Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
168 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C211 | D4 |
| commutator | C4 | C22×C8×C211 |
| Frattini, cosocle | C2×C8 | C23×C211 |
| socle | C22×C211 | C4⋊C8 |
| prime radical | C23×C211 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C211 ◁ C22×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #364×C211 | C2 | direct (N × H) |
| 51. C2×C211 | #364 | direct (N × H) |
| 53. C211 | C2×#364 | direct (N × H) |
| 54. C2×#364 | C211 | direct (N × H) |
| 57. #364 | C2×C211 | direct (N × H) |
| 105. C2 | #364×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M32×C211 | 2 |
| C2×C4×C8×C211 | 2 |
| #364×C211 | 2 |
| C2×#364 | 211 |