Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
650 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 186 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 | #33313 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33289 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #30205 | C2 | 1 | 2 | yes | maximal |
| 4 | #30205 | C2 | 1 | 2 | yes | maximal |
| 5 | #9834 | C2 | 1 | 2 | yes | maximal |
| 6 | #9831 | C2 | 1 | 2 | yes | maximal |
| 7 | #9828 | C2 | 1 | 4 | yes | maximal |
| 8 | #7843 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#498 | C22 | 1 | 2 | yes | — |
| 12 | #2868 | C22 | 1 | 2 | yes | — |
| 13 | #2862 | C22 | 1 | 2 | yes | — |
| 14 | #2072 | — | 2 | 4 | no | — |
| 15 | #2072 | — | 2 | 4 | no | — |
| 16 | #2024 | — | 2 | 4 | no | — |
| 17 | #2024 | — | 2 | 4 | no | — |
| 18 | #1990 | — | 2 | 4 | no | — |
| 19 | #1990 | C22 | 1 | 4 | yes | — |
| 20 | #1990 | C22 | 1 | 4 | yes | — |
| 21 | #1956 | C22 | 1 | 4 | yes | — |
| 22 | #1956 | C22 | 1 | 4 | yes | — |
| 23 | #1846 | — | 2 | 4 | no | — |
| 24 | #1834 | C22 | 1 | 2 | yes | — |
| 25 | #1834 | C22 | 1 | 2 | yes | — |
| 26 | #1744 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1744 | C22 | 1 | 1 | yes | characteristic |
| 28 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#322 | C22 | 1 | 2 | yes | — |
| 30 | C2×#322 | C22 | 1 | 2 | yes | — |
| 31 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 32 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 33 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 34 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 35 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 36 | #500 | — | 2 | 4 | no | — |
| 37 | #500 | — | 2 | 4 | no | — |
| 38 | #498 | — | 2 | 8 | no | — |
| 39 | C2×#107 | — | 2 | 2 | no | — |
| 40 | C2×#107 | — | 2 | 2 | no | — |
| 41 | C2×#106 | — | 2 | 4 | no | — |
| 42 | C2×#103 | — | 2 | 8 | no | — |
| 43 | C2×#103 | — | 2 | 8 | no | — |
| 44 | C2×C4×C8 | D4 | 1 | 4 | yes | — |
| 45 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 46 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 47 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 49 | #391 | — | 2 | 2 | no | — |
| 50 | #391 | — | 2 | 2 | no | — |
| 51 | #391 | — | 2 | 2 | no | — |
| 52 | #391 | — | 2 | 2 | no | — |
| 53 | #384 | — | 2 | 2 | no | — |
| 54 | #384 | — | 2 | 2 | no | — |
| 55 | #384 | — | 2 | 2 | no | — |
| 56 | #384 | C23 | 1 | 4 | yes | — |
| 57 | #382 | — | 2 | 2 | no | — |
| 58 | #382 | — | 2 | 2 | no | — |
| 59 | C43 | C23 | 1 | 1 | yes | characteristic |
| 60 | #340 | — | 2 | 8 | no | — |
| 61 | #340 | — | 2 | 8 | no | — |
| 62 | #340 | — | 2 | 4 | no | — |
| 63 | #340 | — | 2 | 4 | no | — |
| 64 | #336 | D4 | 1 | 2 | yes | — |
| 65 | #336 | D4 | 1 | 4 | yes | — |
| 66 | #336 | D4 | 1 | 2 | yes | — |
| 67 | #322 | — | 2 | 4 | no | — |
| 68 | #322 | — | 2 | 4 | no | — |
| 69 | C22×Q8 | — | 2 | 2 | no | — |
| 70 | C22×Q8 | — | 2 | 2 | no | — |
| 71 | C22×Q8 | — | 4 | 2 | no | — |
| 72 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 73 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 74 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 75 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 76 | C4⋊Q8 | — | 2 | 4 | no | — |
| 77 | C4⋊Q8 | — | 2 | 4 | no | — |
| 78 | C4⋊Q8 | — | 2 | 4 | no | — |
| 79 | C4⋊Q8 | — | 2 | 4 | no | — |
| 80 | C4⋊Q8 | — | 4 | 8 | no | — |
| 81 | C4⋊Q8 | — | 8 | 4 | no | — |
| 82 | C4⋊Q8 | — | 8 | 4 | no | — |
| 83 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 84 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 85 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 86 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 87 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 88 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 89 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 90 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 91 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 92 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 93 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 95 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 97 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 98 | #107 | — | 4 | 4 | no | — |
| 99 | #107 | — | 4 | 4 | no | — |
| 100 | #106 | — | 4 | 8 | no | — |
| 101 | #103 | — | 8 | 8 | no | — |
| 102 | #103 | — | 8 | 8 | no | — |
| 103 | C4×C8 | — | 2 | 8 | no | — |
| 104 | C4×C8 | — | 2 | 2 | no | — |
| 105 | C4×C8 | — | 2 | 2 | no | — |
| 106 | C4×C8 | — | 2 | 2 | no | — |
| 107 | C4×C8 | — | 2 | 2 | no | — |
| 108 | C4×C8 | #41 | 1 | 4 | yes | — |
| 109 | C4×C8 | #41 | 1 | 4 | yes | — |
| 110 | #95 | — | 4 | 2 | no | — |
| 111 | #95 | — | 4 | 2 | no | — |
| 112 | #95 | — | 4 | 2 | no | — |
| 113 | #95 | — | 4 | 2 | no | — |
| 114 | C2×Q8 | — | 4 | 4 | no | — |
| 115 | C2×Q8 | — | 4 | 4 | no | — |
| 116 | C2×Q8 | — | 4 | 4 | no | — |
| 117 | C2×Q8 | — | 4 | 4 | no | — |
| 118 | C2×Q8 | — | 4 | 4 | no | — |
| 119 | C2×Q8 | — | 4 | 4 | no | — |
| 120 | C2×Q8 | — | 8 | 4 | no | — |
| 121 | C2×Q8 | — | 8 | 4 | no | — |
| 122 | C2×Q8 | — | 8 | 4 | no | — |
| 123 | C22×C4 | — | 4 | 2 | no | — |
| 124 | C22×C4 | — | 4 | 2 | no | — |
| 125 | C22×C4 | — | 4 | 4 | no | — |
| 126 | C22×C4 | #127 | 1 | 2 | yes | — |
| 127 | C22×C4 | #127 | 1 | 2 | yes | — |
| 128 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 131 | C2×C8 | — | 2 | 4 | no | — |
| 132 | C2×C8 | — | 2 | 4 | no | — |
| 133 | C2×C8 | — | 2 | 4 | no | — |
| 134 | C2×C8 | — | 2 | 4 | no | — |
| 135 | C2×C8 | — | 2 | 4 | no | — |
| 136 | C2×C8 | — | 2 | 8 | no | — |
| 137 | C2×C8 | — | 2 | 4 | no | — |
| 138 | C2×C8 | #124 | 1 | 4 | yes | — |
| 139 | C2×C8 | #124 | 1 | 4 | yes | — |
| 140 | C4⋊C4 | — | 4 | 2 | no | — |
| 141 | C4⋊C4 | — | 4 | 2 | no | — |
| 142 | C4⋊C4 | — | 4 | 2 | no | — |
| 143 | C4⋊C4 | — | 4 | 2 | no | — |
| 144 | C4⋊C4 | — | 4 | 4 | no | — |
| 145 | C4⋊C4 | — | 4 | 4 | no | — |
| 146 | C4⋊C4 | — | 4 | 8 | no | — |
| 147 | C4⋊C4 | — | 4 | 8 | no | — |
| 148 | C4⋊C4 | — | 4 | 8 | no | — |
| 149 | C4⋊C4 | — | 8 | 4 | no | — |
| 150 | C4⋊C4 | — | 8 | 4 | no | — |
| 151 | C4⋊C4 | — | 8 | 4 | no | — |
| 152 | C42 | — | 2 | 2 | no | — |
| 153 | C42 | — | 2 | 4 | no | — |
| 154 | C42 | — | 2 | 2 | no | — |
| 155 | C42 | #137 | 1 | 8 | yes | — |
| 156 | C42 | C2×#41 | 1 | 2 | yes | — |
| 157 | C42 | C2×#41 | 1 | 2 | yes | — |
| 158 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 159 | Q8 | — | 8 | 8 | no | — |
| 160 | Q8 | — | 8 | 8 | no | — |
| 161 | Q8 | — | 16 | 8 | no | — |
| 162 | C2×C4 | — | 2 | 2 | no | — |
| 163 | C2×C4 | — | 2 | 2 | no | — |
| 164 | C2×C4 | — | 2 | 2 | no | — |
| 165 | C2×C4 | — | 2 | 2 | no | — |
| 166 | C2×C4 | — | 8 | 2 | no | — |
| 167 | C2×C4 | — | 8 | 2 | no | — |
| 168 | C2×C4 | — | 8 | 4 | no | — |
| 169 | C2×C4 | — | 8 | 4 | no | — |
| 170 | C2×C4 | — | 8 | 2 | no | — |
| 171 | C2×C4 | — | 8 | 2 | no | — |
| 172 | C2×C4 | — | 8 | 4 | no | — |
| 173 | C2×C4 | — | 8 | 2 | no | — |
| 174 | C2×C4 | — | 8 | 2 | no | — |
| 175 | C2×C4 | #497 | 1 | 4 | yes | — |
| 176 | C2×C4 | #497 | 1 | 4 | yes | — |
| 177 | C2×C4 | C2×#124 | 1 | 1 | yes | characteristic |
| 178 | C2×C4 | C2×#124 | 1 | 1 | yes | characteristic |
| 179 | C2×C4 | #532 | 1 | 2 | yes | — |
| 180 | C2×C4 | #532 | 1 | 2 | yes | — |
| 181 | C2×C4 | C2×#137 | 1 | 4 | yes | — |
| 182 | C2×C4 | #576 | 1 | 2 | yes | — |
| 183 | C2×C4 | #577 | 1 | 4 | yes | — |
| 184 | C2×C4 | #578 | 1 | 2 | yes | — |
| 185 | C8 | — | 2 | 4 | no | — |
| 186 | C8 | — | 2 | 4 | no | — |
| 187 | C8 | — | 4 | 4 | no | — |
| 188 | C8 | — | 4 | 4 | no | — |
| 189 | C22 | #2858 | 1 | 1 | yes | characteristic |
| 190 | C22 | C2×#497 | 1 | 2 | yes | — |
| 191 | C22 | #3154 | 1 | 2 | yes | — |
| 192 | C22 | #3155 | 1 | 1 | yes | characteristic |
| 193 | C22 | #3156 | 1 | 1 | yes | characteristic |
| 194 | C4 | — | 2 | 4 | no | — |
| 195 | C4 | — | 2 | 4 | no | — |
| 196 | C4 | — | 16 | 2 | no | — |
| 197 | C4 | — | 16 | 2 | no | — |
| 198 | C4 | — | 16 | 4 | no | — |
| 199 | C4 | #3139 | 1 | 2 | yes | — |
| 200 | C4 | #3139 | 1 | 2 | yes | — |
| 201 | C4 | #3173 | 1 | 4 | yes | — |
| 202 | C4 | #3175 | 1 | 4 | yes | — |
| 203 | C2 | #19705 | 1 | 2 | yes | — |
| 204 | C2 | #19706 | 1 | 2 | yes | — |
| 205 | C2 | #33236 | 1 | 1 | yes | characteristic |
| 206 | C2 | #33245 | 1 | 1 | yes | characteristic |
| 207 | C2 | #33247 | 1 | 1 | yes | characteristic |
| 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 |
|---|---|
| #33313 | 2 |
| #33289 | 2 |
| #30205 | 2 |
| #9834 | 2 |
| #9831 | 2 |
| #9828 | 2 |
| #7843 | 2 |