Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
534 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 41 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⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×Q8×C9 | C2×C3 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C22×Q8×C9 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9⋊(C22⋊SL(2,𝔽3)) | C2 | semidirect |
| 3. C22×Q8×D9 | C3 | semidirect |
| 7. C22×Q8×C9 | C2×C3 | semidirect |
| 27. Q8×D9 | A4 | semidirect |
| 46. C22×D9 | SL(2,𝔽3) | semidirect |
| 49. Q8×C9 | C2×A4 | semidirect |
| 77. C22×C9 | C2×SL(2,𝔽3) | semidirect |
| 90. C22×Q8 | AGL(1,ℤ/9) | semidirect |
| 116. D9 | C22⋊SL(2,𝔽3) | semidirect |
| 143. C9 | C2×C22⋊SL(2,𝔽3) | semidirect |
| 152. Q8 | D9⋊A4 | semidirect |
| 172. C22 | D9⋊SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9⋊(C22⋊SL(2,𝔽3)) | 2 |
| D3×C22⋊SL(2,𝔽3) | 3 |
| C22×Q8×D9 | 3 |
| C2×D9⋊A4 | 4 |
| D9⋊SL(2,𝔽3) | 4 |