Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
316 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 186 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×C16 | D3 |
| commutator, hypocenter | C3 | C2×C32×C32 |
| Frattini | C16 | C2×C32×D3 |
| Fitting | C2×C33×C16 | C2 |
| socle, prime radical | C22×C33 | C16 |
| cosocle | C3×C16 | C22×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C32×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C32×C3⋊C32 | C2 | direct (N × H) |
| 4. C2×C3×C3⋊C32 | C3 | direct (N × H) |
| 12. C3×C3⋊C32 | C2×C3 | direct (N × H) |
| 17. C2×C3⋊C32 | C32 | direct (N × H) |
| 34. C3⋊C32 | C2×C32 | direct (N × H) |
| 45. C2×C33 | C32 | semidirect |
| 67. C33 | C2×C32 | semidirect |
| 79. C2×C32 | C3⋊C32 | direct (N × H) |
| 80. C2×C32 | C3×C32 | semidirect |
| 96. C32 | C2×C3⋊C32 | direct (N × H) |
| 97. C32 | C2×C3×C32 | semidirect |
| 103. C2×C3 | C3×C3⋊C32 | direct (N × H) |
| 104. C2×C3 | C32×C32 | semidirect |
| 111. C3 | C2×C3×C3⋊C32 | direct (N × H) |
| 112. C3 | C2×C32×C32 | semidirect |
| 113. C2 | C32×C3⋊C32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×C16 | 2 |
| C32×C3⋊C32 | 2 |
| C2×C32×C32 | 3 |
| C2×C3×C3⋊C32 | 3 |