Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3294 conjugacy classes of subgroups; only the 802 normal subgroups are shown.
This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 802 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1957 | C2 | semidirect |
| 3. C2×C42×C8 | C2 | semidirect |
| 4. #14692 | C2 | semidirect |
| 10. #2867 | C4 | semidirect |
| 33. C2×C4×C8 | D4 | semidirect |
| 97. C2×C8 | #127 | semidirect |
| 100. C42 | #102 | semidirect |
| 143. C4 | #1957 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2867 | 2 |
| C2×#1957 | 2 |
| C2×C42×C8 | 2 |
| #14692 | 2 |