Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
780 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 132 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 | C23×C33 | C22 |
| hypercenter | C23 | D3×C3⋊D3 |
| hypocenter | C33 | #121 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6312 | C2 | semidirect |
| 2. C22×C3×C3⋊Dic3 | C2 | semidirect |
| 3. C2×C32×#67 | C2 | semidirect |
| 4. C2×#6291 | C2 | semidirect |
| 5. #88807 | C2 | semidirect |
| 6. #88804 | C2 | semidirect |
| 7. (C2×D3)⋊(C3⋊Dic3) | C2 | semidirect |
| 26. C22×C3⋊Dic3 | D3 | semidirect |
| 27. C2×C3×#67 | D3 | semidirect |
| 38. C22×C33 | D4 | semidirect |
| 48. C3×C3⋊Dic3 | D4 | semidirect |
| 90. C2×#67 | C3⋊D3 | semidirect |
| 110. C22×C32 | D12 | semidirect |
| 111. C22×C32 | #67 | semidirect |
| 131. C3⋊Dic3 | #67 | semidirect |
| 136. C33 | #121 | semidirect |
| 198. C22×C3 | #626 | semidirect |
| 199. C22×C3 | #638 | semidirect |
| 224. C32 | #930 | semidirect |
| 225. C32 | #939 | semidirect |
| 262. C22 | #6291 | semidirect |
| 269. C3 | #63978 | semidirect |
| 270. C3 | #64148 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6312 | 2 |
| C22×C3×C3⋊Dic3 | 2 |
| C2×C32×#67 | 2 |
| C2×#6291 | 2 |
| #88807 | 2 |
| #88804 | 2 |
| (C2×D3)⋊(C3⋊Dic3) | 2 |
| #64148 | 3 |
| #63978 | 3 |