Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
319 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C32×C22⋊C4 |
| commutator | C22×C3 | C2×C32×C8 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C3 | C32×C22⋊C8 |
| prime radical | C23×H(3,𝔽3) | C8 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C8×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽3)×M32 | C2 | semidirect |
| 3. C32×#349 | C3 | semidirect |
| 10. C32×M32 | C2×C3 | semidirect |
| 42. #349 | H(3,𝔽3) | direct (N × H) |
| 63. M32 | C2×H(3,𝔽3) | semidirect |
| 64. H(3,𝔽3) | #349 | direct (N × H) |
| 99. C32 | C3×#349 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M16×H(3,𝔽3) | 2 |
| H(3,𝔽3)×M32 | 2 |
| C32×#349 | 3 |