Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
913 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 913 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C23×C3⋊D3 |
| commutator, cosocle | C2×H(3,𝔽3) | C24 |
| Fitting | #41×H(3,𝔽3) | C2 |
| hypercenter | C3×#41 | C3⋊D3 |
| hypocenter | H(3,𝔽3) | H(5,𝔽2) |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #41×H(3,𝔽3) | C2 | semidirect |
| 2. #88748 | C2 | semidirect |
| 3. D4×#272 | C2 | semidirect |
| 4. #88740 | C2 | semidirect |
| 5. C2×C4⋊#272 | C2 | semidirect |
| 8. C22×#272 | C22 | semidirect |
| 9. Q8×H(3,𝔽3) | C22 | semidirect |
| 10. D4×H(3,𝔽3) | C22 | semidirect |
| 11. C2×C4×H(3,𝔽3) | C22 | semidirect |
| 12. #6235 | C22 | semidirect |
| 13. C4⋊#272 | C22 | semidirect |
| 14. C4⋊#272 | C22 | semidirect |
| 15. C4×#272 | C22 | semidirect |
| 16. #6229 | C22 | semidirect |
| 17. C32×#41 | D3 | semidirect |
| 18. C32×#41 | D3 | semidirect |
| 38. C32×Q8 | C2×D3 | semidirect |
| 39. C32×Q8 | C2×D3 | semidirect |
| 40. C32×D4 | C2×D3 | semidirect |
| 41. C32×D4 | C2×D3 | semidirect |
| 42. C2×C32×C4 | C2×D3 | semidirect |
| 43. C2×C32×C4 | C2×D3 | semidirect |
| 85. H(3,𝔽3) | H(5,𝔽2) | semidirect |
| 121. #41 | #272 | semidirect |
| 142. C32 | #1009 | semidirect |
| 143. C32 | #1009 | semidirect |
| 146. Q8 | C2×#272 | semidirect |
| 150. D4 | C2×#272 | semidirect |
| 152. C2×C4 | C2×#272 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×H(3,𝔽3) | 2 |
| #88748 | 2 |
| D4×#272 | 2 |
| #88740 | 2 |
| C2×C4⋊#272 | 2 |
| C3×#1009 | 3 |