Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1980 conjugacy classes of subgroups; only the 1098 normal subgroups are shown.
This group has 1980 conjugacy classes of subgroups — too many to draw. Only its 1098 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C42 | D3 |
| commutator, hypocenter | C3 | C32×C43 |
| Frattini | C23 | C22×C32×D3 |
| Fitting | C2×C33×C42 | C2 |
| socle, prime radical | C23×C33 | C23 |
| cosocle | C23×C3 | C23×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C32×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×C42×Dic3 | C3 | direct (N × H) |
| 7. C33×C42 | C4 | semidirect |
| 8. C32×C4×Dic3 | C4 | direct (N × H) |
| 19. C42×Dic3 | C32 | direct (N × H) |
| 27. C32×C42 | Dic3 | direct (N × H) |
| 28. C32×C42 | C3×C4 | semidirect |
| 29. C3×C4×Dic3 | C3×C4 | direct (N × H) |
| 32. C33×C4 | C42 | semidirect |
| 33. C32×Dic3 | C42 | direct (N × H) |
| 60. C3×C42 | C3×Dic3 | direct (N × H) |
| 61. C3×C42 | C32×C4 | semidirect |
| 62. C4×Dic3 | C32×C4 | direct (N × H) |
| 71. C32×C4 | C4×Dic3 | direct (N × H) |
| 72. C32×C4 | C3×C42 | semidirect |
| 73. C3×Dic3 | C3×C42 | direct (N × H) |
| 76. C33 | C43 | semidirect |
| 97. C42 | C32×Dic3 | direct (N × H) |
| 106. C3×C4 | C3×C4×Dic3 | direct (N × H) |
| 107. C3×C4 | C32×C42 | semidirect |
| 108. Dic3 | C32×C42 | direct (N × H) |
| 110. C32 | C42×Dic3 | direct (N × H) |
| 111. C32 | C3×C43 | semidirect |
| 125. C4 | C32×C4×Dic3 | direct (N × H) |
| 127. C3 | C3×C42×Dic3 | direct (N × H) |
| 128. C3 | C32×C43 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×C42 | 2 |
| C2×C32×C4×Dic3 | 2 |
| C32×C43 | 3 |
| C3×C42×Dic3 | 3 |