Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
960 conjugacy classes of subgroups; only the 437 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 437 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C23×C3×C4 |
| Frattini | C2×C3 | C23×C3×D3 |
| Fitting, prime radical | C24×H(3,𝔽3) | C2 |
| socle | C24×C3 | C3×D3 |
| cosocle | C2×C32 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×(C3⋊Dic3)⋊C3 | C2 | direct (N × H) |
| 3. C23×C3⋊Dic3 | C3 | semidirect |
| 5. C23×H(3,𝔽3) | C4 | semidirect |
| 7. C2×(C3⋊Dic3)⋊C3 | C22 | direct (N × H) |
| 11. C22×C3⋊Dic3 | C2×C3 | semidirect |
| 13. C22×H(3,𝔽3) | C2×C4 | semidirect |
| 15. (C3⋊Dic3)⋊C3 | C23 | direct (N × H) |
| 21. C23×C32 | Dic3 | semidirect |
| 22. C23×C32 | C3×C4 | semidirect |
| 25. C2×C3⋊Dic3 | C22×C3 | semidirect |
| 27. C2×H(3,𝔽3) | C22×C4 | semidirect |
| 38. C22×C32 | C2×Dic3 | semidirect |
| 39. C22×C32 | C2×C3×C4 | semidirect |
| 42. C3⋊Dic3 | C23×C3 | semidirect |
| 45. H(3,𝔽3) | C23×C4 | semidirect |
| 59. C2×C32 | C22×Dic3 | semidirect |
| 60. C2×C32 | C22×C3×C4 | semidirect |
| 77. C32 | C23×Dic3 | semidirect |
| 78. C32 | C23×C3×C4 | semidirect |
| 79. C23 | (C3⋊Dic3)⋊C3 | direct (N × H) |
| 90. C22 | C2×(C3⋊Dic3)⋊C3 | direct (N × H) |
| 97. C2 | C22×(C3⋊Dic3)⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×H(3,𝔽3) | 2 |
| C22×(C3⋊Dic3)⋊C3 | 2 |
| C23×C3⋊Dic3 | 3 |
| C23×C3×Dic3 | 3 |