Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
374 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#99 |
| commutator | C23×C3 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×D4 | C22×C32 |
| prime radical | C2×D4×H(3,𝔽3) | C22 |
| LCS | G ▷ C23×C3 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×H(3,𝔽3) ◁ C2×D4×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C32×#352 | C3 | semidirect |
| 6. C22×C4×H(3,𝔽3) | C4 | semidirect |
| 11. C22⋊C4×H(3,𝔽3) | C4 | semidirect |
| 27. C22×C32×C4 | C3×C4 | semidirect |
| 32. C32×C22⋊C4 | C3×C4 | semidirect |
| 57. #352 | H(3,𝔽3) | direct (N × H) |
| 88. H(3,𝔽3) | #352 | direct (N × H) |
| 112. C22×C4 | C4×H(3,𝔽3) | semidirect |
| 117. C22⋊C4 | C4×H(3,𝔽3) | semidirect |
| 136. C32 | C3×#352 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×#123 | 2 |
| H(3,𝔽3)×#100 | 2 |
| H(3,𝔽3)×#99 | 2 |
| C32×#352 | 3 |