Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
190 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 38 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, cosocle | C2×C3×C3⋊D3 | C23 |
| Fitting, socle | C2×C33 | C2×Q8 |
| prime radical | C2×D3×C3⋊D3 | C22 |
| hypocenter | C33 | C22⋊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 |
|---|---|---|
| 1. C2×C3⋊PSU(3,𝔽2) | C2 | semidirect |
| 3. #88945 | C2 | semidirect |
| 4. #88945 | C2 | semidirect |
| 5. C3×#3777 | C2 | semidirect |
| 24. #3777 | D3 | semidirect |
| 25. C2×C3⋊(C3⋊D3) | Q8 | semidirect |
| 27. C2×C32×D3 | Q8 | semidirect |
| 77. C33 | C22⋊Q8 | semidirect |
| 117. C2×D3 | PSU(3,𝔽2) | semidirect |
| 126. C32 | #896 | semidirect |
| 161. C3 | C22⋊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 |
| C2×D3×#157 | 2 |
| #88945 | 2 |
| C3×#3777 | 2 |
| (C2×D3)⋊#157 | 2 |
| C22⋊PSU(3,𝔽2) | 3 |
| #896 | 9 |