Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5515 conjugacy classes of subgroups; only the 2484 normal subgroups are shown.
This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 2484 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C24×C3×C4 |
| Frattini | C2×C3 | C24×C3×D3 |
| Fitting, prime radical | C25×H(3,𝔽3) | C2 |
| socle | C25×C3 | C3×D3 |
| cosocle | C2×C32 | C25×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×(C3⋊Dic3)⋊C3 | C2 | direct (N × H) |
| 3. C24×C3⋊Dic3 | C3 | semidirect |
| 5. C24×H(3,𝔽3) | C4 | semidirect |
| 7. C22×(C3⋊Dic3)⋊C3 | C22 | direct (N × H) |
| 11. C23×C3⋊Dic3 | C2×C3 | semidirect |
| 13. C23×H(3,𝔽3) | C2×C4 | semidirect |
| 15. C2×(C3⋊Dic3)⋊C3 | C23 | direct (N × H) |
| 21. C24×C32 | Dic3 | semidirect |
| 22. C24×C32 | C3×C4 | semidirect |
| 25. C22×C3⋊Dic3 | C22×C3 | semidirect |
| 27. C22×H(3,𝔽3) | C22×C4 | semidirect |
| 29. (C3⋊Dic3)⋊C3 | C24 | direct (N × H) |
| 39. C23×C32 | C2×Dic3 | semidirect |
| 40. C23×C32 | C2×C3×C4 | semidirect |
| 43. C2×C3⋊Dic3 | C23×C3 | semidirect |
| 46. C2×H(3,𝔽3) | C23×C4 | semidirect |
| 61. C22×C32 | C22×Dic3 | semidirect |
| 62. C22×C32 | C22×C3×C4 | semidirect |
| 65. C3⋊Dic3 | C24×C3 | semidirect |
| 69. H(3,𝔽3) | C24×C4 | semidirect |
| 83. C2×C32 | C23×Dic3 | semidirect |
| 84. C2×C32 | C23×C3×C4 | semidirect |
| 87. C24 | (C3⋊Dic3)⋊C3 | direct (N × H) |
| 102. C32 | C24×Dic3 | semidirect |
| 103. C32 | C24×C3×C4 | semidirect |
| 104. C23 | C2×(C3⋊Dic3)⋊C3 | direct (N × H) |
| 115. C22 | C22×(C3⋊Dic3)⋊C3 | direct (N × H) |
| 122. C2 | C23×(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 |
|---|---|
| C25×H(3,𝔽3) | 2 |
| C23×(C3⋊Dic3)⋊C3 | 2 |
| C24×C3⋊Dic3 | 3 |
| C24×C3×Dic3 | 3 |