Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
201 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 34 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 | #13306 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7334 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #7230 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C22×Q32 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×#467 | — | 2 | 1 | no | — |
| 6 | C2×#358 | — | 2 | 1 | no | — |
| 7 | C2×C22⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 8 | #1808 | — | 2 | 1 | no | — |
| 9 | C22⋊C32 | — | 2 | 1 | no | — |
| 10 | #1380 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×Q16 | — | 2 | 1 | no | — |
| 12 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 13 | C2×C22⋊Q8 | — | 4 | 1 | no | — |
| 14 | C2×Q32 | — | 4 | 1 | no | — |
| 15 | C2×Q32 | — | 4 | 2 | no | — |
| 16 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 18 | #467 | — | 4 | 2 | no | — |
| 19 | #467 | — | 4 | 2 | no | — |
| 20 | C2×#107 | — | 2 | 1 | no | — |
| 21 | C2×#107 | — | 4 | 1 | no | — |
| 22 | C2×#103 | — | 4 | 1 | no | — |
| 23 | #380 | — | 4 | 1 | no | — |
| 24 | C2×C32 | — | 4 | 1 | no | — |
| 25 | #358 | — | 4 | 2 | no | — |
| 26 | C22⋊C16 | — | 4 | 1 | no | — |
| 27 | #340 | — | 2 | 2 | no | — |
| 28 | C22×Q8 | — | 4 | 1 | no | — |
| 29 | C23×C4 | D8 | 1 | 1 | yes | characteristic |
| 30 | C2×Q16 | — | 2 | 2 | no | — |
| 31 | C2×Q16 | — | 4 | 1 | no | — |
| 32 | C2×Q16 | — | 4 | 1 | no | — |
| 33 | C2×Q16 | — | 4 | 1 | no | — |
| 34 | C2×Q16 | — | 4 | 1 | no | — |
| 35 | C2×Q16 | — | 4 | 1 | no | — |
| 36 | C22×C8 | — | 2 | 1 | no | — |
| 37 | C22×C8 | — | 2 | 1 | no | — |
| 38 | C22×C8 | — | 4 | 1 | no | — |
| 39 | C22×C8 | — | 4 | 1 | no | — |
| 40 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 41 | C22×C8 | SD16 | 1 | 1 | yes | characteristic |
| 42 | C22⋊Q8 | — | 4 | 4 | no | — |
| 43 | C22⋊Q8 | — | 8 | 2 | no | — |
| 44 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 45 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 46 | C2×C4⋊C4 | — | 8 | 1 | no | — |
| 47 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 48 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 49 | Q32 | — | 4 | 2 | no | — |
| 50 | Q32 | — | 4 | 2 | no | — |
| 51 | C2×C16 | — | 2 | 1 | no | — |
| 52 | C2×C16 | — | 2 | 1 | no | — |
| 53 | C2×C16 | — | 4 | 1 | no | — |
| 54 | C2×C16 | — | 4 | 1 | no | — |
| 55 | #107 | — | 2 | 2 | no | — |
| 56 | #107 | — | 4 | 1 | no | — |
| 57 | #107 | — | 8 | 1 | no | — |
| 58 | #107 | — | 8 | 1 | no | — |
| 59 | #103 | — | 4 | 4 | no | — |
| 60 | #95 | — | 8 | 1 | no | — |
| 61 | C32 | — | 8 | 1 | no | — |
| 62 | C24 | D16 | 1 | 1 | yes | characteristic |
| 63 | C2×Q8 | — | 4 | 2 | no | — |
| 64 | C2×Q8 | — | 8 | 1 | no | — |
| 65 | C2×Q8 | — | 8 | 1 | no | — |
| 66 | C2×Q8 | — | 8 | 1 | no | — |
| 67 | C2×Q8 | — | 8 | 1 | no | — |
| 68 | C2×Q8 | — | 8 | 1 | no | — |
| 69 | C22×C4 | — | 2 | 1 | no | — |
| 70 | C22×C4 | — | 2 | 1 | no | — |
| 71 | C22×C4 | — | 4 | 1 | no | — |
| 72 | C22×C4 | — | 4 | 1 | no | — |
| 73 | C22×C4 | — | 8 | 1 | no | — |
| 74 | C22×C4 | — | 8 | 1 | no | — |
| 75 | C22×C4 | — | 8 | 2 | no | — |
| 76 | C22×C4 | #102 | 1 | 1 | yes | characteristic |
| 77 | C22×C4 | SD32 | 1 | 1 | yes | characteristic |
| 78 | Q16 | — | 4 | 2 | no | — |
| 79 | Q16 | — | 4 | 2 | no | — |
| 80 | Q16 | — | 8 | 1 | no | — |
| 81 | Q16 | — | 8 | 1 | no | — |
| 82 | C2×C8 | — | 2 | 1 | no | — |
| 83 | C2×C8 | — | 2 | 1 | no | — |
| 84 | C2×C8 | — | 2 | 1 | no | — |
| 85 | C2×C8 | — | 4 | 1 | no | — |
| 86 | C2×C8 | — | 4 | 1 | no | — |
| 87 | C2×C8 | — | 4 | 1 | no | — |
| 88 | C2×C8 | — | 8 | 1 | no | — |
| 89 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 90 | C2×C8 | #104 | 1 | 1 | yes | characteristic |
| 91 | C4⋊C4 | — | 4 | 2 | no | — |
| 92 | C4⋊C4 | — | 8 | 1 | no | — |
| 93 | C4⋊C4 | — | 8 | 4 | no | — |
| 94 | C4⋊C4 | — | 8 | 2 | no | — |
| 95 | C4⋊C4 | — | 8 | 2 | no | — |
| 96 | C22⋊C4 | — | 8 | 4 | no | — |
| 97 | C22⋊C4 | — | 16 | 1 | no | — |
| 98 | C22⋊C4 | — | 16 | 1 | no | — |
| 99 | C16 | — | 4 | 1 | no | — |
| 100 | C16 | — | 4 | 1 | no | — |
| 101 | C23 | — | 2 | 1 | no | — |
| 102 | C23 | — | 2 | 1 | no | — |
| 103 | C23 | — | 4 | 1 | no | — |
| 104 | C23 | — | 4 | 1 | no | — |
| 105 | C23 | #357 | 1 | 1 | yes | characteristic |
| 106 | C23 | SD64 | 1 | 1 | yes | characteristic |
| 107 | C23 | Q64 | 1 | 1 | yes | characteristic |
| 108 | Q8 | — | 8 | 2 | no | — |
| 109 | Q8 | — | 8 | 2 | no | — |
| 110 | Q8 | — | 16 | 1 | no | — |
| 111 | Q8 | — | 16 | 1 | no | — |
| 112 | C2×C4 | — | 2 | 1 | no | — |
| 113 | C2×C4 | — | 2 | 1 | no | — |
| 114 | C2×C4 | — | 2 | 1 | no | — |
| 115 | C2×C4 | — | 4 | 1 | no | — |
| 116 | C2×C4 | — | 4 | 1 | no | — |
| 117 | C2×C4 | — | 4 | 1 | no | — |
| 118 | C2×C4 | — | 8 | 1 | no | — |
| 119 | C2×C4 | — | 8 | 2 | no | — |
| 120 | C2×C4 | — | 8 | 2 | no | — |
| 121 | C2×C4 | — | 16 | 1 | no | — |
| 122 | C2×C4 | — | 16 | 1 | no | — |
| 123 | C2×C4 | — | 16 | 1 | no | — |
| 124 | C2×C4 | — | 16 | 1 | no | — |
| 125 | C2×C4 | — | 16 | 2 | no | — |
| 126 | C2×C4 | — | 16 | 2 | no | — |
| 127 | C2×C4 | — | 16 | 2 | no | — |
| 128 | C2×C4 | #327 | 1 | 1 | yes | characteristic |
| 129 | C2×C4 | #360 | 1 | 1 | yes | characteristic |
| 130 | C8 | — | 2 | 1 | no | — |
| 131 | C8 | — | 4 | 1 | no | — |
| 132 | C8 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 133 | C8 | #353 | 1 | 1 | yes | characteristic |
| 134 | C22 | — | 2 | 1 | no | — |
| 135 | C22 | — | 2 | 1 | no | — |
| 136 | C22 | — | 2 | 1 | no | — |
| 137 | C22 | — | 2 | 1 | no | — |
| 138 | C22 | — | 4 | 1 | no | — |
| 139 | C22 | — | 4 | 1 | no | — |
| 140 | C22 | — | 4 | 1 | no | — |
| 141 | C22 | — | 4 | 1 | no | — |
| 142 | C22 | — | 8 | 1 | no | — |
| 143 | C22 | #1347 | 1 | 1 | yes | characteristic |
| 144 | C22 | #1416 | 1 | 1 | yes | characteristic |
| 145 | C22 | #1418 | 1 | 1 | yes | characteristic |
| 146 | C4 | — | 2 | 1 | no | — |
| 147 | C4 | — | 4 | 1 | no | — |
| 148 | C4 | — | 16 | 2 | no | — |
| 149 | C4 | — | 16 | 2 | no | — |
| 150 | C4 | — | 32 | 2 | no | — |
| 151 | C4 | #1339 | 1 | 1 | yes | characteristic |
| 152 | C4 | #1343 | 1 | 1 | yes | characteristic |
| 153 | C2 | — | 2 | 1 | no | — |
| 154 | C2 | — | 2 | 1 | no | — |
| 155 | C2 | — | 4 | 1 | no | — |
| 156 | C2 | — | 4 | 1 | no | — |
| 157 | C2 | #7076 | 1 | 1 | yes | characteristic |
| 158 | C2 | #7419 | 1 | 1 | yes | characteristic |
| 159 | C2 | #7423 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | C2×C22⋊C16 | C22 |
| prime radical | C24 | D16 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C8 ◁ C2×C22⋊C16 ◁ G | |
| LECS | G ▷ C2×C22⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C22×Q32 | C4 | semidirect |
| 10. #1380 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13306 | 2 |
| #7334 | 2 |
| #7230 | 2 |