Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
872 conjugacy classes of subgroups; only the 313 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 313 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D97 |
| commutator | C22×C97 | C23×C4 |
| Fitting, prime radical | C22×#41×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| hypercenter | C22×#41 | D97 |
| hypocenter | C97 | #2341 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×Dic97 | C2 | semidirect |
| 2. #20260638 | C2 | semidirect |
| 3. #20260595 | C2 | semidirect |
| 4. C4×C22⋊Dic97 | C2 | semidirect |
| 14. C2×#41×C97 | C4 | semidirect |
| 40. Q8×C97 | C22⋊C4 | semidirect |
| 52. C2×Dic97 | #41 | semidirect |
| 74. C97 | #2341 | semidirect |
| 81. C2×#41 | Dic97 | semidirect |
| 114. Q8 | C22⋊Dic97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×Dic97 | 2 |
| #20260638 | 2 |
| #20260595 | 2 |
| C4×C22⋊Dic97 | 2 |
| C22×#41×C97 | 2 |
| #2341 | 97 |