Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1029 conjugacy classes of subgroups; only the 543 normal subgroups are shown.
This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 543 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q82 | C2 | semidirect |
| 3. Q8×SD16 | C2 | direct (N × H) |
| 4. C2×#498 | C2 | semidirect |
| 5. C2×#475 | C2 | semidirect |
| 7. C2×C8×Q8 | C2 | semidirect |
| 12. Q82 | C22 | semidirect |
| 23. #498 | C22 | semidirect |
| 25. #475 | C22 | semidirect |
| 26. C8×Q8 | C22 | semidirect |
| 42. C2×SD16 | Q8 | direct (N × H) |
| 82. C2×Q8 | SD16 | direct (N × H) |
| 84. C2×Q8 | C2×Q8 | semidirect |
| 98. SD16 | C2×Q8 | direct (N × H) |
| 100. C2×C8 | C2×Q8 | semidirect |
| 124. Q8 | C2×SD16 | direct (N × H) |
| 125. Q8 | C22×Q8 | semidirect |
| 145. C8 | C22×Q8 | semidirect |
| 163. C2 | Q8×SD16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q82 | 2 |
| C2×D4×Q8 | 2 |
| Q8×SD16 | 2 |
| C2×#498 | 2 |
| C2×#475 | 2 |
| C2×#476 | 2 |
| C2×C8×Q8 | 2 |
| C2×C4×SD16 | 2 |