Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
359 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C4 | C23×A4 |
| Fitting | C22×#131 | C3 |
| socle | C23 | C4×A4 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | #41 | C23×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#131⋊C3 | C2 | direct (N × H) |
| 3. C22×#131 | C3 | semidirect |
| 8. #131⋊C3 | C22 | direct (N × H) |
| 11. C2×#131 | C2×C3 | semidirect |
| 28. #131 | C22×C3 | semidirect |
| 97. C22 | #131⋊C3 | direct (N × H) |
| 106. C2 | C2×#131⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41⋊C3 | 2 |
| C2×#131⋊C3 | 2 |
| C22×#131 | 3 |
| C22×C3×C8 | 4 |