Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
1142 conjugacy classes of subgroups; only the 498 normal subgroups are shown.
This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 498 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×C32 | PSU(3,𝔽2) |
| commutator, cosocle | C3⋊D3 | C24×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | Q8 |
| prime radical | C22×C32×C3⋊D3 | C22 |
| hypocenter | C32 | C22×C32×Q8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C32×PSU(3,𝔽2) | C2 | direct (N × H) |
| 3. C22×C3×PSU(3,𝔽2) | C3 | direct (N × H) |
| 6. C32×PSU(3,𝔽2) | C22 | direct (N × H) |
| 8. C2×C3×PSU(3,𝔽2) | C2×C3 | direct (N × H) |
| 9. C22×C34 | Q8 | semidirect |
| 10. C2×C32×C3⋊D3 | Q8 | semidirect |
| 13. C22×PSU(3,𝔽2) | C32 | direct (N × H) |
| 18. C3×PSU(3,𝔽2) | C22×C3 | direct (N × H) |
| 19. C2×C34 | C2×Q8 | semidirect |
| 20. C32×C3⋊D3 | C2×Q8 | semidirect |
| 23. C2×PSU(3,𝔽2) | C2×C32 | direct (N × H) |
| 28. C22×C33 | C3×Q8 | semidirect |
| 29. C2×C3×C3⋊D3 | C3×Q8 | semidirect |
| 35. C34 | C22×Q8 | semidirect |
| 40. PSU(3,𝔽2) | C22×C32 | direct (N × H) |
| 45. C2×C33 | C2×C3×Q8 | semidirect |
| 46. C3×C3⋊D3 | C2×C3×Q8 | semidirect |
| 58. C22×C32 | C32×Q8 | semidirect |
| 59. C22×C32 | PSU(3,𝔽2) | direct (N × H) |
| 60. C2×C3⋊D3 | C32×Q8 | semidirect |
| 69. C33 | C22×C3×Q8 | semidirect |
| 80. C2×C32 | C2×C32×Q8 | semidirect |
| 81. C2×C32 | C2×PSU(3,𝔽2) | direct (N × H) |
| 82. C3⋊D3 | C2×C32×Q8 | semidirect |
| 92. C22×C3 | C3×PSU(3,𝔽2) | direct (N × H) |
| 100. C32 | C22×C32×Q8 | semidirect |
| 101. C32 | C22×PSU(3,𝔽2) | direct (N × H) |
| 109. C2×C3 | C2×C3×PSU(3,𝔽2) | direct (N × H) |
| 114. C22 | C32×PSU(3,𝔽2) | direct (N × H) |
| 118. C3 | C22×C3×PSU(3,𝔽2) | direct (N × H) |
| 121. C2 | C2×C32×PSU(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×#157 | 2 |
| C2×C32×PSU(3,𝔽2) | 2 |
| C22×C3×PSU(3,𝔽2) | 3 |
| C22×C32×Q8 | 9 |