Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
318 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C9 |
| Frattini | C23×C3 | C2×C3×A4 |
| Fitting, prime radical | C32×B(4,𝔽2) | C3 |
| cosocle | C3×H(5,𝔽2) | C2×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63703 | C2 | semidirect |
| 3. B(4,𝔽2)⋊C9 | C3 | direct (N × H) |
| 7. #63703 | C2×C3 | semidirect |
| 13. C3×B(4,𝔽2) | C9 | semidirect |
| 26. C3×H(5,𝔽2) | C2×C9 | semidirect |
| 48. B(4,𝔽2) | C3×C9 | semidirect |
| 51. C24×C3 | C22⋊C9 | semidirect |
| 78. H(5,𝔽2) | C2×C3×C9 | semidirect |
| 112. C24 | C3×C22⋊C9 | semidirect |
| 171. C3 | B(4,𝔽2)⋊C9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63703 | 2 |
| C32×B(4,𝔽2) | 3 |
| B(4,𝔽2)⋊C9 | 3 |
| C22×C3×C22⋊C9 | 4 |