Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
324 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×#157 |
| commutator, hypocenter | C33 | C22×C8 |
| Frattini | C2 | C2×D3×#157 |
| Fitting, socle | C22×C33 | C2×C4 |
| prime radical | C22×C3⋊(C3⋊D3) | C4 |
| cosocle | C3×C3⋊Dic3 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊#622 | C2 | semidirect |
| 3. C2×C3×#622 | C2 | semidirect |
| 4. #88924 | C2 | direct (N × H) |
| 11. C3⋊#622 | C22 | semidirect |
| 12. C3×#622 | C22 | semidirect |
| 16. C2×#622 | D3 | semidirect |
| 19. C2×C3⋊(C3⋊D3) | C8 | semidirect |
| 37. #622 | C2×D3 | semidirect |
| 38. C2×C33 | C2×C8 | semidirect |
| 40. C3⋊(C3⋊D3) | C2×C8 | semidirect |
| 62. C33 | C22×C8 | semidirect |
| 80. C2×C32 | D3×C8 | semidirect |
| 106. C32 | C2×D3×C8 | semidirect |
| 118. C2×C3 | #3787 | semidirect |
| 131. C3 | C2×#3787 | semidirect |
| 133. C2 | #88924 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6289 | 2 |
| C2×C3⋊#622 | 2 |
| C2×C3×#622 | 2 |
| #88924 | 2 |
| C2×#3787 | 3 |
| C2×D3×C8 | 9 |