Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
226 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×#157 |
| commutator, hypocenter | C22×C32 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C3×C4 |
| prime radical | C2×A4×C3⋊D3 | C2 |
| cosocle | C22×C3⋊D3 | C22×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×#157 | C2 | direct (N × H) |
| 3. C23×#157 | C3 | semidirect |
| 4. C2×C32×A4 | C4 | semidirect |
| 6. A4×C3⋊D3 | C4 | semidirect |
| 12. C22×#157 | C2×C3 | semidirect |
| 15. C32×A4 | C2×C4 | semidirect |
| 18. C23×C32 | C3×C4 | semidirect |
| 23. C22×C3⋊D3 | C3×C4 | semidirect |
| 32. C2×#157 | A4 | direct (N × H) |
| 43. C22×C32 | C2×C3×C4 | semidirect |
| 57. #157 | C2×A4 | direct (N × H) |
| 70. C2×A4 | #157 | direct (N × H) |
| 76. C2×C32 | C4×A4 | semidirect |
| 79. C3⋊D3 | C4×A4 | semidirect |
| 102. A4 | C2×#157 | direct (N × H) |
| 106. C32 | C2×C4×A4 | semidirect |
| 113. C23 | C3×#157 | semidirect |
| 142. C22 | C2×C3×#157 | semidirect |
| 154. C2 | A4×#157 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×A4×C3⋊D3 | 2 |
| A4×#157 | 2 |
| C23×#157 | 3 |
| C2×C3×#157 | 4 |
| C2×C4×A4 | 9 |