Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
158 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 42 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 | C2 | D3×PSU(3,𝔽2) |
| commutator, prime radical, cosocle | C2×C3×C3⋊D3 | C23 |
| Fitting, socle | C2×C33 | C2×Q8 |
| hypocenter | C33 | C4⋊Q8 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 22. C3×#157 | Q8 | semidirect |
| 23. C3⋊(C3⋊Dic3) | Q8 | semidirect |
| 24. C32×Dic3 | Q8 | semidirect |
| 46. #157 | Dic6 | semidirect |
| 53. C33 | C4⋊Q8 | semidirect |
| 83. Dic3 | PSU(3,𝔽2) | semidirect |
| 86. C32 | #888 | semidirect |
| 109. C3 | C4⋊PSU(3,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊PSU(3,𝔽2) | 2 |
| #88946 | 2 |
| C3×#3777 | 2 |
| Dic3⋊#157 | 2 |
| Dic3×#157 | 2 |
| C4⋊PSU(3,𝔽2) | 3 |
| #888 | 9 |