Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
301 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3169 | C2 | semidirect |
| 2. #2252 | C2 | semidirect |
| 3. #2246 | C2 | semidirect |
| 4. #2240 | C2 | semidirect |
| 5. #2215 | C2 | semidirect |
| 6. C4×D16 | C2 | semidirect |
| 7. C2×C4×C16 | C2 | semidirect |
| 76. C16 | #41 | semidirect |
| 90. C2×C4 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3169 | 2 |
| #2252 | 2 |
| #2246 | 2 |
| #2240 | 2 |
| #2215 | 2 |
| C4×D16 | 2 |
| C2×C4×C16 | 2 |