Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
633 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 633 conjugacy classes of subgroups — too many to draw. Only its 167 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 | C22×C4 | C24 |
| prime radical | #2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#492 | C2 | semidirect |
| 2. C2×#492 | C2 | semidirect |
| 3. C2×#460 | C2 | semidirect |
| 4. C2×#461 | C2 | semidirect |
| 5. C2×C4×SD16 | C2 | semidirect |
| 6. #2546 | C2 | semidirect |
| 8. #2014 | C2 | semidirect |
| 9. #2013 | C2 | semidirect |
| 10. #1968 | C2 | semidirect |
| 11. C8×C4⋊C4 | C2 | semidirect |
| 66. C2×SD16 | D4 | semidirect |
| 96. C4⋊C8 | D4 | semidirect |
| 102. C4×C8 | D4 | semidirect |
| 170. C4⋊C4 | SD16 | semidirect |
| 234. C8 | #121 | semidirect |
| 256. C4 | #492 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#492 | 2 |
| C2×#460 | 2 |
| C2×#461 | 2 |
| C2×C4×SD16 | 2 |
| #2546 | 2 |
| #2541 | 2 |
| #2014 | 2 |
| #2013 | 2 |
| #1968 | 2 |
| C8×C4⋊C4 | 2 |