Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
555 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72216364 | C2 | 1 | 3 | yes | maximal |
| 2 | #72216272 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C322 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #151732 | — | 2 | 6 | no | — |
| 5 | #151726 | — | 2 | 6 | no | — |
| 6 | #123404 | C22 | 1 | 3 | yes | — |
| 7 | #123403 | C22 | 1 | 1 | yes | characteristic |
| 8 | C16×C32 | C22 | 1 | 3 | yes | — |
| 9 | #13733 | — | 4 | 12 | no | — |
| 10 | #13727 | — | 4 | 12 | no | — |
| 11 | #12521 | — | 2 | 6 | no | — |
| 12 | #12516 | — | 2 | 3 | no | — |
| 13 | #12515 | — | 2 | 3 | no | — |
| 14 | C8×C32 | D4 | 1 | 6 | yes | — |
| 15 | C162 | C23 | 1 | 1 | yes | characteristic |
| 16 | C2×SD64 | — | 8 | 24 | no | — |
| 17 | #2252 | — | 4 | 12 | no | — |
| 18 | #2247 | — | 4 | 6 | no | — |
| 19 | #2246 | — | 4 | 6 | no | — |
| 20 | #1713 | — | 4 | 3 | no | — |
| 21 | #1712 | — | 4 | 1 | no | — |
| 22 | #1424 | — | 8 | 24 | no | — |
| 23 | C4×C32 | D8 | 1 | 12 | yes | — |
| 24 | C8×C16 | C2×D4 | 1 | 3 | yes | — |
| 25 | C2×Q32 | — | 8 | 12 | no | — |
| 26 | C2×D16 | — | 8 | 12 | no | — |
| 27 | #500 | — | 8 | 6 | no | — |
| 28 | #494 | — | 8 | 3 | no | — |
| 29 | #493 | — | 8 | 3 | no | — |
| 30 | SD64 | — | 16 | 48 | no | — |
| 31 | C2×C32 | D16 | 1 | 24 | yes | — |
| 32 | #366 | — | 8 | 24 | no | — |
| 33 | C4×C16 | C2×D8 | 1 | 6 | yes | — |
| 34 | C82 | #127 | 1 | 1 | yes | characteristic |
| 35 | C2×Q16 | — | 16 | 6 | no | — |
| 36 | C2×D8 | — | 16 | 6 | no | — |
| 37 | C4⋊Q8 | — | 16 | 3 | no | — |
| 38 | #127 | — | 16 | 1 | no | — |
| 39 | Q32 | — | 16 | 24 | no | — |
| 40 | D16 | — | 16 | 24 | no | — |
| 41 | C2×C16 | C2×D16 | 1 | 12 | yes | — |
| 42 | #107 | — | 16 | 12 | no | — |
| 43 | C4×C8 | #493 | 1 | 3 | yes | — |
| 44 | C32 | SD64 | 1 | 48 | yes | complemented |
| 45 | C2×Q8 | — | 32 | 3 | no | — |
| 46 | C2×D4 | — | 32 | 3 | no | — |
| 47 | Q16 | — | 32 | 12 | no | — |
| 48 | D8 | — | 32 | 12 | no | — |
| 49 | C2×C8 | #2246 | 1 | 6 | yes | — |
| 50 | C4⋊C4 | — | 32 | 6 | no | — |
| 51 | C42 | #1712 | 1 | 1 | yes | characteristic |
| 52 | C16 | C2×SD64 | 1 | 24 | yes | — |
| 53 | C23 | — | 64 | 1 | no | — |
| 54 | Q8 | — | 64 | 6 | no | — |
| 55 | D4 | — | 64 | 6 | no | — |
| 56 | C2×C4 | — | 64 | 3 | no | — |
| 57 | C2×C4 | #12515 | 1 | 3 | yes | — |
| 58 | C8 | #13727 | 1 | 12 | yes | — |
| 59 | C22 | — | 128 | 3 | no | — |
| 60 | C22 | #123403 | 1 | 1 | yes | characteristic |
| 61 | C4 | — | 128 | 3 | no | — |
| 62 | C4 | #151726 | 1 | 6 | yes | — |
| 63 | C2 | — | 256 | 1 | no | — |
| 64 | C2 | #72216364 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #123403 |
| commutator, Frattini, cosocle | C162 | C23 |
| prime radical | #123403 | C22 |
| LCS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ C162 ◁ G | |
| LECS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #72216272 | C2 | semidirect |
| 3. C322 | C2 | semidirect |
| 44. C32 | SD64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72216364 | 2 |
| #72216272 | 2 |
| C322 | 2 |