Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
421 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C3 |
| Fitting | C3×2−1+4 | C2×C3 |
| socle | C2×C3 | C23×A4 |
| hypercenter | D4 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×2−1+4⋊C3 | C2 | semidirect |
| 2. #1010⋊C3 | C2 | semidirect |
| 3. D3×#41⋊C3 | C2 | semidirect |
| 4. #1007⋊C3 | C2 | semidirect |
| 5. C2×D3×SL(2,𝔽3) | C2 | semidirect |
| 6. D3×2−1+4 | C3 | semidirect |
| 9. C3×#41⋊C3 | C22 | semidirect |
| 10. C2×C3×SL(2,𝔽3) | C22 | semidirect |
| 12. D3×SL(2,𝔽3) | C22 | semidirect |
| 13. C3⋊(#41⋊C3) | C22 | semidirect |
| 14. C3×2−1+4 | C2×C3 | semidirect |
| 17. #1010 | C2×C3 | semidirect |
| 20. D3×#41 | C2×C3 | semidirect |
| 22. #1007 | C2×C3 | semidirect |
| 24. C2×D3×Q8 | C2×C3 | semidirect |
| 28. 2−1+4⋊C3 | D3 | direct (N × H) |
| 40. C3×#41 | C22×C3 | semidirect |
| 42. C2×C3×Q8 | C22×C3 | semidirect |
| 46. #239 | C22×C3 | semidirect |
| 53. D3×Q8 | C22×C3 | semidirect |
| 75. #41⋊C3 | C2×D3 | semidirect |
| 79. C2×SL(2,𝔽3) | C2×D3 | semidirect |
| 91. 2−1+4 | C3×D3 | semidirect |
| 158. #41 | C2×C3×D3 | semidirect |
| 170. C2×Q8 | C2×C3×D3 | semidirect |
| 244. D3 | 2−1+4⋊C3 | direct (N × H) |
| 260. C3 | C2×2−1+4⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×2−1+4⋊C3 | 2 |
| #1010⋊C3 | 2 |
| D3×#41⋊C3 | 2 |
| #1007⋊C3 | 2 |
| C2×D3×SL(2,𝔽3) | 2 |
| D3×2−1+4 | 3 |
| C2×2−1+4⋊C3 | 3 |
| C3×D3×D4 | 4 |