Finite Group DatabasebetaFinite Group Databasebeta← C22×((C3⋊D3)⋊C3)⋊Q8 #13164710
Order 1728.
3131 conjugacy classes of subgroups; only the 675 normal subgroups are shown.
This group has 3131 conjugacy classes of subgroups — too many to draw. Only its 675 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#6229 | C2 | semidirect |
| 3. C22×#634⋊C3 | C2 | semidirect |
| 5. C22×#6196 | C2 | semidirect |
| 6. C2×((C3⋊D3)⋊C3)⋊Q8 | C2 | direct (N × H) |
| 12. C2×#6229 | C22 | semidirect |
| 16. C2×#634⋊C3 | C22 | semidirect |
| 18. C2×#6196 | C22 | semidirect |
| 19. ((C3⋊D3)⋊C3)⋊Q8 | C22 | direct (N × H) |
| 21. C22×#634 | D3 | semidirect |
| 34. C22×(C3⋊D3)⋊C3 | Q8 | semidirect |
| 38. #6229 | C23 | semidirect |
| 42. #634⋊C3 | C23 | semidirect |
| 44. #6196 | C23 | semidirect |
| 56. C2×#634 | C2×D3 | semidirect |
| 73. C22×H(3,𝔽3) | C2×Q8 | semidirect |
| 75. C2×(C3⋊D3)⋊C3 | C2×Q8 | semidirect |
| 103. C22×C3⋊D3 | Dic6 | semidirect |
| 113. #634 | C22×D3 | semidirect |
| 129. C2×H(3,𝔽3) | C22×Q8 | semidirect |
| 131. (C3⋊D3)⋊C3 | C22×Q8 | semidirect |
| 170. C22×C32 | C2×Dic6 | semidirect |
| 171. C22×C32 | D3×Q8 | semidirect |
| 174. C2×C3⋊D3 | C2×Dic6 | semidirect |
| 194. H(3,𝔽3) | C23×Q8 | semidirect |
| 237. C2×C32 | C22×Dic6 | semidirect |
| 238. C2×C32 | C2×D3×Q8 | semidirect |
| 241. C3⋊D3 | C22×Dic6 | semidirect |
| 283. C32 | C23×Dic6 | semidirect |
| 284. C32 | C22×D3×Q8 | semidirect |
| 310. C22 | ((C3⋊D3)⋊C3)⋊Q8 | direct (N × H) |
| 321. C2 | C2×((C3⋊D3)⋊C3)⋊Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6229 | 2 |
| C22×C4×(C3⋊D3)⋊C3 | 2 |
| C22×#634⋊C3 | 2 |
| C22×((C3⋊D3)⋊C3)⋊C4 | 2 |
| C22×#6196 | 2 |
| C2×((C3⋊D3)⋊C3)⋊Q8 | 2 |
| C22×#3797 | 3 |
| C22×D3×Dic6 | 3 |