Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
366 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1634 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10056 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #19604 | C2 | semidirect |
| 4. #13271 | C2 | semidirect |
| 5. #11956 | C2 | semidirect |
| 7. #7179 | C2 | semidirect |
| 88. C2×Q16 | Q16 | semidirect |
| 90. C2×D8 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33637 | 2 |
| #29154 | 2 |
| #19604 | 2 |
| #13271 | 2 |
| #11956 | 2 |
| C2×#1335 | 2 |
| #7179 | 2 |