Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
728 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 166 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, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2215 | C2 | semidirect |
| 2. C2×C4×D16 | C2 | semidirect |
| 3. C2×D4×C16 | C2 | semidirect |
| 4. #28225 | C2 | semidirect |
| 5. #13381 | C2 | semidirect |
| 6. #12907 | C2 | semidirect |
| 7. #12834 | C2 | semidirect |
| 8. #12813 | C2 | semidirect |
| 38. C2×D16 | D4 | semidirect |
| 71. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2215 | 2 |
| C2×C4×D16 | 2 |
| C2×D4×C16 | 2 |
| #28225 | 2 |
| #13381 | 2 |
| #12907 | 2 |
| #12834 | 2 |
| #12813 | 2 |