Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2800 conjugacy classes of subgroups; only the 731 normal subgroups are shown.
This group has 2800 conjugacy classes of subgroups — too many to draw. Only its 731 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, prime radical, cosocle | C22×C33 | C24 |
| Fitting | C33×C4×Q8 | C2 |
| hypercenter | C4×Q8 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #557 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 44. #6308 | Q8 | semidirect |
| 127. C33 | #557 | semidirect |
| 167. C32 | #5567 | semidirect |
| 191. C3 | #10596038 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C4×Q8 | 2 |
| #11723985 | 2 |
| #11723930 | 2 |
| #11723928 | 2 |
| #11723911 | 2 |
| #11723910 | 2 |
| C4×#6308 | 2 |
| #10596038 | 3 |