Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
563 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 195 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#440 | C2 | semidirect |
| 3. C2×C4×SD16 | C2 | semidirect |
| 5. #2327 | C2 | semidirect |
| 6. #1973 | C2 | semidirect |
| 7. #1968 | C2 | semidirect |
| 9. #1960 | C2 | semidirect |
| 10. #1846 | C2 | semidirect |
| 11. #1726 | C2 | semidirect |
| 18. #492 | C4 | semidirect |
| 48. C2×SD16 | C2×C4 | semidirect |
| 49. C2×SD16 | C2×C4 | semidirect |
| 55. C4⋊Q8 | C2×C4 | semidirect |
| 80. #97 | D4 | semidirect |
| 82. C4×C8 | C2×C4 | semidirect |
| 83. C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#492 | 2 |
| C2×#440 | 2 |
| C2×C4×SD16 | 2 |
| #2328 | 2 |
| #2327 | 2 |
| #1973 | 2 |
| #1968 | 2 |
| #1961 | 2 |
| #1960 | 2 |
| #1846 | 2 |
| #1726 | 2 |