgroups — too many to draw. Only its 230 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 | ((C22⋊C9)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | C22⋊C9 | C24×C3 |
| Frattini | C3 | C23×C3×S4 |
| Fitting | C25×C32 | D3 |
| socle | C25×C3 | C3×D3 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#1094 | C2 | semidirect |
| 2. C22×((C22⋊C9)⋊C2)⋊C3 | C2 | direct (N × H) |
| 4. C23×(C22⋊C9)⋊C2 | C3 | semidirect |
| 6. C22×#1094 | C22 | semidirect |
| 7. C2×((C22⋊C9)⋊C2)⋊C3 | C22 | direct (N × H) |
| 14. C23×C22⋊C9 | C2×C3 | semidirect |
| 15. C22×(C22⋊C9)⋊C2 | C2×C3 | semidirect |
| 18. C2×#1094 | C23 | semidirect |
| 19. ((C22⋊C9)⋊C2)⋊C3 | C23 | direct (N × H) |
| 33. C22×C22⋊C9 | C22×C3 | semidirect |
| 34. C2×(C22⋊C9)⋊C2 | C22×C3 | semidirect |
| 37. #1094 | C24 | semidirect |
| 64. C2×C22⋊C9 | C23×C3 | semidirect |
| 65. (C22⋊C9)⋊C2 | C23×C3 | semidirect |
| 103. C22⋊C9 | C24×C3 | semidirect |
| 105. C25 | AGL(1,ℤ/9) | semidirect |
| 149. C24 | C2×AGL(1,ℤ/9) | semidirect |
| 186. C23 | ((C22⋊C9)⋊C2)⋊C3 | direct (N × H) |
| 187. C23 | C22×AGL(1,ℤ/9) | semidirect |
| 208. C22 | C2×((C22⋊C9)⋊C2)⋊C3 | direct (N × H) |
| 209. C22 | C23×AGL(1,ℤ/9) | semidirect |
| 216. C2 | C22×((C22⋊C9)⋊C2)⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#1094 | 2 |
| C22×((C22⋊C9)⋊C2)⋊C3 | 2 |
| C23×C3×#67 | 3 |
| C23×(C22⋊C9)⋊C2 | 3 |
| C23×AGL(1,ℤ/9) | 4 |