Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
112 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #2213 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ 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 |
|---|---|---|
| 1. #2248 | C2 | semidirect |
| 2. #2214 | C2 | semidirect |
| 4. #2170 | C2 | semidirect |
| 5. #1373 | C2 | semidirect |
| 6. #1358 | C2 | semidirect |
| 7. #1357 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2248 | 2 |
| #2214 | 2 |
| #2213 | 2 |
| #2170 | 2 |
| #1373 | 2 |
| #1358 | 2 |
| #1357 | 2 |