Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
992 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 142 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×C109 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C109 |
| Frattini, cosocle | C22×D4 | C23×C109 |
| prime radical | C2×H(3,𝔽4)×C109 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C23×C109 ◁ C22×D4×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#354×C109 | C2 | semidirect |
| 3. C2×AΣL(1,𝔽16)×C109 | C2 | semidirect |
| 4. C109×#1889 | C2 | semidirect |
| 5. C109×#1885 | C2 | semidirect |
| 8. H(3,𝔽4)×C109 | C4 | semidirect |
| 68. C24×C109 | C22⋊C4 | semidirect |
| 75. C2×Q8×C109 | C22⋊C4 | semidirect |
| 111. C22⋊C4×C109 | C22⋊C4 | semidirect |
| 115. C42×C109 | C22⋊C4 | semidirect |
| 185. #12763 | C109 | direct (N × H) |
| 194. C2×#354 | C2×C109 | semidirect |
| 195. C2×AΣL(1,𝔽16) | C2×C109 | semidirect |
| 196. #1889 | C2×C109 | semidirect |
| 197. #1885 | C2×C109 | semidirect |
| 198. C109 | #12763 | direct (N × H) |
| 201. H(3,𝔽4) | C4×C109 | semidirect |
| 261. C24 | C22⋊C4×C109 | semidirect |
| 268. C2×Q8 | C22⋊C4×C109 | semidirect |
| 304. C22⋊C4 | C22⋊C4×C109 | semidirect |
| 308. C42 | C22⋊C4×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽4)×C109 | 2 |
| C2×#354×C109 | 2 |
| C2×AΣL(1,𝔽16)×C109 | 2 |
| C109×#1889 | 2 |
| C109×#1885 | 2 |
| #12763 | 109 |