Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
146869 conjugacy classes of subgroups.
The subgroup lattice has 210 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C25×C4⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C25×C4⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×#400 | C2 | 1 | 120 | yes | maximal, direct product factor |
| 4 | C24×#95 | C2 | 1 | 2 | yes | maximal |
| 5 | C24×#95 | C2 | 1 | 2 | yes | maximal |
| 6 | C24×#95 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C26×C4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C26×C4 | C22 | 1 | 2 | yes | — |
| 9 | C26×C4 | C22 | 1 | 2 | yes | — |
| 10 | C26×C4 | C22 | 1 | 1 | yes | characteristic |
| 11 | C26×C4 | C22 | 1 | 1 | yes | characteristic |
| 12 | C24×C4⋊C4 | — | 2 | 32 | no | — |
| 13 | C24×C4⋊C4 | — | 2 | 32 | no | — |
| 14 | C24×C4⋊C4 | C22 | 1 | 60 | yes | — |
| 15 | C24×C4⋊C4 | C22 | 1 | 60 | yes | — |
| 16 | C22×#400 | C22 | 1 | 2240 | yes | direct product factor |
| 17 | C23×#95 | C22 | 1 | 120 | yes | — |
| 18 | C23×#95 | C22 | 1 | 120 | yes | — |
| 19 | C23×#95 | C22 | 1 | 60 | yes | — |
| 20 | C27 | C23 | 1 | 1 | yes | characteristic |
| 21 | C25×C4 | — | 2 | 16 | no | — |
| 22 | C25×C4 | — | 2 | 16 | no | — |
| 23 | C25×C4 | — | 2 | 32 | no | — |
| 24 | C25×C4 | — | 2 | 16 | no | — |
| 25 | C25×C4 | — | 2 | 32 | no | — |
| 26 | C25×C4 | — | 2 | 32 | no | — |
| 27 | C25×C4 | — | 2 | 64 | no | — |
| 28 | C25×C4 | — | 2 | 32 | no | — |
| 29 | C25×C4 | — | 2 | 32 | no | — |
| 30 | C25×C4 | — | 2 | 32 | no | — |
| 31 | C25×C4 | D4 | 1 | 32 | yes | — |
| 32 | C25×C4 | Q8 | 1 | 32 | yes | — |
| 33 | C25×C4 | C23 | 1 | 30 | yes | — |
| 34 | C25×C4 | C23 | 1 | 60 | yes | — |
| 35 | C25×C4 | C23 | 1 | 60 | yes | — |
| 36 | C25×C4 | C23 | 1 | 30 | yes | — |
| 37 | C25×C4 | C23 | 1 | 30 | yes | — |
| 38 | C23×C4⋊C4 | — | 2 | 960 | no | — |
| 39 | C23×C4⋊C4 | — | 2 | 960 | no | — |
| 40 | C23×C4⋊C4 | C23 | 1 | 560 | yes | — |
| 41 | C23×C4⋊C4 | C23 | 1 | 560 | yes | — |
| 42 | C2×#400 | C23 | 1 | 7680 | yes | direct product factor |
| 43 | C22×#95 | C23 | 1 | 1120 | yes | — |
| 44 | C22×#95 | C23 | 1 | 1120 | yes | — |
| 45 | C22×#95 | C23 | 1 | 560 | yes | — |
| 46 | C26 | C2×D4 | 1 | 16 | yes | — |
| 47 | C26 | C2×Q8 | 1 | 16 | yes | — |
| 48 | C26 | #41 | 1 | 16 | yes | — |
| 49 | C26 | #41 | 1 | 32 | yes | — |
| 50 | C26 | #41 | 1 | 32 | yes | — |
| 51 | C26 | C24 | 1 | 15 | yes | — |
| 52 | C24×C4 | — | 2 | 512 | no | — |
| 53 | C24×C4 | — | 2 | 512 | no | — |
| 54 | C24×C4 | — | 2 | 240 | no | — |
| 55 | C24×C4 | — | 2 | 240 | no | — |
| 56 | C24×C4 | — | 2 | 480 | no | — |
| 57 | C24×C4 | — | 2 | 240 | no | — |
| 58 | C24×C4 | — | 2 | 480 | no | — |
| 59 | C24×C4 | — | 2 | 480 | no | — |
| 60 | C24×C4 | — | 2 | 960 | no | — |
| 61 | C24×C4 | — | 2 | 480 | no | — |
| 62 | C24×C4 | — | 2 | 480 | no | — |
| 63 | C24×C4 | — | 2 | 480 | no | — |
| 64 | C24×C4 | — | 4 | 256 | no | — |
| 65 | C24×C4 | — | 4 | 512 | no | — |
| 66 | C24×C4 | — | 4 | 512 | no | — |
| 67 | C24×C4 | C2×D4 | 1 | 480 | yes | — |
| 68 | C24×C4 | C2×Q8 | 1 | 480 | yes | — |
| 69 | C24×C4 | C24 | 1 | 140 | yes | — |
| 70 | C24×C4 | C24 | 1 | 280 | yes | — |
| 71 | C24×C4 | C24 | 1 | 280 | yes | — |
| 72 | C24×C4 | C24 | 1 | 140 | yes | — |
| 73 | C24×C4 | C24 | 1 | 140 | yes | — |
| 74 | C22×C4⋊C4 | — | 2 | 4480 | no | — |
| 75 | C22×C4⋊C4 | — | 2 | 4480 | no | — |
| 76 | C22×C4⋊C4 | C24 | 1 | 960 | yes | — |
| 77 | C22×C4⋊C4 | C24 | 1 | 960 | yes | — |
| 78 | #400 | C24 | 1 | 4096 | yes | direct product factor |
| 79 | C2×#95 | C24 | 1 | 1920 | yes | — |
| 80 | C2×#95 | C24 | 1 | 1920 | yes | — |
| 81 | C2×#95 | C24 | 1 | 960 | yes | — |
| 82 | C25 | C22⋊Q8 | 1 | 256 | yes | — |
| 83 | C25 | #123 | 1 | 256 | yes | — |
| 84 | C25 | #124 | 1 | 256 | yes | — |
| 85 | C25 | #125 | 1 | 512 | yes | — |
| 86 | C25 | #126 | 1 | 512 | yes | — |
| 87 | C25 | C22×D4 | 1 | 120 | yes | — |
| 88 | C25 | C22×Q8 | 1 | 120 | yes | — |
| 89 | C25 | C2×#41 | 1 | 120 | yes | — |
| 90 | C25 | C2×#41 | 1 | 240 | yes | — |
| 91 | C25 | C2×#41 | 1 | 240 | yes | — |
| 92 | C25 | C25 | 1 | 35 | yes | — |
| 93 | C23×C4 | — | 2 | 3840 | no | — |
| 94 | C23×C4 | — | 2 | 3840 | no | — |
| 95 | C23×C4 | — | 2 | 560 | no | — |
| 96 | C23×C4 | — | 2 | 560 | no | — |
| 97 | C23×C4 | — | 2 | 1120 | no | — |
| 98 | C23×C4 | — | 2 | 560 | no | — |
| 99 | C23×C4 | — | 2 | 1120 | no | — |
| 100 | C23×C4 | — | 2 | 1120 | no | — |
| 101 | C23×C4 | — | 2 | 2240 | no | — |
| 102 | C23×C4 | — | 2 | 1120 | no | — |
| 103 | C23×C4 | — | 2 | 1120 | no | — |
| 104 | C23×C4 | — | 2 | 1120 | no | — |
| 105 | C23×C4 | — | 4 | 1920 | no | — |
| 106 | C23×C4 | — | 4 | 3840 | no | — |
| 107 | C23×C4 | — | 4 | 3840 | no | — |
| 108 | C23×C4 | C22×D4 | 1 | 1120 | yes | — |
| 109 | C23×C4 | C22×Q8 | 1 | 1120 | yes | — |
| 110 | C23×C4 | C25 | 1 | 120 | yes | — |
| 111 | C23×C4 | C25 | 1 | 240 | yes | — |
| 112 | C23×C4 | C25 | 1 | 240 | yes | — |
| 113 | C23×C4 | C25 | 1 | 120 | yes | — |
| 114 | C23×C4 | C25 | 1 | 120 | yes | — |
| 115 | C2×C4⋊C4 | — | 2 | 3840 | no | — |
| 116 | C2×C4⋊C4 | — | 2 | 3840 | no | — |
| 117 | C2×C4⋊C4 | C25 | 1 | 256 | yes | — |
| 118 | C2×C4⋊C4 | C25 | 1 | 256 | yes | — |
| 119 | #95 | C25 | 1 | 256 | yes | — |
| 120 | #95 | C25 | 1 | 512 | yes | — |
| 121 | #95 | C25 | 1 | 512 | yes | — |
| 122 | C24 | #400 | 1 | 4096 | yes | direct product factor |
| 123 | C24 | C2×C22⋊Q8 | 1 | 960 | yes | — |
| 124 | C24 | C2×#123 | 1 | 960 | yes | — |
| 125 | C24 | C2×#124 | 1 | 960 | yes | — |
| 126 | C24 | C2×#125 | 1 | 1920 | yes | — |
| 127 | C24 | C2×#126 | 1 | 1920 | yes | — |
| 128 | C24 | C23×D4 | 1 | 140 | yes | — |
| 129 | C24 | C23×Q8 | 1 | 140 | yes | — |
| 130 | C24 | C22×#41 | 1 | 140 | yes | — |
| 131 | C24 | C22×#41 | 1 | 280 | yes | — |
| 132 | C24 | C22×#41 | 1 | 280 | yes | — |
| 133 | C24 | C26 | 1 | 15 | yes | — |
| 134 | C22×C4 | — | 2 | 240 | no | — |
| 135 | C22×C4 | — | 2 | 240 | no | — |
| 136 | C22×C4 | — | 2 | 480 | no | — |
| 137 | C22×C4 | — | 2 | 4480 | no | — |
| 138 | C22×C4 | — | 2 | 4480 | no | — |
| 139 | C22×C4 | — | 2 | 240 | no | — |
| 140 | C22×C4 | — | 2 | 480 | no | — |
| 141 | C22×C4 | — | 2 | 480 | no | — |
| 142 | C22×C4 | — | 2 | 960 | no | — |
| 143 | C22×C4 | — | 2 | 480 | no | — |
| 144 | C22×C4 | — | 2 | 480 | no | — |
| 145 | C22×C4 | — | 2 | 480 | no | — |
| 146 | C22×C4 | — | 4 | 2240 | no | — |
| 147 | C22×C4 | — | 4 | 4480 | no | — |
| 148 | C22×C4 | — | 4 | 4480 | no | — |
| 149 | C22×C4 | C23×D4 | 1 | 480 | yes | — |
| 150 | C22×C4 | C23×Q8 | 1 | 480 | yes | — |
| 151 | C22×C4 | C26 | 1 | 32 | yes | — |
| 152 | C22×C4 | C26 | 1 | 16 | yes | — |
| 153 | C22×C4 | C26 | 1 | 32 | yes | — |
| 154 | C22×C4 | C26 | 1 | 16 | yes | — |
| 155 | C22×C4 | C26 | 1 | 16 | yes | — |
| 156 | C4⋊C4 | — | 2 | 512 | no | — |
| 157 | C4⋊C4 | — | 2 | 512 | no | — |
| 158 | C23 | C2×#400 | 1 | 7680 | yes | direct product factor |
| 159 | C23 | C22×C22⋊Q8 | 1 | 560 | yes | — |
| 160 | C23 | C22×#123 | 1 | 560 | yes | — |
| 161 | C23 | C22×#124 | 1 | 560 | yes | — |
| 162 | C23 | C22×#125 | 1 | 1120 | yes | — |
| 163 | C23 | C22×#126 | 1 | 1120 | yes | — |
| 164 | C23 | C24×D4 | 1 | 30 | yes | — |
| 165 | C23 | C24×Q8 | 1 | 30 | yes | — |
| 166 | C23 | C23×#41 | 1 | 30 | yes | — |
| 167 | C23 | C23×#41 | 1 | 60 | yes | — |
| 168 | C23 | C23×#41 | 1 | 60 | yes | — |
| 169 | C23 | C27 | 1 | 1 | yes | characteristic |
| 170 | C2×C4 | — | 2 | 16 | no | — |
| 171 | C2×C4 | — | 2 | 16 | no | — |
| 172 | C2×C4 | — | 2 | 32 | no | — |
| 173 | C2×C4 | — | 2 | 960 | no | — |
| 174 | C2×C4 | — | 2 | 960 | no | — |
| 175 | C2×C4 | — | 2 | 16 | no | — |
| 176 | C2×C4 | — | 2 | 32 | no | — |
| 177 | C2×C4 | — | 2 | 32 | no | — |
| 178 | C2×C4 | — | 2 | 64 | no | — |
| 179 | C2×C4 | — | 2 | 32 | no | — |
| 180 | C2×C4 | — | 2 | 32 | no | — |
| 181 | C2×C4 | — | 2 | 32 | no | — |
| 182 | C2×C4 | — | 4 | 480 | no | — |
| 183 | C2×C4 | — | 4 | 960 | no | — |
| 184 | C2×C4 | — | 4 | 960 | no | — |
| 185 | C2×C4 | C24×D4 | 1 | 32 | yes | — |
| 186 | C2×C4 | C24×Q8 | 1 | 32 | yes | — |
| 187 | C22 | C22×#400 | 1 | 2240 | yes | direct product factor |
| 188 | C22 | C23×C22⋊Q8 | 1 | 60 | yes | — |
| 189 | C22 | C23×#123 | 1 | 60 | yes | — |
| 190 | C22 | C23×#124 | 1 | 60 | yes | — |
| 191 | C22 | C23×#125 | 1 | 120 | yes | — |
| 192 | C22 | C23×#126 | 1 | 120 | yes | — |
| 193 | C22 | C25×D4 | 1 | 1 | yes | characteristic |
| 194 | C22 | C25×Q8 | 1 | 1 | yes | characteristic |
| 195 | C22 | C24×#41 | 1 | 1 | yes | characteristic |
| 196 | C22 | C24×#41 | 1 | 2 | yes | — |
| 197 | C22 | C24×#41 | 1 | 2 | yes | — |
| 198 | C4 | — | 2 | 32 | no | — |
| 199 | C4 | — | 2 | 32 | no | — |
| 200 | C4 | — | 4 | 16 | no | — |
| 201 | C4 | — | 4 | 32 | no | — |
| 202 | C4 | — | 4 | 32 | no | — |
| 203 | C2 | C23×#400 | 1 | 120 | yes | direct product factor |
| 204 | C2 | C24×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 205 | C2 | C24×#123 | 1 | 1 | yes | characteristic |
| 206 | C2 | C24×#124 | 1 | 1 | yes | characteristic |
| 207 | C2 | C24×#125 | 1 | 2 | yes | — |
| 208 | C2 | C24×#126 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C27 | C23 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C23×#400 | C2 | direct (N × H) |
| 16. C22×#400 | C22 | direct (N × H) |
| 42. C2×#400 | C23 | direct (N × H) |
| 78. #400 | C24 | direct (N × H) |
| 122. C24 | #400 | direct (N × H) |
| 158. C23 | C2×#400 | direct (N × H) |
| 187. C22 | C22×#400 | direct (N × H) |
| 203. C2 | C23×#400 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C4⋊C4 | 2 |
| C23×#400 | 2 |
| C24×#95 | 2 |