Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
167 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 22 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 | D9×S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C22 |
| Frattini, socle | C2×C3 | D3×S4 |
| Fitting | Q8×C9 | C2×D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D9×SL(2,𝔽3) | C2 | semidirect |
| 2. C9⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C9×GL(2,𝔽3) | C2 | semidirect |
| 7. C9×SL(2,𝔽3) | C22 | semidirect |
| 11. Q8×D9 | D3 | semidirect |
| 28. Q8×C9 | C2×D3 | semidirect |
| 46. GL(2,𝔽3) | D9 | direct (N × H) |
| 74. SL(2,𝔽3) | C2×D9 | semidirect |
| 86. D9 | GL(2,𝔽3) | direct (N × H) |
| 107. C9 | C2×GL(2,𝔽3) | semidirect |
| 110. Q8 | D3×D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D9×SL(2,𝔽3) | 2 |
| C9⋊GL(2,𝔽3) | 2 |
| C9×GL(2,𝔽3) | 2 |
| D3×GL(2,𝔽3) | 3 |
| SD16×D9 | 3 |
| C2×D3×D9 | 4 |