Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
230 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #29230 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13271 | C2 | 1 | 2 | yes | maximal |
| 3 | #9861 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7273 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7182 | C2 | 1 | 2 | yes | maximal |
| 6 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#364 | C22 | 1 | 2 | yes | — |
| 8 | #2102 | — | 2 | 2 | no | — |
| 9 | #1953 | — | 2 | 2 | no | — |
| 10 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 11 | #1833 | — | 2 | 2 | no | — |
| 12 | #1772 | C22 | 1 | 2 | yes | — |
| 13 | #1493 | — | 2 | 2 | no | — |
| 14 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1389 | — | 2 | 2 | no | — |
| 16 | #1384 | — | 2 | 2 | no | — |
| 17 | #533 | — | 2 | 1 | no | — |
| 18 | C2×M32 | — | 2 | 2 | no | — |
| 19 | #500 | — | 2 | 1 | no | — |
| 20 | #499 | — | 2 | 2 | no | — |
| 21 | #498 | Q8 | 1 | 2 | yes | — |
| 22 | #430 | — | 2 | 2 | no | — |
| 23 | C2×#108 | — | 2 | 2 | no | — |
| 24 | C2×#108 | D4 | 1 | 2 | yes | — |
| 25 | #427 | — | 2 | 2 | no | — |
| 26 | #427 | D4 | 1 | 2 | yes | — |
| 27 | #423 | D4 | 1 | 2 | yes | — |
| 28 | #405 | — | 2 | 2 | no | — |
| 29 | C2×C4×C8 | — | 2 | 1 | no | — |
| 30 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 31 | #364 | — | 2 | 2 | no | — |
| 32 | #364 | — | 2 | 2 | no | — |
| 33 | #344 | — | 2 | 4 | no | — |
| 34 | #339 | — | 2 | 2 | no | — |
| 35 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 36 | Q8⋊C8 | — | 4 | 4 | no | — |
| 37 | #322 | — | 2 | 2 | no | — |
| 38 | C2×M16 | — | 2 | 2 | no | — |
| 39 | C2×M16 | — | 2 | 2 | no | — |
| 40 | C22×C8 | — | 4 | 1 | no | — |
| 41 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 42 | C4⋊Q8 | — | 2 | 2 | no | — |
| 43 | #125 | — | 4 | 1 | no | — |
| 44 | C22⋊Q8 | — | 4 | 2 | no | — |
| 45 | C4×Q8 | — | 4 | 2 | no | — |
| 46 | #117 | — | 2 | 2 | no | — |
| 47 | #117 | — | 4 | 1 | no | — |
| 48 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 49 | M32 | — | 4 | 2 | no | — |
| 50 | M32 | — | 4 | 2 | no | — |
| 51 | C2×C16 | — | 4 | 2 | no | — |
| 52 | #108 | — | 2 | 2 | no | — |
| 53 | #108 | — | 2 | 2 | no | — |
| 54 | #108 | — | 4 | 2 | no | — |
| 55 | #108 | — | 4 | 2 | no | — |
| 56 | #108 | — | 8 | 2 | no | — |
| 57 | #107 | — | 2 | 2 | no | — |
| 58 | #107 | — | 4 | 2 | no | — |
| 59 | #106 | — | 2 | 4 | no | — |
| 60 | #106 | — | 2 | 2 | no | — |
| 61 | C4⋊C8 | — | 4 | 2 | no | — |
| 62 | #97 | — | 4 | 2 | no | — |
| 63 | C4×C8 | — | 2 | 2 | no | — |
| 64 | C4×C8 | — | 4 | 2 | no | — |
| 65 | C4×C8 | — | 4 | 1 | no | — |
| 66 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 67 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 68 | C4×C8 | #41 | 1 | 2 | yes | — |
| 69 | C2×Q8 | — | 4 | 2 | no | — |
| 70 | C22×C4 | — | 2 | 1 | no | — |
| 71 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 72 | M16 | — | 4 | 2 | no | — |
| 73 | M16 | — | 4 | 4 | no | — |
| 74 | M16 | — | 4 | 2 | no | — |
| 75 | C2×C8 | — | 4 | 2 | no | — |
| 76 | C2×C8 | — | 4 | 2 | no | — |
| 77 | C2×C8 | — | 4 | 2 | no | — |
| 78 | C2×C8 | — | 8 | 2 | no | — |
| 79 | C2×C8 | C22⋊Q8 | 1 | 2 | yes | — |
| 80 | C2×C8 | #124 | 1 | 2 | yes | — |
| 81 | C2×C8 | #135 | 1 | 2 | yes | — |
| 82 | C2×C8 | #135 | 1 | 2 | yes | — |
| 83 | C2×C8 | #137 | 1 | 2 | yes | — |
| 84 | C2×C8 | #137 | 1 | 2 | yes | — |
| 85 | C4⋊C4 | — | 4 | 2 | no | — |
| 86 | C4⋊C4 | — | 4 | 2 | no | — |
| 87 | C4⋊C4 | — | 4 | 2 | no | — |
| 88 | C4⋊C4 | — | 8 | 2 | no | — |
| 89 | C22⋊C4 | — | 4 | 2 | no | — |
| 90 | C22⋊C4 | — | 8 | 1 | no | — |
| 91 | C42 | — | 2 | 1 | no | — |
| 92 | C42 | — | 4 | 2 | no | — |
| 93 | C42 | — | 8 | 1 | no | — |
| 94 | C42 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 95 | C42 | #124 | 1 | 1 | yes | characteristic |
| 96 | C16 | — | 8 | 2 | no | — |
| 97 | C23 | #454 | 1 | 1 | yes | characteristic |
| 98 | Q8 | — | 8 | 4 | no | — |
| 99 | C2×C4 | — | 2 | 1 | no | — |
| 100 | C2×C4 | — | 2 | 1 | no | — |
| 101 | C2×C4 | — | 4 | 2 | no | — |
| 102 | C2×C4 | — | 8 | 2 | no | — |
| 103 | C2×C4 | — | 8 | 2 | no | — |
| 104 | C2×C4 | #393 | 1 | 1 | yes | characteristic |
| 105 | C2×C4 | #452 | 1 | 2 | yes | — |
| 106 | C2×C4 | #452 | 1 | 2 | yes | — |
| 107 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 108 | C8 | — | 8 | 2 | no | — |
| 109 | C8 | — | 8 | 2 | no | — |
| 110 | C8 | — | 8 | 2 | no | — |
| 111 | C8 | #479 | 1 | 2 | yes | — |
| 112 | C8 | #479 | 1 | 2 | yes | — |
| 113 | C8 | #488 | 1 | 2 | yes | — |
| 114 | C8 | #491 | 1 | 2 | yes | — |
| 115 | C22 | — | 4 | 1 | no | — |
| 116 | C22 | #1638 | 1 | 2 | yes | — |
| 117 | C22 | #2027 | 1 | 1 | yes | characteristic |
| 118 | C4 | — | 2 | 2 | no | — |
| 119 | C4 | — | 4 | 1 | no | — |
| 120 | C4 | — | 16 | 2 | no | — |
| 121 | C4 | — | 16 | 2 | no | — |
| 122 | C4 | #1630 | 1 | 1 | yes | characteristic |
| 123 | C4 | #1634 | 1 | 1 | yes | characteristic |
| 124 | C4 | #2026 | 1 | 2 | yes | — |
| 125 | C2 | — | 2 | 1 | no | — |
| 126 | C2 | — | 2 | 2 | no | — |
| 127 | C2 | #10352 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1634 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10352 |
| prime radical | C23 | #454 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #29230 | 2 |
| #13271 | 2 |
| #9861 | 2 |
| #7273 | 2 |
| #7182 | 2 |