Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
1152 conjugacy classes of subgroups; only the 320 normal subgroups are shown.
This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 320 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×C3⋊D3 |
| commutator, cosocle | C2×C33 | C24 |
| Frattini | C2 | C22×D3×C3⋊D3 |
| Fitting, socle, prime radical | C23×C33 | C22 |
| hypocenter | C33 | C22×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#6293 | C2 | direct (N × H) |
| 11. #6293 | C22 | direct (N × H) |
| 21. C22×C33 | Q8 | semidirect |
| 40. C2×C33 | C2×Q8 | semidirect |
| 53. C22×C32 | Dic6 | semidirect |
| 54. C22×C32 | Dic6 | semidirect |
| 65. C33 | C22×Q8 | semidirect |
| 80. C2×C32 | C2×Dic6 | semidirect |
| 81. C2×C32 | C2×Dic6 | semidirect |
| 90. C22×C3 | #627 | semidirect |
| 91. C22×C3 | #634 | semidirect |
| 102. C32 | C22×Dic6 | semidirect |
| 103. C32 | C22×Dic6 | semidirect |
| 111. C2×C3 | C2×#627 | semidirect |
| 112. C2×C3 | C2×#634 | semidirect |
| 115. C22 | #6293 | direct (N × H) |
| 121. C3 | C22×#627 | semidirect |
| 122. C3 | C22×#634 | semidirect |
| 123. C2 | C2×#6293 | 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×C3×C3⋊Dic3 | 2 |
| C22×C32×Dic3 | 2 |
| C2×#6293 | 2 |
| C22×#634 | 3 |
| C22×#627 | 3 |