Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1096 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 263 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 | C22×#1113 |
| commutator, socle, prime radical, cosocle | C22×C33 | C24 |
| Fitting | C2×C33×Q8 | C22 |
| hypercenter | C2×Q8 | #1113 |
| hypocenter | C33 | Q82 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 17. C33×Q8 | Q8 | semidirect |
| 22. #6296 | Q8 | direct (N × H) |
| 49. C32×Q8 | Dic6 | semidirect |
| 93. C33 | Q82 | semidirect |
| 96. C3×Q8 | #627 | semidirect |
| 137. C32 | Q8×Dic6 | semidirect |
| 140. Q8 | #6296 | direct (N × H) |
| 159. C3 | Q8×#627 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q8×C3⋊Dic3 | 2 |
| C4⋊#6296 | 2 |
| C4×#6296 | 2 |
| Q8×#627 | 3 |