Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
199 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C9⋊S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C2×C3 |
| Frattini | C2×C3 | C3×C3⋊S4 |
| Fitting | C3×Q8×C9 | D3 |
| socle | C2×C32 | C3⋊S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C9×SL(2,𝔽3) | C2 | semidirect |
| 3. C9⋊GL(2,𝔽3) | C3 | direct (N × H) |
| 8. C3×Q8×C9 | D3 | semidirect |
| 10. C3×Q8⋊C9 | D3 | semidirect |
| 14. C9×SL(2,𝔽3) | C2×C3 | semidirect |
| 33. C3×SL(2,𝔽3) | D9 | semidirect |
| 37. Q8×C9 | C3×D3 | semidirect |
| 40. Q8⋊C9 | C3×D3 | semidirect |
| 70. C3×C9 | GL(2,𝔽3) | semidirect |
| 72. C3×Q8 | C3⋊D9 | semidirect |
| 77. SL(2,𝔽3) | C3×D9 | semidirect |
| 109. C9 | C3×GL(2,𝔽3) | semidirect |
| 110. Q8 | C3×C3⋊D9 | semidirect |
| 126. C3 | C9⋊GL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×SL(2,𝔽3) | 2 |
| C3×C3⋊GL(2,𝔽3) | 3 |
| C9⋊GL(2,𝔽3) | 3 |
| C3×#3689 | 3 |
| C3×#3675 | 3 |
| C2×C3×C3⋊D9 | 4 |