Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
178 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 32 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 | C1 | G |
| commutator, cosocle | C3×#157 | C23 |
| Fitting, socle, hypocenter | C33 | C2×SD16 |
| prime radical | D3×GO+(4,𝔽2) | C2 |
| LCS | G ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D3×PSU(3,𝔽2) | C2 | semidirect |
| 3. AGL(1,𝔽9)⋊D3 | C2 | semidirect |
| 4. #89092 | C2 | semidirect |
| 5. #89091 | C2 | semidirect |
| 6. C3×AΓL(1,𝔽9) | C2 | semidirect |
| 7. D3×AGL(1,𝔽9) | C2 | semidirect |
| 11. C3⋊PSU(3,𝔽2) | C22 | semidirect |
| 12. C3×PSU(3,𝔽2) | C22 | semidirect |
| 18. C3⋊AGL(1,𝔽9) | C22 | semidirect |
| 19. C3×AGL(1,𝔽9) | C22 | semidirect |
| 26. AΓL(1,𝔽9) | D3 | direct (N × H) |
| 43. PSU(3,𝔽2) | C2×D3 | semidirect |
| 48. AGL(1,𝔽9) | C2×D3 | semidirect |
| 49. C3⋊(C3⋊D3) | SD16 | semidirect |
| 53. C32×D3 | SD16 | semidirect |
| 79. C33 | C2×SD16 | semidirect |
| 133. C32 | D3×SD16 | semidirect |
| 159. D3 | AΓL(1,𝔽9) | direct (N × H) |
| 171. C3 | C2×AΓL(1,𝔽9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×PSU(3,𝔽2) | 2 |
| D3×GO+(4,𝔽2) | 2 |
| AGL(1,𝔽9)⋊D3 | 2 |
| #89092 | 2 |
| #89091 | 2 |
| C3×AΓL(1,𝔽9) | 2 |
| D3×AGL(1,𝔽9) | 2 |
| C2×AΓL(1,𝔽9) | 3 |
| D3×SD16 | 9 |