Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2968 conjugacy classes of subgroups; only the 955 normal subgroups are shown.
This group has 2968 conjugacy classes of subgroups — too many to draw. Only its 955 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 | Q82 |
| 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 |
|---|---|---|
| 21. C33×Q8 | Q8 | semidirect |
| 28. #6308 | Q8 | direct (N × H) |
| 50. C32×Q8 | Dic6 | semidirect |
| 82. C33 | Q82 | semidirect |
| 83. C3×Q8 | #634 | semidirect |
| 109. C32 | Q8×Dic6 | semidirect |
| 112. Q8 | #6308 | direct (N × H) |
| 127. C3 | Q8×#634 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C4×Q8 | 2 |
| Q8×C3⋊(C3⋊Dic3) | 2 |
| #11723929 | 2 |
| #11723910 | 2 |
| C4×#6308 | 2 |
| Q8×#634 | 3 |