Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1119 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 503 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 | C2×#550 | C2 |
| 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×#557 | C2 | semidirect |
| 3. #3392 | C2 | direct (N × H) |
| 4. C2×#492 | C2 | semidirect |
| 5. C2×#460 | C2 | semidirect |
| 6. C2×#461 | C2 | semidirect |
| 7. C2×C8×Q8 | C2 | semidirect |
| 8. C2×C4×SD16 | C2 | semidirect |
| 13. #557 | C22 | semidirect |
| 24. #492 | C22 | semidirect |
| 25. #461 | C22 | semidirect |
| 26. #460 | C22 | semidirect |
| 27. C8×Q8 | C22 | semidirect |
| 28. C4×SD16 | C22 | semidirect |
| 80. C2×Q8 | SD16 | semidirect |
| 102. C2×C8 | #41 | semidirect |
| 125. Q8 | C2×SD16 | semidirect |
| 147. C8 | C2×#41 | semidirect |
| 165. C2 | #3392 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#557 | 2 |
| C2×#550 | 2 |
| #3392 | 2 |
| C2×#492 | 2 |
| C2×#460 | 2 |
| C2×#461 | 2 |
| C2×C8×Q8 | 2 |
| C2×C4×SD16 | 2 |