Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
668 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 668 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×C32×D12 | C2 | semidirect |
| 2. C2×#6290 | C2 | semidirect |
| 3. C2×#6290 | C2 | semidirect |
| 4. C2×D3×C3⋊Dic3 | C2 | semidirect |
| 5. #88857 | C2 | semidirect |
| 6. C3×#3752 | C2 | semidirect |
| 7. (C2×D3)⋊(C3⋊Dic3) | C2 | semidirect |
| 25. C2×C3×D12 | D3 | semidirect |
| 30. #3752 | D3 | semidirect |
| 45. C2×C32×D3 | D4 | semidirect |
| 48. C3⋊(C3⋊Dic3) | D4 | semidirect |
| 89. C2×D12 | C3⋊D3 | semidirect |
| 118. C2×C3×D3 | #67 | semidirect |
| 131. C33 | #121 | semidirect |
| 199. C2×D3 | #638 | semidirect |
| 212. C32 | #884 | semidirect |
| 213. C32 | #939 | semidirect |
| 255. C3 | #63885 | semidirect |
| 256. C3 | #64148 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×D12 | 2 |
| C2×#6290 | 2 |
| C2×D3×C3⋊Dic3 | 2 |
| #88857 | 2 |
| C3×#3752 | 2 |
| (C2×D3)⋊(C3⋊Dic3) | 2 |
| #64148 | 3 |
| #63885 | 3 |