Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
241 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #27693 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11485 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #8638 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7440 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 6 | #2916 | C4 | 1 | 1 | yes | characteristic |
| 7 | #2115 | — | 2 | 1 | no | — |
| 8 | #2102 | C22 | 1 | 2 | yes | — |
| 9 | #2098 | C4 | 1 | 2 | yes | — |
| 10 | #1945 | — | 2 | 1 | no | — |
| 11 | #1884 | — | 2 | 1 | no | — |
| 12 | #1868 | — | 2 | 1 | no | — |
| 13 | #1814 | — | 2 | 1 | no | — |
| 14 | #1812 | C4 | 1 | 1 | yes | characteristic |
| 15 | #1567 | — | 2 | 1 | no | — |
| 16 | #1560 | — | 2 | 1 | no | — |
| 17 | #1327 | C22 | 1 | 2 | yes | — |
| 18 | #1321 | — | 2 | 1 | no | — |
| 19 | #1309 | C22 | 1 | 2 | yes | — |
| 20 | #568 | — | 2 | 1 | no | — |
| 21 | C2×#131 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 23 | #532 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×M32 | — | 4 | 1 | no | — |
| 25 | #436 | — | 4 | 1 | no | — |
| 26 | C8×D4 | — | 4 | 1 | no | — |
| 27 | #430 | — | 4 | 1 | no | — |
| 28 | #430 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×#108 | — | 2 | 1 | no | — |
| 30 | C2×#108 | — | 4 | 1 | no | — |
| 31 | #427 | — | 2 | 1 | no | — |
| 32 | #424 | — | 2 | 2 | no | — |
| 33 | #423 | — | 2 | 2 | no | — |
| 34 | #423 | C23 | 1 | 1 | yes | characteristic |
| 35 | #421 | — | 2 | 2 | no | — |
| 36 | #410 | — | 2 | 1 | no | — |
| 37 | #408 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #408 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 40 | #349 | — | 2 | 2 | no | — |
| 41 | #349 | — | 4 | 1 | no | — |
| 42 | #344 | — | 2 | 2 | no | — |
| 43 | #339 | — | 2 | 2 | no | — |
| 44 | H(3,ℤ/4) | C2×C4 | 1 | 2 | yes | — |
| 45 | #335 | — | 4 | 1 | no | — |
| 46 | #335 | — | 4 | 1 | no | — |
| 47 | #333 | — | 4 | 1 | no | — |
| 48 | #331 | — | 4 | 1 | no | — |
| 49 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 50 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 51 | #131 | — | 4 | 1 | no | — |
| 52 | #131 | — | 4 | 1 | no | — |
| 53 | #131 | — | 4 | 1 | no | — |
| 54 | #131 | — | 4 | 1 | no | — |
| 55 | #131 | — | 8 | 2 | no | — |
| 56 | C2×M16 | — | 2 | 1 | no | — |
| 57 | C2×M16 | — | 2 | 1 | no | — |
| 58 | C2×M16 | — | 2 | 2 | no | — |
| 59 | C2×M16 | — | 4 | 2 | no | — |
| 60 | C2×M16 | — | 4 | 2 | no | — |
| 61 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C22×C8 | — | 2 | 1 | no | — |
| 64 | C22×C8 | — | 2 | 1 | no | — |
| 65 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C4⋊Q8 | SD16 | 1 | 1 | yes | characteristic |
| 68 | #127 | SD16 | 1 | 1 | yes | characteristic |
| 69 | #124 | Q16 | 1 | 1 | yes | characteristic |
| 70 | #124 | Q16 | 1 | 1 | yes | characteristic |
| 71 | #121 | — | 4 | 1 | no | — |
| 72 | C4×D4 | — | 4 | 1 | no | — |
| 73 | #117 | — | 2 | 2 | no | — |
| 74 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | M32 | — | 4 | 2 | no | — |
| 76 | M32 | — | 8 | 1 | no | — |
| 77 | C2×C16 | — | 8 | 1 | no | — |
| 78 | #108 | — | 4 | 2 | no | — |
| 79 | #108 | — | 4 | 1 | no | — |
| 80 | #108 | — | 4 | 1 | no | — |
| 81 | #107 | — | 4 | 1 | no | — |
| 82 | #106 | — | 4 | 1 | no | — |
| 83 | C4⋊C8 | — | 4 | 1 | no | — |
| 84 | C4⋊C8 | — | 4 | 2 | no | — |
| 85 | C4⋊C8 | — | 4 | 2 | no | — |
| 86 | C4⋊C8 | — | 8 | 2 | no | — |
| 87 | #104 | — | 8 | 2 | no | — |
| 88 | #99 | — | 4 | 2 | no | — |
| 89 | C22⋊C8 | — | 4 | 1 | no | — |
| 90 | C22⋊C8 | — | 4 | 1 | no | — |
| 91 | C4×C8 | — | 2 | 1 | no | — |
| 92 | C4×C8 | — | 2 | 1 | no | — |
| 93 | #41 | — | 4 | 1 | no | — |
| 94 | #41 | — | 4 | 1 | no | — |
| 95 | #41 | — | 4 | 1 | no | — |
| 96 | #41 | — | 4 | 1 | no | — |
| 97 | C2×Q8 | #103 | 1 | 1 | yes | characteristic |
| 98 | C2×Q8 | #103 | 1 | 1 | yes | characteristic |
| 99 | C2×D4 | — | 2 | 1 | no | — |
| 100 | C2×D4 | — | 2 | 1 | no | — |
| 101 | C2×D4 | — | 4 | 1 | no | — |
| 102 | C2×D4 | #103 | 1 | 1 | yes | characteristic |
| 103 | C2×D4 | #103 | 1 | 1 | yes | characteristic |
| 104 | C22×C4 | — | 2 | 1 | no | — |
| 105 | C22×C4 | — | 2 | 1 | no | — |
| 106 | C22×C4 | — | 2 | 1 | no | — |
| 107 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 108 | M16 | — | 4 | 2 | no | — |
| 109 | M16 | — | 4 | 1 | no | — |
| 110 | M16 | — | 4 | 1 | no | — |
| 111 | M16 | — | 4 | 2 | no | — |
| 112 | M16 | — | 4 | 2 | no | — |
| 113 | M16 | — | 8 | 1 | no | — |
| 114 | M16 | — | 8 | 2 | no | — |
| 115 | M16 | — | 16 | 2 | no | — |
| 116 | C2×C8 | — | 2 | 1 | no | — |
| 117 | C2×C8 | — | 2 | 1 | no | — |
| 118 | C2×C8 | — | 2 | 1 | no | — |
| 119 | C2×C8 | — | 2 | 1 | no | — |
| 120 | C2×C8 | — | 2 | 1 | no | — |
| 121 | C2×C8 | — | 4 | 1 | no | — |
| 122 | C2×C8 | — | 4 | 1 | no | — |
| 123 | C2×C8 | — | 4 | 1 | no | — |
| 124 | C2×C8 | — | 4 | 1 | no | — |
| 125 | C2×C8 | — | 4 | 1 | no | — |
| 126 | C2×C8 | — | 4 | 2 | no | — |
| 127 | C2×C8 | — | 8 | 2 | no | — |
| 128 | C2×C8 | — | 8 | 2 | no | — |
| 129 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 130 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 131 | C4⋊C4 | — | 4 | 1 | no | — |
| 132 | C4⋊C4 | — | 4 | 2 | no | — |
| 133 | C22⋊C4 | — | 4 | 1 | no | — |
| 134 | C22⋊C4 | — | 4 | 1 | no | — |
| 135 | C22⋊C4 | — | 4 | 2 | no | — |
| 136 | C42 | — | 2 | 1 | no | — |
| 137 | C42 | — | 4 | 2 | no | — |
| 138 | C42 | C2×SD16 | 1 | 1 | yes | characteristic |
| 139 | C42 | C2×Q16 | 1 | 1 | yes | characteristic |
| 140 | C16 | — | 8 | 2 | no | — |
| 141 | C23 | — | 2 | 1 | no | — |
| 142 | C23 | — | 2 | 1 | no | — |
| 143 | C23 | #421 | 1 | 1 | yes | characteristic |
| 144 | Q8 | — | 4 | 1 | no | — |
| 145 | Q8 | — | 4 | 1 | no | — |
| 146 | D4 | — | 4 | 1 | no | — |
| 147 | D4 | — | 4 | 1 | no | — |
| 148 | D4 | — | 4 | 2 | no | — |
| 149 | D4 | — | 8 | 1 | no | — |
| 150 | D4 | — | 8 | 1 | no | — |
| 151 | D4 | — | 8 | 1 | no | — |
| 152 | C2×C4 | — | 2 | 1 | no | — |
| 153 | C2×C4 | — | 2 | 1 | no | — |
| 154 | C2×C4 | — | 2 | 1 | no | — |
| 155 | C2×C4 | — | 2 | 1 | no | — |
| 156 | C2×C4 | — | 2 | 1 | no | — |
| 157 | C2×C4 | — | 2 | 1 | no | — |
| 158 | C2×C4 | — | 4 | 1 | no | — |
| 159 | C2×C4 | — | 4 | 1 | no | — |
| 160 | C2×C4 | — | 4 | 1 | no | — |
| 161 | C2×C4 | — | 4 | 1 | no | — |
| 162 | C2×C4 | — | 4 | 1 | no | — |
| 163 | C2×C4 | — | 4 | 1 | no | — |
| 164 | C2×C4 | — | 8 | 2 | no | — |
| 165 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 166 | C2×C4 | C2×#103 | 1 | 1 | yes | characteristic |
| 167 | C8 | — | 2 | 1 | no | — |
| 168 | C8 | — | 4 | 1 | no | — |
| 169 | C8 | — | 4 | 1 | no | — |
| 170 | C8 | — | 4 | 1 | no | — |
| 171 | C8 | — | 8 | 2 | no | — |
| 172 | C8 | — | 16 | 2 | no | — |
| 173 | C8 | #352 | 1 | 1 | yes | characteristic |
| 174 | C8 | #352 | 1 | 1 | yes | characteristic |
| 175 | C22 | — | 2 | 1 | no | — |
| 176 | C22 | — | 4 | 1 | no | — |
| 177 | C22 | — | 4 | 1 | no | — |
| 178 | C22 | — | 4 | 1 | no | — |
| 179 | C22 | — | 8 | 1 | no | — |
| 180 | C22 | — | 8 | 1 | no | — |
| 181 | C22 | #1520 | 1 | 1 | yes | characteristic |
| 182 | C4 | — | 2 | 1 | no | — |
| 183 | C4 | — | 2 | 1 | no | — |
| 184 | C4 | — | 2 | 1 | no | — |
| 185 | C4 | — | 4 | 1 | no | — |
| 186 | C4 | — | 4 | 1 | no | — |
| 187 | C4 | — | 4 | 1 | no | — |
| 188 | C4 | — | 16 | 2 | no | — |
| 189 | C4 | C2×#352 | 1 | 1 | yes | characteristic |
| 190 | C4 | #2134 | 1 | 1 | yes | characteristic |
| 191 | C2 | — | 2 | 1 | no | — |
| 192 | C2 | — | 4 | 1 | no | — |
| 193 | C2 | — | 8 | 1 | no | — |
| 194 | C2 | — | 8 | 1 | no | — |
| 195 | C2 | #11903 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11903 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7440 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27693 | 2 |
| #11485 | 2 |
| #8638 | 2 |
| #7440 | 2 |