Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
337 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle, prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1584 | C2 | direct (N × H) |
| 102. C2×C8 | Q16 | semidirect |
| 155. C8 | C2×Q16 | semidirect |
| 172. C2 | #1584 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#446 | 2 |
| C2×C8×Q8 | 2 |
| C2×C4×Q16 | 2 |
| #1584 | 2 |
| C2×#335 | 2 |
| C2×Q8⋊C8 | 2 |
| C2×#322 | 2 |