Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
139 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 44 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 | C2×D32 |
| commutator, Frattini, cosocle | C2×C32 | C23 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D64 | C2 | semidirect |
| 2. #13725 | C2 | semidirect |
| 3. C4×D32 | C2 | semidirect |
| 4. C4⋊C64 | C2 | semidirect |
| 5. #7537 | C2 | semidirect |
| 25. D32 | D4 | semidirect |
| 93. C4 | D64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D64 | 2 |
| #13725 | 2 |
| C4×D32 | 2 |
| C4⋊C64 | 2 |
| #7537 | 2 |