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×C32×#67 | 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 |
|---|---|---|
| 1. C22×C3⋊(C3⋊Dic3) | C2 | semidirect |
| 3. C3×#3757 | C2 | semidirect |
| 4. #88807 | C2 | semidirect |
| 5. (C2×D3)⋊(C3⋊Dic3) | C2 | semidirect |
| 21. #3757 | D3 | semidirect |
| 89. C33 | #123 | semidirect |
| 144. C32 | #886 | semidirect |
| 145. C32 | #935 | semidirect |
| 173. C3 | #63962 | semidirect |
| 174. C3 | #64144 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊(C3⋊Dic3) | 2 |
| C2×C32×#67 | 2 |
| C3×#3757 | 2 |
| #88807 | 2 |
| (C2×D3)⋊(C3⋊Dic3) | 2 |
| #64144 | 3 |
| #63962 | 3 |