Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
701 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 701 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 | C2×#2070 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#1747 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11312 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11283 | C2 | 1 | 3 | yes | maximal |
| 5 | #11182 | C2 | 1 | 3 | yes | maximal |
| 6 | #10942 | C2 | 1 | 3 | yes | maximal |
| 7 | #10898 | C2 | 1 | 3 | yes | maximal |
| 8 | C2×#430 | C22 | 1 | 3 | yes | — |
| 9 | C2×#419 | C22 | 1 | 3 | yes | — |
| 10 | C22×#103 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×#101 | C22 | 1 | 3 | yes | — |
| 12 | C2×#405 | C22 | 1 | 3 | yes | — |
| 13 | #2070 | — | 2 | 4 | no | — |
| 14 | #1747 | — | 2 | 4 | no | — |
| 15 | C2×#344 | C22 | 1 | 3 | yes | — |
| 16 | C2×#344 | C22 | 1 | 3 | yes | — |
| 17 | C2×#343 | C22 | 1 | 3 | yes | — |
| 18 | #1738 | C22 | 1 | 3 | yes | — |
| 19 | C2×#341 | C22 | 1 | 3 | yes | — |
| 20 | #1736 | C22 | 1 | 3 | yes | — |
| 21 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 23 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 24 | C22×M16 | C23 | 1 | 3 | yes | — |
| 25 | C22×M16 | C23 | 1 | 3 | yes | — |
| 26 | C22×M16 | C23 | 1 | 3 | yes | — |
| 27 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×#117 | C23 | 1 | 3 | yes | — |
| 29 | C22×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 30 | #430 | — | 2 | 6 | no | — |
| 31 | #430 | — | 2 | 6 | no | — |
| 32 | C2×#108 | — | 4 | 3 | no | — |
| 33 | #419 | — | 2 | 6 | no | — |
| 34 | #419 | — | 2 | 6 | no | — |
| 35 | C2×#103 | — | 2 | 3 | no | — |
| 36 | C2×#103 | — | 2 | 3 | no | — |
| 37 | C2×#103 | — | 4 | 3 | no | — |
| 38 | C2×#101 | — | 2 | 6 | no | — |
| 39 | C2×#101 | — | 2 | 6 | no | — |
| 40 | C2×#101 | — | 4 | 3 | no | — |
| 41 | #405 | — | 2 | 6 | no | — |
| 42 | #405 | — | 2 | 6 | no | — |
| 43 | C2×#97 | — | 2 | 3 | no | — |
| 44 | C2×C4×C8 | — | 2 | 3 | no | — |
| 45 | #344 | — | 2 | 6 | no | — |
| 46 | #344 | — | 2 | 6 | no | — |
| 47 | #343 | — | 4 | 3 | no | — |
| 48 | #341 | — | 2 | 3 | no | — |
| 49 | #341 | — | 2 | 3 | no | — |
| 50 | #341 | — | 4 | 3 | no | — |
| 51 | #340 | — | 4 | 3 | no | — |
| 52 | #338 | D4 | 1 | 4 | yes | — |
| 53 | H(3,ℤ/4) | — | 2 | 6 | no | — |
| 54 | C22×Q8 | — | 2 | 1 | no | — |
| 55 | C22×Q8 | — | 2 | 3 | no | — |
| 56 | C22×Q8 | — | 2 | 3 | no | — |
| 57 | C22×Q8 | — | 2 | 3 | no | — |
| 58 | C22×Q8 | — | 2 | 3 | no | — |
| 59 | C22×Q8 | #41 | 1 | 2 | yes | — |
| 60 | C23×C4 | — | 2 | 1 | no | — |
| 61 | C23×C4 | — | 2 | 1 | no | — |
| 62 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 63 | C2×M16 | — | 2 | 3 | no | — |
| 64 | C2×M16 | — | 2 | 3 | no | — |
| 65 | C2×M16 | — | 2 | 3 | no | — |
| 66 | C2×M16 | — | 2 | 3 | no | — |
| 67 | C2×M16 | — | 2 | 3 | no | — |
| 68 | C2×M16 | — | 2 | 3 | no | — |
| 69 | C2×M16 | — | 2 | 3 | no | — |
| 70 | C2×M16 | — | 2 | 3 | no | — |
| 71 | C2×M16 | — | 2 | 3 | no | — |
| 72 | C2×M16 | — | 2 | 3 | no | — |
| 73 | C2×M16 | — | 2 | 3 | no | — |
| 74 | C2×M16 | — | 2 | 3 | no | — |
| 75 | C2×M16 | — | 2 | 3 | no | — |
| 76 | C2×M16 | — | 2 | 3 | no | — |
| 77 | C2×M16 | — | 2 | 3 | no | — |
| 78 | C2×M16 | — | 2 | 3 | no | — |
| 79 | C2×M16 | C2×D4 | 1 | 6 | yes | — |
| 80 | C2×M16 | #41 | 1 | 6 | yes | — |
| 81 | C22×C8 | — | 2 | 1 | no | — |
| 82 | C22×C8 | — | 2 | 3 | no | — |
| 83 | C22×C8 | — | 2 | 3 | no | — |
| 84 | C22×C8 | — | 2 | 3 | no | — |
| 85 | C22×C8 | — | 2 | 3 | no | — |
| 86 | C22×C8 | #41 | 1 | 3 | yes | — |
| 87 | C22×C8 | #41 | 1 | 3 | yes | — |
| 88 | #117 | — | 2 | 3 | no | — |
| 89 | #117 | — | 2 | 3 | no | — |
| 90 | #117 | — | 2 | 3 | no | — |
| 91 | #117 | #41 | 1 | 6 | yes | — |
| 92 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 93 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 94 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 95 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 96 | C2×C42 | — | 2 | 3 | no | — |
| 97 | #108 | — | 4 | 12 | no | — |
| 98 | #103 | — | 4 | 12 | no | — |
| 99 | #103 | — | 4 | 1 | no | — |
| 100 | #103 | — | 4 | 3 | no | — |
| 101 | #101 | — | 4 | 12 | no | — |
| 102 | #97 | — | 2 | 12 | no | — |
| 103 | C4×C8 | — | 2 | 12 | no | — |
| 104 | C24 | 2−1+4 | 1 | 1 | yes | characteristic |
| 105 | C2×Q8 | — | 2 | 6 | no | — |
| 106 | C2×Q8 | — | 2 | 6 | no | — |
| 107 | C2×Q8 | — | 4 | 3 | no | — |
| 108 | C2×Q8 | — | 4 | 3 | no | — |
| 109 | C2×Q8 | — | 4 | 1 | no | — |
| 110 | C2×Q8 | — | 4 | 3 | no | — |
| 111 | C2×Q8 | — | 4 | 3 | no | — |
| 112 | C2×Q8 | — | 4 | 3 | no | — |
| 113 | C2×Q8 | — | 4 | 3 | no | — |
| 114 | C2×Q8 | — | 4 | 3 | no | — |
| 115 | C22×C4 | — | 2 | 2 | no | — |
| 116 | C22×C4 | — | 4 | 3 | no | — |
| 117 | C22×C4 | — | 4 | 3 | no | — |
| 118 | C22×C4 | — | 4 | 3 | no | — |
| 119 | C22×C4 | — | 4 | 3 | no | — |
| 120 | C22×C4 | — | 16 | 1 | no | — |
| 121 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 122 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 123 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 124 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 125 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 126 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 127 | M16 | — | 2 | 6 | no | — |
| 128 | M16 | — | 2 | 6 | no | — |
| 129 | M16 | — | 2 | 6 | no | — |
| 130 | M16 | — | 2 | 6 | no | — |
| 131 | M16 | — | 4 | 3 | no | — |
| 132 | M16 | — | 4 | 3 | no | — |
| 133 | M16 | — | 4 | 3 | no | — |
| 134 | M16 | — | 4 | 3 | no | — |
| 135 | M16 | — | 4 | 3 | no | — |
| 136 | M16 | — | 4 | 3 | no | — |
| 137 | M16 | — | 4 | 3 | no | — |
| 138 | M16 | — | 4 | 3 | no | — |
| 139 | C2×C8 | — | 2 | 3 | no | — |
| 140 | C2×C8 | — | 2 | 3 | no | — |
| 141 | C2×C8 | — | 2 | 3 | no | — |
| 142 | C2×C8 | — | 2 | 3 | no | — |
| 143 | C2×C8 | — | 2 | 6 | no | — |
| 144 | C2×C8 | — | 2 | 6 | no | — |
| 145 | C2×C8 | — | 4 | 3 | no | — |
| 146 | C2×C8 | — | 4 | 3 | no | — |
| 147 | C2×C8 | — | 4 | 3 | no | — |
| 148 | C2×C8 | — | 4 | 3 | no | — |
| 149 | C2×C8 | — | 4 | 3 | no | — |
| 150 | C2×C8 | — | 4 | 3 | no | — |
| 151 | C2×C8 | — | 4 | 3 | no | — |
| 152 | C2×C8 | #126 | 1 | 1 | yes | characteristic |
| 153 | C2×C8 | #126 | 1 | 3 | yes | — |
| 154 | C4⋊C4 | — | 2 | 6 | no | — |
| 155 | C4⋊C4 | — | 4 | 3 | no | — |
| 156 | C4⋊C4 | — | 4 | 1 | no | — |
| 157 | C4⋊C4 | — | 4 | 3 | no | — |
| 158 | C22⋊C4 | — | 2 | 6 | no | — |
| 159 | C22⋊C4 | — | 4 | 3 | no | — |
| 160 | C42 | — | 2 | 6 | no | — |
| 161 | C42 | — | 4 | 3 | no | — |
| 162 | C23 | — | 4 | 1 | no | — |
| 163 | C23 | — | 4 | 1 | no | — |
| 164 | C23 | #542 | 1 | 3 | yes | — |
| 165 | C23 | #548 | 1 | 1 | yes | characteristic |
| 166 | C23 | #552 | 1 | 3 | yes | — |
| 167 | Q8 | — | 4 | 2 | no | — |
| 168 | Q8 | — | 4 | 6 | no | — |
| 169 | Q8 | — | 8 | 1 | no | — |
| 170 | Q8 | — | 8 | 3 | no | — |
| 171 | C2×C4 | — | 4 | 6 | no | — |
| 172 | C2×C4 | — | 4 | 6 | no | — |
| 173 | C2×C4 | — | 8 | 3 | no | — |
| 174 | C2×C4 | — | 8 | 3 | no | — |
| 175 | C2×C4 | — | 8 | 1 | no | — |
| 176 | C2×C4 | — | 8 | 3 | no | — |
| 177 | C2×C4 | — | 8 | 1 | no | — |
| 178 | C2×C4 | — | 16 | 3 | no | — |
| 179 | C2×C4 | #525 | 1 | 3 | yes | — |
| 180 | C2×C4 | C2×#126 | 1 | 1 | yes | characteristic |
| 181 | C2×C4 | #529 | 1 | 3 | yes | — |
| 182 | C2×C4 | #529 | 1 | 3 | yes | — |
| 183 | C2×C4 | #529 | 1 | 3 | yes | — |
| 184 | C2×C4 | #540 | 1 | 3 | yes | — |
| 185 | C2×C4 | #546 | 1 | 3 | yes | — |
| 186 | C2×C4 | #551 | 1 | 3 | yes | — |
| 187 | C2×C4 | #553 | 1 | 3 | yes | — |
| 188 | C2×C4 | #555 | 1 | 3 | yes | — |
| 189 | C8 | — | 2 | 1 | no | — |
| 190 | C8 | — | 2 | 3 | no | — |
| 191 | C8 | — | 4 | 6 | no | — |
| 192 | C8 | — | 4 | 6 | no | — |
| 193 | C8 | — | 8 | 3 | no | — |
| 194 | C22 | — | 2 | 3 | no | — |
| 195 | C22 | — | 2 | 3 | no | — |
| 196 | C22 | — | 4 | 1 | no | — |
| 197 | C22 | — | 4 | 3 | no | — |
| 198 | C22 | #2477 | 1 | 3 | yes | — |
| 199 | C22 | #2529 | 1 | 3 | yes | — |
| 200 | C22 | #2644 | 1 | 1 | yes | characteristic |
| 201 | C4 | — | 8 | 6 | no | — |
| 202 | C4 | — | 8 | 2 | no | — |
| 203 | C4 | — | 16 | 1 | no | — |
| 204 | C4 | #2401 | 1 | 1 | yes | characteristic |
| 205 | C4 | #2476 | 1 | 3 | yes | — |
| 206 | C4 | #2526 | 1 | 3 | yes | — |
| 207 | C4 | #2852 | 1 | 1 | yes | characteristic |
| 208 | C2 | — | 2 | 3 | no | — |
| 209 | C2 | — | 2 | 3 | no | — |
| 210 | C2 | #14899 | 1 | 1 | yes | characteristic |
| 211 | C2 | #30236 | 1 | 1 | yes | characteristic |
| 212 | C2 | #30238 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2070 | 2 |
| C2×#1747 | 2 |
| #11312 | 2 |
| #11283 | 2 |
| #11182 | 2 |
| #10942 | 2 |
| #10898 | 2 |