Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
1061 conjugacy classes of subgroups; only the 233 normal subgroups are shown.
This group has 1061 conjugacy classes of subgroups — too many to draw. Only its 233 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 | #491550 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #483961 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #213274 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #141822 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #141807 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #135802 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #126540 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #33748 | C22 | 1 | 4 | yes | — |
| 9 | #21706 | — | 2 | 2 | no | — |
| 10 | #19603 | C22 | 1 | 2 | yes | — |
| 11 | #17962 | — | 2 | 2 | no | — |
| 12 | #17962 | C22 | 1 | 4 | yes | — |
| 13 | #12594 | C22 | 1 | 2 | yes | — |
| 14 | #12532 | C4 | 1 | 4 | yes | complemented |
| 15 | #12525 | C4 | 1 | 4 | yes | complemented |
| 16 | #12089 | — | 2 | 8 | no | — |
| 17 | #12067 | — | 2 | 8 | no | — |
| 18 | #12020 | C22 | 1 | 4 | yes | — |
| 19 | #11680 | C22 | 1 | 8 | yes | — |
| 20 | #11404 | C22 | 1 | 2 | yes | — |
| 21 | #11402 | C22 | 1 | 1 | yes | characteristic |
| 22 | C8×M32 | C22 | 1 | 1 | yes | characteristic |
| 23 | #9862 | — | 2 | 2 | no | — |
| 24 | #9849 | — | 2 | 2 | no | — |
| 25 | #9849 | C22 | 1 | 4 | yes | — |
| 26 | #8822 | C22 | 1 | 2 | yes | — |
| 27 | #7846 | C22 | 1 | 1 | yes | characteristic |
| 28 | #3169 | — | 2 | 2 | no | — |
| 29 | #3169 | C23 | 1 | 4 | yes | — |
| 30 | #3147 | — | 2 | 2 | no | — |
| 31 | C2×#444 | — | 4 | 4 | no | — |
| 32 | #2931 | — | 2 | 4 | no | — |
| 33 | #2254 | C2×C4 | 1 | 4 | yes | — |
| 34 | #2246 | C2×C4 | 1 | 4 | yes | — |
| 35 | #2240 | C2×C4 | 1 | 8 | yes | — |
| 36 | #2218 | — | 2 | 8 | no | — |
| 37 | #2177 | — | 2 | 8 | no | — |
| 38 | C4×M32 | C23 | 1 | 2 | yes | — |
| 39 | C4×M32 | C23 | 1 | 2 | yes | — |
| 40 | #1975 | — | 4 | 4 | no | — |
| 41 | #1962 | — | 2 | 4 | no | — |
| 42 | #1848 | — | 2 | 2 | no | — |
| 43 | C4×#97 | D4 | 1 | 2 | yes | — |
| 44 | C4×#97 | C23 | 1 | 1 | yes | characteristic |
| 45 | #1713 | C2×C4 | 1 | 2 | yes | — |
| 46 | #1712 | C2×C4 | 1 | 2 | yes | — |
| 47 | #1709 | — | 2 | 2 | no | — |
| 48 | #1706 | — | 2 | 2 | no | — |
| 49 | #1706 | C2×C4 | 1 | 4 | yes | — |
| 50 | #1594 | — | 2 | 8 | no | — |
| 51 | #1594 | — | 4 | 4 | no | — |
| 52 | #1580 | — | 2 | 8 | no | — |
| 53 | #1580 | — | 4 | 4 | no | — |
| 54 | #1566 | — | 2 | 2 | no | — |
| 55 | #1566 | C23 | 1 | 4 | yes | — |
| 56 | #1492 | — | 2 | 4 | no | — |
| 57 | #1492 | — | 2 | 4 | no | — |
| 58 | C2×#322 | D4 | 1 | 2 | yes | — |
| 59 | C2×#322 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C82 | C23 | 1 | 1 | yes | characteristic |
| 61 | #1375 | — | 2 | 4 | no | — |
| 62 | #1368 | — | 2 | 8 | no | — |
| 63 | #1333 | — | 2 | 16 | no | — |
| 64 | #1313 | D4 | 1 | 8 | yes | — |
| 65 | #1312 | C2×C4 | 1 | 4 | yes | — |
| 66 | #1311 | — | 2 | 1 | no | — |
| 67 | C8×C16 | — | 2 | 1 | no | — |
| 68 | C2×#135 | — | 4 | 4 | no | — |
| 69 | C2×#131 | — | 4 | 4 | no | — |
| 70 | #532 | — | 2 | 4 | no | — |
| 71 | C2×D16 | — | 2 | 8 | no | — |
| 72 | C2×M32 | — | 2 | 2 | no | — |
| 73 | C2×M32 | #41 | 1 | 4 | yes | — |
| 74 | #500 | — | 2 | 2 | no | — |
| 75 | #500 | C22×C4 | 1 | 4 | yes | — |
| 76 | #495 | — | 4 | 4 | no | — |
| 77 | #494 | — | 2 | 2 | no | — |
| 78 | #493 | — | 2 | 2 | no | — |
| 79 | #493 | — | 2 | 2 | no | — |
| 80 | #493 | C22×C4 | 1 | 4 | yes | — |
| 81 | #492 | — | 2 | 4 | no | — |
| 82 | #486 | — | 2 | 4 | no | — |
| 83 | #486 | — | 2 | 4 | no | — |
| 84 | #466 | — | 2 | 8 | no | — |
| 85 | #466 | — | 4 | 4 | no | — |
| 86 | #444 | — | 8 | 16 | no | — |
| 87 | C4×D8 | — | 2 | 8 | no | — |
| 88 | C4×D8 | — | 4 | 4 | no | — |
| 89 | #436 | — | 4 | 4 | no | — |
| 90 | #436 | — | 4 | 4 | no | — |
| 91 | C8×D4 | — | 4 | 4 | no | — |
| 92 | C8×D4 | — | 4 | 4 | no | — |
| 93 | C2×#108 | — | 4 | 4 | no | — |
| 94 | #423 | — | 2 | 4 | no | — |
| 95 | C2×#104 | — | 4 | 8 | no | — |
| 96 | #408 | — | 4 | 4 | no | — |
| 97 | C2×#97 | — | 2 | 4 | no | — |
| 98 | C2×#97 | — | 2 | 2 | no | — |
| 99 | C2×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 101 | C2×C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 102 | C2×C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 103 | C43 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | #367 | — | 2 | 8 | no | — |
| 105 | #357 | — | 2 | 16 | no | — |
| 106 | #346 | C2×D4 | 1 | 4 | yes | — |
| 107 | #346 | C2×D4 | 1 | 4 | yes | — |
| 108 | C4×C16 | — | 2 | 2 | no | — |
| 109 | C4×C16 | C22×C4 | 1 | 4 | yes | — |
| 110 | #334 | — | 2 | 8 | no | — |
| 111 | #334 | — | 4 | 4 | no | — |
| 112 | D4⋊C8 | — | 4 | 8 | no | — |
| 113 | D4⋊C8 | — | 4 | 8 | no | — |
| 114 | #322 | — | 2 | 4 | no | — |
| 115 | #322 | C2×D4 | 1 | 4 | yes | — |
| 116 | C82 | C22×C4 | 1 | 2 | yes | — |
| 117 | C82 | #41 | 1 | 1 | yes | characteristic |
| 118 | C82 | #41 | 1 | 1 | yes | characteristic |
| 119 | C2×#41 | — | 4 | 4 | no | — |
| 120 | #135 | — | 8 | 8 | no | — |
| 121 | #135 | — | 8 | 8 | no | — |
| 122 | C2×Q16 | — | 4 | 4 | no | — |
| 123 | C2×SD16 | — | 4 | 8 | no | — |
| 124 | C2×D8 | — | 2 | 8 | no | — |
| 125 | C2×D8 | — | 4 | 4 | no | — |
| 126 | C2×D8 | — | 4 | 4 | no | — |
| 127 | #131 | — | 8 | 16 | no | — |
| 128 | C2×M16 | — | 4 | 4 | no | — |
| 129 | C2×M16 | — | 8 | 4 | no | — |
| 130 | C22×C8 | — | 2 | 2 | no | — |
| 131 | C22×C8 | — | 2 | 1 | no | — |
| 132 | C22×C8 | — | 8 | 4 | no | — |
| 133 | C22×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 134 | C22×C8 | C2×#41 | 1 | 2 | yes | — |
| 135 | C4⋊Q8 | — | 2 | 4 | no | — |
| 136 | #127 | — | 2 | 4 | no | — |
| 137 | #124 | — | 4 | 4 | no | — |
| 138 | #121 | — | 4 | 4 | no | — |
| 139 | #121 | — | 4 | 4 | no | — |
| 140 | C4×D4 | — | 4 | 4 | no | — |
| 141 | C4×D4 | — | 4 | 4 | no | — |
| 142 | C2×C42 | — | 2 | 1 | no | — |
| 143 | C2×C42 | — | 2 | 2 | no | — |
| 144 | C2×C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 145 | D16 | — | 4 | 16 | no | — |
| 146 | M32 | — | 4 | 4 | no | — |
| 147 | M32 | — | 4 | 4 | no | — |
| 148 | C2×C16 | — | 2 | 4 | no | — |
| 149 | C2×C16 | C4×D4 | 1 | 8 | yes | — |
| 150 | #108 | — | 8 | 8 | no | — |
| 151 | #107 | — | 2 | 8 | no | — |
| 152 | #107 | — | 4 | 4 | no | — |
| 153 | C4⋊C8 | — | 4 | 4 | no | — |
| 154 | C4⋊C8 | — | 4 | 4 | no | — |
| 155 | #104 | — | 8 | 16 | no | — |
| 156 | #102 | — | 4 | 8 | no | — |
| 157 | #102 | — | 4 | 8 | no | — |
| 158 | C22⋊C8 | — | 8 | 4 | no | — |
| 159 | C22⋊C8 | — | 8 | 4 | no | — |
| 160 | #97 | — | 4 | 4 | no | — |
| 161 | #97 | — | 4 | 8 | no | — |
| 162 | C4×C8 | — | 2 | 2 | no | — |
| 163 | C4×C8 | — | 2 | 2 | no | — |
| 164 | C4×C8 | C4×D4 | 1 | 2 | yes | — |
| 165 | C4×C8 | C4×D4 | 1 | 2 | yes | — |
| 166 | C4×C8 | #127 | 1 | 4 | yes | — |
| 167 | C4×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 168 | C4×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 169 | C4×C8 | C22×D4 | 1 | 2 | yes | — |
| 170 | C4×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 171 | #41 | — | 8 | 8 | no | — |
| 172 | #41 | — | 8 | 8 | no | — |
| 173 | C2×Q8 | — | 4 | 4 | no | — |
| 174 | C2×D4 | — | 4 | 4 | no | — |
| 175 | C2×D4 | — | 4 | 4 | no | — |
| 176 | C2×D4 | — | 4 | 4 | no | — |
| 177 | C2×D4 | — | 8 | 4 | no | — |
| 178 | C22×C4 | — | 4 | 1 | no | — |
| 179 | C22×C4 | — | 8 | 4 | no | — |
| 180 | C22×C4 | #532 | 1 | 1 | yes | characteristic |
| 181 | C22×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 182 | Q16 | — | 8 | 8 | no | — |
| 183 | SD16 | — | 8 | 16 | no | — |
| 184 | D8 | — | 4 | 16 | no | — |
| 185 | D8 | — | 8 | 8 | no | — |
| 186 | D8 | — | 8 | 8 | no | — |
| 187 | M16 | — | 8 | 8 | no | — |
| 188 | M16 | — | 16 | 8 | no | — |
| 189 | C2×C8 | — | 2 | 8 | no | — |
| 190 | C2×C8 | — | 2 | 4 | no | — |
| 191 | C2×C8 | — | 2 | 4 | no | — |
| 192 | C2×C8 | — | 4 | 4 | no | — |
| 193 | C2×C8 | — | 4 | 2 | no | — |
| 194 | C2×C8 | — | 4 | 1 | no | — |
| 195 | C2×C8 | — | 8 | 4 | no | — |
| 196 | C2×C8 | — | 16 | 8 | no | — |
| 197 | C2×C8 | #442 | 1 | 4 | yes | — |
| 198 | C2×C8 | #442 | 1 | 4 | yes | — |
| 199 | C2×C8 | C2×C4×D4 | 1 | 2 | yes | — |
| 200 | C2×C8 | #532 | 1 | 2 | yes | — |
| 201 | C4⋊C4 | — | 4 | 4 | no | — |
| 202 | C4⋊C4 | — | 4 | 4 | no | — |
| 203 | C22⋊C4 | — | 8 | 4 | no | — |
| 204 | C22⋊C4 | — | 8 | 4 | no | — |
| 205 | C42 | — | 4 | 4 | no | — |
| 206 | C42 | — | 4 | 2 | no | — |
| 207 | C42 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 208 | C42 | C2×#127 | 1 | 1 | yes | characteristic |
| 209 | C42 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 210 | C16 | — | 2 | 8 | no | — |
| 211 | C16 | — | 4 | 4 | no | — |
| 212 | C23 | — | 8 | 4 | no | — |
| 213 | C23 | #3184 | 1 | 1 | yes | characteristic |
| 214 | Q8 | — | 8 | 8 | no | — |
| 215 | D4 | — | 8 | 8 | no | — |
| 216 | D4 | — | 8 | 8 | no | — |
| 217 | D4 | — | 8 | 8 | no | — |
| 218 | D4 | — | 16 | 8 | no | — |
| 219 | C2×C4 | — | 2 | 4 | no | — |
| 220 | C2×C4 | — | 2 | 2 | no | — |
| 221 | C2×C4 | — | 8 | 1 | no | — |
| 222 | C2×C4 | — | 8 | 2 | no | — |
| 223 | C2×C4 | — | 8 | 4 | no | — |
| 224 | C2×C4 | — | 16 | 4 | no | — |
| 225 | C2×C4 | — | 16 | 4 | no | — |
| 226 | C2×C4 | C4×#127 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | C2×#442 | 1 | 2 | yes | — |
| 228 | C2×C4 | #3151 | 1 | 2 | yes | — |
| 229 | C2×C4 | #3184 | 1 | 1 | yes | characteristic |
| 230 | C8 | — | 2 | 4 | no | — |
| 231 | C8 | — | 2 | 2 | no | — |
| 232 | C8 | — | 4 | 4 | no | — |
| 233 | C8 | — | 8 | 1 | no | — |
| 234 | C8 | — | 16 | 4 | no | — |
| 235 | C8 | AGL(1,ℤ/16) | 1 | 8 | yes | — |
| 236 | C8 | #2955 | 1 | 4 | yes | — |
| 237 | C22 | — | 2 | 1 | no | — |
| 238 | C22 | — | 2 | 2 | no | — |
| 239 | C22 | — | 16 | 4 | no | — |
| 240 | C22 | — | 16 | 4 | no | — |
| 241 | C22 | — | 16 | 4 | no | — |
| 242 | C22 | #21690 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 2 | 1 | no | — |
| 244 | C4 | — | 2 | 2 | no | — |
| 245 | C4 | — | 16 | 1 | no | — |
| 246 | C4 | — | 16 | 4 | no | — |
| 247 | C4 | #21709 | 1 | 2 | yes | — |
| 248 | C4 | C2×AGL(1,ℤ/16) | 1 | 4 | yes | — |
| 249 | C2 | — | 4 | 1 | no | — |
| 250 | C2 | — | 32 | 4 | no | — |
| 251 | C2 | #213320 | 1 | 1 | yes | characteristic |
| 252 | C2 | #496124 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #21690 |
| commutator | C4×C8 | C23×C4 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| prime radical | #491550 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #7846 ◁ 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 |
|---|---|---|
| 2. #483961 | C2 | semidirect |
| 3. #213274 | C2 | semidirect |
| 4. #141822 | C2 | semidirect |
| 5. #141807 | C2 | semidirect |
| 6. #135802 | C2 | semidirect |
| 7. #126540 | C2 | semidirect |
| 14. #12532 | C4 | semidirect |
| 15. #12525 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #491550 | 2 |
| #483961 | 2 |
| #213274 | 2 |
| #141822 | 2 |
| #141807 | 2 |
| #135802 | 2 |
| #126540 | 2 |