Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
365 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1982 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12012 | C2 | semidirect |
| 3. #11742 | C2 | semidirect |
| 4. #11645 | C2 | semidirect |
| 5. #8282 | C2 | semidirect |
| 12. #1713 | C4 | semidirect |
| 13. #1712 | C4 | semidirect |
| 16. #1706 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19603 | 2 |
| #12012 | 2 |
| #11742 | 2 |
| #11645 | 2 |
| #8282 | 2 |