Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
572 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 77 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 | C7×#2811 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11727446 | C2 | 1 | 3 | yes | maximal |
| 3 | #11725450 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#6551 | C22 | 1 | 1 | yes | characteristic |
| 5 | C7×#395 | — | 2 | 3 | no | — |
| 6 | C7×#391 | — | 2 | 3 | no | — |
| 7 | C7×#391 | C22 | 1 | 3 | yes | — |
| 8 | C2×#6410 | C22 | 1 | 3 | yes | — |
| 9 | #89738 | — | 2 | 3 | no | — |
| 10 | C2×#6379 | — | 2 | 3 | no | — |
| 11 | #13378 | — | 7 | 1 | no | maximal, Sylow |
| 12 | C22×C7×Q8 | — | 2 | 1 | no | — |
| 13 | C22×C7×Q8 | — | 2 | 1 | no | — |
| 14 | C22×C7×Q8 | C23 | 1 | 1 | yes | characteristic |
| 15 | #6551 | — | 2 | 3 | no | — |
| 16 | #6551 | — | 4 | 2 | no | — |
| 17 | #6551 | Q8 | 1 | 2 | yes | complemented |
| 18 | C2×#1161 | — | 4 | 1 | no | — |
| 19 | C2×Q8×D7 | — | 4 | 1 | no | — |
| 20 | C2×#1157 | — | 2 | 3 | no | — |
| 21 | C2×C7×C4⋊C4 | — | 2 | 3 | no | — |
| 22 | C2×C7×C4⋊C4 | D4 | 1 | 3 | yes | — |
| 23 | C2×C42×C7 | D4 | 1 | 3 | yes | — |
| 24 | C2×#1136 | — | 2 | 3 | no | — |
| 25 | C7×#95 | — | 2 | 3 | no | — |
| 26 | C7×#95 | — | 2 | 6 | no | — |
| 27 | #6410 | — | 2 | 6 | no | — |
| 28 | #6384 | — | 4 | 3 | no | — |
| 29 | #6383 | — | 4 | 3 | no | — |
| 30 | #6379 | — | 4 | 6 | no | — |
| 31 | #2811 | D7 | 1 | 1 | yes | characteristic, complemented |
| 32 | #1793 | — | 7 | 3 | no | — |
| 33 | #1394 | — | 7 | 3 | no | — |
| 34 | C2×C7×Q8 | — | 2 | 4 | no | — |
| 35 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 36 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 37 | C2×C7×Q8 | — | 8 | 1 | no | — |
| 38 | C2×C7×Q8 | — | 8 | 1 | no | — |
| 39 | C2×C7×Q8 | — | 8 | 2 | no | — |
| 40 | C2×C7×Q8 | — | 8 | 1 | no | — |
| 41 | C2×C7×Q8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 42 | C2×C7×Q8 | #41 | 1 | 3 | yes | — |
| 43 | C22×C4×C7 | — | 2 | 3 | no | — |
| 44 | C22×C4×C7 | — | 2 | 3 | no | — |
| 45 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 46 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 47 | #1161 | — | 4 | 2 | no | — |
| 48 | #1161 | — | 4 | 6 | no | — |
| 49 | Q8×D7 | — | 4 | 2 | no | — |
| 50 | Q8×D7 | — | 4 | 6 | no | — |
| 51 | #1157 | — | 2 | 6 | no | — |
| 52 | #1157 | — | 2 | 6 | no | — |
| 53 | #1157 | — | 4 | 3 | no | — |
| 54 | #1157 | — | 4 | 3 | no | — |
| 55 | C2×D28 | — | 2 | 3 | no | — |
| 56 | C2×C4×D7 | — | 4 | 3 | no | — |
| 57 | C2×Dic14 | — | 2 | 3 | no | — |
| 58 | C7×C4⋊C4 | — | 4 | 3 | no | — |
| 59 | C7×C4⋊C4 | — | 8 | 3 | no | — |
| 60 | C42×C7 | — | 4 | 3 | no | — |
| 61 | #1136 | — | 4 | 3 | no | — |
| 62 | #1136 | — | 4 | 3 | no | — |
| 63 | C2×C7⋊C8 | — | 4 | 3 | no | — |
| 64 | C2×2−1+4 | — | 7 | 1 | no | — |
| 65 | C2×#104 | — | 14 | 3 | no | — |
| 66 | #417 | — | 14 | 3 | no | — |
| 67 | C2×#101 | — | 7 | 3 | no | — |
| 68 | #395 | — | 2 | 3 | no | — |
| 69 | #391 | — | 2 | 3 | no | — |
| 70 | #391 | C2×D7 | 1 | 3 | yes | — |
| 71 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 72 | C22×D7 | — | 4 | 1 | no | — |
| 73 | C7×Q8 | — | 4 | 1 | no | — |
| 74 | C7×Q8 | — | 4 | 3 | no | — |
| 75 | C7×Q8 | — | 8 | 4 | no | — |
| 76 | C7×Q8 | — | 8 | 4 | no | — |
| 77 | C2×C4×C7 | — | 2 | 3 | no | — |
| 78 | C2×C4×C7 | — | 2 | 3 | no | — |
| 79 | C2×C4×C7 | — | 2 | 6 | no | — |
| 80 | C2×C4×C7 | — | 4 | 3 | no | — |
| 81 | C2×C4×C7 | — | 4 | 3 | no | — |
| 82 | C2×C4×C7 | — | 4 | 6 | no | — |
| 83 | C2×C4×C7 | — | 4 | 3 | no | — |
| 84 | C2×C4×C7 | C22⋊Q8 | 1 | 3 | yes | — |
| 85 | C2×C4×C7 | #124 | 1 | 3 | yes | — |
| 86 | C22⋊D7 | — | 4 | 2 | no | — |
| 87 | C22⋊D7 | — | 4 | 2 | no | — |
| 88 | C2×Dic7 | — | 4 | 1 | no | — |
| 89 | D28 | — | 2 | 6 | no | — |
| 90 | D28 | — | 4 | 3 | no | — |
| 91 | C4×D7 | — | 4 | 6 | no | — |
| 92 | C4×D7 | — | 4 | 6 | no | — |
| 93 | Dic14 | — | 2 | 6 | no | — |
| 94 | Dic14 | — | 4 | 3 | no | — |
| 95 | C7⋊C8 | — | 8 | 3 | no | — |
| 96 | 2−1+4 | — | 7 | 2 | no | — |
| 97 | 2−1+4 | — | 14 | 3 | no | — |
| 98 | 2−1+4 | — | 28 | 2 | no | — |
| 99 | C2×#41 | — | 14 | 3 | no | — |
| 100 | C2×#41 | — | 28 | 1 | no | — |
| 101 | C22×Q8 | — | 2 | 1 | no | — |
| 102 | C22×Q8 | — | 2 | 1 | no | — |
| 103 | C22×Q8 | — | 28 | 1 | no | — |
| 104 | C22×Q8 | C22×D7 | 1 | 1 | yes | characteristic |
| 105 | C2×M16 | — | 14 | 3 | no | — |
| 106 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 107 | C2×C4⋊C4 | C22⋊D7 | 1 | 3 | yes | — |
| 108 | C2×C42 | C22⋊D7 | 1 | 3 | yes | — |
| 109 | #104 | — | 28 | 6 | no | — |
| 110 | #103 | — | 28 | 3 | no | — |
| 111 | #102 | — | 28 | 3 | no | — |
| 112 | #101 | — | 14 | 6 | no | — |
| 113 | #95 | — | 2 | 3 | no | — |
| 114 | #95 | — | 2 | 6 | no | — |
| 115 | C22×C7 | — | 2 | 1 | no | — |
| 116 | C22×C7 | — | 2 | 1 | no | — |
| 117 | C22×C7 | #393 | 1 | 1 | yes | characteristic |
| 118 | C22×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 119 | C22×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 120 | C2×D7 | — | 4 | 2 | no | — |
| 121 | C2×D7 | — | 8 | 1 | no | — |
| 122 | C2×D7 | — | 8 | 1 | no | — |
| 123 | C4×C7 | — | 2 | 3 | no | — |
| 124 | C4×C7 | — | 2 | 3 | no | — |
| 125 | C4×C7 | — | 4 | 6 | no | — |
| 126 | C4×C7 | — | 8 | 3 | no | — |
| 127 | Dic7 | — | 4 | 2 | no | — |
| 128 | #41 | — | 14 | 6 | no | — |
| 129 | #41 | — | 14 | 6 | no | — |
| 130 | #41 | — | 28 | 3 | no | — |
| 131 | #41 | — | 28 | 2 | no | — |
| 132 | #41 | — | 28 | 6 | no | — |
| 133 | #41 | — | 28 | 3 | no | — |
| 134 | C2×Q8 | — | 2 | 4 | no | — |
| 135 | C2×Q8 | — | 4 | 1 | no | — |
| 136 | C2×Q8 | — | 4 | 1 | no | — |
| 137 | C2×Q8 | — | 8 | 1 | no | — |
| 138 | C2×Q8 | — | 8 | 1 | no | — |
| 139 | C2×Q8 | — | 8 | 2 | no | — |
| 140 | C2×Q8 | — | 8 | 1 | no | — |
| 141 | C2×Q8 | — | 14 | 3 | no | — |
| 142 | C2×Q8 | — | 28 | 2 | no | — |
| 143 | C2×Q8 | — | 28 | 6 | no | — |
| 144 | C2×Q8 | Q8×D7 | 1 | 1 | yes | characteristic |
| 145 | C2×Q8 | #1161 | 1 | 3 | yes | — |
| 146 | C2×D4 | — | 14 | 3 | no | — |
| 147 | C2×D4 | — | 28 | 1 | no | — |
| 148 | C22×C4 | — | 2 | 3 | no | — |
| 149 | C22×C4 | — | 2 | 3 | no | — |
| 150 | C22×C4 | — | 28 | 3 | no | — |
| 151 | C22×C4 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 152 | M16 | — | 28 | 3 | no | — |
| 153 | M16 | — | 28 | 3 | no | — |
| 154 | C2×C8 | — | 28 | 3 | no | — |
| 155 | C4⋊C4 | — | 4 | 3 | no | — |
| 156 | C4⋊C4 | — | 8 | 3 | no | — |
| 157 | C42 | — | 4 | 3 | no | — |
| 158 | C2×C7 | — | 2 | 1 | no | — |
| 159 | C2×C7 | — | 2 | 1 | no | — |
| 160 | C2×C7 | #2029 | 1 | 1 | yes | characteristic |
| 161 | C2×C7 | #2202 | 1 | 1 | yes | characteristic |
| 162 | C2×C7 | #2205 | 1 | 1 | yes | characteristic |
| 163 | D7 | — | 8 | 2 | no | — |
| 164 | C23 | — | 28 | 1 | no | — |
| 165 | C23 | #6516 | 1 | 1 | yes | characteristic |
| 166 | Q8 | — | 4 | 1 | no | — |
| 167 | Q8 | — | 4 | 3 | no | — |
| 168 | Q8 | — | 8 | 4 | no | — |
| 169 | Q8 | — | 8 | 4 | no | — |
| 170 | Q8 | — | 14 | 6 | no | — |
| 171 | Q8 | — | 28 | 3 | no | — |
| 172 | D4 | — | 14 | 6 | no | — |
| 173 | D4 | — | 28 | 3 | no | — |
| 174 | D4 | — | 28 | 2 | no | — |
| 175 | D4 | — | 28 | 2 | no | — |
| 176 | C2×C4 | — | 2 | 3 | no | — |
| 177 | C2×C4 | — | 2 | 3 | no | — |
| 178 | C2×C4 | — | 2 | 6 | no | — |
| 179 | C2×C4 | — | 4 | 3 | no | — |
| 180 | C2×C4 | — | 4 | 3 | no | — |
| 181 | C2×C4 | — | 4 | 6 | no | — |
| 182 | C2×C4 | — | 4 | 3 | no | — |
| 183 | C2×C4 | — | 28 | 6 | no | — |
| 184 | C2×C4 | — | 28 | 6 | no | — |
| 185 | C2×C4 | — | 28 | 1 | no | — |
| 186 | C2×C4 | #6509 | 1 | 3 | yes | — |
| 187 | C2×C4 | #6510 | 1 | 3 | yes | — |
| 188 | C8 | — | 56 | 3 | no | — |
| 189 | C7 | #13378 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 190 | C22 | — | 2 | 1 | no | — |
| 191 | C22 | — | 2 | 1 | no | — |
| 192 | C22 | — | 28 | 2 | no | — |
| 193 | C22 | — | 56 | 1 | no | — |
| 194 | C22 | — | 56 | 1 | no | — |
| 195 | C22 | #90003 | 1 | 1 | yes | characteristic |
| 196 | C22 | #90016 | 1 | 1 | yes | characteristic |
| 197 | C22 | #90016 | 1 | 1 | yes | characteristic |
| 198 | C4 | — | 2 | 3 | no | — |
| 199 | C4 | — | 2 | 3 | no | — |
| 200 | C4 | — | 4 | 6 | no | — |
| 201 | C4 | — | 8 | 3 | no | — |
| 202 | C4 | — | 28 | 2 | no | — |
| 203 | C2 | — | 2 | 1 | no | — |
| 204 | C2 | — | 2 | 1 | no | — |
| 205 | C2 | — | 56 | 2 | no | — |
| 206 | C2 | #11729651 | 1 | 1 | yes | characteristic |
| 207 | C2 | #11729696 | 1 | 1 | yes | characteristic |
| 208 | C2 | #11729705 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2811 | C2 | semidirect |
| 3. #11725450 | C2 | semidirect |
| 17. #6551 | Q8 | semidirect |
| 31. #2811 | D7 | semidirect |
| 189. C7 | #13378 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2811 | 2 |
| #11727446 | 2 |
| #11725450 | 2 |
| #13378 | 7 |