Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1204 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 1204 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 | C2×C4 | C2×C33×C4 |
| Frattini, cosocle | C2×D4 | C22×C33 |
| prime radical | C33×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ 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×#99 | C2 | semidirect |
| 3. C32×#353 | C3 | direct (N × H) |
| 5. C2×C33×D4 | C4 | semidirect |
| 8. C33×C42 | C4 | semidirect |
| 10. C32×#99 | C2×C3 | semidirect |
| 19. C3×#353 | C32 | direct (N × H) |
| 21. C2×C32×D4 | C3×C4 | semidirect |
| 24. C32×C42 | C3×C4 | semidirect |
| 34. C3×#99 | C2×C32 | semidirect |
| 43. #353 | C33 | direct (N × H) |
| 49. C2×C3×D4 | C32×C4 | semidirect |
| 52. C3×C42 | C32×C4 | semidirect |
| 62. #99 | C2×C33 | semidirect |
| 63. C33 | #353 | direct (N × H) |
| 77. C2×D4 | C33×C4 | semidirect |
| 80. C42 | C33×C4 | semidirect |
| 89. C32 | C3×#353 | direct (N × H) |
| 110. C3 | C32×#353 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#127 | 2 |
| C33×#99 | 2 |
| C32×#353 | 3 |