Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
354 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×A4 |
| commutator, cosocle | C22×C4 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | C22×#135 | C3 |
| socle | C23 | C2×C3×D4 |
| hypercenter | #135 | A4 |
| hypocenter | C22 | C3×#135 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×#41 | C2 | semidirect |
| 2. A4×Q16 | C2 | semidirect |
| 3. A4×SD16 | C2 | semidirect |
| 4. A4×D8 | C2 | semidirect |
| 5. C2×C8×A4 | C2 | semidirect |
| 6. C22×#135 | C3 | semidirect |
| 15. C22×#41 | C2×C3 | semidirect |
| 22. C22×Q16 | C2×C3 | semidirect |
| 23. C22×SD16 | C2×C3 | semidirect |
| 24. C22×D8 | C2×C3 | semidirect |
| 25. C23×C8 | C2×C3 | semidirect |
| 57. #135 | A4 | direct (N × H) |
| 96. #41 | C2×A4 | semidirect |
| 121. Q16 | C2×A4 | semidirect |
| 126. SD16 | C2×A4 | semidirect |
| 130. D8 | C2×A4 | semidirect |
| 139. C2×C8 | C2×A4 | semidirect |
| 142. A4 | #135 | direct (N × H) |
| 204. C22 | C3×#135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#41 | 2 |
| A4×Q16 | 2 |
| A4×SD16 | 2 |
| A4×D8 | 2 |
| C2×C8×A4 | 2 |
| C22×#135 | 3 |
| C3×#135 | 4 |