Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
118 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #12311 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7530 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #7092 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #3293 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #2213 | — | 2 | 1 | no | — |
| 6 | #1709 | C22 | 1 | 1 | yes | characteristic |
| 7 | #1408 | — | 2 | 1 | no | — |
| 8 | #1367 | — | 2 | 1 | no | — |
| 9 | #1358 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #1335 | — | 2 | 1 | no | — |
| 11 | #578 | — | 4 | 1 | no | — |
| 12 | #509 | — | 4 | 1 | no | — |
| 13 | #494 | C2×C4 | 1 | 1 | yes | characteristic |
| 14 | #493 | D4 | 1 | 1 | yes | characteristic |
| 15 | #486 | — | 4 | 1 | no | — |
| 16 | #473 | — | 4 | 1 | no | — |
| 17 | #456 | — | 4 | 1 | no | — |
| 18 | #443 | — | 2 | 1 | no | — |
| 19 | #364 | — | 2 | 1 | no | — |
| 20 | C4⋊C16 | — | 4 | 1 | no | — |
| 21 | #361 | — | 4 | 1 | no | — |
| 22 | #353 | — | 4 | 1 | no | — |
| 23 | C82 | D4 | 1 | 1 | yes | characteristic |
| 24 | 2−1+4 | — | 4 | 1 | no | — |
| 25 | #137 | — | 4 | 2 | no | — |
| 26 | #137 | — | 8 | 2 | no | — |
| 27 | #135 | — | 2 | 1 | no | — |
| 28 | #135 | — | 8 | 1 | no | — |
| 29 | C2×Q16 | — | 4 | 1 | no | — |
| 30 | C2×Q16 | — | 8 | 1 | no | — |
| 31 | C2×D8 | — | 4 | 1 | no | — |
| 32 | #131 | — | 4 | 1 | no | — |
| 33 | C4⋊Q8 | — | 2 | 1 | no | — |
| 34 | #127 | — | 2 | 1 | no | — |
| 35 | SD32 | — | 8 | 1 | no | — |
| 36 | D16 | — | 8 | 1 | no | — |
| 37 | M32 | — | 4 | 1 | no | — |
| 38 | C2×C16 | — | 8 | 1 | no | — |
| 39 | #108 | — | 2 | 1 | no | — |
| 40 | #104 | — | 4 | 1 | no | — |
| 41 | #102 | — | 8 | 1 | no | — |
| 42 | #102 | — | 8 | 1 | no | — |
| 43 | #101 | — | 4 | 1 | no | — |
| 44 | #99 | — | 8 | 1 | no | — |
| 45 | C4×C8 | — | 2 | 1 | no | — |
| 46 | C4×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | #41 | — | 4 | 1 | no | — |
| 48 | #41 | — | 8 | 2 | no | — |
| 49 | #41 | — | 16 | 1 | no | — |
| 50 | C2×Q8 | — | 4 | 1 | no | — |
| 51 | C2×Q8 | — | 8 | 2 | no | — |
| 52 | C2×D4 | — | 4 | 1 | no | — |
| 53 | C2×D4 | — | 8 | 1 | no | — |
| 54 | Q16 | — | 2 | 1 | no | — |
| 55 | Q16 | — | 8 | 1 | no | — |
| 56 | Q16 | — | 8 | 1 | no | — |
| 57 | Q16 | — | 8 | 2 | no | — |
| 58 | SD16 | — | 4 | 1 | no | — |
| 59 | SD16 | — | 8 | 2 | no | — |
| 60 | D8 | — | 2 | 1 | no | — |
| 61 | D8 | — | 8 | 1 | no | — |
| 62 | D8 | — | 8 | 1 | no | — |
| 63 | M16 | — | 4 | 1 | no | — |
| 64 | M16 | — | 8 | 1 | no | — |
| 65 | C2×C8 | — | 2 | 1 | no | — |
| 66 | C2×C8 | — | 2 | 1 | no | — |
| 67 | C2×C8 | — | 2 | 1 | no | — |
| 68 | C2×C8 | — | 8 | 1 | no | — |
| 69 | C4⋊C4 | — | 8 | 1 | no | — |
| 70 | C22⋊C4 | — | 16 | 1 | no | — |
| 71 | C42 | #99 | 1 | 1 | yes | characteristic |
| 72 | C16 | — | 8 | 1 | no | — |
| 73 | C16 | — | 16 | 1 | no | — |
| 74 | C23 | — | 8 | 1 | no | — |
| 75 | Q8 | — | 4 | 1 | no | — |
| 76 | Q8 | — | 8 | 2 | no | — |
| 77 | Q8 | — | 16 | 1 | no | — |
| 78 | D4 | — | 4 | 1 | no | — |
| 79 | D4 | — | 8 | 1 | no | — |
| 80 | D4 | — | 16 | 1 | no | — |
| 81 | D4 | — | 16 | 1 | no | — |
| 82 | D4 | — | 16 | 1 | no | — |
| 83 | D4 | — | 16 | 1 | no | — |
| 84 | C2×C4 | — | 2 | 1 | no | — |
| 85 | C2×C4 | — | 8 | 1 | no | — |
| 86 | C2×C4 | — | 8 | 1 | no | — |
| 87 | C2×C4 | — | 16 | 2 | no | — |
| 88 | C2×C4 | — | 32 | 1 | no | — |
| 89 | C2×C4 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 90 | C8 | — | 2 | 1 | no | — |
| 91 | C8 | — | 2 | 1 | no | — |
| 92 | C8 | — | 4 | 1 | no | — |
| 93 | C8 | — | 4 | 1 | no | — |
| 94 | C8 | — | 8 | 1 | no | — |
| 95 | C22 | — | 8 | 1 | no | — |
| 96 | C22 | — | 16 | 1 | no | — |
| 97 | C22 | — | 32 | 1 | no | — |
| 98 | C22 | #1402 | 1 | 1 | yes | characteristic |
| 99 | C4 | — | 2 | 1 | no | — |
| 100 | C4 | — | 2 | 1 | no | — |
| 101 | C4 | — | 2 | 1 | no | — |
| 102 | C4 | — | 8 | 1 | no | — |
| 103 | C4 | — | 16 | 1 | no | — |
| 104 | C4 | — | 64 | 1 | no | — |
| 105 | C2 | — | 2 | 1 | no | — |
| 106 | C2 | — | 16 | 1 | no | — |
| 107 | C2 | — | 32 | 1 | no | — |
| 108 | C2 | #7515 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7515 |
| commutator | #494 | C2×C4 |
| Frattini, cosocle | #1709 | C22 |
| prime radical | #12311 | C2 |
| LCS | G ▷ #494 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1709 ◁ G | |
| LECS | G ▷ #1709 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #7530 | C2 | semidirect |
| 3. #7092 | C2 | semidirect |
| 4. #3293 | C4 | semidirect |
| 9. #1358 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12311 | 2 |
| #7530 | 2 |
| #7092 | 2 |