Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
343 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 343 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 | #2033 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#1667 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ 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. #12033 | C2 | semidirect |
| 3. #11961 | C2 | semidirect |
| 4. #11961 | C2 | semidirect |
| 5. #11629 | C2 | semidirect |
| 6. #8616 | C2 | semidirect |
| 7. #7288 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1667 | 2 |
| #12033 | 2 |
| #11961 | 2 |
| #11629 | 2 |
| #8616 | 2 |
| #7288 | 2 |