Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
266 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1924 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ 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. #12666 | C2 | semidirect |
| 3. #11996 | C2 | semidirect |
| 6. #7168 | C2 | semidirect |
| 7. #7138 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13194 | 2 |
| #12666 | 2 |
| #11996 | 2 |
| #11587 | 2 |
| #8206 | 2 |
| #7168 | 2 |
| #7138 | 2 |