Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
357 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 125 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 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#121 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3103 | C2 | semidirect |
| 3. C2×C8×Q8 | C2 | semidirect |
| 4. #1670 | C2 | semidirect |
| 5. #1668 | C2 | semidirect |
| 6. #1666 | C2 | semidirect |
| 7. #1564 | C2 | semidirect |
| 8. #1540 | C2 | semidirect |
| 108. C2×C8 | SD16 | semidirect |
| 133. Q8 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3147 | 2 |
| #3103 | 2 |
| C2×C8×Q8 | 2 |
| #1670 | 2 |
| #1668 | 2 |
| #1666 | 2 |
| #1564 | 2 |
| #1540 | 2 |