Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
447 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C23 |
| Frattini | C2 | C2×D3×S4 |
| Fitting | C2×C3×Q8 | C2×D3 |
| socle | C22×C3 | C2×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D3×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×C3⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C2×C3×GL(2,𝔽3) | C2 | semidirect |
| 4. D3×GL(2,𝔽3) | C2 | direct (N × H) |
| 8. C2×C3×SL(2,𝔽3) | C22 | semidirect |
| 9. D3×SL(2,𝔽3) | C22 | semidirect |
| 10. C3⋊GL(2,𝔽3) | C22 | semidirect |
| 11. C3×GL(2,𝔽3) | C22 | semidirect |
| 12. C2×D3×Q8 | D3 | semidirect |
| 18. C2×GL(2,𝔽3) | D3 | direct (N × H) |
| 31. C3×SL(2,𝔽3) | C23 | semidirect |
| 35. C2×C3×Q8 | C2×D3 | semidirect |
| 39. D3×Q8 | C2×D3 | semidirect |
| 47. C2×SL(2,𝔽3) | C2×D3 | semidirect |
| 50. GL(2,𝔽3) | C2×D3 | direct (N × H) |
| 89. C3×Q8 | C22×D3 | semidirect |
| 102. SL(2,𝔽3) | C22×D3 | semidirect |
| 114. C2×Q8 | D32 | semidirect |
| 147. C2×D3 | GL(2,𝔽3) | direct (N × H) |
| 161. Q8 | C2×D32 | semidirect |
| 179. C2×C3 | C2×GL(2,𝔽3) | semidirect |
| 185. D3 | C2×GL(2,𝔽3) | direct (N × H) |
| 199. C3 | C22×GL(2,𝔽3) | semidirect |
| 203. C2 | D3×GL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×SL(2,𝔽3) | 2 |
| C2×C3⋊GL(2,𝔽3) | 2 |
| C2×C3×GL(2,𝔽3) | 2 |
| D3×GL(2,𝔽3) | 2 |
| C22×GL(2,𝔽3) | 3 |
| C2×D3×SD16 | 3 |
| C22×D32 | 4 |