Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
401 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #327 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7126 | C2 | semidirect |
| 5. #7111 | C2 | semidirect |
| 31. #470 | C8 | semidirect |
| 74. C2×Q16 | C2×C8 | semidirect |
| 117. #106 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#470 | 2 |
| C4×#349 | 2 |
| #7860 | 2 |
| #7126 | 2 |
| #7111 | 2 |