Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
522 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle, prime radical | C23×C32 | C2×D3 |
| hypocenter | C3×C9 | C22×Q8 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#6189 | C2 | direct (N × H) |
| 11. #6189 | C22 | direct (N × H) |
| 21. C22×C3×C9 | Q8 | semidirect |
| 40. C2×C3×C9 | C2×Q8 | semidirect |
| 58. C22×C9 | Dic6 | semidirect |
| 65. C3×C9 | C22×Q8 | semidirect |
| 82. C2×C9 | C2×Dic6 | semidirect |
| 90. C22×C3 | Dic18 | semidirect |
| 103. C9 | C22×Dic6 | semidirect |
| 111. C2×C3 | C2×Dic18 | semidirect |
| 115. C22 | #6189 | direct (N × H) |
| 121. C3 | C22×Dic18 | semidirect |
| 123. C2 | C2×#6189 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊Dic9 | 2 |
| C22×C9×Dic3 | 2 |
| C22×C3×Dic9 | 2 |
| C2×#6189 | 2 |
| C22×#627 | 3 |
| C22×Dic18 | 3 |