Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7602 conjugacy classes of subgroups; only the 1858 normal subgroups are shown.
This group has 7602 conjugacy classes of subgroups — too many to draw. Only its 1858 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C23×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C4⋊C8 | C2 | semidirect |
| 3. C22×#331 | C2 | direct (N × H) |
| 12. C22×C4⋊C8 | C22 | semidirect |
| 13. C2×#331 | C22 | direct (N × H) |
| 34. C2×C4⋊C8 | C23 | semidirect |
| 35. #331 | C23 | direct (N × H) |
| 67. C4⋊C8 | C24 | semidirect |
| 103. C23 | #331 | direct (N × H) |
| 125. C22 | C2×#331 | direct (N × H) |
| 135. C2 | C22×#331 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#127 | 2 |
| C23×C4⋊C8 | 2 |
| C22×#331 | 2 |