Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
572 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C23 |
| Fitting | C2×C33×C4 | C22 |
| prime radical | C2×#6290 | C2 |
| hypercenter | C2×C4 | D3×C3⋊D3 |
| hypocenter | C33 | #123 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C32×C4×D3 | C2 | semidirect |
| 3. #88857 | C2 | semidirect |
| 4. C3×#3752 | C2 | semidirect |
| 5. #88807 | C2 | semidirect |
| 6. #88803 | C2 | semidirect |
| 7. (C2×D3)⋊(C3⋊Dic3) | C2 | semidirect |
| 21. C2×C3×C4×D3 | D3 | semidirect |
| 25. #3752 | D3 | semidirect |
| 77. C2×C4×D3 | C3⋊D3 | semidirect |
| 118. C33 | #123 | semidirect |
| 189. C32 | #883 | semidirect |
| 190. C32 | #929 | semidirect |
| 226. C3 | #63905 | semidirect |
| 227. C3 | #64138 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×C4×D3 | 2 |
| C2×#6290 | 2 |
| #88857 | 2 |
| C3×#3752 | 2 |
| #88807 | 2 |
| #88803 | 2 |
| (C2×D3)⋊(C3⋊Dic3) | 2 |
| #64138 | 3 |
| #63905 | 3 |