Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
558 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 558 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 | #23814 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #17856 | C2 | 1 | 2 | yes | maximal |
| 3 | #9671 | C2 | 1 | 2 | yes | maximal |
| 4 | #9100 | C2 | 1 | 4 | yes | maximal |
| 5 | #9088 | C2 | 1 | 4 | yes | maximal |
| 6 | #8782 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #8141 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #2982 | C22 | 1 | 4 | yes | — |
| 9 | C2×#441 | C22 | 1 | 2 | yes | — |
| 10 | C2×C4×Q16 | C22 | 1 | 2 | yes | — |
| 11 | #2327 | C22 | 1 | 2 | yes | — |
| 12 | #2097 | C4 | 1 | 4 | yes | — |
| 13 | #2016 | C4 | 1 | 4 | yes | — |
| 14 | #1746 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1579 | — | 2 | 4 | no | — |
| 16 | C8×Q16 | — | 2 | 4 | no | — |
| 17 | #1564 | C22 | 1 | 2 | yes | — |
| 18 | #1479 | C22 | 1 | 4 | yes | — |
| 19 | #1479 | C22 | 1 | 4 | yes | — |
| 20 | #1461 | C22 | 1 | 4 | yes | — |
| 21 | #1452 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1281 | C22 | 1 | 2 | yes | — |
| 24 | #1275 | C22 | 1 | 2 | yes | — |
| 25 | #1274 | C22 | 1 | 2 | yes | — |
| 26 | #1274 | C22 | 1 | 2 | yes | — |
| 27 | C22×Q16 | C2×C4 | 1 | 2 | yes | — |
| 28 | C22×Q16 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 31 | C8×Q8 | — | 2 | 8 | no | — |
| 32 | C8×Q8 | — | 4 | 4 | no | — |
| 33 | #441 | — | 4 | 4 | no | — |
| 34 | C4×Q16 | — | 2 | 2 | no | — |
| 35 | C4×Q16 | — | 2 | 2 | no | — |
| 36 | C4×Q16 | — | 2 | 2 | no | — |
| 37 | C4×Q16 | — | 2 | 2 | no | — |
| 38 | #431 | — | 4 | 4 | no | — |
| 39 | #431 | C23 | 1 | 4 | yes | — |
| 40 | #431 | C23 | 1 | 2 | yes | — |
| 41 | #431 | C23 | 1 | 2 | yes | — |
| 42 | C2×#107 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×#106 | C2×C4 | 1 | 2 | yes | — |
| 44 | C2×#103 | C2×C4 | 1 | 4 | yes | — |
| 45 | C2×#103 | C2×C4 | 1 | 4 | yes | — |
| 46 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 48 | #391 | C2×C4 | 1 | 4 | yes | — |
| 49 | #387 | — | 4 | 4 | no | — |
| 50 | #382 | C2×C4 | 1 | 2 | yes | — |
| 51 | #376 | C23 | 1 | 1 | yes | characteristic |
| 52 | #336 | C2×C4 | 1 | 4 | yes | — |
| 53 | #335 | — | 2 | 2 | no | — |
| 54 | #334 | — | 2 | 2 | no | — |
| 55 | Q8⋊C8 | — | 2 | 8 | no | — |
| 56 | Q8⋊C8 | — | 4 | 4 | no | — |
| 57 | #324 | — | 4 | 4 | no | — |
| 58 | #322 | — | 2 | 2 | no | — |
| 59 | #322 | D4 | 1 | 2 | yes | — |
| 60 | C82 | D4 | 1 | 2 | yes | — |
| 61 | C22×Q8 | C22×C4 | 1 | 4 | yes | — |
| 62 | C2×Q16 | — | 2 | 8 | no | — |
| 63 | C2×Q16 | — | 2 | 8 | no | — |
| 64 | C2×Q16 | — | 2 | 2 | no | — |
| 65 | C2×Q16 | — | 2 | 2 | no | — |
| 66 | C2×Q16 | — | 4 | 2 | no | — |
| 67 | C22×C8 | C22×C4 | 1 | 2 | yes | — |
| 68 | C22×C8 | C22×C4 | 1 | 2 | yes | — |
| 69 | C4⋊Q8 | — | 2 | 8 | no | — |
| 70 | C4×Q8 | — | 2 | 4 | no | — |
| 71 | C4×Q8 | — | 2 | 4 | no | — |
| 72 | C4×Q8 | — | 2 | 4 | no | — |
| 73 | C4×Q8 | — | 2 | 4 | no | — |
| 74 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 75 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 76 | C2×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 77 | C2×C42 | — | 2 | 4 | no | — |
| 78 | C2×C42 | C22×C4 | 1 | 2 | yes | — |
| 79 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 80 | #107 | — | 2 | 2 | no | — |
| 81 | #107 | — | 2 | 2 | no | — |
| 82 | #106 | — | 4 | 2 | no | — |
| 83 | C4⋊C8 | — | 2 | 4 | no | — |
| 84 | C4⋊C8 | — | 2 | 2 | no | — |
| 85 | C4⋊C8 | — | 2 | 2 | no | — |
| 86 | C4⋊C8 | — | 4 | 8 | no | — |
| 87 | #103 | — | 2 | 4 | no | — |
| 88 | #103 | — | 2 | 4 | no | — |
| 89 | #103 | — | 4 | 4 | no | — |
| 90 | C22⋊C8 | — | 2 | 2 | no | — |
| 91 | C22⋊C8 | — | 2 | 2 | no | — |
| 92 | C22⋊C8 | — | 2 | 4 | no | — |
| 93 | #97 | — | 2 | 2 | no | — |
| 94 | C4×C8 | — | 2 | 2 | no | — |
| 95 | C4×C8 | — | 2 | 4 | no | — |
| 96 | C4×C8 | — | 4 | 8 | no | — |
| 97 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 98 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C4×C8 | #41 | 1 | 2 | yes | — |
| 101 | #95 | — | 2 | 2 | no | — |
| 102 | #95 | — | 4 | 2 | no | — |
| 103 | #95 | — | 4 | 4 | no | — |
| 104 | C2×Q8 | — | 2 | 4 | no | — |
| 105 | C2×Q8 | — | 2 | 4 | no | — |
| 106 | C2×Q8 | — | 2 | 4 | no | — |
| 107 | C2×Q8 | — | 2 | 4 | no | — |
| 108 | C2×Q8 | — | 2 | 4 | no | — |
| 109 | C2×Q8 | #131 | 1 | 8 | yes | — |
| 110 | C22×C4 | — | 2 | 4 | no | — |
| 111 | C22×C4 | — | 2 | 2 | no | — |
| 112 | C22×C4 | — | 2 | 4 | no | — |
| 113 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 114 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 115 | Q16 | — | 2 | 8 | no | — |
| 116 | Q16 | — | 2 | 8 | no | — |
| 117 | Q16 | — | 4 | 8 | no | — |
| 118 | C2×C8 | — | 2 | 4 | no | — |
| 119 | C2×C8 | — | 2 | 2 | no | — |
| 120 | C2×C8 | — | 2 | 4 | no | — |
| 121 | C2×C8 | — | 2 | 4 | no | — |
| 122 | C2×C8 | — | 4 | 4 | no | — |
| 123 | C2×C8 | — | 4 | 2 | no | — |
| 124 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 125 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 126 | C4⋊C4 | — | 2 | 4 | no | — |
| 127 | C4⋊C4 | — | 2 | 4 | no | — |
| 128 | C4⋊C4 | — | 4 | 8 | no | — |
| 129 | C4⋊C4 | — | 4 | 4 | no | — |
| 130 | C4⋊C4 | — | 4 | 4 | no | — |
| 131 | C42 | — | 2 | 4 | no | — |
| 132 | C42 | — | 4 | 4 | no | — |
| 133 | C42 | — | 4 | 4 | no | — |
| 134 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 136 | C42 | H(5,𝔽2) | 1 | 2 | yes | — |
| 137 | C23 | #518 | 1 | 1 | yes | characteristic |
| 138 | Q8 | — | 2 | 8 | no | — |
| 139 | Q8 | — | 2 | 8 | no | — |
| 140 | Q8 | — | 4 | 4 | no | — |
| 141 | Q8 | — | 4 | 4 | no | — |
| 142 | C2×C4 | — | 2 | 8 | no | — |
| 143 | C2×C4 | — | 4 | 2 | no | — |
| 144 | C2×C4 | — | 4 | 4 | no | — |
| 145 | C2×C4 | — | 4 | 4 | no | — |
| 146 | C2×C4 | — | 4 | 4 | no | — |
| 147 | C2×C4 | — | 4 | 4 | no | — |
| 148 | C2×C4 | — | 4 | 2 | no | — |
| 149 | C2×C4 | — | 8 | 4 | no | — |
| 150 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 151 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 152 | C2×C4 | #535 | 1 | 2 | yes | — |
| 153 | C2×C4 | #540 | 1 | 2 | yes | — |
| 154 | C2×C4 | C2×#131 | 1 | 4 | yes | — |
| 155 | C8 | — | 2 | 2 | no | — |
| 156 | C8 | — | 2 | 2 | no | — |
| 157 | C8 | — | 4 | 4 | no | — |
| 158 | C8 | — | 4 | 2 | no | — |
| 159 | C8 | — | 8 | 4 | no | — |
| 160 | C8 | — | 8 | 2 | no | — |
| 161 | C22 | — | 2 | 2 | no | — |
| 162 | C22 | #2328 | 1 | 1 | yes | characteristic |
| 163 | C22 | #2977 | 1 | 2 | yes | — |
| 164 | C4 | — | 2 | 2 | no | — |
| 165 | C4 | — | 2 | 2 | no | — |
| 166 | C4 | — | 4 | 8 | no | — |
| 167 | C4 | — | 8 | 2 | no | — |
| 168 | C4 | — | 8 | 4 | no | — |
| 169 | C4 | #2929 | 1 | 2 | yes | — |
| 170 | C4 | #2977 | 1 | 2 | yes | — |
| 171 | C2 | — | 2 | 2 | no | — |
| 172 | C2 | #22261 | 1 | 1 | yes | characteristic |
| 173 | C2 | #22285 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #518 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ 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 |
|---|---|
| #23814 | 2 |
| #17856 | 2 |
| #9671 | 2 |
| #9100 | 2 |
| #9088 | 2 |
| #8782 | 2 |
| #8141 | 2 |