Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1594 conjugacy classes of subgroups; only the 610 normal subgroups are shown.
This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 610 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 | C2×Q8×Q16 | C2 | 1 | 4 | yes | maximal |
| 2 | Q162 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×#1713 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#1676 | C2 | 1 | 4 | yes | maximal |
| 5 | C2×#1633 | C2 | 1 | 4 | yes | maximal |
| 6 | C2×C8×Q16 | C2 | 1 | 2 | yes | maximal |
| 7 | C2×Q82 | C22 | 1 | 4 | yes | — |
| 8 | Q8×Q16 | — | 2 | 16 | no | — |
| 9 | Q8×Q16 | C22 | 1 | 32 | yes | — |
| 10 | C2×#500 | — | 2 | 4 | no | — |
| 11 | C2×#500 | C22 | 1 | 4 | yes | — |
| 12 | C2×#494 | C22 | 1 | 2 | yes | — |
| 13 | C2×#477 | — | 2 | 8 | no | — |
| 14 | C2×#477 | C22 | 1 | 8 | yes | — |
| 15 | C2×C8×Q8 | C22 | 1 | 4 | yes | — |
| 16 | C2×C4×Q16 | — | 2 | 4 | no | — |
| 17 | C2×C4×Q16 | C22 | 1 | 2 | yes | — |
| 18 | #1713 | C22 | 1 | 8 | yes | — |
| 19 | #1676 | C22 | 1 | 32 | yes | — |
| 20 | #1633 | C22 | 1 | 32 | yes | — |
| 21 | C8×Q16 | C22 | 1 | 16 | yes | — |
| 22 | C2×#335 | C22 | 1 | 2 | yes | — |
| 23 | C2×#333 | C22 | 1 | 4 | yes | — |
| 24 | C2×Q8⋊C8 | C22 | 1 | 4 | yes | — |
| 25 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 26 | C22×Q16 | — | 4 | 2 | no | — |
| 27 | C22×Q16 | D4 | 1 | 2 | yes | — |
| 28 | Q82 | — | 2 | 16 | no | — |
| 29 | Q82 | — | 4 | 4 | no | — |
| 30 | Q82 | C23 | 1 | 16 | yes | — |
| 31 | C2×C4⋊Q8 | — | 2 | 8 | no | — |
| 32 | C2×C4⋊Q8 | — | 4 | 4 | no | — |
| 33 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 34 | C2×C4×Q8 | — | 2 | 8 | no | — |
| 35 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 36 | #500 | — | 2 | 32 | no | — |
| 37 | #500 | — | 2 | 8 | no | — |
| 38 | #500 | C23 | 1 | 16 | yes | — |
| 39 | #494 | — | 4 | 2 | no | — |
| 40 | #494 | C23 | 1 | 8 | yes | — |
| 41 | #477 | — | 2 | 16 | no | — |
| 42 | #477 | — | 2 | 64 | no | — |
| 43 | #477 | C23 | 1 | 32 | yes | — |
| 44 | C8×Q8 | — | 2 | 8 | no | — |
| 45 | C8×Q8 | C23 | 1 | 16 | yes | — |
| 46 | C4×Q16 | — | 2 | 32 | no | — |
| 47 | C4×Q16 | D4 | 1 | 8 | yes | — |
| 48 | C4×Q16 | C23 | 1 | 8 | yes | — |
| 49 | C2×#107 | — | 2 | 4 | no | — |
| 50 | C2×#107 | — | 2 | 4 | no | — |
| 51 | C2×#107 | — | 2 | 8 | no | — |
| 52 | C2×#107 | D4 | 1 | 2 | yes | — |
| 53 | C2×C4⋊C8 | — | 2 | 4 | no | — |
| 54 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 55 | C2×#103 | — | 2 | 8 | no | — |
| 56 | C2×#103 | D4 | 1 | 4 | yes | — |
| 57 | C2×C4×C8 | — | 2 | 4 | no | — |
| 58 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 59 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 60 | #335 | C23 | 1 | 8 | yes | — |
| 61 | #333 | C23 | 1 | 16 | yes | — |
| 62 | Q8⋊C8 | C23 | 1 | 16 | yes | — |
| 63 | C82 | C23 | 1 | 4 | yes | — |
| 64 | C22×Q8 | — | 2 | 4 | no | — |
| 65 | C22×Q8 | — | 4 | 4 | no | — |
| 66 | C22×Q8 | C2×D4 | 1 | 4 | yes | — |
| 67 | C2×Q16 | — | 4 | 8 | no | — |
| 68 | C2×Q16 | — | 8 | 8 | no | — |
| 69 | C2×Q16 | Q16 | 1 | 16 | yes | direct product factor |
| 70 | C2×Q16 | C2×D4 | 1 | 8 | yes | — |
| 71 | C22×C8 | — | 2 | 1 | no | — |
| 72 | C22×C8 | — | 2 | 4 | no | — |
| 73 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 74 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 75 | C4⋊Q8 | — | 2 | 32 | no | — |
| 76 | C4⋊Q8 | — | 2 | 8 | no | — |
| 77 | C4⋊Q8 | — | 2 | 4 | no | — |
| 78 | C4⋊Q8 | — | 4 | 16 | no | — |
| 79 | C4⋊Q8 | — | 4 | 16 | no | — |
| 80 | C4⋊Q8 | — | 4 | 16 | no | — |
| 81 | C4⋊Q8 | C24 | 1 | 8 | yes | — |
| 82 | C4×Q8 | — | 2 | 4 | no | — |
| 83 | C4×Q8 | — | 2 | 4 | no | — |
| 84 | C4×Q8 | — | 2 | 32 | no | — |
| 85 | C4×Q8 | — | 4 | 16 | no | — |
| 86 | C4×Q8 | C2×D4 | 1 | 8 | yes | — |
| 87 | C4×Q8 | C24 | 1 | 8 | yes | — |
| 88 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 89 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 90 | C2×C4⋊C4 | — | 4 | 8 | no | — |
| 91 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 92 | C2×C42 | — | 2 | 4 | no | — |
| 93 | C2×C42 | — | 4 | 4 | no | — |
| 94 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 95 | #107 | — | 2 | 16 | no | — |
| 96 | #107 | — | 2 | 16 | no | — |
| 97 | #107 | — | 2 | 32 | no | — |
| 98 | #107 | C2×D4 | 1 | 8 | yes | — |
| 99 | C4⋊C8 | — | 2 | 16 | no | — |
| 100 | C4⋊C8 | — | 2 | 4 | no | — |
| 101 | C4⋊C8 | C24 | 1 | 8 | yes | — |
| 102 | #103 | — | 2 | 32 | no | — |
| 103 | #103 | C2×D4 | 1 | 16 | yes | — |
| 104 | C4×C8 | — | 2 | 16 | no | — |
| 105 | C4×C8 | — | 2 | 1 | no | — |
| 106 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 107 | C4×C8 | C24 | 1 | 2 | yes | — |
| 108 | C4×C8 | C24 | 1 | 4 | yes | — |
| 109 | C2×Q8 | — | 2 | 4 | no | — |
| 110 | C2×Q8 | — | 2 | 8 | no | — |
| 111 | C2×Q8 | — | 2 | 8 | no | — |
| 112 | C2×Q8 | — | 4 | 16 | no | — |
| 113 | C2×Q8 | — | 4 | 4 | no | — |
| 114 | C2×Q8 | — | 8 | 8 | no | — |
| 115 | C2×Q8 | — | 16 | 8 | no | — |
| 116 | C2×Q8 | C2×Q16 | 1 | 16 | yes | — |
| 117 | C2×Q8 | C22×D4 | 1 | 8 | yes | — |
| 118 | C22×C4 | — | 2 | 4 | no | — |
| 119 | C22×C4 | — | 2 | 4 | no | — |
| 120 | C22×C4 | — | 4 | 4 | no | — |
| 121 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 122 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 123 | Q16 | — | 8 | 16 | no | — |
| 124 | Q16 | C2×Q16 | 1 | 32 | yes | direct product factor |
| 125 | C2×C8 | — | 2 | 4 | no | — |
| 126 | C2×C8 | — | 2 | 4 | no | — |
| 127 | C2×C8 | — | 2 | 2 | no | — |
| 128 | C2×C8 | — | 2 | 8 | no | — |
| 129 | C2×C8 | — | 4 | 8 | no | — |
| 130 | C2×C8 | C2×Q16 | 1 | 8 | yes | — |
| 131 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 132 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 133 | C4⋊C4 | — | 2 | 8 | no | — |
| 134 | C4⋊C4 | — | 2 | 4 | no | — |
| 135 | C4⋊C4 | — | 2 | 8 | no | — |
| 136 | C4⋊C4 | — | 2 | 16 | no | — |
| 137 | C4⋊C4 | — | 4 | 32 | no | — |
| 138 | C4⋊C4 | — | 4 | 8 | no | — |
| 139 | C4⋊C4 | C22×D4 | 1 | 8 | yes | — |
| 140 | C42 | — | 2 | 8 | no | — |
| 141 | C42 | — | 2 | 8 | no | — |
| 142 | C42 | — | 4 | 16 | no | — |
| 143 | C42 | C22×D4 | 1 | 2 | yes | — |
| 144 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 145 | C42 | C25 | 1 | 1 | yes | characteristic |
| 146 | C23 | D42 | 1 | 1 | yes | characteristic |
| 147 | Q8 | — | 2 | 16 | no | — |
| 148 | Q8 | — | 2 | 8 | no | — |
| 149 | Q8 | — | 8 | 8 | no | — |
| 150 | Q8 | — | 16 | 4 | no | — |
| 151 | Q8 | — | 16 | 16 | no | — |
| 152 | Q8 | C22×Q16 | 1 | 16 | yes | — |
| 153 | C2×C4 | — | 2 | 8 | no | — |
| 154 | C2×C4 | — | 2 | 16 | no | — |
| 155 | C2×C4 | — | 2 | 2 | no | — |
| 156 | C2×C4 | — | 2 | 8 | no | — |
| 157 | C2×C4 | — | 4 | 8 | no | — |
| 158 | C2×C4 | — | 4 | 8 | no | — |
| 159 | C2×C4 | — | 8 | 8 | no | — |
| 160 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 161 | C2×C4 | D42 | 1 | 2 | yes | — |
| 162 | C2×C4 | C22×Q16 | 1 | 4 | yes | — |
| 163 | C2×C4 | #578 | 1 | 4 | yes | — |
| 164 | C2×C4 | C23×D4 | 1 | 2 | yes | — |
| 165 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 166 | C8 | — | 2 | 4 | no | — |
| 167 | C8 | — | 2 | 4 | no | — |
| 168 | C8 | — | 4 | 8 | no | — |
| 169 | C8 | C22×Q16 | 1 | 8 | yes | — |
| 170 | C22 | D4×Q16 | 1 | 4 | yes | — |
| 171 | C22 | #3292 | 1 | 2 | yes | — |
| 172 | C22 | C2×D42 | 1 | 1 | yes | characteristic |
| 173 | C4 | — | 2 | 16 | no | — |
| 174 | C4 | — | 2 | 1 | no | — |
| 175 | C4 | — | 2 | 1 | no | — |
| 176 | C4 | — | 4 | 8 | no | — |
| 177 | C4 | — | 8 | 8 | no | — |
| 178 | C4 | D4×Q16 | 1 | 4 | yes | — |
| 179 | C4 | C23×Q16 | 1 | 2 | yes | — |
| 180 | C4 | C2×#578 | 1 | 2 | yes | — |
| 181 | C2 | Q162 | 1 | 4 | yes | direct product factor |
| 182 | C2 | C2×D4×Q16 | 1 | 2 | yes | — |
| 183 | C2 | C2×#3292 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q162 | C2 | direct (N × H) |
| 69. C2×Q16 | Q16 | direct (N × H) |
| 124. Q16 | C2×Q16 | direct (N × H) |
| 181. C2 | Q162 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×Q16 | 2 |
| Q162 | 2 |
| C2×#1713 | 2 |
| C2×#1676 | 2 |
| C2×#1633 | 2 |
| C2×C8×Q16 | 2 |