Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
567 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 223 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 | C22×D8 | C22 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 3. D4×C4⋊C4 | C2 | semidirect |
| 4. #1942 | C2 | semidirect |
| 5. #1862 | C2 | semidirect |
| 6. #1845 | C2 | semidirect |
| 7. #1802 | C2 | semidirect |
| 8. C8×C4⋊C4 | C2 | semidirect |
| 14. C4×D8 | C4 | semidirect |
| 42. C4×D4 | C2×C4 | semidirect |
| 57. #107 | C2×C4 | semidirect |
| 60. #102 | C2×C4 | semidirect |
| 62. C4×C8 | C2×C4 | semidirect |
| 85. D8 | C4⋊C4 | direct (N × H) |
| 102. C4⋊C4 | D8 | direct (N × H) |
| 121. D4 | C2×C4⋊C4 | semidirect |
| 150. C8 | C2×C4⋊C4 | semidirect |
| 170. C4 | C4×D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| D4×C4⋊C4 | 2 |
| #1942 | 2 |
| #1862 | 2 |
| #1845 | 2 |
| #1802 | 2 |
| C8×C4⋊C4 | 2 |