Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
224 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 224 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 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | #102 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C16 | C2 | semidirect |
| 3. C2×C3×#108 | C2 | semidirect |
| 4. C3×#359 | C2 | direct (N × H) |
| 5. C2×#359 | C3 | direct (N × H) |
| 18. C2×C3×C16 | C22 | semidirect |
| 20. C3×#108 | C22 | semidirect |
| 22. C22×C16 | C2×C3 | semidirect |
| 23. C2×#108 | C2×C3 | semidirect |
| 24. #359 | C2×C3 | direct (N × H) |
| 62. C2×C16 | C22×C3 | semidirect |
| 64. #108 | C22×C3 | semidirect |
| 148. C2×C3 | #359 | direct (N × H) |
| 161. C3 | C2×#359 | direct (N × H) |
| 165. C2 | C3×#359 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#135 | 2 |
| C22×C3×C16 | 2 |
| C2×C3×#108 | 2 |
| C3×#359 | 2 |
| C2×#359 | 3 |