Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
715 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2920 | C2 | semidirect |
| 3. C2×C43 | C2 | semidirect |
| 4. #1994 | C2 | semidirect |
| 5. #1962 | C2 | semidirect |
| 6. #1896 | C2 | semidirect |
| 7. C4×#104 | C2 | semidirect |
| 8. #533 | C4 | semidirect |
| 10. #532 | C4 | semidirect |
| 14. #529 | C4 | semidirect |
| 38. C4⋊Q8 | C2×C4 | semidirect |
| 39. C4⋊Q8 | C2×C4 | semidirect |
| 40. #127 | C2×C4 | semidirect |
| 43. #125 | C2×C4 | semidirect |
| 45. #124 | C2×C4 | semidirect |
| 57. C4×Q8 | C2×C4 | semidirect |
| 63. C4×D4 | C2×C4 | semidirect |
| 142. C42 | #41 | semidirect |
| 169. C2×C4 | #104 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3444 | 2 |
| #2920 | 2 |
| C2×C43 | 2 |
| #1994 | 2 |
| #1962 | 2 |
| #1896 | 2 |
| C4×#104 | 2 |