Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
565 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2240 | C2 | semidirect |
| 3. C2×#1706 | C2 | semidirect |
| 4. #13282 | C2 | semidirect |
| 5. #13202 | C2 | semidirect |
| 6. #9824 | C2 | semidirect |
| 7. #7136 | C2 | semidirect |
| 131. C2×C16 | SD16 | semidirect |
| 202. C2×C8 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2246 | 2 |
| C2×#2240 | 2 |
| C2×#1706 | 2 |
| #13282 | 2 |
| #13202 | 2 |
| #9824 | 2 |
| #7136 | 2 |