Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
842 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 842 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 | C23×C32 | C2×D3 |
| hypocenter | C3×C9 | C22×D4 |
| 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 |
|---|---|---|
| 1. C23×C3⋊D9 | C2 | semidirect |
| 2. C23×C3×D9 | C2 | semidirect |
| 3. C22×C9×Dic3 | C2 | semidirect |
| 4. C2×#6192 | C2 | direct (N × H) |
| 9. C22×C3⋊D9 | C22 | semidirect |
| 11. C22×C3×D9 | C22 | semidirect |
| 12. C2×C9×Dic3 | C22 | semidirect |
| 13. #6192 | C22 | direct (N × H) |
| 20. C23×D9 | D3 | semidirect |
| 23. C22×C3×C9 | D4 | semidirect |
| 26. C2×C3⋊D9 | C23 | semidirect |
| 28. C2×C3×D9 | C23 | semidirect |
| 29. C9×Dic3 | C23 | semidirect |
| 46. C22×D9 | C2×D3 | semidirect |
| 49. C2×C3×C9 | C2×D4 | semidirect |
| 60. C22×Dic3 | D9 | semidirect |
| 71. C22×C9 | #67 | semidirect |
| 78. C2×D9 | C22×D3 | semidirect |
| 81. C3×C9 | C22×D4 | semidirect |
| 97. C2×Dic3 | C2×D9 | semidirect |
| 105. C2×C9 | C2×#67 | semidirect |
| 118. C22×C3 | D36 | semidirect |
| 131. Dic3 | C22×D9 | semidirect |
| 134. C9 | C22×#67 | semidirect |
| 145. C2×C3 | C2×D36 | semidirect |
| 157. C22 | #6192 | direct (N × H) |
| 161. C3 | C22×D36 | semidirect |
| 165. C2 | C2×#6192 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3⋊D9 | 2 |
| C23×C3×D9 | 2 |
| C22×C9×Dic3 | 2 |
| C2×#6192 | 2 |
| C22×#626 | 3 |
| C22×D36 | 3 |