Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
508 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1990 | C2 | semidirect |
| 3. #1803 | C2 | semidirect |
| 5. C2×#332 | C2 | semidirect |
| 6. C2×Q8⋊C8 | C2 | semidirect |
| 12. D4×Q8 | C4 | semidirect |
| 181. Q8 | #102 | semidirect |
| 185. D4 | #103 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×Q8 | 2 |
| #1990 | 2 |
| #1803 | 2 |
| #1802 | 2 |
| C2×#332 | 2 |
| C2×Q8⋊C8 | 2 |
| C2×D4⋊C8 | 2 |