Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
174 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 116 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×C16 | C22 |
| commutator | C2 | C22×C3×C16 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C2×C3×D4 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C16 | C2 | semidirect |
| 3. C3×C4⋊C16 | C2 | semidirect |
| 4. C3×C22⋊C16 | C2 | semidirect |
| 5. C3×C4×C16 | C2 | semidirect |
| 6. D4×C16 | C3 | direct (N × H) |
| 16. C22×C16 | C2×C3 | semidirect |
| 18. C4⋊C16 | C2×C3 | semidirect |
| 19. C22⋊C16 | C2×C3 | semidirect |
| 20. C4×C16 | C2×C3 | semidirect |
| 33. C3×C16 | D4 | direct (N × H) |
| 44. C3×D4 | C16 | direct (N × H) |
| 67. C16 | C3×D4 | direct (N × H) |
| 70. C22×C3 | C2×C16 | semidirect |
| 74. C3×C4 | C2×C16 | semidirect |
| 78. D4 | C3×C16 | direct (N × H) |
| 95. C22 | C2×C3×C16 | semidirect |
| 99. C4 | C2×C3×C16 | semidirect |
| 102. C3 | D4×C16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C16 | 2 |
| C3×C8×D4 | 2 |
| C3×C4⋊C16 | 2 |
| C3×C22⋊C16 | 2 |
| C3×C4×C16 | 2 |
| D4×C16 | 3 |