Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
384 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 67 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 | #23765476 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #23763064 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #23761441 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #23761376 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C79×#3291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19098998 | — | 2 | 2 | no | — |
| 7 | #19098940 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | #19098939 | — | 2 | 1 | no | — |
| 9 | #19098911 | — | 2 | 1 | no | — |
| 10 | #19098384 | C22 | 1 | 2 | yes | complemented |
| 11 | #19098375 | — | 2 | 2 | no | — |
| 12 | #19098284 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #577×C79 | — | 2 | 2 | no | — |
| 14 | #492×C79 | C22 | 1 | 1 | yes | characteristic |
| 15 | #471×C79 | — | 2 | 2 | no | — |
| 16 | #456×C79 | — | 2 | 1 | no | — |
| 17 | #453×C79 | — | 2 | 1 | no | — |
| 18 | #443×C79 | C22 | 1 | 2 | yes | — |
| 19 | C2×D632 | — | 2 | 2 | no | — |
| 20 | C2×#11929609 | — | 4 | 2 | no | — |
| 21 | #14579158 | — | 2 | 1 | no | — |
| 22 | #14579104 | — | 4 | 2 | no | — |
| 23 | #14579102 | — | 4 | 2 | no | — |
| 24 | #14579101 | — | 2 | 2 | no | — |
| 25 | #14579081 | — | 4 | 2 | no | — |
| 26 | C4×C79⋊C8 | — | 2 | 1 | no | — |
| 27 | 2−1+4×C79 | — | 2 | 1 | no | — |
| 28 | H(5,𝔽2)×C79 | — | 2 | 1 | no | — |
| 29 | #137×C79 | — | 2 | 2 | no | — |
| 30 | #137×C79 | — | 4 | 2 | no | — |
| 31 | AGL(1,ℤ/8)×C79 | — | 2 | 2 | no | — |
| 32 | AGL(1,ℤ/8)×C79 | — | 4 | 2 | no | — |
| 33 | #135×C79 | — | 4 | 2 | no | — |
| 34 | #135×C79 | D4 | 1 | 2 | yes | — |
| 35 | C2×SD16×C79 | — | 2 | 2 | no | — |
| 36 | C2×SD16×C79 | — | 4 | 2 | no | — |
| 37 | #131×C79 | — | 2 | 2 | no | — |
| 38 | C4⋊Q8×C79 | D4 | 1 | 1 | yes | characteristic |
| 39 | #127×C79 | D4 | 1 | 1 | yes | characteristic |
| 40 | #108×C79 | D4 | 1 | 2 | yes | — |
| 41 | #104×C79 | — | 2 | 2 | no | — |
| 42 | #101×C79 | — | 2 | 1 | no | — |
| 43 | #100×C79 | — | 2 | 1 | no | — |
| 44 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×D316 | — | 4 | 1 | no | — |
| 46 | C2×D316 | — | 4 | 1 | no | — |
| 47 | C2×D316 | — | 4 | 1 | no | — |
| 48 | D632 | — | 4 | 2 | no | — |
| 49 | D632 | — | 4 | 2 | no | — |
| 50 | #11929609 | — | 8 | 4 | no | — |
| 51 | C2×C79⋊C8 | — | 2 | 2 | no | — |
| 52 | C79⋊C16 | — | 4 | 2 | no | — |
| 53 | #41×C79 | — | 2 | 2 | no | — |
| 54 | #41×C79 | — | 4 | 2 | no | — |
| 55 | #41×C79 | — | 4 | 2 | no | — |
| 56 | #41×C79 | — | 8 | 1 | no | — |
| 57 | C2×Q8×C79 | — | 2 | 1 | no | — |
| 58 | C2×Q8×C79 | — | 4 | 2 | no | — |
| 59 | C2×D4×C79 | — | 2 | 1 | no | — |
| 60 | C2×D4×C79 | — | 4 | 1 | no | — |
| 61 | C2×D4×C79 | — | 4 | 2 | no | — |
| 62 | C2×D4×C79 | — | 8 | 1 | no | — |
| 63 | Q16×C79 | — | 4 | 2 | no | — |
| 64 | Q16×C79 | D8 | 1 | 2 | yes | complemented |
| 65 | SD16×C79 | — | 2 | 2 | no | — |
| 66 | SD16×C79 | — | 4 | 2 | no | — |
| 67 | SD16×C79 | — | 4 | 2 | no | — |
| 68 | SD16×C79 | — | 4 | 2 | no | — |
| 69 | SD16×C79 | — | 4 | 2 | no | — |
| 70 | D8×C79 | — | 4 | 2 | no | — |
| 71 | D8×C79 | D8 | 1 | 2 | yes | complemented |
| 72 | M16×C79 | — | 2 | 2 | no | — |
| 73 | M16×C79 | — | 4 | 2 | no | — |
| 74 | C2×C8×C79 | — | 4 | 2 | no | — |
| 75 | C2×C8×C79 | C2×D4 | 1 | 2 | yes | — |
| 76 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 77 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C23×C79 | — | 4 | 1 | no | — |
| 79 | C23×C79 | — | 8 | 1 | no | — |
| 80 | C22×D79 | — | 8 | 1 | no | — |
| 81 | Q8×C79 | — | 2 | 2 | no | — |
| 82 | Q8×C79 | — | 4 | 2 | no | — |
| 83 | Q8×C79 | — | 8 | 1 | no | — |
| 84 | D4×C79 | — | 2 | 2 | no | — |
| 85 | D4×C79 | — | 4 | 2 | no | — |
| 86 | D4×C79 | — | 4 | 2 | no | — |
| 87 | D4×C79 | — | 8 | 2 | no | — |
| 88 | D4×C79 | — | 8 | 2 | no | — |
| 89 | D4×C79 | — | 8 | 1 | no | — |
| 90 | C2×C4×C79 | — | 2 | 1 | no | — |
| 91 | C2×C4×C79 | — | 4 | 2 | no | — |
| 92 | C2×C4×C79 | — | 4 | 1 | no | — |
| 93 | C2×C4×C79 | — | 8 | 1 | no | — |
| 94 | C2×C4×C79 | — | 8 | 2 | no | — |
| 95 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 96 | D316 | — | 8 | 2 | no | — |
| 97 | D316 | — | 8 | 2 | no | — |
| 98 | D316 | — | 8 | 2 | no | — |
| 99 | C8×C79 | — | 4 | 2 | no | — |
| 100 | C8×C79 | C2×D8 | 1 | 2 | yes | — |
| 101 | C8×C79 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 102 | C79⋊C8 | — | 4 | 2 | no | — |
| 103 | C22×C79 | — | 4 | 2 | no | — |
| 104 | C22×C79 | — | 8 | 1 | no | — |
| 105 | C22×C79 | — | 16 | 1 | no | — |
| 106 | C22×C79 | — | 32 | 1 | no | — |
| 107 | C22×C79 | #453 | 1 | 1 | yes | characteristic |
| 108 | C2×D79 | — | 16 | 1 | no | — |
| 109 | C2×D79 | — | 16 | 1 | no | — |
| 110 | C2×D79 | — | 16 | 1 | no | — |
| 111 | C4×C79 | — | 2 | 2 | no | — |
| 112 | C4×C79 | — | 4 | 2 | no | — |
| 113 | C4×C79 | — | 8 | 1 | no | — |
| 114 | C4×C79 | #447 | 1 | 2 | yes | — |
| 115 | #12314 | — | 79 | 1 | no | maximal, Sylow |
| 116 | C2×C79 | — | 2 | 1 | no | — |
| 117 | C2×C79 | — | 8 | 2 | no | — |
| 118 | C2×C79 | — | 16 | 1 | no | — |
| 119 | C2×C79 | #1619 | 1 | 1 | yes | characteristic |
| 120 | D79 | — | 32 | 1 | no | — |
| 121 | #3291 | D79 | 1 | 1 | yes | characteristic, complemented |
| 122 | #2213 | — | 79 | 2 | no | — |
| 123 | #1721 | — | 79 | 1 | no | — |
| 124 | #1359 | — | 79 | 1 | no | — |
| 125 | #1336 | — | 79 | 2 | no | — |
| 126 | C79 | #12314 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 127 | #577 | — | 2 | 2 | no | — |
| 128 | #509 | — | 158 | 2 | no | — |
| 129 | #493 | — | 79 | 1 | no | — |
| 130 | #493 | — | 158 | 1 | no | — |
| 131 | #492 | C2×D79 | 1 | 1 | yes | characteristic |
| 132 | #486 | — | 158 | 1 | no | — |
| 133 | #471 | — | 2 | 2 | no | — |
| 134 | #456 | — | 2 | 1 | no | — |
| 135 | #453 | — | 2 | 1 | no | — |
| 136 | #443 | C2×D79 | 1 | 2 | yes | — |
| 137 | #364 | — | 79 | 2 | no | — |
| 138 | #361 | — | 158 | 2 | no | — |
| 139 | #322 | — | 79 | 1 | no | — |
| 140 | 2−1+4 | — | 2 | 1 | no | — |
| 141 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 142 | #137 | — | 2 | 2 | no | — |
| 143 | #137 | — | 4 | 2 | no | — |
| 144 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 145 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 146 | #135 | — | 4 | 2 | no | — |
| 147 | #135 | C22⋊D79 | 1 | 2 | yes | — |
| 148 | C2×SD16 | — | 2 | 2 | no | — |
| 149 | C2×SD16 | — | 4 | 2 | no | — |
| 150 | C2×D8 | — | 316 | 2 | no | — |
| 151 | C2×D8 | — | 158 | 2 | no | — |
| 152 | #131 | — | 2 | 2 | no | — |
| 153 | C4⋊Q8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 154 | #127 | — | 158 | 1 | no | — |
| 155 | #127 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 156 | SD32 | — | 316 | 2 | no | — |
| 157 | D16 | — | 316 | 2 | no | — |
| 158 | M32 | — | 158 | 2 | no | — |
| 159 | #108 | C22⋊D79 | 1 | 2 | yes | — |
| 160 | #104 | — | 2 | 2 | no | — |
| 161 | #102 | — | 316 | 2 | no | — |
| 162 | #101 | — | 2 | 1 | no | — |
| 163 | #100 | — | 2 | 1 | no | — |
| 164 | C4×C8 | — | 158 | 1 | no | — |
| 165 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 166 | #41 | — | 2 | 2 | no | — |
| 167 | #41 | — | 4 | 2 | no | — |
| 168 | #41 | — | 4 | 2 | no | — |
| 169 | #41 | — | 8 | 1 | no | — |
| 170 | C2×Q8 | — | 2 | 1 | no | — |
| 171 | C2×Q8 | — | 4 | 2 | no | — |
| 172 | C2×D4 | — | 2 | 1 | no | — |
| 173 | C2×D4 | — | 4 | 1 | no | — |
| 174 | C2×D4 | — | 4 | 2 | no | — |
| 175 | C2×D4 | — | 8 | 1 | no | — |
| 176 | C2×D4 | — | 316 | 1 | no | — |
| 177 | C2×D4 | — | 316 | 1 | no | — |
| 178 | C2×D4 | — | 316 | 1 | no | — |
| 179 | Q16 | — | 4 | 2 | no | — |
| 180 | Q16 | #11929609 | 1 | 2 | yes | complemented |
| 181 | SD16 | — | 2 | 2 | no | — |
| 182 | SD16 | — | 4 | 2 | no | — |
| 183 | SD16 | — | 4 | 2 | no | — |
| 184 | SD16 | — | 4 | 2 | no | — |
| 185 | SD16 | — | 4 | 2 | no | — |
| 186 | D8 | — | 4 | 2 | no | — |
| 187 | D8 | — | 316 | 2 | no | — |
| 188 | D8 | — | 316 | 2 | no | — |
| 189 | D8 | — | 632 | 4 | no | — |
| 190 | D8 | #11929609 | 1 | 2 | yes | complemented |
| 191 | M16 | — | 2 | 2 | no | — |
| 192 | M16 | — | 4 | 2 | no | — |
| 193 | C2×C8 | — | 4 | 2 | no | — |
| 194 | C2×C8 | — | 158 | 2 | no | — |
| 195 | C2×C8 | C2×C22⋊D79 | 1 | 2 | yes | — |
| 196 | C4⋊C4 | — | 4 | 1 | no | — |
| 197 | C42 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 198 | C16 | — | 316 | 2 | no | — |
| 199 | C23 | — | 4 | 1 | no | — |
| 200 | C23 | — | 8 | 1 | no | — |
| 201 | C23 | — | 632 | 1 | no | — |
| 202 | Q8 | — | 2 | 2 | no | — |
| 203 | Q8 | — | 4 | 2 | no | — |
| 204 | Q8 | — | 8 | 1 | no | — |
| 205 | D4 | — | 2 | 2 | no | — |
| 206 | D4 | — | 4 | 2 | no | — |
| 207 | D4 | — | 4 | 2 | no | — |
| 208 | D4 | — | 8 | 2 | no | — |
| 209 | D4 | — | 8 | 2 | no | — |
| 210 | D4 | — | 8 | 1 | no | — |
| 211 | D4 | — | 632 | 2 | no | — |
| 212 | D4 | — | 632 | 2 | no | — |
| 213 | D4 | — | 632 | 2 | no | — |
| 214 | C2×C4 | — | 2 | 1 | no | — |
| 215 | C2×C4 | — | 4 | 2 | no | — |
| 216 | C2×C4 | — | 4 | 1 | no | — |
| 217 | C2×C4 | — | 8 | 1 | no | — |
| 218 | C2×C4 | — | 8 | 2 | no | — |
| 219 | C2×C4 | #14579130 | 1 | 1 | yes | characteristic |
| 220 | C8 | — | 4 | 2 | no | — |
| 221 | C8 | — | 316 | 2 | no | — |
| 222 | C8 | C2×#11929609 | 1 | 2 | yes | — |
| 223 | C8 | #14579187 | 1 | 2 | yes | — |
| 224 | C22 | — | 4 | 2 | no | — |
| 225 | C22 | — | 8 | 1 | no | — |
| 226 | C22 | — | 16 | 1 | no | — |
| 227 | C22 | — | 32 | 1 | no | — |
| 228 | C22 | — | 1264 | 1 | no | — |
| 229 | C22 | — | 1264 | 1 | no | — |
| 230 | C22 | — | 1264 | 1 | no | — |
| 231 | C22 | #19098731 | 1 | 1 | yes | characteristic |
| 232 | C4 | — | 2 | 2 | no | — |
| 233 | C4 | — | 4 | 2 | no | — |
| 234 | C4 | — | 8 | 1 | no | — |
| 235 | C4 | #19098697 | 1 | 2 | yes | — |
| 236 | C2 | — | 2 | 1 | no | — |
| 237 | C2 | — | 8 | 2 | no | — |
| 238 | C2 | — | 16 | 1 | no | — |
| 239 | C2 | — | 2528 | 1 | no | — |
| 240 | C2 | #23762454 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23765476 | C2 | semidirect |
| 2. #23763064 | C2 | semidirect |
| 3. #23761441 | C2 | semidirect |
| 4. #23761376 | C2 | semidirect |
| 5. C79×#3291 | C2 | semidirect |
| 7. #19098940 | C22 | semidirect |
| 10. #19098384 | C22 | semidirect |
| 12. #19098284 | C22 | semidirect |
| 64. Q16×C79 | D8 | semidirect |
| 71. D8×C79 | D8 | semidirect |
| 121. #3291 | D79 | semidirect |
| 126. C79 | #12314 | semidirect |
| 180. Q16 | #11929609 | semidirect |
| 190. D8 | #11929609 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23765476 | 2 |
| #23763064 | 2 |
| #23761441 | 2 |
| #23761376 | 2 |
| C79×#3291 | 2 |
| #12314 | 79 |