Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
818 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 296 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×C3 | D32 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini | C2 | C22×C3×D32 |
| Fitting, socle, prime radical | C23×C33 | C22 |
| hypocenter | C32 | C22×C3×Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×#627 | C2 | direct (N × H) |
| 5. C22×#627 | C3 | direct (N × H) |
| 9. C3×#627 | C22 | direct (N × H) |
| 18. C2×#627 | C2×C3 | direct (N × H) |
| 19. C22×C33 | Q8 | semidirect |
| 38. #627 | C22×C3 | direct (N × H) |
| 39. C2×C33 | C2×Q8 | semidirect |
| 56. C22×C32 | Dic6 | semidirect |
| 57. C22×C32 | C3×Q8 | semidirect |
| 68. C33 | C22×Q8 | semidirect |
| 89. C2×C32 | C2×Dic6 | semidirect |
| 90. C2×C32 | C2×C3×Q8 | semidirect |
| 102. C22×C3 | #627 | direct (N × H) |
| 103. C22×C3 | C3×Dic6 | semidirect |
| 116. C32 | C22×Dic6 | semidirect |
| 117. C32 | C22×C3×Q8 | semidirect |
| 128. C2×C3 | C2×#627 | direct (N × H) |
| 129. C2×C3 | C2×C3×Dic6 | semidirect |
| 132. C22 | C3×#627 | direct (N × H) |
| 139. C3 | C22×#627 | direct (N × H) |
| 140. C3 | C22×C3×Dic6 | semidirect |
| 141. C2 | C2×C3×#627 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C3⋊Dic3 | 2 |
| C22×C32×Dic3 | 2 |
| C2×C3×#627 | 2 |
| C22×C3×Dic6 | 3 |
| C22×#627 | 3 |