Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
418 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 80 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 | #33716 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33594 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #13196 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11922 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #7161 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #3414 | — | 2 | 2 | no | — |
| 7 | C2×#508 | — | 2 | 1 | no | — |
| 8 | C2×#508 | — | 2 | 1 | no | — |
| 9 | #3147 | C22 | 1 | 2 | yes | — |
| 10 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 11 | #2251 | — | 2 | 2 | no | — |
| 12 | #2250 | — | 2 | 2 | no | — |
| 13 | #2214 | — | 2 | 2 | no | — |
| 14 | #2213 | — | 2 | 2 | no | — |
| 15 | #2212 | — | 2 | 4 | no | — |
| 16 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#364 | C22 | 1 | 2 | yes | — |
| 18 | #2148 | — | 2 | 2 | no | — |
| 19 | C2×#350 | — | 2 | 1 | no | — |
| 20 | #2113 | — | 2 | 1 | no | — |
| 21 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1359 | — | 2 | 4 | no | — |
| 23 | C2×#135 | — | 2 | 2 | no | — |
| 24 | C2×#135 | — | 2 | 2 | no | — |
| 25 | C2×#135 | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×#131 | — | 2 | 1 | no | — |
| 27 | #532 | — | 2 | 2 | no | — |
| 28 | #510 | — | 2 | 4 | no | — |
| 29 | #510 | — | 4 | 2 | no | — |
| 30 | #509 | — | 2 | 4 | no | — |
| 31 | #509 | — | 4 | 2 | no | — |
| 32 | #508 | — | 4 | 2 | no | — |
| 33 | #508 | — | 4 | 2 | no | — |
| 34 | #508 | — | 4 | 4 | no | — |
| 35 | #508 | — | 4 | 2 | no | — |
| 36 | #508 | — | 4 | 2 | no | — |
| 37 | C2×Q32 | — | 2 | 1 | no | — |
| 38 | C2×Q32 | — | 2 | 1 | no | — |
| 39 | C2×SD32 | — | 2 | 2 | no | — |
| 40 | C2×SD32 | — | 2 | 2 | no | — |
| 41 | C2×D16 | — | 2 | 1 | no | — |
| 42 | C2×D16 | — | 2 | 1 | no | — |
| 43 | C2×M32 | — | 2 | 2 | no | — |
| 44 | C2×M32 | — | 2 | 1 | no | — |
| 45 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 46 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 47 | #495 | — | 2 | 2 | no | — |
| 48 | #495 | — | 2 | 2 | no | — |
| 49 | #494 | D4 | 1 | 2 | yes | — |
| 50 | #493 | D4 | 1 | 2 | yes | — |
| 51 | #492 | — | 2 | 2 | no | — |
| 52 | #443 | — | 2 | 2 | no | — |
| 53 | #443 | — | 2 | 2 | no | — |
| 54 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 55 | C2×#104 | — | 2 | 1 | no | — |
| 56 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 57 | #364 | — | 2 | 2 | no | — |
| 58 | #364 | — | 2 | 2 | no | — |
| 59 | #364 | D4 | 1 | 4 | yes | — |
| 60 | #362 | — | 2 | 4 | no | — |
| 61 | #361 | — | 2 | 4 | no | — |
| 62 | #350 | — | 4 | 2 | no | — |
| 63 | C22⋊C16 | — | 4 | 2 | no | — |
| 64 | #346 | — | 2 | 2 | no | — |
| 65 | C2×#41 | — | 2 | 1 | no | — |
| 66 | C2×#41 | — | 4 | 2 | no | — |
| 67 | #135 | — | 2 | 2 | no | — |
| 68 | #135 | — | 2 | 2 | no | — |
| 69 | #135 | — | 4 | 2 | no | — |
| 70 | #135 | — | 4 | 2 | no | — |
| 71 | #135 | — | 4 | 2 | no | — |
| 72 | #135 | — | 4 | 2 | no | — |
| 73 | #135 | — | 4 | 2 | no | — |
| 74 | #135 | — | 4 | 2 | no | — |
| 75 | #135 | — | 4 | 2 | no | — |
| 76 | #135 | — | 4 | 2 | no | — |
| 77 | C2×Q16 | — | 2 | 2 | no | — |
| 78 | C2×Q16 | — | 2 | 2 | no | — |
| 79 | C2×Q16 | D8 | 1 | 1 | yes | characteristic |
| 80 | C2×SD16 | — | 2 | 1 | no | — |
| 81 | C2×SD16 | — | 4 | 2 | no | — |
| 82 | C2×SD16 | — | 4 | 2 | no | — |
| 83 | C2×D8 | — | 2 | 2 | no | — |
| 84 | C2×D8 | — | 2 | 2 | no | — |
| 85 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 86 | #131 | — | 4 | 1 | no | — |
| 87 | #131 | — | 4 | 1 | no | — |
| 88 | #131 | — | 4 | 1 | no | — |
| 89 | #131 | — | 4 | 1 | no | — |
| 90 | C2×M16 | — | 2 | 1 | no | — |
| 91 | C2×M16 | — | 4 | 1 | no | — |
| 92 | C22×C8 | — | 4 | 1 | no | — |
| 93 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C4⋊Q8 | — | 2 | 2 | no | — |
| 96 | #127 | — | 2 | 2 | no | — |
| 97 | #124 | — | 4 | 2 | no | — |
| 98 | #121 | — | 8 | 2 | no | — |
| 99 | C4×D4 | — | 8 | 2 | no | — |
| 100 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 101 | Q32 | — | 4 | 2 | no | — |
| 102 | Q32 | — | 4 | 2 | no | — |
| 103 | Q32 | — | 4 | 4 | no | — |
| 104 | SD32 | — | 4 | 4 | no | — |
| 105 | SD32 | — | 4 | 4 | no | — |
| 106 | SD32 | — | 4 | 4 | no | — |
| 107 | SD32 | — | 4 | 4 | no | — |
| 108 | D16 | — | 4 | 2 | no | — |
| 109 | D16 | — | 4 | 2 | no | — |
| 110 | D16 | — | 4 | 4 | no | — |
| 111 | M32 | — | 2 | 4 | no | — |
| 112 | M32 | — | 4 | 2 | no | — |
| 113 | M32 | — | 4 | 1 | no | — |
| 114 | M32 | — | 4 | 1 | no | — |
| 115 | C2×C16 | — | 2 | 1 | no | — |
| 116 | C2×C16 | — | 2 | 1 | no | — |
| 117 | C2×C16 | — | 2 | 1 | no | — |
| 118 | C2×C16 | — | 2 | 1 | no | — |
| 119 | C2×C16 | — | 4 | 2 | no | — |
| 120 | C2×C16 | — | 4 | 1 | no | — |
| 121 | C2×C16 | D8 | 1 | 2 | yes | — |
| 122 | C2×C16 | D8 | 1 | 2 | yes | — |
| 123 | #108 | — | 2 | 2 | no | — |
| 124 | #104 | — | 4 | 2 | no | — |
| 125 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 126 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 127 | #41 | — | 4 | 1 | no | — |
| 128 | #41 | — | 4 | 1 | no | — |
| 129 | #41 | — | 4 | 1 | no | — |
| 130 | #41 | — | 4 | 1 | no | — |
| 131 | #41 | — | 8 | 2 | no | — |
| 132 | #41 | — | 8 | 2 | no | — |
| 133 | #41 | — | 8 | 2 | no | — |
| 134 | #41 | — | 8 | 2 | no | — |
| 135 | C2×Q8 | — | 2 | 1 | no | — |
| 136 | C2×Q8 | — | 4 | 2 | no | — |
| 137 | C2×D4 | — | 2 | 1 | no | — |
| 138 | C2×D4 | — | 4 | 1 | no | — |
| 139 | C2×D4 | — | 4 | 2 | no | — |
| 140 | C2×D4 | — | 8 | 2 | no | — |
| 141 | C2×D4 | — | 8 | 2 | no | — |
| 142 | C22×C4 | — | 2 | 1 | no | — |
| 143 | C22×C4 | — | 4 | 1 | no | — |
| 144 | C22×C4 | — | 8 | 2 | no | — |
| 145 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 146 | Q16 | — | 2 | 2 | no | — |
| 147 | Q16 | — | 4 | 2 | no | — |
| 148 | Q16 | — | 4 | 2 | no | — |
| 149 | Q16 | — | 4 | 2 | no | — |
| 150 | Q16 | — | 4 | 2 | no | — |
| 151 | SD16 | — | 4 | 2 | no | — |
| 152 | SD16 | — | 8 | 2 | no | — |
| 153 | SD16 | — | 8 | 2 | no | — |
| 154 | SD16 | — | 8 | 2 | no | — |
| 155 | SD16 | — | 8 | 2 | no | — |
| 156 | D8 | — | 2 | 2 | no | — |
| 157 | D8 | — | 4 | 2 | no | — |
| 158 | D8 | — | 4 | 2 | no | — |
| 159 | D8 | — | 4 | 2 | no | — |
| 160 | D8 | — | 4 | 2 | no | — |
| 161 | M16 | — | 4 | 1 | no | — |
| 162 | M16 | — | 4 | 1 | no | — |
| 163 | M16 | — | 8 | 1 | no | — |
| 164 | M16 | — | 8 | 1 | no | — |
| 165 | C2×C8 | — | 4 | 1 | no | — |
| 166 | C2×C8 | — | 8 | 1 | no | — |
| 167 | C2×C8 | — | 8 | 1 | no | — |
| 168 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 169 | C2×C8 | #120 | 1 | 2 | yes | — |
| 170 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 175 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 176 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 177 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 178 | C4⋊C4 | — | 8 | 2 | no | — |
| 179 | C22⋊C4 | — | 8 | 4 | no | — |
| 180 | C42 | — | 2 | 1 | no | — |
| 181 | C42 | #121 | 1 | 2 | yes | — |
| 182 | C16 | — | 2 | 2 | no | — |
| 183 | C16 | — | 2 | 2 | no | — |
| 184 | C16 | — | 4 | 4 | no | — |
| 185 | C16 | — | 8 | 1 | no | — |
| 186 | C23 | — | 4 | 1 | no | — |
| 187 | C23 | — | 8 | 2 | no | — |
| 188 | C23 | #472 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 4 | 1 | no | — |
| 190 | Q8 | — | 4 | 1 | no | — |
| 191 | Q8 | — | 8 | 2 | no | — |
| 192 | Q8 | — | 8 | 2 | no | — |
| 193 | D4 | — | 4 | 1 | no | — |
| 194 | D4 | — | 4 | 1 | no | — |
| 195 | D4 | — | 8 | 1 | no | — |
| 196 | D4 | — | 8 | 1 | no | — |
| 197 | D4 | — | 8 | 2 | no | — |
| 198 | D4 | — | 8 | 2 | no | — |
| 199 | D4 | — | 16 | 4 | no | — |
| 200 | D4 | — | 16 | 2 | no | — |
| 201 | D4 | — | 16 | 2 | no | — |
| 202 | C2×C4 | — | 2 | 2 | no | — |
| 203 | C2×C4 | — | 4 | 1 | no | — |
| 204 | C2×C4 | — | 8 | 1 | no | — |
| 205 | C2×C4 | — | 8 | 1 | no | — |
| 206 | C2×C4 | — | 8 | 1 | no | — |
| 207 | C2×C4 | — | 8 | 2 | no | — |
| 208 | C2×C4 | — | 16 | 2 | no | — |
| 209 | C2×C4 | — | 16 | 2 | no | — |
| 210 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #466 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | #469 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | #472 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 8 | 1 | no | — |
| 217 | C8 | #447 | 1 | 2 | yes | — |
| 218 | C8 | #447 | 1 | 1 | yes | characteristic |
| 219 | C8 | #447 | 1 | 1 | yes | characteristic |
| 220 | C8 | #449 | 1 | 2 | yes | — |
| 221 | C8 | #493 | 1 | 1 | yes | characteristic |
| 222 | C8 | #496 | 1 | 1 | yes | characteristic |
| 223 | C22 | — | 4 | 1 | no | — |
| 224 | C22 | — | 8 | 2 | no | — |
| 225 | C22 | — | 8 | 1 | no | — |
| 226 | C22 | — | 16 | 4 | no | — |
| 227 | C22 | — | 16 | 2 | no | — |
| 228 | C22 | #1667 | 1 | 1 | yes | characteristic |
| 229 | C22 | #1687 | 1 | 1 | yes | characteristic |
| 230 | C22 | #2018 | 1 | 1 | yes | characteristic |
| 231 | C4 | — | 4 | 2 | no | — |
| 232 | C4 | — | 8 | 1 | no | — |
| 233 | C4 | — | 16 | 2 | no | — |
| 234 | C4 | #1667 | 1 | 1 | yes | characteristic |
| 235 | C4 | #1687 | 1 | 1 | yes | characteristic |
| 236 | C4 | #1999 | 1 | 1 | yes | characteristic |
| 237 | C4 | #2011 | 1 | 1 | yes | characteristic |
| 238 | C2 | — | 2 | 1 | no | — |
| 239 | C2 | — | 2 | 1 | no | — |
| 240 | C2 | — | 2 | 1 | no | — |
| 241 | C2 | — | 16 | 1 | no | — |
| 242 | C2 | — | 32 | 2 | no | — |
| 243 | C2 | #10260 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1687 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10260 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33716 | C2 | semidirect |
| 2. #33594 | C2 | semidirect |
| 3. #13196 | C2 | semidirect |
| 4. #11922 | C2 | semidirect |
| 5. #7161 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33716 | 2 |
| #33594 | 2 |
| #13196 | 2 |
| #11922 | 2 |
| #7161 | 2 |