Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
436 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C211 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C211 |
| prime radical | C22×D4×C211 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊Q8×C211 | C2 | semidirect |
| 3. #489×C211 | C2 | direct (N × H) |
| 4. C2×#103×C211 | C2 | semidirect |
| 6. C2×#97×C211 | C2 | semidirect |
| 12. C4⋊Q8×C211 | C22 | semidirect |
| 20. #103×C211 | C22 | semidirect |
| 22. #97×C211 | C22 | semidirect |
| 83. C2×C211 | #489 | direct (N × H) |
| 87. C211 | C2×#489 | direct (N × H) |
| 88. C2×#489 | C211 | direct (N × H) |
| 89. C2×C4⋊Q8 | C2×C211 | semidirect |
| 91. #489 | C2×C211 | direct (N × H) |
| 92. C2×#103 | C2×C211 | semidirect |
| 94. C2×#97 | C2×C211 | semidirect |
| 100. C4⋊Q8 | C22×C211 | semidirect |
| 108. #103 | C22×C211 | semidirect |
| 110. #97 | C22×C211 | semidirect |
| 171. C2 | #489×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊Q8×C211 | 2 |
| C2×#124×C211 | 2 |
| #489×C211 | 2 |
| C2×#103×C211 | 2 |
| C2×#102×C211 | 2 |
| C2×#97×C211 | 2 |
| C2×#489 | 211 |