Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3826 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.
This group has 3826 conjugacy classes of subgroups — too many to draw. Only its 1346 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| 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 |
|---|---|---|
| 3. C22×#330 | C2 | direct (N × H) |
| 13. C2×#330 | C22 | direct (N × H) |
| 32. #330 | C23 | direct (N × H) |
| 81. C23 | #330 | direct (N × H) |
| 96. C22 | C2×#330 | direct (N × H) |
| 105. C2 | C22×#330 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#125 | 2 |
| C23×#97 | 2 |
| C22×#330 | 2 |