Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
610 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33610 | C2 | semidirect |
| 2. C2×#2178 | C2 | semidirect |
| 3. C2×#2170 | C2 | semidirect |
| 4. #28245 | C2 | semidirect |
| 5. #13385 | C2 | semidirect |
| 6. #12907 | C2 | semidirect |
| 7. #12842 | C2 | semidirect |
| 8. #12820 | C2 | semidirect |
| 79. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33610 | 2 |
| C2×#2178 | 2 |
| C2×#2170 | 2 |
| #28245 | 2 |
| #13385 | 2 |
| #12907 | 2 |
| #12842 | 2 |
| #12820 | 2 |