Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
1092 conjugacy classes of subgroups; only the 319 normal subgroups are shown.
This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 319 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 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting, prime radical | C2×C33×D4 | C2 |
| hypercenter | C2×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #123 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3⋊(C3⋊Dic3) | C2 | semidirect |
| 3. #88864 | C2 | semidirect |
| 4. #88864 | C2 | semidirect |
| 5. #88857 | C2 | semidirect |
| 53. C33 | #123 | semidirect |
| 78. C32 | #935 | semidirect |
| 98. C3 | #64144 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×D4 | 2 |
| C22×C3⋊(C3⋊Dic3) | 2 |
| #88864 | 2 |
| #88857 | 2 |
| #64144 | 3 |