Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
702 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 702 conjugacy classes of subgroups — too many to draw. Only its 144 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 | C2×C32×Q8 | C22 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | #532 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3832 | C2 | semidirect |
| 2. C2×#3796 | C2 | semidirect |
| 3. C3×#947 | C2 | semidirect |
| 4. C3×#944 | C2 | semidirect |
| 5. #63911 | C2 | semidirect |
| 6. #63910 | C2 | semidirect |
| 7. C4×#624 | C2 | semidirect |
| 8. #63881 | C2 | semidirect |
| 9. #63864 | C2 | semidirect |
| 39. #947 | D3 | semidirect |
| 40. #944 | D3 | semidirect |
| 60. C4×C3⋊D3 | D4 | semidirect |
| 127. C3×Dic3 | #41 | semidirect |
| 244. Dic3 | #239 | semidirect |
| 245. C32 | #532 | semidirect |
| 294. C3 | #5668 | semidirect |
| 295. C3 | #5725 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3832 | 2 |
| C2×#3796 | 2 |
| C3×#947 | 2 |
| C3×#944 | 2 |
| #63911 | 2 |
| #63910 | 2 |
| C4×#624 | 2 |
| #63881 | 2 |
| #63864 | 2 |
| #5725 | 3 |
| #5668 | 3 |