Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2555 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 2555 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×Q16 | C2 | semidirect |
| 3. C23×SD16 | C2 | semidirect |
| 4. C22×C22⋊Q8 | C2 | semidirect |
| 5. C2×#452 | C2 | direct (N × H) |
| 6. C22×#103 | C2 | semidirect |
| 7. C22×#102 | C2 | semidirect |
| 8. C22×C22⋊C8 | C2 | semidirect |
| 20. C22×Q16 | C22 | semidirect |
| 22. C22×SD16 | C22 | semidirect |
| 26. C2×C22⋊Q8 | C22 | semidirect |
| 30. #452 | C22 | direct (N × H) |
| 31. C2×#103 | C22 | semidirect |
| 32. C2×#102 | C22 | semidirect |
| 33. C2×C22⋊C8 | C22 | semidirect |
| 67. C2×Q16 | C23 | semidirect |
| 69. C2×SD16 | C23 | semidirect |
| 75. C22⋊Q8 | C23 | semidirect |
| 80. #103 | C23 | semidirect |
| 81. #102 | C23 | semidirect |
| 82. C22⋊C8 | C23 | semidirect |
| 192. C22 | #452 | direct (N × H) |
| 206. C2 | C2×#452 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C23×Q16 | 2 |
| C23×SD16 | 2 |
| C22×C22⋊Q8 | 2 |
| C2×#452 | 2 |
| C22×#103 | 2 |
| C22×#102 | 2 |
| C22×C22⋊C8 | 2 |