Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
503 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 162 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×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting, prime radical | C2×C32×D4 | C22 |
| hypercenter | C2×D4 | D32 |
| hypocenter | C32 | #554 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #63982 | C2 | semidirect |
| 4. C2×#3724 | C2 | semidirect |
| 5. #63964 | C2 | semidirect |
| 6. #63919 | C2 | semidirect |
| 7. C4×#627 | C2 | semidirect |
| 8. #63869 | C2 | semidirect |
| 39. C32×D4 | Q8 | semidirect |
| 102. C3×D4 | Dic6 | semidirect |
| 165. C32 | #554 | semidirect |
| 169. D4 | #627 | semidirect |
| 199. C3 | #5539 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C3⋊Dic3 | 2 |
| C3×D4×Dic3 | 2 |
| #63982 | 2 |
| C2×#3724 | 2 |
| #63964 | 2 |
| #63919 | 2 |
| C4×#627 | 2 |
| #63869 | 2 |
| #5539 | 3 |