Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
480 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#441 | C2 | direct (N × H) |
| 7. C2×#441 | C3 | direct (N × H) |
| 9. C2×C3×Q16 | C4 | semidirect |
| 24. #441 | C2×C3 | direct (N × H) |
| 36. C3×Q16 | C2×C4 | semidirect |
| 48. C2×Q16 | C3×C4 | semidirect |
| 85. Q16 | C2×C3×C4 | semidirect |
| 121. C2×C3 | #441 | direct (N × H) |
| 133. C3 | C2×#441 | direct (N × H) |
| 134. C2 | C3×#441 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q16 | 2 |
| C2×C3×C4×Q8 | 2 |
| C3×#441 | 2 |
| C2×C3×#106 | 2 |
| C2×C3×#103 | 2 |
| C2×C3×#97 | 2 |
| C2×#441 | 3 |