Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
421 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C3 |
| Frattini | C2 | C22×D3×A4 |
| Fitting | C2×C3×#41 | C2×C3 |
| socle | C22×C3 | C22×A4 |
| hypercenter | C2×C4 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#41⋊C3 | C2 | semidirect |
| 2. #1010⋊C3 | C2 | direct (N × H) |
| 3. C2×D3×SL(2,𝔽3) | C2 | semidirect |
| 4. C2×#1010 | C3 | semidirect |
| 7. C3×#41⋊C3 | C22 | semidirect |
| 9. D3×SL(2,𝔽3) | C22 | semidirect |
| 10. C2×C3×#41 | C2×C3 | semidirect |
| 12. #1010 | C2×C3 | semidirect |
| 13. C2×D3×Q8 | C2×C3 | semidirect |
| 19. C2×#41⋊C3 | D3 | semidirect |
| 27. C3×#41 | C22×C3 | semidirect |
| 35. D3×Q8 | C22×C3 | semidirect |
| 47. #41⋊C3 | C2×D3 | semidirect |
| 59. C2×#41 | C3×D3 | semidirect |
| 95. #41 | C2×C3×D3 | semidirect |
| 150. C2×C3 | 2−1+4⋊C3 | semidirect |
| 164. C3 | C2×2−1+4⋊C3 | semidirect |
| 167. C2 | #1010⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#41⋊C3 | 2 |
| #1010⋊C3 | 2 |
| C2×D3×SL(2,𝔽3) | 2 |
| C2×#1010 | 3 |
| C2×2−1+4⋊C3 | 3 |
| C2×C3×D12 | 4 |