Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1008 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2176 | C2 | semidirect |
| 3. #22451 | C2 | semidirect |
| 4. #13239 | C2 | semidirect |
| 5. #13013 | C2 | semidirect |
| 6. #12835 | C2 | semidirect |
| 7. #12733 | C2 | semidirect |
| 8. #12611 | C2 | semidirect |
| 11. C2×#508 | C4 | semidirect |
| 112. Q32 | C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#508 | 2 |
| C2×#2176 | 2 |
| #22451 | 2 |
| #13239 | 2 |
| #13013 | 2 |
| #12835 | 2 |
| #12733 | 2 |
| #12611 | 2 |