Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
303 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 110 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 | C4×C22⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1752 |
| prime radical | C2×#41 | M16 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ 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. C4×#347 | C2 | semidirect |
| 4. #11414 | C2 | semidirect |
| 6. #7164 | C2 | semidirect |
| 7. #7149 | C2 | semidirect |
| 9. Q8×C16 | C4 | semidirect |
| 11. #2169 | C4 | semidirect |
| 23. Q8⋊C16 | C4 | semidirect |
| 24. D4⋊C16 | C4 | semidirect |
| 174. C16 | #104 | semidirect |
| 178. Q8 | #347 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33485 | 2 |
| C4×#347 | 2 |
| #11615 | 2 |
| #11414 | 2 |
| #8192 | 2 |
| #7164 | 2 |
| #7149 | 2 |