Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2538 conjugacy classes of subgroups; only the 906 normal subgroups are shown.
This group has 2538 conjugacy classes of subgroups — too many to draw. Only its 906 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C43 | D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C43 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C42×M16 | C2 | semidirect |
| 4. C44 | C2 | semidirect |
| 5. C4×#339 | C2 | semidirect |
| 9. C42×Q8 | C4 | semidirect |
| 11. C42×D4 | C4 | semidirect |
| 18. C4×#104 | C4 | direct (N × H) |
| 39. C4×M16 | C2×C4 | semidirect |
| 52. C43 | C2×C4 | semidirect |
| 55. #339 | C2×C4 | semidirect |
| 70. C4×Q8 | C42 | semidirect |
| 73. C4×D4 | C42 | semidirect |
| 98. #104 | C42 | direct (N × H) |
| 127. M16 | C2×C42 | semidirect |
| 140. C42 | #104 | direct (N × H) |
| 142. C42 | C2×C42 | semidirect |
| 152. Q8 | C43 | semidirect |
| 155. D4 | C43 | semidirect |
| 184. C4 | C4×#104 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C42×#41 | 2 |
| C2×C4×#104 | 2 |
| C42×M16 | 2 |
| C44 | 2 |
| C4×#339 | 2 |