Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
228 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 33 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 | C2×#87838 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#87837 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11717033 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4×#6037 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11716897 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11716798 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11716797 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C4×C5×D5 | — | 2 | 1 | no | — |
| 9 | C2×#6039 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#6037 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#6037 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#6034 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#6034 | C22 | 1 | 1 | yes | characteristic |
| 14 | #87838 | — | 2 | 2 | no | — |
| 15 | #87837 | — | 2 | 2 | no | — |
| 16 | C4×#1039 | C22 | 1 | 1 | yes | characteristic |
| 17 | C5×#685 | — | 2 | 1 | no | — |
| 18 | #87781 | C22 | 1 | 1 | yes | characteristic |
| 19 | #87778 | — | 2 | 1 | no | — |
| 20 | #87778 | — | 2 | 1 | no | — |
| 21 | C22×C5×D5 | — | 2 | 1 | no | — |
| 22 | C22×C5×D5 | — | 2 | 1 | no | — |
| 23 | C2×C4×C52 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×#1039 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×C5×Dic5 | — | 2 | 1 | no | — |
| 27 | C2×C5×Dic5 | — | 2 | 1 | no | — |
| 28 | C4×C5×D5 | — | 4 | 2 | no | — |
| 29 | #6039 | — | 2 | 1 | no | — |
| 30 | #6039 | — | 2 | 1 | no | — |
| 31 | #6037 | — | 2 | 1 | no | — |
| 32 | #6037 | — | 2 | 1 | no | — |
| 33 | #6037 | — | 2 | 1 | no | — |
| 34 | #6037 | D4 | 1 | 2 | yes | — |
| 35 | #6034 | — | 2 | 1 | no | — |
| 36 | #6034 | — | 2 | 1 | no | — |
| 37 | C4×C22⋊D5 | — | 10 | 1 | no | — |
| 38 | #4031 | — | 10 | 1 | no | — |
| 39 | #4004 | — | 10 | 1 | no | — |
| 40 | C22×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 41 | C2×C5×D5 | — | 2 | 2 | no | — |
| 42 | C2×C5×D5 | — | 4 | 1 | no | — |
| 43 | C2×C5×D5 | — | 4 | 1 | no | — |
| 44 | C2×C5×D5 | — | 4 | 1 | no | — |
| 45 | C2×C5×D5 | — | 4 | 1 | no | — |
| 46 | C2×C5×D5 | — | 4 | 1 | no | — |
| 47 | C4×C52 | — | 2 | 1 | no | — |
| 48 | #1039 | — | 2 | 1 | no | — |
| 49 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | #1039 | #41 | 1 | 1 | yes | characteristic |
| 51 | C5×Dic5 | — | 4 | 1 | no | — |
| 52 | C5×Dic5 | — | 4 | 1 | no | — |
| 53 | C22×C4×C5 | — | 10 | 1 | no | — |
| 54 | C2×C22⋊D5 | — | 10 | 1 | no | — |
| 55 | C2×C22⋊D5 | — | 10 | 1 | no | — |
| 56 | C2×C4×D5 | — | 2 | 1 | no | — |
| 57 | C2×Dic10 | — | 10 | 1 | no | — |
| 58 | C2×#170 | — | 10 | 1 | no | — |
| 59 | C2×#170 | — | 10 | 1 | no | — |
| 60 | C5×C22⋊C4 | — | 10 | 1 | no | — |
| 61 | C22⋊Dic5 | — | 10 | 1 | no | — |
| 62 | C22⋊Dic5 | — | 10 | 1 | no | — |
| 63 | #685 | — | 2 | 1 | no | — |
| 64 | #685 | — | 10 | 1 | no | — |
| 65 | #685 | — | 10 | 1 | no | — |
| 66 | #683 | — | 10 | 1 | no | — |
| 67 | C4×Dic5 | — | 10 | 1 | no | — |
| 68 | C4×Dic5 | — | 10 | 1 | no | — |
| 69 | C4×Dic5 | — | 10 | 1 | no | — |
| 70 | #463 | — | 25 | 1 | no | maximal, Sylow |
| 71 | C2×C52 | #121 | 1 | 1 | yes | characteristic |
| 72 | C2×C52 | #135 | 1 | 1 | yes | characteristic |
| 73 | C2×C52 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 74 | C5×D5 | — | 4 | 2 | no | — |
| 75 | C5×D5 | — | 8 | 1 | no | — |
| 76 | C23×C5 | — | 10 | 1 | no | — |
| 77 | C23×C5 | — | 10 | 1 | no | — |
| 78 | C22×D5 | — | 2 | 1 | no | — |
| 79 | C22×D5 | — | 2 | 1 | no | — |
| 80 | C2×C4×C5 | — | 2 | 1 | no | — |
| 81 | C2×C4×C5 | — | 2 | 1 | no | — |
| 82 | C2×C4×C5 | — | 2 | 1 | no | — |
| 83 | C2×C4×C5 | — | 10 | 1 | no | — |
| 84 | C2×C4×C5 | — | 10 | 1 | no | — |
| 85 | C2×C4×C5 | — | 20 | 2 | no | — |
| 86 | C22⋊D5 | — | 10 | 2 | no | — |
| 87 | C22⋊D5 | — | 10 | 2 | no | — |
| 88 | C22⋊D5 | — | 20 | 1 | no | — |
| 89 | C22⋊D5 | — | 20 | 1 | no | — |
| 90 | C2×Dic5 | — | 2 | 1 | no | — |
| 91 | C2×Dic5 | — | 2 | 1 | no | — |
| 92 | C2×Dic5 | — | 10 | 1 | no | — |
| 93 | C2×Dic5 | — | 10 | 1 | no | — |
| 94 | C2×Dic5 | — | 10 | 1 | no | — |
| 95 | C2×Dic5 | — | 10 | 1 | no | — |
| 96 | C2×Dic5 | — | 10 | 1 | no | — |
| 97 | C2×Dic5 | — | 10 | 1 | no | — |
| 98 | C4×D5 | — | 4 | 2 | no | — |
| 99 | Dic10 | — | 20 | 1 | no | — |
| 100 | Dic10 | — | 20 | 1 | no | — |
| 101 | #170 | — | 20 | 1 | no | — |
| 102 | #170 | — | 20 | 1 | no | — |
| 103 | C2×SD16 | — | 25 | 1 | no | — |
| 104 | C2×D8 | — | 25 | 1 | no | — |
| 105 | #124 | — | 25 | 1 | no | — |
| 106 | C4×D4 | — | 25 | 1 | no | — |
| 107 | C4⋊C8 | — | 25 | 1 | no | — |
| 108 | #103 | — | 25 | 1 | no | — |
| 109 | #102 | — | 25 | 1 | no | — |
| 110 | C52 | #463 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 111 | C22×C5 | — | 2 | 1 | no | — |
| 112 | C22×C5 | — | 2 | 1 | no | — |
| 113 | C22×C5 | — | 2 | 1 | no | — |
| 114 | C22×C5 | — | 10 | 2 | no | — |
| 115 | C22×C5 | — | 20 | 1 | no | — |
| 116 | C22×C5 | — | 20 | 1 | no | — |
| 117 | C22×C5 | — | 20 | 1 | no | — |
| 118 | C22×C5 | — | 20 | 1 | no | — |
| 119 | C22×C5 | — | 20 | 1 | no | — |
| 120 | C2×D5 | — | 2 | 2 | no | — |
| 121 | C2×D5 | — | 4 | 1 | no | — |
| 122 | C2×D5 | — | 4 | 1 | no | — |
| 123 | C2×D5 | — | 4 | 1 | no | — |
| 124 | C2×D5 | — | 4 | 1 | no | — |
| 125 | C2×D5 | — | 4 | 1 | no | — |
| 126 | C4×C5 | — | 2 | 2 | no | — |
| 127 | C4×C5 | — | 4 | 1 | no | — |
| 128 | C4×C5 | — | 4 | 1 | no | — |
| 129 | C4×C5 | — | 20 | 1 | no | — |
| 130 | C4×C5 | — | 20 | 1 | no | — |
| 131 | Dic5 | — | 4 | 1 | no | — |
| 132 | Dic5 | — | 4 | 1 | no | — |
| 133 | Dic5 | — | 10 | 1 | no | — |
| 134 | Dic5 | — | 10 | 1 | no | — |
| 135 | Dic5 | — | 10 | 1 | no | — |
| 136 | Dic5 | — | 10 | 1 | no | — |
| 137 | Dic5 | — | 10 | 1 | no | — |
| 138 | Dic5 | — | 10 | 1 | no | — |
| 139 | Dic5 | — | 10 | 2 | no | — |
| 140 | Dic5 | — | 20 | 1 | no | — |
| 141 | Dic5 | — | 20 | 1 | no | — |
| 142 | C2×Q8 | — | 25 | 1 | no | — |
| 143 | C2×D4 | — | 25 | 1 | no | — |
| 144 | C2×D4 | — | 25 | 1 | no | — |
| 145 | C22×C4 | — | 10 | 1 | no | — |
| 146 | SD16 | — | 50 | 2 | no | — |
| 147 | D8 | — | 50 | 2 | no | — |
| 148 | C2×C8 | — | 25 | 1 | no | — |
| 149 | C2×C8 | — | 25 | 1 | no | — |
| 150 | C4⋊C4 | — | 25 | 1 | no | — |
| 151 | C22⋊C4 | — | 10 | 1 | no | — |
| 152 | C22⋊C4 | — | 50 | 1 | no | — |
| 153 | C22⋊C4 | — | 50 | 1 | no | — |
| 154 | C42 | — | 25 | 1 | no | — |
| 155 | C2×C5 | — | 2 | 1 | no | — |
| 156 | C2×C5 | — | 2 | 1 | no | — |
| 157 | C2×C5 | — | 2 | 1 | no | — |
| 158 | C2×C5 | — | 2 | 1 | no | — |
| 159 | C2×C5 | — | 2 | 1 | no | — |
| 160 | C2×C5 | — | 2 | 1 | no | — |
| 161 | C2×C5 | — | 2 | 1 | no | — |
| 162 | C2×C5 | — | 2 | 1 | no | — |
| 163 | C2×C5 | — | 2 | 1 | no | — |
| 164 | C2×C5 | — | 20 | 2 | no | — |
| 165 | C2×C5 | — | 40 | 1 | no | — |
| 166 | D5 | — | 4 | 2 | no | — |
| 167 | D5 | — | 8 | 1 | no | — |
| 168 | C23 | — | 10 | 1 | no | — |
| 169 | C23 | — | 10 | 1 | no | — |
| 170 | Q8 | — | 50 | 1 | no | — |
| 171 | Q8 | — | 50 | 1 | no | — |
| 172 | D4 | — | 25 | 2 | no | — |
| 173 | D4 | — | 50 | 1 | no | — |
| 174 | D4 | — | 50 | 1 | no | — |
| 175 | D4 | — | 50 | 1 | no | — |
| 176 | C2×C4 | — | 10 | 1 | no | — |
| 177 | C2×C4 | — | 10 | 1 | no | — |
| 178 | C2×C4 | — | 20 | 2 | no | — |
| 179 | C2×C4 | — | 25 | 1 | no | — |
| 180 | C2×C4 | — | 25 | 1 | no | — |
| 181 | C2×C4 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 182 | C8 | — | 50 | 1 | no | — |
| 183 | C8 | — | 50 | 1 | no | — |
| 184 | C5 | — | 2 | 1 | no | — |
| 185 | C5 | — | 2 | 1 | no | — |
| 186 | C5 | — | 2 | 1 | no | — |
| 187 | C22 | — | 10 | 2 | no | — |
| 188 | C22 | — | 20 | 1 | no | — |
| 189 | C22 | — | 20 | 1 | no | — |
| 190 | C22 | — | 20 | 1 | no | — |
| 191 | C22 | — | 20 | 1 | no | — |
| 192 | C22 | — | 20 | 1 | no | — |
| 193 | C22 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 194 | C4 | — | 2 | 1 | no | — |
| 195 | C4 | — | 20 | 1 | no | — |
| 196 | C4 | — | 20 | 1 | no | — |
| 197 | C4 | — | 25 | 1 | no | — |
| 198 | C4 | — | 25 | 1 | no | — |
| 199 | C4 | — | 50 | 1 | no | — |
| 200 | C2 | — | 20 | 2 | no | — |
| 201 | C2 | — | 40 | 1 | no | — |
| 202 | C2 | #11717381 | 1 | 1 | yes | characteristic |
| 203 | C2 | #11717388 | 1 | 1 | yes | characteristic |
| 204 | C2 | #11717394 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#87838 | C2 | semidirect |
| 3. #11717033 | C2 | semidirect |
| 4. C4×#6037 | C2 | semidirect |
| 5. #11716897 | C2 | semidirect |
| 6. #11716798 | C2 | semidirect |
| 7. #11716797 | C2 | semidirect |
| 110. C52 | #463 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87838 | 2 |
| C2×#87837 | 2 |
| #11717033 | 2 |
| C4×#6037 | 2 |
| #11716897 | 2 |
| #11716798 | 2 |
| #11716797 | 2 |
| #463 | 25 |