Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
438 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 84 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 |
|---|---|---|
| 2. C2×C3×#1088 | C2 | direct (N × H) |
| 3. C22×#1088 | C3 | direct (N × H) |
| 4. C22×C3×H(3,𝔽3) | C4 | semidirect |
| 5. C2×C3×#272 | C4 | semidirect |
| 7. C3×#1088 | C22 | direct (N × H) |
| 10. C2×#1088 | C2×C3 | direct (N × H) |
| 11. C2×C3×H(3,𝔽3) | C2×C4 | semidirect |
| 12. C3×#272 | C2×C4 | semidirect |
| 19. C22×H(3,𝔽3) | C3×C4 | semidirect |
| 20. C2×#272 | C3×C4 | semidirect |
| 22. #1088 | C22×C3 | direct (N × H) |
| 23. C3×H(3,𝔽3) | C22×C4 | semidirect |
| 32. C2×H(3,𝔽3) | C2×C3×C4 | semidirect |
| 33. #272 | C2×C3×C4 | semidirect |
| 50. H(3,𝔽3) | C22×C3×C4 | semidirect |
| 73. C22×C3 | #1088 | direct (N × H) |
| 91. C2×C3 | C2×#1088 | direct (N × H) |
| 97. C22 | C3×#1088 | direct (N × H) |
| 101. C3 | C22×#1088 | direct (N × H) |
| 105. C2 | C2×C3×#1088 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#272 | 2 |
| C2×C3×#1088 | 2 |
| C22×#1088 | 3 |
| C22×C32×C4 | 9 |