Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
266 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 104 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 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ 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×#423 | C2 | semidirect |
| 3. C3×C4×M16 | C2 | semidirect |
| 4. C3×Q8⋊C8 | C2 | semidirect |
| 5. C3×D4⋊C8 | C2 | semidirect |
| 6. #1489 | C3 | direct (N × H) |
| 25. #423 | C2×C3 | semidirect |
| 26. C4×M16 | C2×C3 | semidirect |
| 27. Q8⋊C8 | C2×C3 | semidirect |
| 28. D4⋊C8 | C2×C3 | semidirect |
| 78. C3×Q8 | M16 | semidirect |
| 81. C3×D4 | M16 | semidirect |
| 152. Q8 | C3×M16 | semidirect |
| 155. D4 | C3×M16 | semidirect |
| 199. C3 | #1489 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×#41 | 2 |
| C3×#423 | 2 |
| C3×C4×M16 | 2 |
| C3×Q8⋊C8 | 2 |
| C3×D4⋊C8 | 2 |
| #1489 | 3 |