Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
390 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C23 |
| commutator | C22 | C23×C3×C9 |
| Frattini, cosocle | C22×C3 | C23×C32 |
| socle | C22×C32 | C23×C3 |
| prime radical | C2×C32×D4 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C4×C9 | C2 | semidirect |
| 3. C3×C9×C4⋊C4 | C2 | semidirect |
| 4. C3×C9×C22⋊C4 | C2 | semidirect |
| 5. C3×C9×C22⋊C4 | C2 | semidirect |
| 7. C9×#123 | C3 | direct (N × H) |
| 21. C22×C4×C9 | C2×C3 | semidirect |
| 22. C9×C4⋊C4 | C2×C3 | semidirect |
| 23. C9×C22⋊C4 | C2×C3 | semidirect |
| 24. C9×C22⋊C4 | C2×C3 | semidirect |
| 33. C3×#123 | C9 | direct (N × H) |
| 55. C22×C3×C4 | C2×C9 | semidirect |
| 57. C3×C4⋊C4 | C2×C9 | semidirect |
| 59. C3×C22⋊C4 | C2×C9 | semidirect |
| 60. C3×C22⋊C4 | C2×C9 | semidirect |
| 79. #123 | C3×C9 | direct (N × H) |
| 80. C3×C9 | #123 | direct (N × H) |
| 104. C22×C4 | C2×C3×C9 | semidirect |
| 105. C4⋊C4 | C2×C3×C9 | semidirect |
| 106. C22⋊C4 | C2×C3×C9 | semidirect |
| 107. C22⋊C4 | C2×C3×C9 | semidirect |
| 125. C9 | C3×#123 | direct (N × H) |
| 149. C3 | C9×#123 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D4×C9 | 2 |
| C22×C3×C4×C9 | 2 |
| C3×C9×C4⋊C4 | 2 |
| C3×C9×C22⋊C4 | 2 |
| C32×#123 | 3 |
| C9×#123 | 3 |