Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
980 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 980 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 | C2×C33×D4 | C22 |
| 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 |
|---|---|---|
| 3. C32×#354 | C3 | direct (N × H) |
| 4. C2×C33×Q8 | C4 | semidirect |
| 8. C33×C42 | C4 | semidirect |
| 18. C3×#354 | C32 | direct (N × H) |
| 19. C2×C32×Q8 | C3×C4 | semidirect |
| 23. C32×C42 | C3×C4 | semidirect |
| 40. #354 | C33 | direct (N × H) |
| 44. C2×C3×Q8 | C32×C4 | semidirect |
| 48. C3×C42 | C32×C4 | semidirect |
| 58. C33 | #354 | direct (N × H) |
| 69. C2×Q8 | C33×C4 | semidirect |
| 73. C42 | C33×C4 | semidirect |
| 81. C32 | C3×#354 | direct (N × H) |
| 99. C3 | C32×#354 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#124 | 2 |
| C33×#99 | 2 |
| C32×#354 | 3 |