Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1416 conjugacy classes of subgroups; only the 472 normal subgroups are shown.
This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 472 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 | #157 |
| commutator, hypocenter | C32 | C23×C3×C8 |
| Frattini | C2 | C23×C3×#157 |
| Fitting, socle, prime radical | C24×C33 | C4 |
| cosocle | C3⋊Dic3 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C24×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×#622 | C2 | direct (N × H) |
| 3. C23×#622 | C3 | direct (N × H) |
| 7. C2×C3×#622 | C22 | direct (N × H) |
| 10. C22×#622 | C2×C3 | direct (N × H) |
| 11. C23×C33 | C8 | semidirect |
| 15. C3×#622 | C23 | direct (N × H) |
| 24. C2×#622 | C22×C3 | direct (N × H) |
| 25. C22×C33 | C2×C8 | semidirect |
| 36. C23×C32 | C3×C8 | semidirect |
| 42. #622 | C23×C3 | direct (N × H) |
| 44. C2×C33 | C22×C8 | semidirect |
| 58. C22×C32 | C2×C3×C8 | semidirect |
| 66. C33 | C23×C8 | semidirect |
| 71. C23×C3 | #622 | direct (N × H) |
| 82. C2×C32 | C22×C3×C8 | semidirect |
| 92. C22×C3 | C2×#622 | direct (N × H) |
| 100. C32 | C23×C3×C8 | semidirect |
| 101. C23 | C3×#622 | direct (N × H) |
| 110. C2×C3 | C22×#622 | direct (N × H) |
| 112. C22 | C2×C3×#622 | direct (N × H) |
| 118. C3 | C23×#622 | direct (N × H) |
| 119. C2 | C22×C3×#622 | 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⋊Dic3 | 2 |
| C22×C3×#622 | 2 |
| C23×#622 | 3 |
| C23×C3×C8 | 9 |