Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
1060 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #10620193 |
| commutator, hypocenter | C34 | C22×C8 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C8 |
| prime radical | C22×C3⋊(C3⋊(C3⋊D3)) | C4 |
| cosocle | #66472 | C23 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#10620193 | C2 | direct (N × H) |
| 6. #10620193 | C22 | direct (N × H) |
| 7. C22×C34 | C8 | semidirect |
| 8. C2×C3⋊(C3⋊(C3⋊D3)) | C8 | semidirect |
| 15. C2×C34 | C2×C8 | semidirect |
| 16. C3⋊(C3⋊(C3⋊D3)) | C2×C8 | semidirect |
| 27. C34 | C22×C8 | semidirect |
| 50. C22×C32 | AGL(1,𝔽9) | semidirect |
| 77. C2×C32 | C2×AGL(1,𝔽9) | semidirect |
| 103. C32 | C22×AGL(1,𝔽9) | semidirect |
| 116. C22 | #10620193 | direct (N × H) |
| 123. C2 | C2×#10620193 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#66472 | 2 |
| C2×#10620193 | 2 |
| C22×AGL(1,𝔽9) | 9 |