Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
239 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 36 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 | #32872 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13429 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #12301 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #12111 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7439 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7401 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7377 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3199 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2238 | — | 2 | 1 | no | — |
| 10 | #2185 | — | 2 | 1 | no | — |
| 11 | #2134 | — | 2 | 1 | no | — |
| 12 | #2103 | — | 2 | 1 | no | — |
| 13 | #2098 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2082 | — | 2 | 1 | no | — |
| 15 | #2070 | — | 2 | 1 | no | — |
| 16 | #2049 | — | 2 | 1 | no | — |
| 17 | #2047 | — | 2 | 1 | no | — |
| 18 | #2020 | C22 | 1 | 1 | yes | characteristic |
| 19 | #1914 | — | 2 | 1 | no | — |
| 20 | #1902 | — | 2 | 1 | no | — |
| 21 | #1858 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1392 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1349 | — | 2 | 1 | no | — |
| 24 | #1339 | — | 2 | 1 | no | — |
| 25 | #1321 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1315 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#137 | — | 2 | 1 | no | — |
| 28 | C22×Q16 | D4 | 1 | 1 | yes | characteristic |
| 29 | #568 | D4 | 1 | 1 | yes | characteristic |
| 30 | #535 | — | 2 | 1 | no | — |
| 31 | C2×SD32 | — | 4 | 1 | no | — |
| 32 | #482 | — | 4 | 1 | no | — |
| 33 | #469 | — | 2 | 1 | no | — |
| 34 | #466 | — | 2 | 1 | no | — |
| 35 | #447 | — | 4 | 1 | no | — |
| 36 | #430 | — | 2 | 1 | no | — |
| 37 | #430 | — | 2 | 1 | no | — |
| 38 | #430 | D4 | 1 | 1 | yes | characteristic |
| 39 | #424 | — | 2 | 1 | no | — |
| 40 | #421 | — | 2 | 1 | no | — |
| 41 | #419 | — | 2 | 1 | no | — |
| 42 | #419 | — | 2 | 1 | no | — |
| 43 | #413 | — | 2 | 1 | no | — |
| 44 | #413 | D4 | 1 | 1 | yes | characteristic |
| 45 | C2×#101 | — | 2 | 1 | no | — |
| 46 | C2×#101 | — | 2 | 1 | no | — |
| 47 | C2×#101 | — | 2 | 1 | no | — |
| 48 | C2×#100 | — | 4 | 1 | no | — |
| 49 | #408 | C23 | 1 | 1 | yes | characteristic |
| 50 | #365 | — | 4 | 1 | no | — |
| 51 | #361 | — | 4 | 1 | no | — |
| 52 | #358 | — | 4 | 1 | no | — |
| 53 | #355 | — | 4 | 2 | no | — |
| 54 | #349 | — | 2 | 1 | no | — |
| 55 | C22⋊C16 | — | 2 | 1 | no | — |
| 56 | #344 | — | 2 | 1 | no | — |
| 57 | #344 | — | 2 | 1 | no | — |
| 58 | #344 | — | 4 | 1 | no | — |
| 59 | #341 | — | 2 | 1 | no | — |
| 60 | #341 | D4 | 1 | 1 | yes | characteristic |
| 61 | #338 | — | 2 | 1 | no | — |
| 62 | H(3,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C22×Q8 | — | 2 | 1 | no | — |
| 65 | C22×D4 | — | 2 | 1 | no | — |
| 66 | #137 | — | 8 | 1 | no | — |
| 67 | #137 | — | 8 | 1 | no | — |
| 68 | C2×Q16 | — | 4 | 1 | no | — |
| 69 | C2×Q16 | — | 4 | 1 | no | — |
| 70 | C2×Q16 | — | 8 | 1 | no | — |
| 71 | C2×SD16 | — | 4 | 1 | no | — |
| 72 | C2×D8 | — | 2 | 1 | no | — |
| 73 | C2×D8 | — | 2 | 1 | no | — |
| 74 | #131 | — | 8 | 1 | no | — |
| 75 | C2×M16 | — | 2 | 1 | no | — |
| 76 | C2×M16 | — | 2 | 1 | no | — |
| 77 | C2×M16 | — | 2 | 1 | no | — |
| 78 | C2×M16 | — | 2 | 1 | no | — |
| 79 | C2×M16 | — | 4 | 1 | no | — |
| 80 | C2×M16 | — | 4 | 1 | no | — |
| 81 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | #127 | — | 4 | 1 | no | — |
| 84 | #124 | — | 4 | 1 | no | — |
| 85 | #121 | — | 4 | 1 | no | — |
| 86 | #121 | — | 4 | 1 | no | — |
| 87 | #120 | — | 4 | 1 | no | — |
| 88 | #120 | — | 4 | 1 | no | — |
| 89 | #117 | — | 2 | 1 | no | — |
| 90 | #117 | — | 2 | 1 | no | — |
| 91 | SD32 | — | 8 | 2 | no | — |
| 92 | M32 | — | 4 | 1 | no | — |
| 93 | C2×C16 | — | 4 | 1 | no | — |
| 94 | #108 | — | 4 | 1 | no | — |
| 95 | #108 | — | 8 | 1 | no | — |
| 96 | #108 | — | 8 | 1 | no | — |
| 97 | #107 | — | 2 | 1 | no | — |
| 98 | #106 | — | 2 | 1 | no | — |
| 99 | C4⋊C8 | — | 8 | 1 | no | — |
| 100 | #104 | — | 8 | 1 | no | — |
| 101 | #103 | — | 8 | 1 | no | — |
| 102 | #103 | — | 8 | 1 | no | — |
| 103 | #102 | — | 4 | 1 | no | — |
| 104 | #102 | — | 4 | 1 | no | — |
| 105 | #101 | — | 2 | 1 | no | — |
| 106 | #101 | — | 4 | 1 | no | — |
| 107 | #101 | — | 8 | 1 | no | — |
| 108 | #101 | — | 8 | 1 | no | — |
| 109 | #101 | — | 8 | 1 | no | — |
| 110 | #100 | — | 2 | 1 | no | — |
| 111 | #100 | — | 4 | 1 | no | — |
| 112 | #100 | — | 8 | 2 | no | — |
| 113 | C22⋊C8 | — | 4 | 1 | no | — |
| 114 | C24 | — | 4 | 1 | no | — |
| 115 | #41 | — | 4 | 1 | no | — |
| 116 | #41 | — | 4 | 1 | no | — |
| 117 | C2×Q8 | — | 4 | 1 | no | — |
| 118 | C2×Q8 | — | 4 | 1 | no | — |
| 119 | C2×Q8 | — | 4 | 1 | no | — |
| 120 | C2×Q8 | — | 4 | 1 | no | — |
| 121 | C2×Q8 | — | 4 | 1 | no | — |
| 122 | C2×Q8 | — | 4 | 1 | no | — |
| 123 | C2×Q8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 124 | C2×D4 | — | 2 | 1 | no | — |
| 125 | C2×D4 | — | 4 | 1 | no | — |
| 126 | C2×D4 | — | 4 | 1 | no | — |
| 127 | C2×D4 | — | 8 | 1 | no | — |
| 128 | C2×D4 | — | 8 | 1 | no | — |
| 129 | C2×D4 | — | 8 | 1 | no | — |
| 130 | C2×D4 | — | 8 | 1 | no | — |
| 131 | C2×D4 | #135 | 1 | 1 | yes | characteristic |
| 132 | C22×C4 | — | 2 | 1 | no | — |
| 133 | C22×C4 | — | 4 | 1 | no | — |
| 134 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 135 | Q16 | — | 4 | 2 | no | — |
| 136 | Q16 | — | 8 | 1 | no | — |
| 137 | Q16 | — | 8 | 2 | no | — |
| 138 | SD16 | — | 8 | 2 | no | — |
| 139 | D8 | — | 8 | 1 | no | — |
| 140 | D8 | — | 8 | 1 | no | — |
| 141 | M16 | — | 4 | 1 | no | — |
| 142 | M16 | — | 4 | 1 | no | — |
| 143 | M16 | — | 4 | 1 | no | — |
| 144 | M16 | — | 4 | 1 | no | — |
| 145 | M16 | — | 4 | 1 | no | — |
| 146 | M16 | — | 4 | 1 | no | — |
| 147 | M16 | — | 4 | 2 | no | — |
| 148 | M16 | — | 4 | 1 | no | — |
| 149 | M16 | — | 4 | 1 | no | — |
| 150 | M16 | — | 4 | 1 | no | — |
| 151 | M16 | — | 8 | 1 | no | — |
| 152 | M16 | — | 8 | 1 | no | — |
| 153 | C2×C8 | — | 2 | 1 | no | — |
| 154 | C2×C8 | — | 2 | 1 | no | — |
| 155 | C2×C8 | — | 4 | 1 | no | — |
| 156 | C2×C8 | — | 4 | 1 | no | — |
| 157 | C2×C8 | — | 4 | 1 | no | — |
| 158 | C2×C8 | — | 4 | 1 | no | — |
| 159 | C2×C8 | — | 4 | 1 | no | — |
| 160 | C2×C8 | — | 8 | 1 | no | — |
| 161 | C2×C8 | — | 8 | 1 | no | — |
| 162 | C4⋊C4 | — | 4 | 1 | no | — |
| 163 | C4⋊C4 | — | 4 | 1 | no | — |
| 164 | C22⋊C4 | — | 4 | 1 | no | — |
| 165 | C22⋊C4 | — | 4 | 1 | no | — |
| 166 | C22⋊C4 | — | 8 | 1 | no | — |
| 167 | C22⋊C4 | — | 8 | 1 | no | — |
| 168 | C42 | — | 4 | 1 | no | — |
| 169 | C42 | — | 4 | 1 | no | — |
| 170 | C16 | — | 8 | 1 | no | — |
| 171 | C16 | — | 8 | 1 | no | — |
| 172 | C23 | — | 2 | 1 | no | — |
| 173 | C23 | — | 8 | 2 | no | — |
| 174 | C23 | — | 8 | 1 | no | — |
| 175 | C23 | — | 32 | 1 | no | — |
| 176 | C23 | #454 | 1 | 1 | yes | characteristic |
| 177 | Q8 | — | 4 | 1 | no | — |
| 178 | Q8 | — | 8 | 1 | no | — |
| 179 | Q8 | — | 8 | 1 | no | — |
| 180 | Q8 | — | 8 | 1 | no | — |
| 181 | Q8 | — | 8 | 1 | no | — |
| 182 | D4 | — | 4 | 1 | no | — |
| 183 | D4 | — | 8 | 1 | no | — |
| 184 | D4 | — | 8 | 1 | no | — |
| 185 | D4 | — | 8 | 1 | no | — |
| 186 | D4 | — | 16 | 1 | no | — |
| 187 | D4 | — | 32 | 1 | no | — |
| 188 | D4 | — | 32 | 1 | no | — |
| 189 | C2×C4 | — | 2 | 1 | no | — |
| 190 | C2×C4 | — | 2 | 1 | no | — |
| 191 | C2×C4 | — | 2 | 1 | no | — |
| 192 | C2×C4 | — | 4 | 1 | no | — |
| 193 | C2×C4 | — | 4 | 1 | no | — |
| 194 | C2×C4 | — | 8 | 1 | no | — |
| 195 | C2×C4 | — | 8 | 1 | no | — |
| 196 | C2×C4 | — | 8 | 1 | no | — |
| 197 | C2×C4 | — | 8 | 1 | no | — |
| 198 | C2×C4 | — | 8 | 1 | no | — |
| 199 | C2×C4 | #449 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 201 | C8 | — | 4 | 1 | no | — |
| 202 | C8 | — | 4 | 1 | no | — |
| 203 | C8 | — | 8 | 1 | no | — |
| 204 | C8 | — | 8 | 1 | no | — |
| 205 | C8 | — | 8 | 1 | no | — |
| 206 | C8 | — | 8 | 1 | no | — |
| 207 | C8 | — | 16 | 1 | no | — |
| 208 | C22 | — | 2 | 1 | no | — |
| 209 | C22 | — | 4 | 1 | no | — |
| 210 | C22 | — | 4 | 1 | no | — |
| 211 | C22 | — | 8 | 1 | no | — |
| 212 | C22 | — | 16 | 1 | no | — |
| 213 | C22 | — | 16 | 1 | no | — |
| 214 | C22 | — | 16 | 1 | no | — |
| 215 | C22 | — | 32 | 2 | no | — |
| 216 | C22 | #1597 | 1 | 1 | yes | characteristic |
| 217 | C4 | — | 2 | 1 | no | — |
| 218 | C4 | — | 4 | 1 | no | — |
| 219 | C4 | — | 16 | 1 | no | — |
| 220 | C4 | — | 16 | 1 | no | — |
| 221 | C4 | — | 16 | 1 | no | — |
| 222 | C4 | #2200 | 1 | 1 | yes | characteristic |
| 223 | C4 | #2200 | 1 | 1 | yes | characteristic |
| 224 | C2 | — | 2 | 1 | no | — |
| 225 | C2 | — | 4 | 1 | no | — |
| 226 | C2 | — | 8 | 1 | no | — |
| 227 | C2 | — | 32 | 1 | no | — |
| 228 | C2 | #12137 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12137 |
| commutator, Frattini, cosocle | #408 | C23 |
| prime radical | #3199 | C22 |
| LCS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #32872 | C2 | semidirect |
| 5. #7439 | C2 | semidirect |
| 6. #7401 | C2 | semidirect |
| 7. #7377 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #32872 | 2 |
| #13429 | 2 |
| #12301 | 2 |
| #12111 | 2 |
| #7439 | 2 |
| #7401 | 2 |
| #7377 | 2 |