Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
435 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#460 | C2 | semidirect |
| 3. C2×C8×Q8 | C2 | semidirect |
| 4. #1668 | C2 | direct (N × H) |
| 5. C2×#335 | C2 | semidirect |
| 6. C2×#331 | C2 | semidirect |
| 15. #460 | C22 | semidirect |
| 17. C8×Q8 | C22 | semidirect |
| 24. #335 | C22 | semidirect |
| 25. #331 | C22 | semidirect |
| 64. C2×Q8 | D8 | semidirect |
| 90. C2×C8 | SD16 | semidirect |
| 110. Q8 | C2×D8 | semidirect |
| 137. C8 | C2×SD16 | semidirect |
| 157. C2 | #1668 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#493 | 2 |
| C2×#460 | 2 |
| C2×C8×Q8 | 2 |
| #1668 | 2 |
| C2×#335 | 2 |
| C2×#331 | 2 |