Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
900 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×C3⋊D3 |
| commutator, Fitting, socle, cosocle | C2×C34 | C23 |
| prime radical | C2×C3⋊(C3⋊(C3⋊D3)) | C22 |
| hypocenter | C34 | C2×Q8 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊#6296 | C2 | semidirect |
| 4. C3×#6293 | C2 | semidirect |
| 5. C32×#627 | C2 | semidirect |
| 18. #6293 | D3 | semidirect |
| 26. C3×#627 | D3 | semidirect |
| 28. C3⋊(C3⋊(C3⋊D3)) | Q8 | semidirect |
| 51. C34 | C2×Q8 | semidirect |
| 66. #627 | C3⋊D3 | semidirect |
| 116. C33 | D3×Q8 | semidirect |
| 117. C33 | D3×Q8 | semidirect |
| 176. C32 | #3797 | semidirect |
| 177. C32 | #3797 | semidirect |
| 178. C32 | Q8×C3⋊D3 | semidirect |
| 202. C3 | #88958 | semidirect |
| 203. C3 | #89019 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊#6296 | 2 |
| #10620150 | 2 |
| #10620144 | 2 |
| C3×#6293 | 2 |
| C32×#627 | 2 |
| #89019 | 3 |
| #88958 | 3 |