Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
371 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C22⋊C8 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ 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 |
|---|---|---|
| 3. #11402 | C2 | semidirect |
| 4. C2×#1311 | C2 | semidirect |
| 5. #7141 | C2 | semidirect |
| 71. C2×C16 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #17335 | 2 |
| C4×#350 | 2 |
| #11402 | 2 |
| C2×#1311 | 2 |
| #7141 | 2 |