Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
342 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2631 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3140 | C2 | semidirect |
| 2. #2956 | C2 | semidirect |
| 3. #2082 | C2 | semidirect |
| 4. #2081 | C2 | semidirect |
| 5. #2031 | C2 | semidirect |
| 6. #2031 | C2 | semidirect |
| 7. #2009 | C2 | semidirect |
| 9. #2004 | C2 | semidirect |
| 10. #2003 | C2 | semidirect |
| 11. #1910 | C2 | semidirect |
| 12. #1909 | C2 | semidirect |
| 13. #1903 | C2 | semidirect |
| 14. #1902 | C2 | semidirect |
| 15. #1841 | C2 | semidirect |
| 122. #97 | D4 | semidirect |
| 123. C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3140 | 2 |
| #2956 | 2 |
| #2082 | 2 |
| #2081 | 2 |
| #2031 | 2 |
| #2009 | 2 |
| #2008 | 2 |
| #2004 | 2 |
| #2003 | 2 |
| #1910 | 2 |
| #1909 | 2 |
| #1903 | 2 |
| #1902 | 2 |
| #1841 | 2 |