Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
336 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D9 |
| commutator, cosocle | C22×C3×C9 | C23 |
| Frattini | C22×C3 | C2×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle | C22×C32 | C22×D3 |
| hypercenter | C23 | D3×D9 |
| hypocenter | C3×C9 | #121 |
| LCS | G ▷ C22×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3⋊Dic9 | C2 | semidirect |
| 2. C2×C9×#67 | C2 | semidirect |
| 3. C2×C3×#611 | C2 | semidirect |
| 4. C2×#6194 | C2 | semidirect |
| 5. (C2×D3)⋊Dic9 | C2 | semidirect |
| 6. (C2×D9)⋊Dic3 | C2 | semidirect |
| 7. #88442 | C2 | semidirect |
| 30. C2×#611 | D3 | semidirect |
| 38. C22×C3×C9 | D4 | semidirect |
| 47. C3⋊Dic9 | D4 | semidirect |
| 89. C2×#67 | D9 | semidirect |
| 122. C22×C9 | #67 | semidirect |
| 134. C3×C9 | #121 | semidirect |
| 198. C22×C3 | #611 | semidirect |
| 220. C9 | #939 | semidirect |
| 256. C22 | #6194 | semidirect |
| 263. C3 | #63501 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊Dic9 | 2 |
| C2×C9×#67 | 2 |
| C2×C3×#611 | 2 |
| C2×#6194 | 2 |
| (C2×D3)⋊Dic9 | 2 |
| (C2×D9)⋊Dic3 | 2 |
| #88442 | 2 |
| #63980 | 3 |
| #63501 | 3 |