Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
500 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 229 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×D97 |
| commutator, socle | C22×C97 | C23×C4 |
| Fitting, prime radical | #532×C97 | C2 |
| cosocle | C2×C4×C97 | C24 |
| hypercenter | #532 | D97 |
| hypocenter | C97 | #2931 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690817 | C2 | semidirect |
| 2. #20260813 | C2 | semidirect |
| 3. D4×#11715895 | C2 | semidirect |
| 4. #20260759 | C2 | semidirect |
| 5. #20260702 | C2 | semidirect |
| 6. C2×C4×#11715895 | C2 | semidirect |
| 29. C2×#11715895 | D4 | semidirect |
| 76. C4×C97 | #131 | semidirect |
| 84. C97 | #2931 | semidirect |
| 160. C4 | #15690817 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690817 | 2 |
| #20260813 | 2 |
| D4×#11715895 | 2 |
| #20260759 | 2 |
| #20260702 | 2 |
| C2×C4×#11715895 | 2 |
| #532×C97 | 2 |
| #2931 | 97 |