Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
271 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 50 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 | #20021 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #12515 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #12425 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #7096 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7069 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #3396 | — | 2 | 2 | no | — |
| 7 | #2246 | — | 2 | 2 | no | — |
| 8 | #2207 | — | 2 | 2 | no | — |
| 9 | #1712 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1712 | C22 | 1 | 1 | yes | characteristic |
| 11 | #1668 | — | 2 | 2 | no | — |
| 12 | #1620 | — | 2 | 2 | no | — |
| 13 | C8×Q16 | C22 | 1 | 2 | yes | — |
| 14 | #1370 | C22 | 1 | 2 | yes | — |
| 15 | #1361 | — | 2 | 2 | no | — |
| 16 | C8×C16 | C22 | 1 | 1 | yes | characteristic |
| 17 | #559 | — | 2 | 2 | no | — |
| 18 | C2×SD32 | — | 4 | 2 | no | — |
| 19 | C2×D16 | — | 4 | 4 | no | — |
| 20 | #495 | — | 4 | 2 | no | — |
| 21 | #493 | — | 2 | 2 | no | — |
| 22 | #493 | — | 2 | 2 | no | — |
| 23 | #493 | — | 2 | 1 | no | — |
| 24 | #493 | D4 | 1 | 2 | yes | — |
| 25 | #463 | — | 4 | 4 | no | — |
| 26 | #460 | — | 2 | 4 | no | — |
| 27 | C8×Q8 | — | 2 | 2 | no | — |
| 28 | C4×Q16 | D4 | 1 | 2 | yes | — |
| 29 | C4×D8 | — | 4 | 2 | no | — |
| 30 | C4⋊C16 | — | 2 | 2 | no | — |
| 31 | #357 | — | 4 | 2 | no | — |
| 32 | C4×C16 | — | 2 | 1 | no | — |
| 33 | #335 | D4 | 1 | 2 | yes | — |
| 34 | #331 | — | 2 | 2 | no | — |
| 35 | Q8⋊C8 | — | 2 | 2 | no | — |
| 36 | C82 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×Q16 | D8 | 1 | 2 | yes | — |
| 38 | C2×SD16 | — | 4 | 4 | no | — |
| 39 | C2×D8 | — | 4 | 2 | no | — |
| 40 | C2×D8 | — | 4 | 2 | no | — |
| 41 | C2×D8 | — | 4 | 2 | no | — |
| 42 | C2×D8 | — | 4 | 2 | no | — |
| 43 | C2×D8 | — | 4 | 2 | no | — |
| 44 | C2×D8 | — | 4 | 2 | no | — |
| 45 | #127 | — | 2 | 2 | no | — |
| 46 | #127 | — | 4 | 1 | no | — |
| 47 | #124 | — | 4 | 4 | no | — |
| 48 | #123 | — | 8 | 2 | no | — |
| 49 | C4×Q8 | — | 2 | 2 | no | — |
| 50 | C4×D4 | — | 4 | 4 | no | — |
| 51 | SD32 | — | 8 | 4 | no | — |
| 52 | D16 | — | 8 | 8 | no | — |
| 53 | C2×C16 | — | 2 | 2 | no | — |
| 54 | C2×C16 | — | 4 | 2 | no | — |
| 55 | #107 | D8 | 1 | 2 | yes | — |
| 56 | C4⋊C8 | — | 2 | 2 | no | — |
| 57 | C4⋊C8 | — | 4 | 2 | no | — |
| 58 | #103 | — | 2 | 2 | no | — |
| 59 | #102 | — | 4 | 4 | no | — |
| 60 | C4×C8 | — | 4 | 2 | no | — |
| 61 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 63 | C2×Q8 | — | 2 | 2 | no | — |
| 64 | C2×D4 | — | 4 | 4 | no | — |
| 65 | C2×D4 | — | 4 | 2 | no | — |
| 66 | C2×D4 | — | 8 | 2 | no | — |
| 67 | C2×D4 | — | 8 | 1 | no | — |
| 68 | C22×C4 | — | 8 | 2 | no | — |
| 69 | Q16 | — | 2 | 2 | no | — |
| 70 | Q16 | — | 2 | 2 | no | — |
| 71 | SD16 | — | 16 | 4 | no | — |
| 72 | D8 | — | 8 | 4 | no | — |
| 73 | D8 | — | 8 | 4 | no | — |
| 74 | D8 | — | 8 | 2 | no | — |
| 75 | D8 | — | 8 | 2 | no | — |
| 76 | D8 | — | 8 | 2 | no | — |
| 77 | D8 | — | 8 | 2 | no | — |
| 78 | D8 | — | 16 | 2 | no | — |
| 79 | D8 | — | 16 | 2 | no | — |
| 80 | C2×C8 | — | 2 | 1 | no | — |
| 81 | C2×C8 | — | 4 | 2 | no | — |
| 82 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 83 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 84 | C4⋊C4 | — | 2 | 2 | no | — |
| 85 | C4⋊C4 | — | 4 | 2 | no | — |
| 86 | C22⋊C4 | — | 8 | 4 | no | — |
| 87 | C22⋊C4 | — | 8 | 2 | no | — |
| 88 | C42 | — | 4 | 2 | no | — |
| 89 | C42 | #120 | 1 | 1 | yes | characteristic |
| 90 | C16 | — | 2 | 4 | no | — |
| 91 | C16 | — | 8 | 2 | no | — |
| 92 | C23 | — | 8 | 2 | no | — |
| 93 | C23 | — | 16 | 1 | no | — |
| 94 | Q8 | — | 4 | 2 | no | — |
| 95 | Q8 | — | 4 | 2 | no | — |
| 96 | D4 | — | 8 | 4 | no | — |
| 97 | D4 | — | 8 | 4 | no | — |
| 98 | D4 | — | 16 | 2 | no | — |
| 99 | D4 | — | 16 | 2 | no | — |
| 100 | D4 | — | 16 | 2 | no | — |
| 101 | D4 | — | 16 | 2 | no | — |
| 102 | C2×C4 | — | 4 | 2 | no | — |
| 103 | C2×C4 | — | 4 | 2 | no | — |
| 104 | C2×C4 | — | 32 | 2 | no | — |
| 105 | C2×C4 | #447 | 1 | 2 | yes | — |
| 106 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 107 | C8 | — | 2 | 2 | no | — |
| 108 | C8 | — | 2 | 2 | no | — |
| 109 | C8 | — | 8 | 2 | no | — |
| 110 | C8 | #508 | 1 | 2 | yes | — |
| 111 | C8 | #509 | 1 | 2 | yes | — |
| 112 | C22 | — | 16 | 2 | no | — |
| 113 | C22 | — | 16 | 4 | no | — |
| 114 | C22 | — | 32 | 1 | no | — |
| 115 | C22 | — | 32 | 2 | no | — |
| 116 | C22 | #1619 | 1 | 1 | yes | characteristic |
| 117 | C4 | — | 2 | 1 | no | — |
| 118 | C4 | — | 8 | 2 | no | — |
| 119 | C4 | — | 8 | 2 | no | — |
| 120 | C4 | #2186 | 1 | 2 | yes | — |
| 121 | C4 | #2192 | 1 | 2 | yes | — |
| 122 | C2 | — | 32 | 2 | no | — |
| 123 | C2 | — | 64 | 1 | no | — |
| 124 | C2 | #12310 | 1 | 1 | yes | characteristic |
| 125 | C2 | #12317 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | #12515 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20021 | C2 | semidirect |
| 3. #12425 | C2 | semidirect |
| 4. #7096 | C2 | semidirect |
| 5. #7069 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20021 | 2 |
| #12515 | 2 |
| #12425 | 2 |
| #7096 | 2 |
| #7069 | 2 |