Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
432 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, hypocenter | H(3,𝔽3) | C2×C42 |
| Frattini | C22×C3 | C2×D32 |
| Fitting, prime radical | C23×H(3,𝔽3) | C22 |
| socle | C23×C3 | D32 |
| cosocle | C22×H(3,𝔽3) | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #88447 | C2 | direct (N × H) |
| 6. C2×#1084 | C4 | semidirect |
| 8. C2×(C3⋊Dic3)⋊C3 | C4 | semidirect |
| 18. #1084 | C2×C4 | semidirect |
| 19. (C3⋊Dic3)⋊C3 | C2×C4 | semidirect |
| 24. C2×C3⋊Dic3 | Dic3 | semidirect |
| 32. C2×H(3,𝔽3) | C42 | semidirect |
| 50. C3⋊Dic3 | C2×Dic3 | semidirect |
| 55. H(3,𝔽3) | C2×C42 | semidirect |
| 74. C2×C32 | C4×Dic3 | semidirect |
| 98. C32 | C2×C4×Dic3 | semidirect |
| 122. C2 | #88447 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#1084 | 2 |
| C22×(C3⋊Dic3)⋊C3 | 2 |
| #88447 | 2 |
| C2×Dic32 | 3 |