Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
164 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 87 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×C8 | C22×D3 |
| commutator, socle | C22×C3 | C22×C8 |
| Fitting | C3×C8×D4 | C2 |
| prime radical | C2×C3×D4 | C8 |
| cosocle | C2×C3×C8 | C23 |
| hypercenter | C8×D4 | D3 |
| hypocenter | C3 | #2168 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ C8×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#812 | C2 | semidirect |
| 3. C22×C3⋊C16 | C2 | semidirect |
| 4. #4549 | C2 | semidirect |
| 5. #4549 | C2 | semidirect |
| 6. C4⋊(C3⋊C16) | C2 | semidirect |
| 7. #4461 | C2 | semidirect |
| 47. C3⋊C16 | D4 | semidirect |
| 102. C22×C3 | M32 | semidirect |
| 137. C22 | #812 | semidirect |
| 145. C3 | #2168 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C8×D4 | 2 |
| C2×#812 | 2 |
| C22×C3⋊C16 | 2 |
| #4549 | 2 |
| C4⋊(C3⋊C16) | 2 |
| #4461 | 2 |
| #2168 | 3 |