Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
373 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #2184 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D32 | C2 | semidirect |
| 2. C2×#2221 | C2 | semidirect |
| 3. #33607 | C2 | semidirect |
| 4. C2×#1419 | C2 | semidirect |
| 5. #13617 | C2 | semidirect |
| 6. C2×#1401 | C2 | semidirect |
| 7. #7473 | C2 | semidirect |
| 45. C2×D16 | D4 | semidirect |
| 68. C2×C32 | D4 | semidirect |
| 70. #368 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D32 | 2 |
| C2×#2221 | 2 |
| #33607 | 2 |
| C2×#1419 | 2 |
| #13617 | 2 |
| C2×#1401 | 2 |
| #7473 | 2 |