Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
370 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 74 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 | #33981 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33716 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13620 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13242 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7466 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#508 | — | 2 | 1 | no | — |
| 7 | C2×#508 | C22 | 1 | 2 | yes | — |
| 8 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 9 | #3147 | — | 2 | 1 | no | — |
| 10 | #2264 | — | 2 | 2 | no | — |
| 11 | #2264 | — | 2 | 2 | no | — |
| 12 | #2263 | — | 2 | 2 | no | — |
| 13 | #2263 | — | 2 | 2 | no | — |
| 14 | #2262 | — | 2 | 4 | no | — |
| 15 | C2×M64 | C22 | 1 | 2 | yes | — |
| 16 | #2251 | — | 2 | 2 | no | — |
| 17 | #2250 | — | 2 | 2 | no | — |
| 18 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#359 | — | 2 | 1 | no | — |
| 20 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1974 | — | 2 | 1 | no | — |
| 22 | #1417 | — | 2 | 4 | no | — |
| 23 | C2×#135 | — | 2 | 2 | no | — |
| 24 | C2×#135 | — | 2 | 1 | no | — |
| 25 | C2×#135 | — | 4 | 1 | no | — |
| 26 | #532 | — | 4 | 1 | no | — |
| 27 | #508 | — | 2 | 2 | no | — |
| 28 | #508 | — | 2 | 2 | no | — |
| 29 | #508 | — | 2 | 2 | no | — |
| 30 | #508 | — | 2 | 2 | no | — |
| 31 | #508 | — | 2 | 2 | no | — |
| 32 | #508 | — | 2 | 2 | no | — |
| 33 | #508 | — | 4 | 2 | no | — |
| 34 | #508 | — | 4 | 2 | no | — |
| 35 | C2×Q32 | — | 2 | 1 | no | — |
| 36 | C2×Q32 | D4 | 1 | 2 | yes | — |
| 37 | C2×Q32 | D4 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2×SD32 | — | 2 | 2 | no | — |
| 39 | C2×SD32 | — | 2 | 2 | no | — |
| 40 | C2×SD32 | — | 2 | 1 | no | — |
| 41 | C2×D16 | — | 2 | 1 | no | — |
| 42 | C2×D16 | D4 | 1 | 2 | yes | — |
| 43 | C2×D16 | D4 | 1 | 1 | yes | characteristic, complemented |
| 44 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 45 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 46 | #495 | — | 2 | 2 | no | — |
| 47 | #495 | — | 4 | 1 | no | — |
| 48 | #494 | — | 2 | 1 | no | — |
| 49 | #493 | — | 2 | 1 | no | — |
| 50 | #492 | — | 4 | 1 | no | — |
| 51 | C2×#108 | — | 2 | 1 | no | — |
| 52 | C2×#104 | — | 4 | 1 | no | — |
| 53 | #408 | — | 4 | 1 | no | — |
| 54 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 55 | Q64 | — | 2 | 4 | no | — |
| 56 | SD64 | — | 2 | 4 | no | — |
| 57 | SD64 | — | 2 | 4 | no | — |
| 58 | D32 | — | 2 | 4 | no | — |
| 59 | M64 | — | 2 | 2 | no | — |
| 60 | M64 | — | 2 | 2 | no | — |
| 61 | C2×C32 | D4 | 1 | 4 | yes | complemented |
| 62 | #368 | — | 2 | 2 | no | — |
| 63 | #359 | — | 4 | 2 | no | — |
| 64 | #346 | — | 2 | 2 | no | — |
| 65 | C2×#41 | — | 4 | 2 | no | — |
| 66 | C2×#41 | — | 4 | 1 | no | — |
| 67 | #135 | — | 4 | 1 | no | — |
| 68 | #135 | — | 4 | 2 | no | — |
| 69 | #135 | — | 4 | 2 | no | — |
| 70 | #135 | — | 4 | 1 | no | — |
| 71 | #135 | — | 4 | 1 | no | — |
| 72 | #135 | — | 4 | 2 | no | — |
| 73 | #135 | — | 4 | 1 | no | — |
| 74 | #135 | — | 4 | 2 | no | — |
| 75 | #135 | — | 8 | 2 | no | — |
| 76 | #135 | — | 8 | 2 | no | — |
| 77 | C2×Q16 | — | 2 | 2 | no | — |
| 78 | C2×Q16 | — | 2 | 1 | no | — |
| 79 | C2×Q16 | — | 4 | 1 | no | — |
| 80 | C2×SD16 | — | 4 | 2 | no | — |
| 81 | C2×SD16 | — | 4 | 1 | no | — |
| 82 | C2×SD16 | — | 4 | 2 | no | — |
| 83 | C2×D8 | — | 2 | 2 | no | — |
| 84 | C2×D8 | — | 2 | 1 | no | — |
| 85 | C2×D8 | — | 4 | 1 | no | — |
| 86 | C2×M16 | — | 4 | 1 | no | — |
| 87 | C22×C8 | D8 | 1 | 1 | yes | characteristic |
| 88 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C4⋊Q8 | — | 4 | 1 | no | — |
| 90 | #127 | — | 4 | 1 | no | — |
| 91 | #124 | — | 4 | 2 | no | — |
| 92 | #121 | — | 8 | 2 | no | — |
| 93 | C4×D4 | — | 8 | 2 | no | — |
| 94 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 95 | Q32 | — | 2 | 2 | no | — |
| 96 | Q32 | — | 2 | 2 | no | — |
| 97 | Q32 | — | 2 | 2 | no | — |
| 98 | Q32 | — | 4 | 2 | no | — |
| 99 | SD32 | — | 4 | 4 | no | — |
| 100 | SD32 | — | 4 | 2 | no | — |
| 101 | SD32 | — | 4 | 2 | no | — |
| 102 | SD32 | — | 4 | 2 | no | — |
| 103 | D16 | — | 2 | 2 | no | — |
| 104 | D16 | — | 2 | 2 | no | — |
| 105 | D16 | — | 2 | 2 | no | — |
| 106 | D16 | — | 4 | 2 | no | — |
| 107 | C2×C16 | — | 2 | 1 | no | — |
| 108 | C2×C16 | — | 2 | 1 | no | — |
| 109 | C2×C16 | D8 | 1 | 2 | yes | — |
| 110 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 112 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 113 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 114 | #108 | — | 4 | 1 | no | — |
| 115 | #108 | — | 4 | 1 | no | — |
| 116 | #104 | — | 8 | 2 | no | — |
| 117 | C22⋊C8 | — | 8 | 2 | no | — |
| 118 | C4×C8 | — | 2 | 1 | no | — |
| 119 | C4×C8 | — | 2 | 1 | no | — |
| 120 | C32 | — | 2 | 4 | no | — |
| 121 | #41 | — | 8 | 1 | no | — |
| 122 | #41 | — | 8 | 2 | no | — |
| 123 | #41 | — | 8 | 1 | no | — |
| 124 | #41 | — | 8 | 2 | no | — |
| 125 | #41 | — | 8 | 1 | no | — |
| 126 | #41 | — | 8 | 2 | no | — |
| 127 | #41 | — | 8 | 2 | no | — |
| 128 | #41 | — | 8 | 1 | no | — |
| 129 | C2×Q8 | — | 4 | 2 | no | — |
| 130 | C2×Q8 | — | 4 | 1 | no | — |
| 131 | C2×D4 | — | 4 | 2 | no | — |
| 132 | C2×D4 | — | 4 | 1 | no | — |
| 133 | C2×D4 | — | 8 | 2 | no | — |
| 134 | C2×D4 | — | 8 | 2 | no | — |
| 135 | C2×D4 | — | 8 | 1 | no | — |
| 136 | C22×C4 | — | 2 | 1 | no | — |
| 137 | C22×C4 | — | 8 | 1 | no | — |
| 138 | C22×C4 | — | 8 | 2 | no | — |
| 139 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 140 | Q16 | — | 4 | 2 | no | — |
| 141 | Q16 | — | 4 | 1 | no | — |
| 142 | Q16 | — | 4 | 2 | no | — |
| 143 | Q16 | — | 4 | 1 | no | — |
| 144 | Q16 | — | 8 | 2 | no | — |
| 145 | SD16 | — | 8 | 4 | no | — |
| 146 | SD16 | — | 8 | 1 | no | — |
| 147 | SD16 | — | 8 | 2 | no | — |
| 148 | SD16 | — | 8 | 1 | no | — |
| 149 | SD16 | — | 8 | 2 | no | — |
| 150 | D8 | — | 4 | 2 | no | — |
| 151 | D8 | — | 4 | 1 | no | — |
| 152 | D8 | — | 4 | 2 | no | — |
| 153 | D8 | — | 4 | 1 | no | — |
| 154 | D8 | — | 8 | 2 | no | — |
| 155 | M16 | — | 8 | 1 | no | — |
| 156 | M16 | — | 8 | 1 | no | — |
| 157 | C2×C8 | — | 2 | 1 | no | — |
| 158 | C2×C8 | — | 2 | 1 | no | — |
| 159 | C2×C8 | — | 8 | 1 | no | — |
| 160 | C2×C8 | D16 | 1 | 2 | yes | — |
| 161 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 162 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 163 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 164 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 165 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 166 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 167 | C4⋊C4 | — | 8 | 2 | no | — |
| 168 | C22⋊C4 | — | 8 | 4 | no | — |
| 169 | C42 | — | 2 | 1 | no | — |
| 170 | C42 | D16 | 1 | 2 | yes | — |
| 171 | C16 | — | 2 | 2 | no | — |
| 172 | C16 | #120 | 1 | 2 | yes | — |
| 173 | C16 | #121 | 1 | 2 | yes | — |
| 174 | C23 | — | 8 | 2 | no | — |
| 175 | C23 | — | 8 | 1 | no | — |
| 176 | C23 | #509 | 1 | 1 | yes | characteristic |
| 177 | Q8 | — | 8 | 2 | no | — |
| 178 | Q8 | — | 8 | 1 | no | — |
| 179 | Q8 | — | 8 | 1 | no | — |
| 180 | Q8 | — | 8 | 2 | no | — |
| 181 | D4 | — | 8 | 2 | no | — |
| 182 | D4 | — | 8 | 1 | no | — |
| 183 | D4 | — | 8 | 2 | no | — |
| 184 | D4 | — | 8 | 1 | no | — |
| 185 | D4 | — | 16 | 4 | no | — |
| 186 | D4 | — | 16 | 1 | no | — |
| 187 | D4 | — | 16 | 2 | no | — |
| 188 | D4 | — | 16 | 2 | no | — |
| 189 | D4 | — | 16 | 1 | no | — |
| 190 | C2×C4 | — | 2 | 2 | no | — |
| 191 | C2×C4 | — | 8 | 1 | no | — |
| 192 | C2×C4 | — | 8 | 2 | no | — |
| 193 | C2×C4 | — | 8 | 1 | no | — |
| 194 | C2×C4 | — | 16 | 1 | no | — |
| 195 | C2×C4 | — | 16 | 2 | no | — |
| 196 | C2×C4 | — | 16 | 1 | no | — |
| 197 | C2×C4 | — | 16 | 2 | no | — |
| 198 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | #509 | 1 | 1 | yes | characteristic |
| 204 | C8 | — | 2 | 2 | no | — |
| 205 | C8 | — | 16 | 1 | no | — |
| 206 | C8 | #392 | 1 | 1 | yes | characteristic |
| 207 | C8 | #447 | 1 | 1 | yes | characteristic |
| 208 | C8 | #459 | 1 | 1 | yes | characteristic |
| 209 | C8 | #496 | 1 | 1 | yes | characteristic |
| 210 | C22 | — | 4 | 1 | no | — |
| 211 | C22 | — | 16 | 4 | no | — |
| 212 | C22 | — | 16 | 2 | no | — |
| 213 | C22 | — | 16 | 1 | no | — |
| 214 | C22 | — | 16 | 2 | no | — |
| 215 | C22 | #2184 | 1 | 1 | yes | characteristic |
| 216 | C22 | #2206 | 1 | 1 | yes | characteristic |
| 217 | C22 | #2250 | 1 | 1 | yes | characteristic |
| 218 | C4 | — | 4 | 2 | no | — |
| 219 | C4 | — | 16 | 2 | no | — |
| 220 | C4 | — | 16 | 1 | no | — |
| 221 | C4 | #2011 | 1 | 1 | yes | characteristic |
| 222 | C4 | #2184 | 1 | 1 | yes | characteristic |
| 223 | C4 | #2206 | 1 | 1 | yes | characteristic |
| 224 | C4 | #2246 | 1 | 1 | yes | characteristic |
| 225 | C2 | — | 2 | 1 | no | — |
| 226 | C2 | — | 2 | 1 | no | — |
| 227 | C2 | — | 2 | 1 | no | — |
| 228 | C2 | — | 32 | 2 | no | — |
| 229 | C2 | — | 32 | 1 | no | — |
| 230 | C2 | #13325 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2246 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13325 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33981 | C2 | semidirect |
| 2. #33716 | C2 | semidirect |
| 3. #13620 | C2 | semidirect |
| 4. #13242 | C2 | semidirect |
| 5. #7466 | C2 | semidirect |
| 37. C2×Q32 | D4 | semidirect |
| 43. C2×D16 | D4 | semidirect |
| 61. C2×C32 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33981 | 2 |
| #33716 | 2 |
| #13620 | 2 |
| #13242 | 2 |
| #7466 | 2 |