Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
448 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 118 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 | #21294 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #13519 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13466 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13444 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #13137 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13117 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13027 | C2 | 1 | 2 | yes | maximal |
| 8 | #12951 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12884 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3414 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×#135 | C22 | 1 | 2 | yes | — |
| 12 | #2923 | — | 2 | 2 | no | — |
| 13 | #2161 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2155 | C22 | 1 | 2 | yes | — |
| 15 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2148 | C22 | 1 | 2 | yes | — |
| 18 | #2145 | C22 | 1 | 2 | yes | — |
| 19 | #2143 | C22 | 1 | 2 | yes | — |
| 20 | #1974 | C22 | 1 | 2 | yes | — |
| 21 | #1974 | C22 | 1 | 2 | yes | — |
| 22 | #1945 | C22 | 1 | 2 | yes | — |
| 23 | #1945 | C22 | 1 | 2 | yes | — |
| 24 | #1938 | — | 2 | 2 | no | — |
| 25 | #1938 | — | 2 | 2 | no | — |
| 26 | C4×#108 | C22 | 1 | 2 | yes | — |
| 27 | #1759 | — | 2 | 2 | no | — |
| 28 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1389 | C22 | 1 | 2 | yes | — |
| 30 | #1388 | C22 | 1 | 2 | yes | — |
| 31 | #1385 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1384 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1376 | C22 | 1 | 2 | yes | — |
| 34 | #1376 | C22 | 1 | 2 | yes | — |
| 35 | C2×#135 | C23 | 1 | 2 | yes | — |
| 36 | C4×#41 | — | 2 | 2 | no | — |
| 37 | #510 | — | 2 | 2 | no | — |
| 38 | #509 | — | 2 | 2 | no | — |
| 39 | #508 | — | 4 | 1 | no | — |
| 40 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×M32 | C23 | 1 | 2 | yes | — |
| 42 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 43 | C4×Q16 | — | 2 | 2 | no | — |
| 44 | C4×SD16 | — | 2 | 4 | no | — |
| 45 | C4×D8 | — | 2 | 2 | no | — |
| 46 | #433 | — | 4 | 4 | no | — |
| 47 | #432 | — | 4 | 4 | no | — |
| 48 | C2×#108 | C23 | 1 | 2 | yes | — |
| 49 | C2×#108 | C23 | 1 | 2 | yes | — |
| 50 | #427 | C23 | 1 | 2 | yes | — |
| 51 | #424 | — | 2 | 2 | no | — |
| 52 | #424 | — | 2 | 2 | no | — |
| 53 | C2×#104 | — | 2 | 2 | no | — |
| 54 | C2×#104 | — | 2 | 2 | no | — |
| 55 | #416 | — | 2 | 2 | no | — |
| 56 | #408 | — | 2 | 2 | no | — |
| 57 | #408 | — | 2 | 2 | no | — |
| 58 | C4×M16 | — | 2 | 2 | no | — |
| 59 | C2×#97 | D4 | 1 | 2 | yes | — |
| 60 | C2×#97 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 62 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | #376 | D4 | 1 | 1 | yes | characteristic |
| 64 | #368 | — | 4 | 1 | no | — |
| 65 | #365 | — | 2 | 4 | no | — |
| 66 | C4⋊C16 | — | 4 | 1 | no | — |
| 67 | C4⋊C16 | — | 4 | 1 | no | — |
| 68 | #362 | — | 2 | 2 | no | — |
| 69 | #361 | — | 2 | 2 | no | — |
| 70 | #360 | — | 2 | 4 | no | — |
| 71 | #359 | — | 4 | 2 | no | — |
| 72 | #341 | — | 2 | 2 | no | — |
| 73 | #339 | — | 2 | 2 | no | — |
| 74 | #339 | — | 2 | 2 | no | — |
| 75 | C2×#41 | — | 2 | 2 | no | — |
| 76 | #135 | — | 2 | 2 | no | — |
| 77 | #135 | — | 2 | 2 | no | — |
| 78 | #135 | — | 2 | 2 | no | — |
| 79 | #135 | — | 2 | 2 | no | — |
| 80 | C2×Q16 | #41 | 1 | 2 | yes | — |
| 81 | C2×SD16 | — | 2 | 2 | no | — |
| 82 | C2×D8 | #41 | 1 | 2 | yes | — |
| 83 | C2×M16 | — | 2 | 2 | no | — |
| 84 | C2×M16 | — | 2 | 2 | no | — |
| 85 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 86 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 88 | C4×Q8 | — | 2 | 4 | no | — |
| 89 | C4×D4 | — | 2 | 4 | no | — |
| 90 | C4×D4 | — | 4 | 4 | no | — |
| 91 | #117 | — | 2 | 2 | no | — |
| 92 | #117 | — | 4 | 2 | no | — |
| 93 | C2×C42 | — | 4 | 2 | no | — |
| 94 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 96 | Q32 | — | 4 | 1 | no | — |
| 97 | SD32 | — | 4 | 2 | no | — |
| 98 | D16 | — | 4 | 1 | no | — |
| 99 | M32 | — | 2 | 2 | no | — |
| 100 | M32 | — | 2 | 2 | no | — |
| 101 | M32 | — | 2 | 2 | no | — |
| 102 | M32 | — | 2 | 2 | no | — |
| 103 | C2×C16 | — | 2 | 2 | no | — |
| 104 | C2×C16 | — | 2 | 1 | no | — |
| 105 | C2×C16 | — | 2 | 1 | no | — |
| 106 | #108 | — | 2 | 2 | no | — |
| 107 | #108 | — | 2 | 2 | no | — |
| 108 | #108 | — | 2 | 2 | no | — |
| 109 | #108 | — | 2 | 2 | no | — |
| 110 | #107 | — | 2 | 2 | no | — |
| 111 | #106 | #41 | 1 | 4 | yes | — |
| 112 | C4⋊C8 | — | 4 | 4 | no | — |
| 113 | C4⋊C8 | — | 4 | 4 | no | — |
| 114 | #104 | — | 4 | 4 | no | — |
| 115 | #104 | — | 4 | 2 | no | — |
| 116 | #104 | — | 4 | 2 | no | — |
| 117 | #103 | — | 2 | 4 | no | — |
| 118 | #102 | — | 2 | 4 | no | — |
| 119 | C22⋊C8 | — | 4 | 4 | no | — |
| 120 | C22⋊C8 | — | 4 | 4 | no | — |
| 121 | #97 | — | 2 | 2 | no | — |
| 122 | #97 | — | 2 | 2 | no | — |
| 123 | #97 | — | 2 | 2 | no | — |
| 124 | #97 | — | 4 | 2 | no | — |
| 125 | C4×C8 | — | 2 | 2 | no | — |
| 126 | C4×C8 | — | 2 | 2 | no | — |
| 127 | C4×C8 | — | 2 | 1 | no | — |
| 128 | C4×C8 | — | 2 | 1 | no | — |
| 129 | C4×C8 | — | 4 | 2 | no | — |
| 130 | #95 | — | 4 | 1 | no | — |
| 131 | #41 | — | 4 | 2 | no | — |
| 132 | #41 | — | 4 | 2 | no | — |
| 133 | #41 | — | 4 | 2 | no | — |
| 134 | #41 | — | 4 | 2 | no | — |
| 135 | C2×Q8 | — | 2 | 2 | no | — |
| 136 | C2×D4 | — | 2 | 2 | no | — |
| 137 | C2×D4 | — | 4 | 2 | no | — |
| 138 | C22×C4 | — | 2 | 1 | no | — |
| 139 | C22×C4 | — | 2 | 2 | no | — |
| 140 | C22×C4 | — | 4 | 4 | no | — |
| 141 | C22×C4 | — | 4 | 2 | no | — |
| 142 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 143 | Q16 | — | 2 | 2 | no | — |
| 144 | Q16 | — | 2 | 2 | no | — |
| 145 | SD16 | — | 4 | 2 | no | — |
| 146 | SD16 | — | 4 | 2 | no | — |
| 147 | D8 | — | 2 | 2 | no | — |
| 148 | D8 | — | 2 | 2 | no | — |
| 149 | M16 | — | 4 | 2 | no | — |
| 150 | M16 | — | 4 | 2 | no | — |
| 151 | M16 | — | 4 | 2 | no | — |
| 152 | M16 | — | 4 | 2 | no | — |
| 153 | C2×C8 | — | 2 | 2 | no | — |
| 154 | C2×C8 | — | 2 | 1 | no | — |
| 155 | C2×C8 | — | 2 | 2 | no | — |
| 156 | C2×C8 | — | 2 | 2 | no | — |
| 157 | C2×C8 | — | 2 | 2 | no | — |
| 158 | C2×C8 | — | 4 | 2 | no | — |
| 159 | C2×C8 | — | 4 | 2 | no | — |
| 160 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 161 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 162 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 163 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 164 | C4⋊C4 | — | 2 | 4 | no | — |
| 165 | C4⋊C4 | — | 4 | 4 | no | — |
| 166 | C22⋊C4 | — | 4 | 4 | no | — |
| 167 | C22⋊C4 | — | 4 | 2 | no | — |
| 168 | C42 | — | 2 | 2 | no | — |
| 169 | C42 | — | 2 | 1 | no | — |
| 170 | C42 | — | 4 | 4 | no | — |
| 171 | C42 | — | 4 | 2 | no | — |
| 172 | C42 | #135 | 1 | 2 | yes | — |
| 173 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 174 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 175 | C16 | — | 4 | 1 | no | — |
| 176 | C16 | — | 4 | 1 | no | — |
| 177 | C16 | — | 4 | 2 | no | — |
| 178 | C23 | — | 4 | 2 | no | — |
| 179 | C23 | #575 | 1 | 1 | yes | characteristic |
| 180 | Q8 | — | 4 | 2 | no | — |
| 181 | Q8 | — | 4 | 2 | no | — |
| 182 | D4 | — | 4 | 2 | no | — |
| 183 | D4 | — | 4 | 2 | no | — |
| 184 | D4 | — | 8 | 2 | no | — |
| 185 | D4 | — | 8 | 2 | no | — |
| 186 | C2×C4 | — | 2 | 2 | no | — |
| 187 | C2×C4 | — | 2 | 2 | no | — |
| 188 | C2×C4 | — | 2 | 2 | no | — |
| 189 | C2×C4 | — | 4 | 4 | no | — |
| 190 | C2×C4 | — | 4 | 2 | no | — |
| 191 | C2×C4 | — | 8 | 1 | no | — |
| 192 | C2×C4 | — | 8 | 2 | no | — |
| 193 | C2×C4 | — | 8 | 2 | no | — |
| 194 | C2×C4 | — | 8 | 2 | no | — |
| 195 | C2×C4 | — | 16 | 4 | no | — |
| 196 | C2×C4 | #525 | 1 | 2 | yes | — |
| 197 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 198 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 200 | C8 | — | 4 | 2 | no | — |
| 201 | C8 | — | 4 | 1 | no | — |
| 202 | C8 | — | 8 | 2 | no | — |
| 203 | C8 | — | 8 | 2 | no | — |
| 204 | C8 | #551 | 1 | 1 | yes | characteristic |
| 205 | C8 | #552 | 1 | 2 | yes | — |
| 206 | C8 | #555 | 1 | 1 | yes | characteristic |
| 207 | C22 | — | 4 | 1 | no | — |
| 208 | C22 | — | 8 | 4 | no | — |
| 209 | C22 | — | 8 | 2 | no | — |
| 210 | C22 | #3114 | 1 | 2 | yes | — |
| 211 | C22 | #3250 | 1 | 1 | yes | characteristic |
| 212 | C4 | — | 4 | 2 | no | — |
| 213 | C4 | — | 4 | 2 | no | — |
| 214 | C4 | — | 4 | 2 | no | — |
| 215 | C4 | — | 8 | 4 | no | — |
| 216 | C4 | — | 8 | 2 | no | — |
| 217 | C4 | #2609 | 1 | 1 | yes | characteristic |
| 218 | C4 | #3106 | 1 | 2 | yes | — |
| 219 | C4 | #3253 | 1 | 1 | yes | characteristic |
| 220 | C2 | — | 2 | 2 | no | — |
| 221 | C2 | — | 2 | 1 | no | — |
| 222 | C2 | — | 16 | 2 | no | — |
| 223 | C2 | #29533 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3253 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #29533 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21294 | C2 | semidirect |
| 2. #13519 | C2 | semidirect |
| 3. #13466 | C2 | semidirect |
| 5. #13137 | C2 | semidirect |
| 6. #13117 | C2 | semidirect |
| 8. #12951 | C2 | semidirect |
| 9. #12884 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21294 | 2 |
| #13519 | 2 |
| #13466 | 2 |
| #13444 | 2 |
| #13137 | 2 |
| #13117 | 2 |
| #13027 | 2 |
| #12951 | 2 |
| #12884 | 2 |