Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
716 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #26012326 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26012063 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #26011682 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #26011681 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #26011680 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #26011463 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #458×C173 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #19149726 | — | 2 | 3 | no | — |
| 9 | #19149724 | — | 2 | 3 | no | — |
| 10 | #19149724 | C22 | 1 | 3 | yes | — |
| 11 | C2×#14583825 | — | 2 | 3 | no | — |
| 12 | C2×#14583825 | C22 | 1 | 3 | yes | — |
| 13 | #41×D173 | — | 2 | 3 | no | — |
| 14 | C22×Dic346 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #19149678 | — | 2 | 3 | no | — |
| 16 | #19149665 | — | 2 | 3 | no | — |
| 17 | #19149663 | C22 | 1 | 3 | yes | complemented |
| 18 | #19149662 | C22 | 1 | 3 | yes | complemented |
| 19 | #19149660 | C22 | 1 | 3 | yes | complemented |
| 20 | #19149657 | — | 2 | 3 | no | — |
| 21 | #19149656 | — | 2 | 3 | no | — |
| 22 | #19149635 | C22 | 1 | 3 | yes | complemented |
| 23 | #19149594 | C22 | 1 | 3 | yes | complemented |
| 24 | #19149593 | C22 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2)×C173 | C22 | 1 | 1 | yes | characteristic |
| 26 | #123×C173 | C22 | 1 | 3 | yes | complemented |
| 27 | #99×C173 | C22 | 1 | 3 | yes | complemented |
| 28 | #14583827 | — | 2 | 1 | no | — |
| 29 | #14583826 | — | 2 | 6 | no | — |
| 30 | #14583826 | — | 2 | 3 | no | — |
| 31 | #14583825 | — | 2 | 3 | no | — |
| 32 | #14583825 | — | 2 | 6 | no | — |
| 33 | #14583825 | — | 2 | 6 | no | — |
| 34 | #14583825 | — | 2 | 6 | no | — |
| 35 | #14583825 | — | 2 | 6 | no | — |
| 36 | C2×Dic346 | — | 2 | 3 | no | — |
| 37 | C2×Dic346 | — | 2 | 3 | no | — |
| 38 | C2×Dic346 | — | 2 | 3 | no | — |
| 39 | C2×Dic346 | D4 | 1 | 3 | yes | — |
| 40 | Q8×D173 | — | 2 | 3 | no | — |
| 41 | C2×C22⋊D173 | — | 2 | 3 | no | — |
| 42 | C2×C22⋊D173 | — | 2 | 3 | no | — |
| 43 | C2×C22⋊D173 | D4 | 1 | 3 | yes | — |
| 44 | C2×C22⋊D173 | C23 | 1 | 3 | yes | — |
| 45 | D4×D173 | — | 2 | 3 | no | — |
| 46 | D4×D173 | — | 2 | 6 | no | — |
| 47 | C22×Dic173 | — | 2 | 1 | no | — |
| 48 | C22×Dic173 | D4 | 1 | 3 | yes | — |
| 49 | C22×Dic173 | C23 | 1 | 1 | yes | characteristic |
| 50 | C22×Dic173 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×D173 | — | 2 | 3 | no | — |
| 52 | C2×C4×D173 | D4 | 1 | 3 | yes | — |
| 53 | C4⋊Dic173 | — | 2 | 3 | no | — |
| 54 | #14583805 | — | 2 | 3 | no | — |
| 55 | #14583805 | — | 2 | 3 | no | — |
| 56 | #14583805 | — | 2 | 3 | no | — |
| 57 | C22⋊Dic173 | — | 2 | 3 | no | — |
| 58 | C22⋊Dic173 | C23 | 1 | 3 | yes | — |
| 59 | #14583803 | — | 2 | 3 | no | — |
| 60 | #14583803 | — | 2 | 3 | no | — |
| 61 | C4×Dic173 | — | 2 | 3 | no | — |
| 62 | #41×C173 | — | 2 | 3 | no | — |
| 63 | C2×D4×C173 | — | 2 | 3 | no | — |
| 64 | C2×D4×C173 | C23 | 1 | 3 | yes | — |
| 65 | C22×C4×C173 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4⋊C4×C173 | — | 2 | 3 | no | — |
| 67 | C22⋊C4×C173 | — | 2 | 3 | no | — |
| 68 | C22⋊C4×C173 | C23 | 1 | 3 | yes | — |
| 69 | C22×D173 | — | 2 | 3 | no | — |
| 70 | C22×D173 | C2×D4 | 1 | 3 | yes | — |
| 71 | Dic346 | — | 2 | 3 | no | — |
| 72 | Dic346 | — | 2 | 6 | no | — |
| 73 | Dic346 | — | 4 | 1 | no | — |
| 74 | Dic346 | — | 4 | 3 | no | — |
| 75 | D692 | — | 2 | 3 | no | — |
| 76 | C22⋊D173 | — | 2 | 6 | no | — |
| 77 | C22⋊D173 | — | 2 | 6 | no | — |
| 78 | C22⋊D173 | — | 2 | 6 | no | — |
| 79 | C2×Dic173 | — | 2 | 3 | no | — |
| 80 | C2×Dic173 | — | 2 | 3 | no | — |
| 81 | C2×Dic173 | — | 2 | 3 | no | — |
| 82 | C2×Dic173 | — | 2 | 3 | no | — |
| 83 | C2×Dic173 | — | 2 | 3 | no | — |
| 84 | C2×Dic173 | C2×D4 | 1 | 3 | yes | — |
| 85 | C2×Dic173 | C2×D4 | 1 | 3 | yes | — |
| 86 | C2×Dic173 | C2×D4 | 1 | 3 | yes | — |
| 87 | C4×D173 | — | 2 | 6 | no | — |
| 88 | C4×D173 | — | 2 | 3 | no | — |
| 89 | C4×D173 | — | 2 | 3 | no | — |
| 90 | C23×C173 | — | 2 | 1 | no | — |
| 91 | C23×C173 | C2×D4 | 1 | 3 | yes | — |
| 92 | C23×C173 | C24 | 1 | 1 | yes | characteristic |
| 93 | Q8×C173 | — | 2 | 1 | no | — |
| 94 | D4×C173 | — | 2 | 3 | no | — |
| 95 | D4×C173 | — | 2 | 6 | no | — |
| 96 | C2×C4×C173 | — | 2 | 3 | no | — |
| 97 | C2×C4×C173 | — | 2 | 3 | no | — |
| 98 | C2×C4×C173 | — | 2 | 3 | no | — |
| 99 | C2×C4×C173 | C2×D4 | 1 | 3 | yes | — |
| 100 | C22×C173 | — | 2 | 3 | no | — |
| 101 | C22×C173 | — | 4 | 1 | no | — |
| 102 | C22×C173 | — | 4 | 3 | no | — |
| 103 | C22×C173 | — | 8 | 1 | no | — |
| 104 | C22×C173 | C22×D4 | 1 | 3 | yes | — |
| 105 | C2×D173 | — | 2 | 3 | no | — |
| 106 | C2×D173 | — | 4 | 3 | no | — |
| 107 | C2×D173 | — | 8 | 3 | no | — |
| 108 | C4×C173 | — | 2 | 3 | no | — |
| 109 | C4×C173 | — | 4 | 3 | no | — |
| 110 | C4×C173 | — | 4 | 1 | no | — |
| 111 | Dic173 | — | 2 | 3 | no | — |
| 112 | Dic173 | — | 4 | 3 | no | — |
| 113 | Dic173 | — | 4 | 1 | no | — |
| 114 | Dic173 | #120 | 1 | 1 | yes | characteristic |
| 115 | Dic173 | #120 | 1 | 3 | yes | — |
| 116 | C2×C173 | — | 2 | 3 | no | — |
| 117 | C2×C173 | — | 4 | 3 | no | — |
| 118 | C2×C173 | C2×#120 | 1 | 1 | yes | characteristic |
| 119 | D173 | — | 4 | 3 | no | — |
| 120 | C173 | #3028 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 121 | #3028 | — | 173 | 1 | no | maximal, Sylow |
| 122 | #585 | — | 173 | 1 | no | — |
| 123 | #536 | — | 173 | 3 | no | — |
| 124 | #458 | — | 173 | 1 | no | — |
| 125 | #458 | — | 173 | 3 | no | — |
| 126 | #458 | — | 173 | 3 | no | — |
| 127 | #458 | D173 | 1 | 1 | yes | characteristic, complemented |
| 128 | #410 | — | 173 | 3 | no | — |
| 129 | 2−1+4 | — | 346 | 3 | no | — |
| 130 | H(5,𝔽2) | — | 346 | 3 | no | — |
| 131 | H(5,𝔽2) | — | 173 | 3 | no | — |
| 132 | H(5,𝔽2) | C2×D173 | 1 | 1 | yes | characteristic |
| 133 | C2×#41 | — | 346 | 3 | no | — |
| 134 | C2×#41 | — | 346 | 3 | no | — |
| 135 | C2×#41 | — | 173 | 3 | no | — |
| 136 | C22×Q8 | — | 173 | 1 | no | — |
| 137 | C4⋊Q8 | — | 346 | 3 | no | — |
| 138 | #124 | — | 346 | 3 | no | — |
| 139 | #123 | — | 173 | 3 | no | — |
| 140 | #123 | — | 173 | 3 | no | — |
| 141 | #123 | — | 173 | 3 | no | — |
| 142 | #123 | C2×D173 | 1 | 3 | yes | complemented |
| 143 | C22⋊Q8 | — | 346 | 3 | no | — |
| 144 | C22⋊Q8 | — | 346 | 3 | no | — |
| 145 | #117 | — | 173 | 3 | no | — |
| 146 | #99 | — | 173 | 3 | no | — |
| 147 | #99 | — | 173 | 6 | no | — |
| 148 | #99 | C2×D173 | 1 | 3 | yes | complemented |
| 149 | #41 | — | 2 | 3 | no | — |
| 150 | #41 | — | 346 | 6 | no | — |
| 151 | #41 | — | 346 | 3 | no | — |
| 152 | #41 | — | 346 | 6 | no | — |
| 153 | #41 | — | 346 | 1 | no | — |
| 154 | #41 | — | 346 | 6 | no | — |
| 155 | #41 | — | 346 | 6 | no | — |
| 156 | #41 | — | 346 | 6 | no | — |
| 157 | #41 | — | 346 | 3 | no | — |
| 158 | C2×Q8 | — | 346 | 3 | no | — |
| 159 | C2×Q8 | — | 346 | 3 | no | — |
| 160 | C2×Q8 | — | 346 | 3 | no | — |
| 161 | C2×Q8 | — | 346 | 3 | no | — |
| 162 | C2×Q8 | — | 173 | 3 | no | — |
| 163 | C2×D4 | — | 2 | 3 | no | — |
| 164 | C2×D4 | — | 346 | 3 | no | — |
| 165 | C2×D4 | — | 346 | 6 | no | — |
| 166 | C2×D4 | — | 346 | 3 | no | — |
| 167 | C2×D4 | — | 346 | 3 | no | — |
| 168 | C2×D4 | — | 173 | 3 | no | — |
| 169 | C2×D4 | — | 173 | 3 | no | — |
| 170 | C2×D4 | C22×D173 | 1 | 3 | yes | — |
| 171 | C22×C4 | — | 346 | 3 | no | — |
| 172 | C22×C4 | — | 346 | 1 | no | — |
| 173 | C22×C4 | — | 173 | 1 | no | — |
| 174 | C22×C4 | — | 173 | 3 | no | — |
| 175 | C22×C4 | — | 173 | 3 | no | — |
| 176 | C22×C4 | — | 173 | 1 | no | — |
| 177 | C22×C4 | C22×D173 | 1 | 1 | yes | characteristic |
| 178 | C4⋊C4 | — | 2 | 3 | no | — |
| 179 | C4⋊C4 | — | 346 | 3 | no | — |
| 180 | C4⋊C4 | — | 346 | 3 | no | — |
| 181 | C4⋊C4 | — | 346 | 3 | no | — |
| 182 | C4⋊C4 | — | 346 | 3 | no | — |
| 183 | C22⋊C4 | — | 2 | 3 | no | — |
| 184 | C22⋊C4 | — | 346 | 3 | no | — |
| 185 | C22⋊C4 | — | 346 | 3 | no | — |
| 186 | C22⋊C4 | — | 346 | 3 | no | — |
| 187 | C22⋊C4 | — | 173 | 3 | no | — |
| 188 | C22⋊C4 | C22×D173 | 1 | 3 | yes | — |
| 189 | C42 | — | 346 | 3 | no | — |
| 190 | C23 | — | 2 | 1 | no | — |
| 191 | C23 | — | 346 | 3 | no | — |
| 192 | C23 | — | 173 | 3 | no | — |
| 193 | C23 | D4×D173 | 1 | 3 | yes | — |
| 194 | C23 | C23×D173 | 1 | 1 | yes | characteristic |
| 195 | Q8 | — | 2 | 1 | no | — |
| 196 | Q8 | — | 346 | 3 | no | — |
| 197 | Q8 | — | 346 | 6 | no | — |
| 198 | Q8 | — | 692 | 1 | no | — |
| 199 | Q8 | — | 692 | 3 | no | — |
| 200 | D4 | — | 2 | 3 | no | — |
| 201 | D4 | — | 2 | 6 | no | — |
| 202 | D4 | — | 346 | 3 | no | — |
| 203 | D4 | — | 346 | 6 | no | — |
| 204 | D4 | — | 346 | 6 | no | — |
| 205 | D4 | — | 346 | 6 | no | — |
| 206 | C2×C4 | — | 2 | 3 | no | — |
| 207 | C2×C4 | — | 2 | 3 | no | — |
| 208 | C2×C4 | — | 2 | 3 | no | — |
| 209 | C2×C4 | — | 346 | 6 | no | — |
| 210 | C2×C4 | — | 346 | 3 | no | — |
| 211 | C2×C4 | — | 346 | 3 | no | — |
| 212 | C2×C4 | — | 346 | 3 | no | — |
| 213 | C2×C4 | — | 346 | 3 | no | — |
| 214 | C2×C4 | — | 346 | 3 | no | — |
| 215 | C2×C4 | — | 346 | 3 | no | — |
| 216 | C2×C4 | — | 346 | 3 | no | — |
| 217 | C2×C4 | — | 173 | 3 | no | — |
| 218 | C2×C4 | — | 173 | 3 | no | — |
| 219 | C2×C4 | — | 173 | 3 | no | — |
| 220 | C2×C4 | D4×D173 | 1 | 3 | yes | — |
| 221 | C22 | — | 2 | 3 | no | — |
| 222 | C22 | — | 4 | 1 | no | — |
| 223 | C22 | — | 4 | 3 | no | — |
| 224 | C22 | — | 8 | 1 | no | — |
| 225 | C22 | — | 346 | 3 | no | — |
| 226 | C22 | — | 1384 | 3 | no | — |
| 227 | C22 | — | 692 | 3 | no | — |
| 228 | C22 | C2×D4×D173 | 1 | 3 | yes | — |
| 229 | C4 | — | 2 | 3 | no | — |
| 230 | C4 | — | 4 | 3 | no | — |
| 231 | C4 | — | 4 | 1 | no | — |
| 232 | C4 | — | 346 | 3 | no | — |
| 233 | C4 | — | 173 | 1 | no | — |
| 234 | C4 | — | 173 | 3 | no | — |
| 235 | C4 | — | 692 | 3 | no | — |
| 236 | C4 | — | 692 | 1 | no | — |
| 237 | C2 | — | 2 | 3 | no | — |
| 238 | C2 | — | 4 | 3 | no | — |
| 239 | C2 | — | 692 | 3 | no | — |
| 240 | C2 | #120×D173 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #26012063 | C2 | semidirect |
| 3. #26011682 | C2 | semidirect |
| 4. #26011681 | C2 | semidirect |
| 5. #26011680 | C2 | semidirect |
| 6. #26011463 | C2 | semidirect |
| 7. #458×C173 | C2 | semidirect |
| 14. C22×Dic346 | C22 | semidirect |
| 17. #19149663 | C22 | semidirect |
| 18. #19149662 | C22 | semidirect |
| 19. #19149660 | C22 | semidirect |
| 22. #19149635 | C22 | semidirect |
| 23. #19149594 | C22 | semidirect |
| 24. #19149593 | C22 | semidirect |
| 26. #123×C173 | C22 | semidirect |
| 27. #99×C173 | C22 | semidirect |
| 120. C173 | #3028 | semidirect |
| 127. #458 | D173 | semidirect |
| 142. #123 | C2×D173 | semidirect |
| 148. #99 | C2×D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26012326 | 2 |
| #26012063 | 2 |
| #26011682 | 2 |
| #26011681 | 2 |
| #26011680 | 2 |
| #26011463 | 2 |
| #458×C173 | 2 |
| #3028 | 173 |