Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4242 conjugacy classes of subgroups; only the 756 normal subgroups are shown.
This group has 4242 conjugacy classes of subgroups — too many to draw. Only its 756 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 |
| prime radical | C23×C3⋊(C3⋊D3) | C22 |
| hypocenter | C33 | C23×Q8 |
| 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 |
| 3. C22×C3×#627 | C2 | semidirect |
| 4. C2×#88958 | C2 | direct (N × H) |
| 9. C2×#6296 | C22 | semidirect |
| 11. C2×C3×#627 | C22 | semidirect |
| 12. #88958 | C22 | direct (N × H) |
| 16. C22×#627 | D3 | semidirect |
| 21. C22×C3⋊(C3⋊D3) | Q8 | semidirect |
| 25. #6296 | C23 | semidirect |
| 27. C3×#627 | C23 | semidirect |
| 41. C2×#627 | C2×D3 | semidirect |
| 45. C22×C33 | C2×Q8 | semidirect |
| 47. C2×C3⋊(C3⋊D3) | C2×Q8 | semidirect |
| 76. #627 | C22×D3 | semidirect |
| 80. C2×C33 | C22×Q8 | semidirect |
| 82. C3⋊(C3⋊D3) | C22×Q8 | semidirect |
| 103. C22×C32 | D3×Q8 | semidirect |
| 121. C33 | C23×Q8 | semidirect |
| 147. C2×C32 | C2×D3×Q8 | semidirect |
| 163. C22×C3 | #3797 | semidirect |
| 179. C32 | C22×D3×Q8 | semidirect |
| 192. C2×C3 | C2×#3797 | semidirect |
| 199. C22 | #88958 | direct (N × H) |
| 205. C3 | C22×#3797 | semidirect |
| 207. C2 | C2×#88958 | 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×#6289 | 2 |
| C22×C3×#627 | 2 |
| C2×#88958 | 2 |
| C22×#3797 | 3 |