Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
379 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #513308 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #512945 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #153560 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #152295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #94764 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#2262 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#2258 | C22 | 1 | 1 | yes | characteristic |
| 8 | #33610 | — | 2 | 1 | no | — |
| 9 | #13737 | — | 2 | 4 | no | — |
| 10 | C2×M128 | — | 2 | 2 | no | — |
| 11 | C2×M128 | C22 | 1 | 2 | yes | — |
| 12 | #13698 | C4 | 1 | 2 | yes | — |
| 13 | #13697 | C4 | 1 | 2 | yes | — |
| 14 | #13696 | — | 2 | 2 | no | — |
| 15 | C2×#1425 | C22 | 1 | 2 | yes | — |
| 16 | C2×#1424 | C22 | 1 | 1 | yes | characteristic |
| 17 | #13616 | — | 2 | 1 | no | — |
| 18 | #12840 | — | 2 | 1 | no | — |
| 19 | #7539 | — | 2 | 4 | no | — |
| 20 | #7476 | — | 2 | 1 | no | — |
| 21 | C2×#508 | — | 2 | 1 | no | — |
| 22 | C2×#504 | D4 | 1 | 1 | yes | characteristic |
| 23 | #3061 | — | 4 | 1 | no | — |
| 24 | #2262 | — | 2 | 2 | no | — |
| 25 | #2262 | — | 2 | 2 | no | — |
| 26 | C2×Q64 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C2×SD64 | — | 2 | 1 | no | — |
| 28 | C2×D32 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #2258 | — | 2 | 2 | no | — |
| 30 | #2258 | — | 2 | 2 | no | — |
| 31 | C2×M64 | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×M64 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×C32 | D4 | 1 | 2 | yes | — |
| 34 | C22×C32 | C23 | 1 | 1 | yes | characteristic |
| 35 | #2220 | — | 4 | 2 | no | — |
| 36 | #2216 | — | 2 | 2 | no | — |
| 37 | #2215 | — | 2 | 2 | no | — |
| 38 | C2×#368 | — | 2 | 2 | no | — |
| 39 | C2×#366 | — | 2 | 1 | no | — |
| 40 | #2139 | — | 4 | 1 | no | — |
| 41 | #1813 | — | 4 | 1 | no | — |
| 42 | M128 | — | 2 | 4 | no | — |
| 43 | M128 | — | 2 | 2 | no | — |
| 44 | M128 | — | 2 | 2 | no | — |
| 45 | C2×C64 | — | 2 | 4 | no | — |
| 46 | C2×C64 | D4 | 1 | 4 | yes | complemented |
| 47 | #1425 | — | 2 | 2 | no | — |
| 48 | #1425 | — | 2 | 2 | no | — |
| 49 | #1424 | — | 2 | 1 | no | — |
| 50 | #1424 | C2×C4 | 1 | 2 | yes | — |
| 51 | #1416 | — | 2 | 2 | no | — |
| 52 | #1415 | — | 2 | 2 | no | — |
| 53 | #1385 | — | 4 | 1 | no | — |
| 54 | C2×#135 | — | 4 | 1 | no | — |
| 55 | C2×#131 | D8 | 1 | 1 | yes | characteristic |
| 56 | #537 | — | 8 | 1 | no | — |
| 57 | #508 | — | 4 | 2 | no | — |
| 58 | #508 | — | 4 | 2 | no | — |
| 59 | C2×Q32 | — | 2 | 1 | no | — |
| 60 | C2×SD32 | — | 4 | 1 | no | — |
| 61 | C2×D16 | — | 2 | 1 | no | — |
| 62 | #504 | — | 2 | 2 | no | — |
| 63 | #504 | — | 2 | 2 | no | — |
| 64 | C2×M32 | D8 | 1 | 1 | yes | characteristic |
| 65 | C2×M32 | SD16 | 1 | 2 | yes | — |
| 66 | C22×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 67 | C22×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | #468 | — | 8 | 2 | no | — |
| 69 | #467 | — | 4 | 2 | no | — |
| 70 | #466 | — | 4 | 2 | no | — |
| 71 | C2×#108 | — | 4 | 2 | no | — |
| 72 | C2×#107 | — | 4 | 1 | no | — |
| 73 | #417 | — | 8 | 1 | no | — |
| 74 | #413 | — | 8 | 1 | no | — |
| 75 | Q64 | — | 2 | 2 | no | — |
| 76 | SD64 | — | 4 | 1 | no | — |
| 77 | SD64 | — | 4 | 1 | no | — |
| 78 | D32 | — | 2 | 2 | no | — |
| 79 | M64 | — | 2 | 2 | no | — |
| 80 | M64 | — | 2 | 4 | no | — |
| 81 | C2×C32 | — | 2 | 2 | no | — |
| 82 | C2×C32 | — | 2 | 2 | no | — |
| 83 | C2×C32 | C22⋊C4 | 1 | 2 | yes | — |
| 84 | C2×C32 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C32 | C2×D4 | 1 | 2 | yes | — |
| 86 | C2×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C32 | #41 | 1 | 2 | yes | — |
| 88 | #368 | — | 4 | 2 | no | — |
| 89 | #368 | — | 4 | 2 | no | — |
| 90 | #366 | — | 2 | 2 | no | — |
| 91 | #366 | — | 4 | 1 | no | — |
| 92 | #358 | — | 4 | 2 | no | — |
| 93 | #357 | — | 4 | 2 | no | — |
| 94 | #338 | — | 8 | 1 | no | — |
| 95 | C64 | — | 2 | 4 | no | — |
| 96 | C2×#41 | — | 8 | 1 | no | — |
| 97 | C2×#41 | D16 | 1 | 1 | yes | characteristic |
| 98 | #135 | — | 8 | 2 | no | — |
| 99 | #135 | — | 8 | 2 | no | — |
| 100 | C2×Q16 | — | 4 | 1 | no | — |
| 101 | C2×SD16 | — | 8 | 1 | no | — |
| 102 | C2×D8 | — | 4 | 1 | no | — |
| 103 | #131 | — | 2 | 2 | no | — |
| 104 | #131 | — | 2 | 2 | no | — |
| 105 | C2×M16 | — | 8 | 2 | no | — |
| 106 | C2×M16 | D16 | 1 | 1 | yes | characteristic |
| 107 | C2×M16 | SD32 | 1 | 2 | yes | — |
| 108 | C22×C8 | #102 | 1 | 2 | yes | — |
| 109 | C22×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 110 | C22⋊Q8 | — | 8 | 2 | no | — |
| 111 | C22⋊Q8 | — | 16 | 1 | no | — |
| 112 | #121 | — | 8 | 2 | no | — |
| 113 | #121 | — | 16 | 1 | no | — |
| 114 | #121 | — | 16 | 2 | no | — |
| 115 | C2×C4⋊C4 | — | 8 | 1 | no | — |
| 116 | Q32 | — | 4 | 1 | no | — |
| 117 | Q32 | — | 4 | 1 | no | — |
| 118 | SD32 | — | 8 | 2 | no | — |
| 119 | D16 | — | 4 | 1 | no | — |
| 120 | D16 | — | 4 | 1 | no | — |
| 121 | M32 | — | 2 | 2 | no | — |
| 122 | M32 | — | 2 | 4 | no | — |
| 123 | C2×C16 | — | 2 | 2 | no | — |
| 124 | C2×C16 | — | 2 | 2 | no | — |
| 125 | C2×C16 | #102 | 1 | 2 | yes | — |
| 126 | C2×C16 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 127 | C2×C16 | #121 | 1 | 2 | yes | — |
| 128 | C2×C16 | C2×SD16 | 1 | 1 | yes | characteristic |
| 129 | C2×C16 | #135 | 1 | 2 | yes | — |
| 130 | #108 | — | 8 | 2 | no | — |
| 131 | #108 | — | 8 | 2 | no | — |
| 132 | #107 | — | 4 | 2 | no | — |
| 133 | #107 | — | 8 | 1 | no | — |
| 134 | #103 | — | 8 | 2 | no | — |
| 135 | #102 | — | 8 | 2 | no | — |
| 136 | #101 | — | 16 | 1 | no | — |
| 137 | #100 | — | 16 | 1 | no | — |
| 138 | C32 | — | 2 | 2 | no | — |
| 139 | C32 | C4×D4 | 1 | 2 | yes | — |
| 140 | C32 | #120 | 1 | 1 | yes | characteristic |
| 141 | C32 | #123 | 1 | 1 | yes | characteristic |
| 142 | #41 | — | 2 | 2 | no | — |
| 143 | #41 | — | 2 | 2 | no | — |
| 144 | #41 | — | 16 | 2 | no | — |
| 145 | #41 | — | 16 | 2 | no | — |
| 146 | C2×Q8 | — | 8 | 1 | no | — |
| 147 | C2×Q8 | D32 | 1 | 1 | yes | characteristic |
| 148 | C2×D4 | — | 8 | 1 | no | — |
| 149 | C2×D4 | — | 16 | 1 | no | — |
| 150 | C2×D4 | — | 16 | 2 | no | — |
| 151 | C2×D4 | D32 | 1 | 1 | yes | characteristic |
| 152 | C2×D4 | SD64 | 1 | 2 | yes | — |
| 153 | C22×C4 | — | 16 | 1 | no | — |
| 154 | C22×C4 | — | 16 | 1 | no | — |
| 155 | C22×C4 | #357 | 1 | 2 | yes | — |
| 156 | C22×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 157 | Q16 | — | 8 | 1 | no | — |
| 158 | Q16 | — | 8 | 1 | no | — |
| 159 | SD16 | — | 16 | 2 | no | — |
| 160 | D8 | — | 8 | 1 | no | — |
| 161 | D8 | — | 8 | 1 | no | — |
| 162 | M16 | — | 2 | 2 | no | — |
| 163 | M16 | — | 2 | 4 | no | — |
| 164 | M16 | — | 16 | 2 | no | — |
| 165 | M16 | — | 16 | 2 | no | — |
| 166 | C2×C8 | — | 2 | 2 | no | — |
| 167 | C2×C8 | — | 2 | 2 | no | — |
| 168 | C2×C8 | — | 16 | 2 | no | — |
| 169 | C2×C8 | #357 | 1 | 2 | yes | — |
| 170 | C2×C8 | C2×#102 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | #466 | 1 | 2 | yes | — |
| 172 | C2×C8 | C2×SD32 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | #508 | 1 | 2 | yes | — |
| 174 | C4⋊C4 | — | 8 | 2 | no | — |
| 175 | C4⋊C4 | — | 16 | 2 | no | — |
| 176 | C4⋊C4 | — | 16 | 1 | no | — |
| 177 | C22⋊C4 | — | 16 | 2 | no | — |
| 178 | C22⋊C4 | — | 16 | 2 | no | — |
| 179 | C16 | — | 2 | 2 | no | — |
| 180 | C16 | #386 | 1 | 1 | yes | characteristic |
| 181 | C16 | #416 | 1 | 1 | yes | characteristic |
| 182 | C16 | #465 | 1 | 2 | yes | — |
| 183 | C23 | — | 16 | 1 | no | — |
| 184 | C23 | #1415 | 1 | 2 | yes | — |
| 185 | C23 | C2×SD64 | 1 | 1 | yes | characteristic |
| 186 | Q8 | — | 2 | 2 | no | — |
| 187 | Q8 | — | 16 | 1 | no | — |
| 188 | Q8 | — | 16 | 1 | no | — |
| 189 | D4 | — | 2 | 4 | no | — |
| 190 | D4 | — | 2 | 2 | no | — |
| 191 | D4 | — | 16 | 1 | no | — |
| 192 | D4 | — | 16 | 1 | no | — |
| 193 | D4 | — | 32 | 2 | no | — |
| 194 | D4 | — | 64 | 2 | no | — |
| 195 | C2×C4 | — | 2 | 2 | no | — |
| 196 | C2×C4 | — | 2 | 2 | no | — |
| 197 | C2×C4 | — | 16 | 2 | no | — |
| 198 | C2×C4 | — | 16 | 1 | no | — |
| 199 | C2×C4 | — | 32 | 2 | no | — |
| 200 | C2×C4 | — | 64 | 1 | no | — |
| 201 | C2×C4 | #1415 | 1 | 2 | yes | — |
| 202 | C2×C4 | C2×#357 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | #2215 | 1 | 2 | yes | — |
| 204 | C2×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | #2262 | 1 | 2 | yes | — |
| 206 | C8 | — | 2 | 2 | no | — |
| 207 | C8 | — | 32 | 2 | no | — |
| 208 | C8 | #1893 | 1 | 1 | yes | characteristic |
| 209 | C8 | #2138 | 1 | 1 | yes | characteristic |
| 210 | C8 | #2217 | 1 | 2 | yes | — |
| 211 | C22 | — | 2 | 2 | no | — |
| 212 | C22 | — | 4 | 1 | no | — |
| 213 | C22 | — | 4 | 2 | no | — |
| 214 | C22 | — | 32 | 1 | no | — |
| 215 | C22 | — | 32 | 1 | no | — |
| 216 | C22 | — | 32 | 1 | no | — |
| 217 | C22 | C2×#1415 | 1 | 1 | yes | characteristic |
| 218 | C22 | #13696 | 1 | 2 | yes | — |
| 219 | C4 | — | 2 | 2 | no | — |
| 220 | C4 | — | 32 | 2 | no | — |
| 221 | C4 | — | 32 | 1 | no | — |
| 222 | C4 | #13059 | 1 | 1 | yes | characteristic |
| 223 | C4 | #13615 | 1 | 1 | yes | characteristic |
| 224 | C4 | #13694 | 1 | 2 | yes | — |
| 225 | C2 | — | 2 | 1 | no | — |
| 226 | C2 | — | 2 | 2 | no | — |
| 227 | C2 | — | 4 | 2 | no | — |
| 228 | C2 | — | 64 | 1 | no | — |
| 229 | C2 | #152629 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152629 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | #512945 | C2 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #513308 | C2 | semidirect |
| 3. #153560 | C2 | semidirect |
| 4. #152295 | C2 | semidirect |
| 5. #94764 | C2 | semidirect |
| 46. C2×C64 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #513308 | 2 |
| #512945 | 2 |
| #153560 | 2 |
| #152295 | 2 |
| #94764 | 2 |