Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
131 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×A4 |
| commutator, cosocle | C4×Q8 | C22×C3 |
| Frattini | C2×C4 | C22×A4 |
| Fitting | #3358 | C3 |
| hypercenter | #102 | A4 |
| hypocenter | Q8 | C3×D8 |
| LCS | G ▷ C4×Q8 ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊(#41⋊C3) | C2 | semidirect |
| 2. D4×SL(2,𝔽3) | C2 | semidirect |
| 3. C8×SL(2,𝔽3) | C2 | semidirect |
| 4. #3358 | C3 | semidirect |
| 9. #550 | C2×C3 | semidirect |
| 14. C8×Q8 | C2×C3 | semidirect |
| 43. SL(2,𝔽3) | D8 | semidirect |
| 86. D4 | #41⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊(#41⋊C3) | 2 |
| D4×SL(2,𝔽3) | 2 |
| C8×SL(2,𝔽3) | 2 |
| #3358 | 3 |
| C3×#102 | 4 |