Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4322 conjugacy classes of subgroups; only the 716 normal subgroups are shown.
This group has 4322 conjugacy classes of subgroups — too many to draw. Only its 716 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D33 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C22×D33 |
| Fitting, socle | C23×C33 | C23 |
| hypocenter | C33 | C22×#41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#6296 | C2 | semidirect |
| 2. C22×D3×C3⋊Dic3 | C2 | semidirect |
| 3. C22×C3×#625 | C2 | semidirect |
| 4. C2×#88963 | C2 | direct (N × H) |
| 9. C2×#6296 | C22 | semidirect |
| 10. C2×D3×C3⋊Dic3 | C22 | semidirect |
| 11. C2×C3×#625 | C22 | semidirect |
| 12. #88963 | C22 | direct (N × H) |
| 16. C22×#625 | D3 | semidirect |
| 25. #6296 | C23 | semidirect |
| 26. D3×C3⋊Dic3 | C23 | semidirect |
| 27. C3×#625 | C23 | semidirect |
| 41. C2×#625 | C2×D3 | semidirect |
| 45. C22×C33 | #41 | semidirect |
| 76. #625 | C22×D3 | semidirect |
| 80. C2×C33 | C2×#41 | semidirect |
| 105. C22×C32 | #237 | semidirect |
| 121. C33 | C22×#41 | semidirect |
| 148. C2×C32 | C2×#237 | semidirect |
| 167. C22×C3 | #3805 | semidirect |
| 179. C32 | C22×#237 | semidirect |
| 194. C2×C3 | C2×#3805 | semidirect |
| 199. C22 | #88963 | direct (N × H) |
| 205. C3 | C22×#3805 | semidirect |
| 207. C2 | C2×#88963 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6296 | 2 |
| C22×D3×C3⋊Dic3 | 2 |
| C22×C3×#625 | 2 |
| C2×#88963 | 2 |
| C22×#3805 | 3 |