Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
670 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 84 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 | D173⋊#99 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #26012439 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | H(5,𝔽2)×D173 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C23×#10621924 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×C22⋊#10621924 | — | 2 | 3 | no | — |
| 6 | C2×C22⋊#10621924 | C22 | 1 | 3 | yes | complemented |
| 7 | #19149738 | — | 2 | 3 | no | — |
| 8 | #19149737 | — | 2 | 3 | no | — |
| 9 | (C2×#10621924)⋊C4 | — | 2 | 3 | no | — |
| 10 | #19149725 | — | 2 | 3 | no | — |
| 11 | #19149724 | — | 2 | 3 | no | — |
| 12 | #19149724 | C4 | 1 | 3 | yes | complemented |
| 13 | #41×D173 | — | 2 | 3 | no | — |
| 14 | C2×D4×D173 | — | 2 | 3 | no | — |
| 15 | C2×D4×D173 | C22 | 1 | 3 | yes | — |
| 16 | H(5,𝔽2)×C173 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×#10621924 | — | 2 | 3 | no | — |
| 18 | C22×#10621924 | — | 2 | 3 | no | — |
| 19 | C22×#10621924 | — | 4 | 2 | no | — |
| 20 | C22⋊#10621924 | — | 2 | 6 | no | — |
| 21 | C22⋊#10621924 | — | 4 | 6 | no | — |
| 22 | C23×D173 | — | 2 | 1 | no | — |
| 23 | C23×D173 | D4 | 1 | 3 | yes | — |
| 24 | C23×D173 | C23 | 1 | 1 | yes | characteristic |
| 25 | #14583827 | — | 2 | 3 | no | — |
| 26 | #14583826 | — | 2 | 6 | no | — |
| 27 | #14583826 | — | 2 | 3 | no | — |
| 28 | #14583825 | — | 2 | 3 | no | — |
| 29 | #14583825 | — | 2 | 6 | no | — |
| 30 | Q8×D173 | — | 2 | 1 | no | — |
| 31 | C2×D692 | — | 2 | 3 | no | — |
| 32 | C2×D692 | D4 | 1 | 3 | yes | — |
| 33 | C2×C22⋊D173 | — | 2 | 3 | no | — |
| 34 | C2×C22⋊D173 | — | 2 | 3 | no | — |
| 35 | C2×C22⋊D173 | C2×C4 | 1 | 3 | yes | — |
| 36 | C2×C22⋊D173 | D4 | 1 | 3 | yes | — |
| 37 | D4×D173 | — | 2 | 3 | no | — |
| 38 | D4×D173 | — | 2 | 6 | no | — |
| 39 | D4×D173 | — | 2 | 6 | no | — |
| 40 | D4×D173 | — | 2 | 6 | no | — |
| 41 | D4×D173 | — | 2 | 6 | no | — |
| 42 | C2×C4×D173 | — | 2 | 3 | no | — |
| 43 | C2×C4×D173 | D4 | 1 | 3 | yes | — |
| 44 | #41×C173 | — | 2 | 3 | no | — |
| 45 | C2×D4×C173 | — | 2 | 3 | no | — |
| 46 | C2×D4×C173 | C2×C4 | 1 | 3 | yes | — |
| 47 | C2×#10621924 | — | 2 | 6 | no | — |
| 48 | C2×#10621924 | — | 4 | 3 | no | — |
| 49 | C2×#10621924 | — | 4 | 1 | no | — |
| 50 | C2×#10621924 | — | 4 | 3 | no | — |
| 51 | C2×#10621924 | — | 4 | 3 | no | — |
| 52 | C2×#10621924 | — | 4 | 3 | no | — |
| 53 | C22×D173 | — | 2 | 3 | no | — |
| 54 | C22×D173 | — | 2 | 3 | no | — |
| 55 | C22×D173 | — | 2 | 3 | no | — |
| 56 | C22×D173 | — | 4 | 1 | no | — |
| 57 | C22×D173 | — | 4 | 1 | no | — |
| 58 | C22×D173 | — | 4 | 3 | no | — |
| 59 | C22×D173 | — | 4 | 3 | no | — |
| 60 | C22×D173 | — | 8 | 1 | no | — |
| 61 | C22×D173 | — | 8 | 1 | no | — |
| 62 | C22×D173 | C22⋊C4 | 1 | 3 | yes | complemented |
| 63 | C22×D173 | C2×D4 | 1 | 3 | yes | — |
| 64 | C22×D173 | C2×D4 | 1 | 3 | yes | — |
| 65 | Dic346 | — | 2 | 3 | no | — |
| 66 | D692 | — | 2 | 3 | no | — |
| 67 | D692 | — | 2 | 6 | no | — |
| 68 | C22⋊D173 | — | 2 | 6 | no | — |
| 69 | C22⋊D173 | — | 2 | 6 | no | — |
| 70 | C22⋊D173 | — | 2 | 6 | no | — |
| 71 | C2×Dic173 | — | 2 | 3 | no | — |
| 72 | C2×Dic173 | C22⋊C4 | 1 | 3 | yes | complemented |
| 73 | C4×D173 | — | 2 | 6 | no | — |
| 74 | C4×D173 | — | 2 | 3 | no | — |
| 75 | C4×D173 | — | 2 | 3 | no | — |
| 76 | C23×C173 | — | 2 | 1 | no | — |
| 77 | C23×C173 | C22⋊C4 | 1 | 3 | yes | complemented |
| 78 | C23×C173 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | Q8×C173 | — | 2 | 1 | no | — |
| 80 | D4×C173 | — | 2 | 3 | no | — |
| 81 | D4×C173 | — | 2 | 6 | no | — |
| 82 | C2×C4×C173 | — | 2 | 3 | no | — |
| 83 | C2×C4×C173 | C22⋊C4 | 1 | 3 | yes | complemented |
| 84 | C22×C173 | — | 2 | 3 | no | — |
| 85 | C22×C173 | — | 4 | 1 | no | — |
| 86 | C22×C173 | — | 4 | 3 | no | — |
| 87 | C22×C173 | — | 8 | 1 | no | — |
| 88 | C22×C173 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 89 | C2×D173 | — | 2 | 3 | no | — |
| 90 | C2×D173 | — | 2 | 3 | no | — |
| 91 | C2×D173 | — | 2 | 3 | no | — |
| 92 | C2×D173 | — | 4 | 3 | no | — |
| 93 | C2×D173 | — | 4 | 3 | no | — |
| 94 | C2×D173 | — | 4 | 3 | no | — |
| 95 | C2×D173 | — | 4 | 3 | no | — |
| 96 | C2×D173 | — | 8 | 3 | no | — |
| 97 | C2×D173 | — | 8 | 3 | no | — |
| 98 | C2×D173 | #120 | 1 | 3 | yes | — |
| 99 | C2×D173 | #120 | 1 | 1 | yes | characteristic |
| 100 | #10621924 | — | 4 | 2 | no | — |
| 101 | #10621924 | — | 8 | 3 | no | — |
| 102 | C4×C173 | — | 2 | 3 | no | — |
| 103 | Dic173 | — | 2 | 3 | no | — |
| 104 | C2×C173 | — | 2 | 3 | no | — |
| 105 | C2×C173 | — | 4 | 3 | no | — |
| 106 | C2×C173 | #379 | 1 | 1 | yes | characteristic |
| 107 | D173 | — | 2 | 3 | no | — |
| 108 | D173 | — | 4 | 3 | no | — |
| 109 | D173 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 110 | D173 | #458 | 1 | 1 | yes | characteristic |
| 111 | C173 | #1790 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 112 | #1790 | — | 173 | 1 | no | maximal, Sylow |
| 113 | C2×H(5,𝔽2) | — | 173 | 1 | no | — |
| 114 | C2×#99 | — | 173 | 3 | no | — |
| 115 | #386 | — | 173 | 3 | no | — |
| 116 | H(5,𝔽2) | — | 346 | 3 | no | — |
| 117 | H(5,𝔽2) | — | 346 | 3 | no | — |
| 118 | H(5,𝔽2) | — | 173 | 3 | no | — |
| 119 | H(5,𝔽2) | #10621924 | 1 | 1 | yes | characteristic, complemented |
| 120 | C2×#41 | — | 346 | 3 | no | — |
| 121 | C22×D4 | — | 346 | 3 | no | — |
| 122 | C22×D4 | — | 173 | 3 | no | — |
| 123 | C23×C4 | — | 173 | 1 | no | — |
| 124 | C2×C22⋊C4 | — | 346 | 3 | no | — |
| 125 | C2×C22⋊C4 | — | 173 | 3 | no | — |
| 126 | #99 | — | 346 | 3 | no | — |
| 127 | #99 | — | 346 | 3 | no | — |
| 128 | #95 | — | 346 | 3 | no | — |
| 129 | C24 | — | 346 | 1 | no | — |
| 130 | C24 | — | 173 | 3 | no | — |
| 131 | C24 | — | 173 | 1 | no | — |
| 132 | #41 | — | 2 | 3 | no | — |
| 133 | #41 | — | 346 | 6 | no | — |
| 134 | #41 | — | 346 | 3 | no | — |
| 135 | #41 | — | 346 | 3 | no | — |
| 136 | #41 | — | 346 | 6 | no | — |
| 137 | #41 | — | 346 | 3 | no | — |
| 138 | C2×Q8 | — | 346 | 1 | no | — |
| 139 | C2×D4 | — | 2 | 3 | no | — |
| 140 | C2×D4 | — | 346 | 3 | no | — |
| 141 | C2×D4 | — | 346 | 3 | no | — |
| 142 | C2×D4 | — | 346 | 6 | no | — |
| 143 | C2×D4 | — | 346 | 6 | no | — |
| 144 | C2×D4 | — | 346 | 6 | no | — |
| 145 | C2×D4 | — | 346 | 6 | no | — |
| 146 | C2×D4 | — | 346 | 3 | no | — |
| 147 | C2×D4 | — | 346 | 3 | no | — |
| 148 | C2×D4 | — | 173 | 3 | no | — |
| 149 | C2×D4 | — | 173 | 3 | no | — |
| 150 | C2×D4 | — | 173 | 3 | no | — |
| 151 | C2×D4 | C2×#10621924 | 1 | 3 | yes | — |
| 152 | C22×C4 | — | 346 | 3 | no | — |
| 153 | C22×C4 | — | 346 | 3 | no | — |
| 154 | C22×C4 | — | 346 | 3 | no | — |
| 155 | C22×C4 | — | 173 | 3 | no | — |
| 156 | C22×C4 | — | 692 | 2 | no | — |
| 157 | C22⋊C4 | — | 346 | 6 | no | — |
| 158 | C22⋊C4 | — | 692 | 6 | no | — |
| 159 | C23 | — | 2 | 1 | no | — |
| 160 | C23 | — | 346 | 3 | no | — |
| 161 | C23 | — | 346 | 3 | no | — |
| 162 | C23 | — | 346 | 3 | no | — |
| 163 | C23 | — | 1384 | 1 | no | — |
| 164 | C23 | — | 1384 | 1 | no | — |
| 165 | C23 | — | 173 | 3 | no | — |
| 166 | C23 | — | 173 | 3 | no | — |
| 167 | C23 | — | 173 | 3 | no | — |
| 168 | C23 | — | 692 | 1 | no | — |
| 169 | C23 | — | 692 | 1 | no | — |
| 170 | C23 | — | 692 | 3 | no | — |
| 171 | C23 | — | 692 | 3 | no | — |
| 172 | C23 | C22⋊#10621924 | 1 | 3 | yes | complemented |
| 173 | C23 | C22×#10621924 | 1 | 1 | yes | characteristic |
| 174 | Q8 | — | 2 | 1 | no | — |
| 175 | Q8 | — | 346 | 3 | no | — |
| 176 | D4 | — | 2 | 3 | no | — |
| 177 | D4 | — | 2 | 6 | no | — |
| 178 | D4 | — | 346 | 3 | no | — |
| 179 | D4 | — | 346 | 6 | no | — |
| 180 | D4 | — | 346 | 6 | no | — |
| 181 | D4 | — | 346 | 6 | no | — |
| 182 | D4 | — | 346 | 6 | no | — |
| 183 | C2×C4 | — | 2 | 3 | no | — |
| 184 | C2×C4 | — | 346 | 6 | no | — |
| 185 | C2×C4 | — | 346 | 3 | no | — |
| 186 | C2×C4 | — | 346 | 3 | no | — |
| 187 | C2×C4 | — | 346 | 3 | no | — |
| 188 | C2×C4 | — | 346 | 6 | no | — |
| 189 | C2×C4 | — | 173 | 3 | no | — |
| 190 | C2×C4 | — | 692 | 3 | no | — |
| 191 | C2×C4 | — | 692 | 3 | no | — |
| 192 | C2×C4 | — | 692 | 3 | no | — |
| 193 | C2×C4 | — | 692 | 3 | no | — |
| 194 | C2×C4 | — | 692 | 1 | no | — |
| 195 | C2×C4 | C22⋊#10621924 | 1 | 3 | yes | complemented |
| 196 | C22 | — | 2 | 3 | no | — |
| 197 | C22 | — | 4 | 1 | no | — |
| 198 | C22 | — | 4 | 3 | no | — |
| 199 | C22 | — | 8 | 1 | no | — |
| 200 | C22 | — | 346 | 3 | no | — |
| 201 | C22 | — | 346 | 3 | no | — |
| 202 | C22 | — | 346 | 3 | no | — |
| 203 | C22 | — | 1384 | 3 | no | — |
| 204 | C22 | — | 1384 | 3 | no | — |
| 205 | C22 | — | 173 | 3 | no | — |
| 206 | C22 | — | 173 | 1 | no | — |
| 207 | C22 | — | 692 | 3 | no | — |
| 208 | C22 | — | 692 | 3 | no | — |
| 209 | C22 | — | 692 | 3 | no | — |
| 210 | C22 | — | 692 | 3 | no | — |
| 211 | C22 | C2×C22⋊#10621924 | 1 | 3 | yes | — |
| 212 | C4 | — | 2 | 3 | no | — |
| 213 | C4 | — | 346 | 3 | no | — |
| 214 | C4 | — | 1384 | 3 | no | — |
| 215 | C4 | — | 692 | 2 | no | — |
| 216 | C2 | — | 2 | 3 | no | — |
| 217 | C2 | — | 4 | 3 | no | — |
| 218 | C2 | — | 346 | 3 | no | — |
| 219 | C2 | — | 173 | 1 | no | — |
| 220 | C2 | — | 173 | 1 | no | — |
| 221 | C2 | — | 692 | 3 | no | — |
| 222 | C2 | #26012432 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D173⋊#99 | C2 | semidirect |
| 2. #26012439 | C2 | semidirect |
| 4. C23×#10621924 | C22 | semidirect |
| 6. C2×C22⋊#10621924 | C22 | semidirect |
| 12. #19149724 | C4 | semidirect |
| 16. H(5,𝔽2)×C173 | C4 | semidirect |
| 62. C22×D173 | C22⋊C4 | semidirect |
| 72. C2×Dic173 | C22⋊C4 | semidirect |
| 77. C23×C173 | C22⋊C4 | semidirect |
| 83. C2×C4×C173 | C22⋊C4 | semidirect |
| 111. C173 | #1790 | semidirect |
| 119. H(5,𝔽2) | #10621924 | semidirect |
| 172. C23 | C22⋊#10621924 | semidirect |
| 195. C2×C4 | C22⋊#10621924 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D173⋊#99 | 2 |
| #26012439 | 2 |
| H(5,𝔽2)×D173 | 2 |
| #1790 | 173 |