Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
138 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 26 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 | #10585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7367 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #7266 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #2845 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×#347 | — | 2 | 1 | no | — |
| 6 | #1847 | — | 2 | 1 | no | — |
| 7 | #1836 | C22 | 1 | 1 | yes | characteristic |
| 8 | #1325 | — | 2 | 1 | no | — |
| 9 | #1295 | — | 2 | 1 | no | — |
| 10 | #1295 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×M32 | — | 4 | 1 | no | — |
| 12 | C2×#107 | — | 4 | 1 | no | — |
| 13 | C2×#106 | — | 4 | 1 | no | — |
| 14 | C2×#97 | — | 2 | 1 | no | — |
| 15 | C2×#97 | D4 | 1 | 1 | yes | characteristic |
| 16 | #398 | — | 4 | 1 | no | — |
| 17 | #398 | — | 4 | 1 | no | — |
| 18 | #398 | — | 4 | 1 | no | — |
| 19 | #384 | D4 | 1 | 1 | yes | characteristic |
| 20 | #382 | — | 2 | 1 | no | — |
| 21 | #382 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | #347 | — | 2 | 2 | no | — |
| 23 | #347 | — | 4 | 1 | no | — |
| 24 | #340 | — | 4 | 1 | no | — |
| 25 | #324 | — | 4 | 1 | no | — |
| 26 | C22×C8 | — | 2 | 1 | no | — |
| 27 | C22×C8 | — | 4 | 1 | no | — |
| 28 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 29 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 30 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 31 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 32 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 33 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 34 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 35 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 36 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 37 | M32 | — | 4 | 2 | no | — |
| 38 | M32 | — | 4 | 2 | no | — |
| 39 | C2×C16 | — | 4 | 2 | no | — |
| 40 | #107 | — | 8 | 2 | no | — |
| 41 | #106 | — | 8 | 2 | no | — |
| 42 | C22⋊C8 | — | 8 | 1 | no | — |
| 43 | #97 | — | 2 | 2 | no | — |
| 44 | #97 | — | 2 | 1 | no | — |
| 45 | #97 | — | 4 | 1 | no | — |
| 46 | #97 | SD16 | 1 | 1 | yes | characteristic |
| 47 | #97 | Q16 | 1 | 1 | yes | characteristic |
| 48 | #95 | — | 4 | 1 | no | — |
| 49 | #95 | — | 4 | 1 | no | — |
| 50 | #95 | — | 4 | 1 | no | — |
| 51 | #95 | — | 4 | 1 | no | — |
| 52 | C22×C4 | — | 2 | 1 | no | — |
| 53 | C22×C4 | — | 4 | 1 | no | — |
| 54 | C22×C4 | — | 4 | 1 | no | — |
| 55 | C22×C4 | — | 4 | 1 | no | — |
| 56 | C22×C4 | — | 4 | 1 | no | — |
| 57 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 58 | C2×C8 | — | 2 | 1 | no | — |
| 59 | C2×C8 | — | 2 | 1 | no | — |
| 60 | C2×C8 | — | 4 | 2 | no | — |
| 61 | C2×C8 | — | 4 | 1 | no | — |
| 62 | C2×C8 | — | 4 | 1 | no | — |
| 63 | C2×C8 | — | 4 | 2 | no | — |
| 64 | C2×C8 | — | 4 | 2 | no | — |
| 65 | C2×C8 | — | 16 | 1 | no | — |
| 66 | C4⋊C4 | — | 8 | 1 | no | — |
| 67 | C4⋊C4 | — | 8 | 1 | no | — |
| 68 | C4⋊C4 | — | 8 | 1 | no | — |
| 69 | C4⋊C4 | — | 8 | 1 | no | — |
| 70 | C4⋊C4 | — | 8 | 1 | no | — |
| 71 | C4⋊C4 | — | 8 | 1 | no | — |
| 72 | C4⋊C4 | — | 16 | 1 | no | — |
| 73 | C4⋊C4 | — | 16 | 1 | no | — |
| 74 | C4⋊C4 | — | 16 | 1 | no | — |
| 75 | C42 | — | 2 | 1 | no | — |
| 76 | C42 | #103 | 1 | 1 | yes | characteristic |
| 77 | C42 | #104 | 1 | 1 | yes | characteristic |
| 78 | C16 | — | 4 | 4 | no | — |
| 79 | C23 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 80 | C2×C4 | — | 2 | 1 | no | — |
| 81 | C2×C4 | — | 2 | 1 | no | — |
| 82 | C2×C4 | — | 2 | 1 | no | — |
| 83 | C2×C4 | — | 2 | 1 | no | — |
| 84 | C2×C4 | — | 4 | 1 | no | — |
| 85 | C2×C4 | — | 4 | 1 | no | — |
| 86 | C2×C4 | — | 8 | 1 | no | — |
| 87 | C2×C4 | — | 8 | 1 | no | — |
| 88 | C2×C4 | — | 8 | 1 | no | — |
| 89 | C2×C4 | — | 8 | 1 | no | — |
| 90 | C2×C4 | — | 8 | 1 | no | — |
| 91 | C2×C4 | — | 8 | 1 | no | — |
| 92 | C2×C4 | — | 8 | 1 | no | — |
| 93 | C2×C4 | — | 8 | 1 | no | — |
| 94 | C2×C4 | — | 8 | 1 | no | — |
| 95 | C2×C4 | — | 8 | 1 | no | — |
| 96 | C2×C4 | — | 16 | 1 | no | — |
| 97 | C2×C4 | #328 | 1 | 1 | yes | characteristic |
| 98 | C2×C4 | #356 | 1 | 1 | yes | characteristic |
| 99 | C8 | — | 4 | 1 | no | — |
| 100 | C8 | — | 4 | 1 | no | — |
| 101 | C8 | — | 4 | 4 | no | — |
| 102 | C8 | — | 32 | 1 | no | — |
| 103 | C22 | — | 2 | 1 | no | — |
| 104 | C22 | — | 2 | 1 | no | — |
| 105 | C22 | #1354 | 1 | 1 | yes | characteristic |
| 106 | C22 | #1403 | 1 | 1 | yes | characteristic |
| 107 | C22 | #1404 | 1 | 1 | yes | characteristic |
| 108 | C4 | — | 2 | 1 | no | — |
| 109 | C4 | — | 2 | 1 | no | — |
| 110 | C4 | — | 4 | 1 | no | — |
| 111 | C4 | — | 4 | 1 | no | — |
| 112 | C4 | — | 16 | 1 | no | — |
| 113 | C4 | — | 16 | 1 | no | — |
| 114 | C4 | — | 16 | 1 | no | — |
| 115 | C4 | — | 16 | 1 | no | — |
| 116 | C2 | — | 2 | 1 | no | — |
| 117 | C2 | — | 2 | 1 | no | — |
| 118 | C2 | #7442 | 1 | 1 | yes | characteristic |
| 119 | C2 | #7523 | 1 | 1 | yes | characteristic |
| 120 | C2 | #7526 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1404 |
| commutator | #382 | C2×C4 |
| Frattini, cosocle | #1836 | C22 |
| prime radical | C23 | AΣL(1,𝔽16) |
| LCS | G ▷ #382 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1836 ◁ G | |
| LECS | G ▷ #1836 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10585 | 2 |
| #7367 | 2 |
| #7266 | 2 |