Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
657 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 255 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×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C8×Q8 | C2 | semidirect |
| 3. Q8×C4⋊C4 | C2 | semidirect |
| 4. #1990 | C2 | semidirect |
| 5. #1988 | C2 | semidirect |
| 10. D4×Q8 | C4 | semidirect |
| 56. #102 | Q8 | direct (N × H) |
| 87. C2×C8 | C2×Q8 | semidirect |
| 97. C4⋊C4 | C2×Q8 | semidirect |
| 111. Q8 | #102 | direct (N × H) |
| 115. D4 | C4×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×Q8 | 2 |
| C2×C8×Q8 | 2 |
| Q8×C4⋊C4 | 2 |
| #1990 | 2 |
| #1988 | 2 |
| #1802 | 2 |
| C4×#102 | 2 |