Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
159 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C2×C4 |
| Frattini | C22×C3 | C2×S4 |
| Fitting | C22×C3×Q8 | D3 |
| socle, prime radical | C23×C3 | S4 |
| cosocle | C2×Q8⋊C9 | C22 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #63421 | C2 | direct (N × H) |
| 5. C2×Q8⋊C9 | C4 | semidirect |
| 15. Q8⋊C9 | C2×C4 | semidirect |
| 55. C2×Q8 | Dic9 | semidirect |
| 76. Q8 | C2×Dic9 | semidirect |
| 97. C2 | #63421 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8⋊C9 | 2 |
| #63421 | 2 |
| C2×#835 | 3 |
| C22×Dic9 | 4 |