Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
536 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23×C3 |
| Frattini | C2×C3 | C22×C3×S4 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | C3×Q8 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63700 | C2 | semidirect |
| 2. C3×#63691 | C2 | semidirect |
| 3. C3×#63690 | C2 | semidirect |
| 5. #10593099 | C3 | direct (N × H) |
| 8. C3×#3689 | C22 | semidirect |
| 9. C3×#3688 | C22 | semidirect |
| 18. #63700 | C2×C3 | semidirect |
| 19. #63691 | C2×C3 | semidirect |
| 20. #63690 | C2×C3 | semidirect |
| 44. #3689 | C22×C3 | semidirect |
| 45. #3688 | C22×C3 | semidirect |
| 51. C3×H(5,𝔽2) | D9 | semidirect |
| 129. H(5,𝔽2) | C3×D9 | semidirect |
| 238. C3 | #10593099 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63700 | 2 |
| C3×#63691 | 2 |
| C3×#63690 | 2 |
| C3×#5836 | 3 |
| #10593099 | 3 |
| C3×Q8×D9 | 4 |