Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
768 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 232 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 | C2×C33 | C22×C4 |
| Frattini | C22 | C2×D3×C3⋊D3 |
| Fitting, socle, prime radical | C23×C33 | C22 |
| cosocle | C22×C33 | C23 |
| hypocenter | C33 | C2×C4⋊C4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #88806 | C2 | direct (N × H) |
| 8. C2×C3⋊(C3⋊Dic3) | C4 | semidirect |
| 27. C3⋊(C3⋊Dic3) | C2×C4 | semidirect |
| 49. C2×C33 | C4⋊C4 | semidirect |
| 82. C33 | C2×C4⋊C4 | semidirect |
| 104. C2×C32 | Dic3⋊C4 | semidirect |
| 105. C2×C32 | Dic3⋊C4 | semidirect |
| 136. C32 | C2×Dic3⋊C4 | semidirect |
| 137. C32 | C2×Dic3⋊C4 | semidirect |
| 149. C2×C3 | #3724 | semidirect |
| 150. C2×C3 | #3750 | semidirect |
| 165. C3 | C2×#3724 | semidirect |
| 166. C3 | C2×#3750 | semidirect |
| 167. C2 | #88806 | 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 |
| #88806 | 2 |
| C2×#3750 | 3 |
| C2×#3724 | 3 |