Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
312 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C22×C3×D4 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#431 | C2 | semidirect |
| 3. C3×#323 | C2 | semidirect |
| 4. #1460 | C3 | direct (N × H) |
| 16. #431 | C2×C3 | semidirect |
| 17. #323 | C2×C3 | semidirect |
| 21. C2×C3×D4 | C8 | semidirect |
| 67. C2×D4 | C3×C8 | semidirect |
| 136. C3 | #1460 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×D4 | 2 |
| C3×#431 | 2 |
| C3×#323 | 2 |
| #1460 | 3 |