Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
460 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #89026 |
| commutator, cosocle | C33×C4 | C23 |
| Frattini | C4 | C2×D3×C3⋊D3 |
| Fitting | C33×C8 | C22 |
| socle | C2×C33 | C2×D4 |
| hypercenter | C8 | D3×C3⋊D3 |
| hypocenter | C33 | #135 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #89017 | C2 | semidirect |
| 2. C8×C3⋊(C3⋊D3) | C2 | semidirect |
| 3. C3×#3745 | C2 | semidirect |
| 4. C32×Dic12 | C2 | semidirect |
| 5. #88793 | C2 | semidirect |
| 19. #3745 | D3 | semidirect |
| 23. C3×Dic12 | D3 | semidirect |
| 59. Dic12 | C3⋊D3 | semidirect |
| 78. C33 | #135 | semidirect |
| 123. C32 | #912 | semidirect |
| 124. C32 | #919 | semidirect |
| 146. C3 | #63810 | semidirect |
| 147. C3 | #64128 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #89017 | 2 |
| C8×C3⋊(C3⋊D3) | 2 |
| C3×#3745 | 2 |
| C32×Dic12 | 2 |
| #88793 | 2 |
| #64128 | 3 |
| #63810 | 3 |