Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
308 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 308 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×C8 | C32×D4 |
| commutator | C3×C4 | C2×C32×C8 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C3 | C32×C22⋊C8 |
| prime radical | #41×H(3,𝔽3) | C4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ 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. C2×C16×H(3,𝔽3) | C2 | semidirect |
| 4. C32×#350 | C3 | semidirect |
| 13. C32×M32 | C2×C3 | semidirect |
| 14. C2×C32×C16 | C2×C3 | semidirect |
| 51. #350 | H(3,𝔽3) | direct (N × H) |
| 75. M32 | C2×H(3,𝔽3) | semidirect |
| 76. C2×C16 | C2×H(3,𝔽3) | semidirect |
| 77. H(3,𝔽3) | #350 | direct (N × H) |
| 114. C32 | C3×#350 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×#131 | 2 |
| H(3,𝔽3)×M32 | 2 |
| C2×C16×H(3,𝔽3) | 2 |
| C32×#350 | 3 |