Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
168 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 56 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×C3×C8 | #157 |
| commutator, hypocenter | C32 | C2×C3×C32 |
| Frattini | C8 | C2×C3×#157 |
| Fitting | C2×C33×C8 | C4 |
| socle, prime radical | C22×C33 | C16 |
| cosocle | C3⋊(C3⋊C16) | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#63540 | C2 | direct (N × H) |
| 3. C2×#63540 | C3 | direct (N × H) |
| 9. #63540 | C2×C3 | direct (N × H) |
| 37. C2×C33 | C32 | semidirect |
| 57. C33 | C2×C32 | semidirect |
| 71. C2×C32 | C3×C32 | semidirect |
| 87. C32 | C2×C3×C32 | semidirect |
| 95. C2×C3 | #63540 | direct (N × H) |
| 102. C3 | C2×#63540 | direct (N × H) |
| 103. C2 | C3×#63540 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C3⋊(C3⋊C16) | 2 |
| C3×#63540 | 2 |
| C2×#63540 | 3 |
| C2×C3×C32 | 9 |