Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
455 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C4×D4 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C8×D4 | C2 | semidirect |
| 3. C8×D8 | C2 | direct (N × H) |
| 4. C2×#335 | C2 | semidirect |
| 5. C2×D4⋊C8 | C2 | semidirect |
| 6. C2×C82 | C2 | semidirect |
| 13. C8×D4 | C22 | semidirect |
| 21. #335 | C22 | semidirect |
| 22. D4⋊C8 | C22 | semidirect |
| 23. C82 | C22 | semidirect |
| 26. C2×D8 | C8 | direct (N × H) |
| 57. C2×D4 | C2×C8 | semidirect |
| 67. D8 | C2×C8 | direct (N × H) |
| 75. C2×C8 | C2×C8 | semidirect |
| 76. C2×C8 | D8 | direct (N × H) |
| 96. D4 | C22×C8 | semidirect |
| 117. C8 | C22×C8 | semidirect |
| 118. C8 | C2×D8 | direct (N × H) |
| 137. C2 | C8×D8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| C2×C8×D4 | 2 |
| C8×D8 | 2 |
| C2×#335 | 2 |
| C2×D4⋊C8 | 2 |
| C2×C82 | 2 |