Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
754 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 754 conjugacy classes of subgroups — too many to draw. Only its 170 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 | #22248 | C2 | 1 | 4 | yes | maximal |
| 2 | #21691 | C2 | 1 | 2 | yes | maximal |
| 3 | #10686 | C2 | 1 | 4 | yes | maximal |
| 4 | #10309 | C2 | 1 | 4 | yes | maximal |
| 5 | #8134 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#500 | C22 | 1 | 4 | yes | — |
| 7 | C2×#494 | C22 | 1 | 2 | yes | — |
| 8 | C2×#441 | — | 2 | 4 | no | — |
| 9 | C4×C4⋊Q8 | C22 | 1 | 4 | yes | — |
| 10 | #1990 | C22 | 1 | 8 | yes | — |
| 11 | #1971 | — | 2 | 4 | no | — |
| 12 | #1956 | C22 | 1 | 4 | yes | — |
| 13 | #1921 | — | 2 | 16 | no | — |
| 14 | #1844 | C22 | 1 | 2 | yes | — |
| 15 | #1834 | C22 | 1 | 4 | yes | — |
| 16 | #1744 | C22 | 1 | 2 | yes | — |
| 17 | C4×#97 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1273 | C22 | 1 | 4 | yes | — |
| 19 | C22×Q16 | — | 2 | 4 | no | — |
| 20 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 21 | C2×C4×Q8 | — | 2 | 4 | no | — |
| 22 | #500 | — | 2 | 16 | no | — |
| 23 | #494 | — | 2 | 8 | no | — |
| 24 | #441 | — | 4 | 16 | no | — |
| 25 | C2×#107 | — | 2 | 8 | no | — |
| 26 | C2×#106 | — | 2 | 4 | no | — |
| 27 | C2×#103 | — | 2 | 8 | no | — |
| 28 | C2×#97 | D4 | 1 | 4 | yes | — |
| 29 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 30 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 31 | #391 | — | 2 | 4 | no | — |
| 32 | #389 | — | 2 | 8 | no | — |
| 33 | #384 | C23 | 1 | 4 | yes | — |
| 34 | C4×C4⋊C4 | — | 2 | 8 | no | — |
| 35 | C43 | C23 | 1 | 1 | yes | characteristic |
| 36 | #340 | — | 2 | 8 | no | — |
| 37 | #340 | — | 2 | 16 | no | — |
| 38 | #336 | D4 | 1 | 8 | yes | — |
| 39 | C22×Q8 | — | 2 | 4 | no | — |
| 40 | C2×Q16 | — | 2 | 16 | no | — |
| 41 | C2×Q16 | — | 4 | 8 | no | — |
| 42 | C2×Q16 | — | 4 | 8 | no | — |
| 43 | C22×C8 | C2×D4 | 1 | 8 | yes | — |
| 44 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 45 | C4⋊Q8 | — | 2 | 8 | no | — |
| 46 | C4⋊Q8 | — | 2 | 4 | no | — |
| 47 | C4⋊Q8 | #41 | 1 | 8 | yes | — |
| 48 | C4×Q8 | — | 4 | 16 | no | — |
| 49 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 50 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 51 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 52 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 53 | C2×C42 | — | 4 | 4 | no | — |
| 54 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 55 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 56 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 57 | #107 | — | 2 | 32 | no | — |
| 58 | #106 | — | 4 | 8 | no | — |
| 59 | #103 | — | 4 | 16 | no | — |
| 60 | #97 | — | 2 | 8 | no | — |
| 61 | C4×C8 | — | 2 | 4 | no | — |
| 62 | C4×C8 | — | 2 | 8 | no | — |
| 63 | #95 | — | 4 | 8 | no | — |
| 64 | C2×Q8 | — | 2 | 8 | no | — |
| 65 | C2×Q8 | — | 4 | 4 | no | — |
| 66 | C2×Q8 | — | 4 | 8 | no | — |
| 67 | C2×Q8 | — | 4 | 8 | no | — |
| 68 | C22×C4 | — | 4 | 4 | no | — |
| 69 | C22×C4 | — | 4 | 4 | no | — |
| 70 | C22×C4 | #127 | 1 | 4 | yes | — |
| 71 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 72 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 73 | Q16 | — | 4 | 32 | no | — |
| 74 | C2×C8 | — | 2 | 4 | no | — |
| 75 | C2×C8 | — | 2 | 4 | no | — |
| 76 | C2×C8 | — | 2 | 8 | no | — |
| 77 | C2×C8 | — | 2 | 16 | no | — |
| 78 | C2×C8 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 79 | C4⋊C4 | — | 2 | 16 | no | — |
| 80 | C4⋊C4 | — | 4 | 8 | no | — |
| 81 | C4⋊C4 | — | 4 | 8 | no | — |
| 82 | C4⋊C4 | — | 4 | 16 | no | — |
| 83 | C4⋊C4 | — | 8 | 8 | no | — |
| 84 | C42 | — | 2 | 4 | no | — |
| 85 | C42 | — | 2 | 8 | no | — |
| 86 | C42 | — | 8 | 8 | no | — |
| 87 | C42 | C2×#41 | 1 | 4 | yes | — |
| 88 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 89 | Q8 | — | 4 | 16 | no | — |
| 90 | Q8 | — | 8 | 8 | no | — |
| 91 | C2×C4 | — | 2 | 4 | no | — |
| 92 | C2×C4 | — | 2 | 8 | no | — |
| 93 | C2×C4 | — | 4 | 8 | no | — |
| 94 | C2×C4 | — | 8 | 4 | no | — |
| 95 | C2×C4 | — | 8 | 4 | no | — |
| 96 | C2×C4 | — | 8 | 4 | no | — |
| 97 | C2×C4 | — | 8 | 8 | no | — |
| 98 | C2×C4 | #525 | 1 | 2 | yes | — |
| 99 | C2×C4 | #525 | 1 | 4 | yes | — |
| 100 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 101 | C2×C4 | #578 | 1 | 8 | yes | — |
| 102 | C8 | — | 2 | 8 | no | — |
| 103 | C8 | — | 4 | 8 | no | — |
| 104 | C22 | #2397 | 1 | 1 | yes | characteristic |
| 105 | C22 | #3151 | 1 | 2 | yes | — |
| 106 | C22 | #3156 | 1 | 4 | yes | — |
| 107 | C4 | — | 4 | 4 | no | — |
| 108 | C4 | — | 8 | 8 | no | — |
| 109 | C4 | — | 16 | 4 | no | — |
| 110 | C4 | #3109 | 1 | 4 | yes | — |
| 111 | C4 | #3127 | 1 | 8 | yes | — |
| 112 | C2 | #19731 | 1 | 4 | yes | — |
| 113 | C2 | #23802 | 1 | 1 | yes | characteristic |
| 114 | C2 | #23814 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22248 | 2 |
| #21691 | 2 |
| #10686 | 2 |
| #10309 | 2 |
| #8134 | 2 |