Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
424 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 79 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 | #120 |
| 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. C23×C3×D9 | C2 | semidirect |
| 2. C2×C9×#67 | C2 | semidirect |
| 3. C2×#6194 | C2 | semidirect |
| 4. #88551 | C2 | semidirect |
| 5. (C2×D9)⋊Dic3 | C2 | semidirect |
| 9. C22×D3×C9 | C22 | semidirect |
| 13. C2×C3⋊Dic9 | C22 | semidirect |
| 15. C2×C9×Dic3 | C22 | semidirect |
| 20. C23×D9 | D3 | semidirect |
| 29. C22×C3×C9 | D4 | semidirect |
| 36. C2×C3×D9 | D4 | semidirect |
| 71. C2×#67 | D9 | semidirect |
| 94. C22×C9 | #67 | semidirect |
| 99. C2×D9 | #67 | semidirect |
| 104. C3×C9 | #120 | semidirect |
| 115. C22×D3 | C2×D9 | semidirect |
| 124. C2×Dic3 | C2×D9 | semidirect |
| 154. C22×C3 | #611 | semidirect |
| 171. C9 | #953 | semidirect |
| 201. C22 | #6194 | semidirect |
| 206. C3 | #63499 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D9 | 2 |
| C2×C9×#67 | 2 |
| C2×#6194 | 2 |
| #88551 | 2 |
| (C2×D9)⋊Dic3 | 2 |
| #63976 | 3 |
| #63499 | 3 |