Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
199 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #102 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1334 | C2 | direct (N × H) |
| 23. C2×Q16 | C8 | semidirect |
| 59. Q16 | C2×C8 | semidirect |
| 117. C2 | #1334 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Q16 | 2 |
| C2×#346 | 2 |
| C2×#334 | 2 |
| #1334 | 2 |