Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
549 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini, socle | C2×C3 | C23×C3×D3 |
| Fitting | C2×Q8×H(3,𝔽3) | C2 |
| hypercenter | C2×Q8 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×2−1+4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×H(3,𝔽3) | C2 | semidirect |
| 2. #88722 | C2 | semidirect |
| 3. Q8×(C3⋊D3)⋊C3 | C2 | semidirect |
| 4. #3828⋊C3 | C2 | semidirect |
| 5. #64364 | C3 | semidirect |
| 14. C2×C32×Q8 | D3 | semidirect |
| 15. C2×C32×Q8 | C2×C3 | semidirect |
| 16. #3832 | C2×C3 | semidirect |
| 17. Q8×C3⋊D3 | C2×C3 | semidirect |
| 18. #3828 | C2×C3 | semidirect |
| 70. H(3,𝔽3) | 2−1+4 | semidirect |
| 102. C2×Q8 | (C3⋊D3)⋊C3 | semidirect |
| 118. C32 | #1007 | semidirect |
| 119. C32 | C3×2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×H(3,𝔽3) | 2 |
| #88722 | 2 |
| Q8×(C3⋊D3)⋊C3 | 2 |
| #3828⋊C3 | 2 |
| #64364 | 3 |
| C3×#1007 | 3 |