Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1416 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 250 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 | D3×#157 |
| commutator, hypocenter | C33 | C23×C8 |
| Frattini | C2 | C22×D3×#157 |
| Fitting, socle | C23×C33 | C2×C4 |
| prime radical | C23×C3⋊(C3⋊D3) | C4 |
| cosocle | C3×C3⋊Dic3 | C24 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3⋊#622 | C2 | semidirect |
| 3. C22×C3×#622 | C2 | semidirect |
| 4. C2×#88924 | C2 | direct (N × H) |
| 11. C2×C3⋊#622 | C22 | semidirect |
| 12. C2×C3×#622 | C22 | semidirect |
| 13. #88924 | C22 | direct (N × H) |
| 17. C22×#622 | D3 | semidirect |
| 20. C22×C3⋊(C3⋊D3) | C8 | semidirect |
| 27. C3⋊#622 | C23 | semidirect |
| 28. C3×#622 | C23 | semidirect |
| 42. C2×#622 | C2×D3 | semidirect |
| 44. C22×C33 | C2×C8 | semidirect |
| 46. C2×C3⋊(C3⋊D3) | C2×C8 | semidirect |
| 76. #622 | C22×D3 | semidirect |
| 78. C2×C33 | C22×C8 | semidirect |
| 80. C3⋊(C3⋊D3) | C22×C8 | semidirect |
| 99. C22×C32 | D3×C8 | semidirect |
| 117. C33 | C23×C8 | semidirect |
| 143. C2×C32 | C2×D3×C8 | semidirect |
| 160. C22×C3 | #3787 | semidirect |
| 176. C32 | C22×D3×C8 | semidirect |
| 191. C2×C3 | C2×#3787 | semidirect |
| 198. C22 | #88924 | direct (N × H) |
| 205. C3 | C22×#3787 | semidirect |
| 207. C2 | C2×#88924 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6289 | 2 |
| C22×C3⋊#622 | 2 |
| C22×C3×#622 | 2 |
| C2×#88924 | 2 |
| C22×#3787 | 3 |
| C22×D3×C8 | 9 |