Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1316 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 364 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #99 |
| commutator | C23 | C2×C33×C4 |
| Frattini, cosocle | C2×D4 | C22×C33 |
| prime radical | C33×#120 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C22×C33 ◁ C2×C33×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C33×#100 | C2 | semidirect |
| 3. C33×#99 | C2 | semidirect |
| 4. C32×AΣL(1,𝔽16) | C3 | direct (N × H) |
| 5. C24×C33 | C4 | semidirect |
| 11. C33×C22⋊C4 | C4 | semidirect |
| 13. C32×#100 | C2×C3 | semidirect |
| 14. C32×#99 | C2×C3 | semidirect |
| 26. C3×AΣL(1,𝔽16) | C32 | direct (N × H) |
| 27. C24×C32 | C3×C4 | semidirect |
| 33. C32×C22⋊C4 | C3×C4 | semidirect |
| 48. C3×#100 | C2×C32 | semidirect |
| 49. C3×#99 | C2×C32 | semidirect |
| 61. AΣL(1,𝔽16) | C33 | direct (N × H) |
| 67. C24×C3 | C32×C4 | semidirect |
| 73. C3×C22⋊C4 | C32×C4 | semidirect |
| 88. #100 | C2×C33 | semidirect |
| 89. #99 | C2×C33 | semidirect |
| 90. C33 | AΣL(1,𝔽16) | direct (N × H) |
| 107. C24 | C33×C4 | semidirect |
| 113. C22⋊C4 | C33×C4 | semidirect |
| 127. C32 | C3×AΣL(1,𝔽16) | direct (N × H) |
| 157. C3 | C32×AΣL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#120 | 2 |
| C33×#100 | 2 |
| C33×#99 | 2 |
| C32×AΣL(1,𝔽16) | 3 |