Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
780 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting | C2×C33×C4 | C22 |
| hypercenter | C2×C4 | D3×C3⋊D3 |
| hypocenter | C33 | #121 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 2. C2×#6292 | C2 | semidirect |
| 3. C2×#6292 | C2 | semidirect |
| 4. C2×#6289 | C2 | semidirect |
| 5. C3×#3752 | C2 | semidirect |
| 6. C32×Dic3⋊C4 | C2 | semidirect |
| 7. #88804 | C2 | semidirect |
| 32. #3752 | D3 | semidirect |
| 34. C3×Dic3⋊C4 | D3 | semidirect |
| 42. C2×C3⋊(C3⋊D3) | D4 | semidirect |
| 49. C32×Dic3 | D4 | semidirect |
| 98. Dic3⋊C4 | C3⋊D3 | semidirect |
| 134. C3×Dic3 | D12 | semidirect |
| 136. C33 | #121 | semidirect |
| 222. Dic3 | C4⋊(C3⋊D3) | semidirect |
| 224. C32 | #884 | semidirect |
| 225. C32 | #895 | semidirect |
| 269. C3 | #63910 | semidirect |
| 270. C3 | #64104 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊(C3⋊(C3⋊D3)) | 2 |
| C2×#6292 | 2 |
| C2×#6289 | 2 |
| C3×#3752 | 2 |
| C32×Dic3⋊C4 | 2 |
| #88804 | 2 |
| #64104 | 3 |
| #63910 | 3 |