Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1464 conjugacy classes of subgroups; only the 434 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 434 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C23×C8 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1839 | C2 | semidirect |
| 3. C2×C42×C8 | C2 | semidirect |
| 5. #10874 | C2 | semidirect |
| 26. C2×#126 | C8 | semidirect |
| 135. C2×C8 | #126 | semidirect |
| 145. C42 | C22⋊C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#126 | 2 |
| C2×#1839 | 2 |
| C2×C42×C8 | 2 |
| #10921 | 2 |
| #10874 | 2 |