Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×C8×Q8 | C2 |
| prime radical | C2×#239 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | C8×Q8 | D3 |
| hypocenter | C3 | #2991 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C8×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C8×Q8 | C2 | semidirect |
| 3. Q8×C3⋊C8 | C2 | semidirect |
| 4. #4838 | C2 | semidirect |
| 5. #4837 | C2 | semidirect |
| 6. #4688 | C2 | semidirect |
| 7. #4685 | C2 | semidirect |
| 27. C8×Q8 | D3 | semidirect |
| 67. C3×Q8 | M16 | semidirect |
| 113. Q8 | ΓL(1,𝔽25) | semidirect |
| 144. C3 | #2991 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#239 | 2 |
| C3×C8×Q8 | 2 |
| Q8×C3⋊C8 | 2 |
| #4838 | 2 |
| #4837 | 2 |
| #4688 | 2 |
| #4685 | 2 |
| #2991 | 3 |