Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1746 conjugacy classes of subgroups; only the 550 normal subgroups are shown.
This group has 1746 conjugacy classes of subgroups — too many to draw. Only its 550 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C33 |
| Frattini | C2 | C23×C32×A4 |
| Fitting | C32×#585 | C3 |
| socle | C2×C32 | C23×A4 |
| hypercenter | C32×#41 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C32×C4 ◁ C32×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×2−1+4⋊C3 | C2 | semidirect |
| 2. C32×#994 | C2 | semidirect |
| 3. C2×C32×#41⋊C3 | C2 | semidirect |
| 4. C32×#585 | C3 | semidirect |
| 5. C3×#5945 | C3 | direct (N × H) |
| 7. C32×#41⋊C3 | C22 | semidirect |
| 8. C32×#41⋊C3 | C22 | semidirect |
| 9. C2×C32×SL(2,𝔽3) | C22 | semidirect |
| 11. C32×2−1+4 | C2×C3 | semidirect |
| 13. C32×H(5,𝔽2) | C2×C3 | semidirect |
| 16. C2×C32×#41 | C2×C3 | semidirect |
| 17. C3×2−1+4⋊C3 | C2×C3 | semidirect |
| 18. C3×#994 | C2×C3 | semidirect |
| 19. C2×C3×#41⋊C3 | C2×C3 | semidirect |
| 24. C3×#585 | C32 | semidirect |
| 25. #5945 | C32 | direct (N × H) |
| 46. C3×#41⋊C3 | C22×C3 | semidirect |
| 47. C3×#41⋊C3 | C22×C3 | semidirect |
| 48. C2×C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 53. C3×2−1+4 | C2×C32 | semidirect |
| 55. C3×H(5,𝔽2) | C2×C32 | semidirect |
| 58. C2×C3×#41 | C2×C32 | semidirect |
| 59. 2−1+4⋊C3 | C2×C32 | semidirect |
| 60. #994 | C2×C32 | semidirect |
| 61. C2×#41⋊C3 | C2×C32 | semidirect |
| 85. #585 | C33 | semidirect |
| 108. #41⋊C3 | C22×C32 | semidirect |
| 109. #41⋊C3 | C22×C32 | semidirect |
| 110. C2×SL(2,𝔽3) | C22×C32 | semidirect |
| 124. 2−1+4 | C2×C33 | semidirect |
| 126. H(5,𝔽2) | C2×C33 | semidirect |
| 129. C2×#41 | C2×C33 | semidirect |
| 192. C32 | #5945 | direct (N × H) |
| 228. C3 | C3×#5945 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×2−1+4⋊C3 | 2 |
| C32×#994 | 2 |
| C2×C32×#41⋊C3 | 2 |
| C32×#585 | 3 |
| C3×#5945 | 3 |
| C33×#41 | 4 |