Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
810 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #23798 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #21681 | C2 | semidirect |
| 3. #9644 | C2 | semidirect |
| 4. #8951 | C2 | semidirect |
| 5. #8787 | C2 | semidirect |
| 6. #8133 | C2 | semidirect |
| 101. C4×C8 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23798 | 2 |
| #21681 | 2 |
| #9644 | 2 |
| #8951 | 2 |
| #8787 | 2 |
| #8133 | 2 |