Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
169 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 169 conjugacy classes of subgroups — too many to draw. Only its 24 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 | #10216 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7507 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #7115 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #3255 | C4 | 1 | 1 | yes | characteristic |
| 5 | #3235 | — | 2 | 1 | no | — |
| 6 | #2132 | — | 2 | 1 | no | — |
| 7 | #1963 | C22 | 1 | 1 | yes | characteristic |
| 8 | #1506 | — | 2 | 1 | no | — |
| 9 | #1307 | C4 | 1 | 1 | yes | characteristic |
| 10 | #1274 | — | 2 | 1 | no | — |
| 11 | #533 | — | 2 | 1 | no | — |
| 12 | #532 | D4 | 1 | 1 | yes | characteristic |
| 13 | #501 | — | 4 | 1 | no | — |
| 14 | #490 | — | 4 | 1 | no | — |
| 15 | #482 | — | 4 | 2 | no | — |
| 16 | #476 | — | 4 | 1 | no | — |
| 17 | #475 | — | 4 | 1 | no | — |
| 18 | #474 | — | 4 | 1 | no | — |
| 19 | #470 | — | 4 | 1 | no | — |
| 20 | #462 | — | 4 | 1 | no | — |
| 21 | #432 | — | 2 | 1 | no | — |
| 22 | #431 | — | 4 | 1 | no | — |
| 23 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×#104 | — | 4 | 1 | no | — |
| 25 | #413 | — | 4 | 1 | no | — |
| 26 | #376 | D4 | 1 | 1 | yes | characteristic |
| 27 | #356 | — | 4 | 1 | no | — |
| 28 | #355 | — | 2 | 2 | no | — |
| 29 | #328 | — | 4 | 2 | no | — |
| 30 | C2×#41 | — | 2 | 1 | no | — |
| 31 | C2×Q16 | — | 4 | 1 | no | — |
| 32 | C2×M16 | — | 2 | 1 | no | — |
| 33 | C2×M16 | — | 8 | 1 | no | — |
| 34 | C4⋊Q8 | — | 2 | 1 | no | — |
| 35 | C4⋊Q8 | — | 2 | 1 | no | — |
| 36 | C4⋊Q8 | D8 | 1 | 1 | yes | characteristic |
| 37 | #127 | SD16 | 1 | 1 | yes | characteristic |
| 38 | #125 | — | 4 | 1 | no | — |
| 39 | #124 | — | 2 | 1 | no | — |
| 40 | C22⋊Q8 | — | 4 | 1 | no | — |
| 41 | C22⋊Q8 | — | 4 | 1 | no | — |
| 42 | #121 | — | 4 | 1 | no | — |
| 43 | C4×Q8 | — | 4 | 1 | no | — |
| 44 | C4×Q8 | — | 4 | 1 | no | — |
| 45 | C4×D4 | — | 4 | 1 | no | — |
| 46 | #117 | — | 4 | 1 | no | — |
| 47 | C2×C42 | — | 2 | 1 | no | — |
| 48 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 49 | #107 | — | 4 | 2 | no | — |
| 50 | #106 | — | 4 | 2 | no | — |
| 51 | #106 | — | 4 | 1 | no | — |
| 52 | C4⋊C8 | — | 4 | 1 | no | — |
| 53 | C4⋊C8 | — | 4 | 1 | no | — |
| 54 | C4⋊C8 | — | 8 | 1 | no | — |
| 55 | #104 | — | 8 | 1 | no | — |
| 56 | #104 | — | 8 | 1 | no | — |
| 57 | #103 | — | 4 | 2 | no | — |
| 58 | #102 | — | 4 | 2 | no | — |
| 59 | #102 | — | 4 | 2 | no | — |
| 60 | #101 | — | 4 | 2 | no | — |
| 61 | #100 | — | 8 | 1 | no | — |
| 62 | C22⋊C8 | — | 2 | 2 | no | — |
| 63 | C22⋊C8 | — | 8 | 1 | no | — |
| 64 | #97 | — | 4 | 1 | no | — |
| 65 | C4×C8 | — | 4 | 2 | no | — |
| 66 | #95 | — | 4 | 1 | no | — |
| 67 | #41 | — | 8 | 1 | no | — |
| 68 | #41 | — | 8 | 1 | no | — |
| 69 | C2×Q8 | — | 2 | 1 | no | — |
| 70 | C2×Q8 | — | 4 | 1 | no | — |
| 71 | C2×Q8 | — | 4 | 1 | no | — |
| 72 | C2×D4 | — | 2 | 1 | no | — |
| 73 | C2×D4 | — | 4 | 1 | no | — |
| 74 | C2×D4 | — | 4 | 1 | no | — |
| 75 | C22×C4 | — | 2 | 1 | no | — |
| 76 | C22×C4 | — | 4 | 1 | no | — |
| 77 | C22×C4 | — | 4 | 1 | no | — |
| 78 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 79 | Q16 | — | 16 | 1 | no | — |
| 80 | M16 | — | 8 | 1 | no | — |
| 81 | M16 | — | 8 | 2 | no | — |
| 82 | M16 | — | 16 | 1 | no | — |
| 83 | C2×C8 | — | 4 | 2 | no | — |
| 84 | C2×C8 | — | 4 | 1 | no | — |
| 85 | C2×C8 | — | 8 | 2 | no | — |
| 86 | C2×C8 | — | 16 | 1 | no | — |
| 87 | C4⋊C4 | — | 4 | 1 | no | — |
| 88 | C4⋊C4 | — | 4 | 1 | no | — |
| 89 | C4⋊C4 | — | 4 | 1 | no | — |
| 90 | C4⋊C4 | — | 4 | 1 | no | — |
| 91 | C4⋊C4 | — | 4 | 1 | no | — |
| 92 | C4⋊C4 | — | 8 | 1 | no | — |
| 93 | C4⋊C4 | — | 8 | 1 | no | — |
| 94 | C22⋊C4 | — | 8 | 1 | no | — |
| 95 | C22⋊C4 | — | 8 | 1 | no | — |
| 96 | C42 | — | 2 | 1 | no | — |
| 97 | C42 | — | 2 | 1 | no | — |
| 98 | C42 | — | 2 | 1 | no | — |
| 99 | C42 | — | 4 | 1 | no | — |
| 100 | C42 | — | 8 | 1 | no | — |
| 101 | C42 | #102 | 1 | 1 | yes | characteristic |
| 102 | C42 | #104 | 1 | 1 | yes | characteristic |
| 103 | C23 | — | 4 | 1 | no | — |
| 104 | C23 | #353 | 1 | 1 | yes | characteristic |
| 105 | Q8 | — | 8 | 1 | no | — |
| 106 | Q8 | — | 8 | 1 | no | — |
| 107 | Q8 | — | 8 | 1 | no | — |
| 108 | Q8 | — | 16 | 1 | no | — |
| 109 | D4 | — | 8 | 1 | no | — |
| 110 | D4 | — | 8 | 2 | no | — |
| 111 | D4 | — | 16 | 1 | no | — |
| 112 | C2×C4 | — | 2 | 1 | no | — |
| 113 | C2×C4 | — | 2 | 1 | no | — |
| 114 | C2×C4 | — | 2 | 1 | no | — |
| 115 | C2×C4 | — | 2 | 1 | no | — |
| 116 | C2×C4 | — | 4 | 1 | no | — |
| 117 | C2×C4 | — | 4 | 1 | no | — |
| 118 | C2×C4 | — | 4 | 1 | no | — |
| 119 | C2×C4 | — | 8 | 1 | no | — |
| 120 | C2×C4 | — | 8 | 1 | no | — |
| 121 | C2×C4 | — | 8 | 1 | no | — |
| 122 | C2×C4 | — | 8 | 1 | no | — |
| 123 | C2×C4 | — | 8 | 1 | no | — |
| 124 | C2×C4 | — | 16 | 1 | no | — |
| 125 | C2×C4 | #327 | 1 | 1 | yes | characteristic |
| 126 | C2×C4 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 127 | C8 | — | 8 | 2 | no | — |
| 128 | C8 | — | 8 | 1 | no | — |
| 129 | C8 | — | 16 | 2 | no | — |
| 130 | C8 | — | 16 | 2 | no | — |
| 131 | C22 | — | 2 | 1 | no | — |
| 132 | C22 | — | 4 | 1 | no | — |
| 133 | C22 | — | 8 | 1 | no | — |
| 134 | C22 | — | 16 | 1 | no | — |
| 135 | C22 | #1343 | 1 | 1 | yes | characteristic |
| 136 | C4 | — | 2 | 1 | no | — |
| 137 | C4 | — | 4 | 1 | no | — |
| 138 | C4 | — | 4 | 1 | no | — |
| 139 | C4 | — | 8 | 1 | no | — |
| 140 | C4 | — | 8 | 1 | no | — |
| 141 | C4 | — | 8 | 1 | no | — |
| 142 | C4 | — | 16 | 1 | no | — |
| 143 | C4 | — | 16 | 1 | no | — |
| 144 | C4 | #1402 | 1 | 1 | yes | characteristic |
| 145 | C4 | #1404 | 1 | 1 | yes | characteristic |
| 146 | C2 | — | 2 | 1 | no | — |
| 147 | C2 | — | 4 | 1 | no | — |
| 148 | C2 | — | 16 | 1 | no | — |
| 149 | C2 | #7408 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7408 |
| commutator | #423 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| prime radical | #532 | D4 |
| LCS | G ▷ #423 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ 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 |
|---|---|
| #10216 | 2 |
| #7507 | 2 |
| #7115 | 2 |