Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
802 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #1113 |
| commutator | C2×C33 | C22×C4 |
| Frattini | C22 | C2×#1113 |
| Fitting, socle | C23×C33 | C22 |
| prime radical | C23×C3×C3⋊D3 | C2 |
| cosocle | C22×C33 | C23 |
| hypocenter | C33 | C2×C22⋊C4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C3⋊Dic3 | C2 | semidirect |
| 3. #88814 | C2 | direct (N × H) |
| 8. C22×C3×C3⋊D3 | C4 | semidirect |
| 11. C2×C3×C3⋊Dic3 | C22 | semidirect |
| 15. C22×C3⋊Dic3 | D3 | semidirect |
| 26. C2×C3×C3⋊D3 | C2×C4 | semidirect |
| 36. C22×C3⋊D3 | Dic3 | semidirect |
| 46. C2×C3⋊Dic3 | C2×D3 | semidirect |
| 53. C2×C33 | C22⋊C4 | semidirect |
| 85. C2×C3⋊D3 | C2×Dic3 | semidirect |
| 98. C33 | C2×C22⋊C4 | semidirect |
| 132. C2×C32 | (C2×D3)⋊C4 | semidirect |
| 133. C2×C32 | #217 | semidirect |
| 179. C32 | C2×(C2×D3)⋊C4 | semidirect |
| 180. C32 | C2×#217 | semidirect |
| 198. C2×C3 | (C2×D3)⋊Dic3 | semidirect |
| 199. C2×C3 | #3722 | semidirect |
| 217. C3 | C2×(C2×D3)⋊Dic3 | semidirect |
| 218. C3 | C2×#3722 | semidirect |
| 220. C2 | #88814 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C3⋊D3 | 2 |
| C22×C3×C3⋊Dic3 | 2 |
| #88814 | 2 |
| C2×#3722 | 3 |
| C2×(C2×D3)⋊Dic3 | 3 |