Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
684 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting | C23×C33 | C22 |
| prime radical | C2×#6312 | C2 |
| hypercenter | C23 | D3×C3⋊D3 |
| hypocenter | C33 | #123 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C3⋊Dic3 | C2 | semidirect |
| 3. C32×#217 | C2 | semidirect |
| 4. Dic3⋊(C3⋊Dic3) | C2 | semidirect |
| 5. #88804 | C2 | semidirect |
| 18. C22×C3⋊Dic3 | D3 | semidirect |
| 23. C3×#217 | D3 | semidirect |
| 64. #217 | C3⋊D3 | semidirect |
| 92. C33 | #123 | semidirect |
| 151. C32 | #886 | semidirect |
| 152. C32 | #929 | semidirect |
| 181. C3 | #63966 | semidirect |
| 182. C3 | #64095 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6312 | 2 |
| C22×C3×C3⋊Dic3 | 2 |
| C32×#217 | 2 |
| Dic3⋊(C3⋊Dic3) | 2 |
| #88804 | 2 |
| #64095 | 3 |
| #63966 | 3 |