Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
200 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ 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. #3287 | C2 | semidirect |
| 2. #2246 | C2 | semidirect |
| 3. #2207 | C2 | semidirect |
| 4. #1667 | C2 | semidirect |
| 5. #1361 | C2 | semidirect |
| 6. #1332 | C2 | semidirect |
| 7. D4⋊C16 | C2 | semidirect |
| 71. Q16 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3287 | 2 |
| #2246 | 2 |
| #2207 | 2 |
| #1667 | 2 |
| #1361 | 2 |
| #1332 | 2 |
| D4⋊C16 | 2 |