Finite Group DatabasebetaFinite Group Databasebeta← C23×C3×(C22⋊C9)⋊C2 #13180079
Order 1728.
2714 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 2714 conjugacy classes of subgroups — too many to draw. Only its 262 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, cosocle, hypocenter | C22⋊C9 | C24×C3 |
| Frattini | C3 | C23×C3×S4 |
| Fitting, socle | C25×C32 | D3 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×C22⋊C9 | C2 | semidirect |
| 2. C22×C3×(C22⋊C9)⋊C2 | C2 | direct (N × H) |
| 4. C23×(C22⋊C9)⋊C2 | C3 | direct (N × H) |
| 6. C22×C3×C22⋊C9 | C22 | semidirect |
| 7. C2×C3×(C22⋊C9)⋊C2 | C22 | direct (N × H) |
| 14. C23×C22⋊C9 | C2×C3 | semidirect |
| 15. C22×(C22⋊C9)⋊C2 | C2×C3 | direct (N × H) |
| 18. C2×C3×C22⋊C9 | C23 | semidirect |
| 19. C3×(C22⋊C9)⋊C2 | C23 | direct (N × H) |
| 33. C22×C22⋊C9 | C22×C3 | semidirect |
| 34. C2×(C22⋊C9)⋊C2 | C22×C3 | direct (N × H) |
| 37. C3×C22⋊C9 | C24 | semidirect |
| 40. C25×C3 | D9 | semidirect |
| 65. C2×C22⋊C9 | C23×C3 | semidirect |
| 66. (C22⋊C9)⋊C2 | C23×C3 | direct (N × H) |
| 80. C24×C3 | C2×D9 | semidirect |
| 105. C22⋊C9 | C24×C3 | semidirect |
| 107. C25 | C3×D9 | semidirect |
| 129. C23×C3 | (C22⋊C9)⋊C2 | direct (N × H) |
| 130. C23×C3 | C22×D9 | semidirect |
| 154. C24 | C2×C3×D9 | semidirect |
| 175. C22×C3 | C2×(C22⋊C9)⋊C2 | direct (N × H) |
| 176. C22×C3 | C23×D9 | semidirect |
| 194. C23 | C3×(C22⋊C9)⋊C2 | direct (N × H) |
| 195. C23 | C22×C3×D9 | semidirect |
| 207. C2×C3 | C22×(C22⋊C9)⋊C2 | direct (N × H) |
| 217. C22 | C2×C3×(C22⋊C9)⋊C2 | direct (N × H) |
| 218. C22 | C23×C3×D9 | semidirect |
| 221. C3 | C23×(C22⋊C9)⋊C2 | direct (N × H) |
| 226. C2 | C22×C3×(C22⋊C9)⋊C2 | 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 |
| C22×C3×(C22⋊C9)⋊C2 | 2 |
| C23×C3×#67 | 3 |
| C23×(C22⋊C9)⋊C2 | 3 |
| C23×C3×D9 | 4 |