Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1866 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊PSU(3,𝔽2) |
| commutator, cosocle | C3×C3⋊D3 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C23×C33 | Q8 |
| prime radical | C23×C3×C3⋊D3 | C22 |
| hypocenter | C33 | C23×Q8 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C3⋊PSU(3,𝔽2) | C2 | direct (N × H) |
| 8. C2×C3⋊PSU(3,𝔽2) | C22 | direct (N × H) |
| 12. C23×C33 | Q8 | semidirect |
| 13. C22×C3×C3⋊D3 | Q8 | semidirect |
| 17. C3⋊PSU(3,𝔽2) | C23 | direct (N × H) |
| 24. C22×C33 | C2×Q8 | semidirect |
| 25. C2×C3×C3⋊D3 | C2×Q8 | semidirect |
| 34. C23×C32 | Dic6 | semidirect |
| 35. C22×C3⋊D3 | Dic6 | semidirect |
| 43. C2×C33 | C22×Q8 | semidirect |
| 44. C3×C3⋊D3 | C22×Q8 | semidirect |
| 53. C22×C32 | C2×Dic6 | semidirect |
| 54. C2×C3⋊D3 | C2×Dic6 | semidirect |
| 63. C33 | C23×Q8 | semidirect |
| 68. C23×C3 | PSU(3,𝔽2) | semidirect |
| 76. C2×C32 | C22×Dic6 | semidirect |
| 77. C3⋊D3 | C22×Dic6 | semidirect |
| 89. C22×C3 | C2×PSU(3,𝔽2) | semidirect |
| 96. C32 | C23×Dic6 | semidirect |
| 99. C23 | C3⋊PSU(3,𝔽2) | direct (N × H) |
| 107. C2×C3 | C22×PSU(3,𝔽2) | semidirect |
| 112. C22 | C2×C3⋊PSU(3,𝔽2) | direct (N × H) |
| 117. C3 | C23×PSU(3,𝔽2) | semidirect |
| 120. C2 | C22×C3⋊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 |
|---|---|
| C23×C3⋊#157 | 2 |
| C23×C3×#157 | 2 |
| C22×C3⋊PSU(3,𝔽2) | 2 |
| C23×PSU(3,𝔽2) | 3 |
| C23×Dic6 | 9 |