Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
192 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 42 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 | #33821 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13718 | C2 | 1 | 2 | yes | maximal |
| 3 | #13519 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7423 | C2 | 1 | 2 | yes | maximal |
| 5 | #7364 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #3170 | — | 2 | 1 | no | — |
| 7 | #2225 | — | 2 | 2 | no | — |
| 8 | #2219 | — | 2 | 1 | no | — |
| 9 | #2219 | C22 | 1 | 2 | yes | — |
| 10 | #2209 | — | 2 | 2 | no | — |
| 11 | #2161 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1945 | — | 2 | 1 | no | — |
| 14 | #1426 | — | 2 | 2 | no | — |
| 15 | #1420 | — | 2 | 2 | no | — |
| 16 | #1400 | C22 | 1 | 2 | yes | — |
| 17 | #1389 | — | 2 | 1 | no | — |
| 18 | #1384 | C22 | 1 | 1 | yes | characteristic |
| 19 | #533 | — | 4 | 1 | no | — |
| 20 | C2×Q32 | — | 2 | 1 | no | — |
| 21 | C2×Q32 | D4 | 1 | 2 | yes | — |
| 22 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 23 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 24 | #500 | — | 2 | 1 | no | — |
| 25 | #494 | — | 2 | 1 | no | — |
| 26 | #487 | — | 4 | 1 | no | — |
| 27 | #467 | — | 2 | 2 | no | — |
| 28 | #467 | — | 4 | 1 | no | — |
| 29 | C4×Q16 | — | 2 | 2 | no | — |
| 30 | C4×Q16 | — | 4 | 1 | no | — |
| 31 | C2×#108 | — | 2 | 1 | no | — |
| 32 | #427 | — | 2 | 1 | no | — |
| 33 | #424 | — | 4 | 1 | no | — |
| 34 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 35 | M64 | — | 2 | 2 | no | — |
| 36 | #368 | — | 2 | 2 | no | — |
| 37 | #367 | — | 2 | 1 | no | — |
| 38 | #367 | D4 | 1 | 2 | yes | — |
| 39 | C4⋊C16 | — | 2 | 2 | no | — |
| 40 | #358 | — | 2 | 2 | no | — |
| 41 | #358 | — | 2 | 2 | no | — |
| 42 | #339 | — | 4 | 1 | no | — |
| 43 | C2×Q16 | — | 2 | 2 | no | — |
| 44 | C2×Q16 | — | 4 | 1 | no | — |
| 45 | C2×M16 | — | 4 | 1 | no | — |
| 46 | C22×C8 | D8 | 1 | 1 | yes | characteristic |
| 47 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C4⋊Q8 | — | 4 | 1 | no | — |
| 49 | C4⋊Q8 | — | 4 | 1 | no | — |
| 50 | #125 | — | 8 | 1 | no | — |
| 51 | C22⋊Q8 | — | 4 | 2 | no | — |
| 52 | C22⋊Q8 | — | 8 | 1 | no | — |
| 53 | C4×Q8 | — | 4 | 2 | no | — |
| 54 | C4×Q8 | — | 8 | 1 | no | — |
| 55 | #117 | — | 4 | 1 | no | — |
| 56 | #117 | — | 8 | 1 | no | — |
| 57 | C2×C42 | D8 | 1 | 1 | yes | characteristic |
| 58 | Q32 | — | 4 | 2 | no | — |
| 59 | Q32 | — | 8 | 1 | no | — |
| 60 | M32 | — | 4 | 1 | no | — |
| 61 | M32 | — | 4 | 1 | no | — |
| 62 | C2×C16 | — | 2 | 1 | no | — |
| 63 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 64 | #108 | — | 4 | 1 | no | — |
| 65 | #108 | — | 4 | 1 | no | — |
| 66 | #107 | — | 2 | 2 | no | — |
| 67 | #107 | — | 4 | 1 | no | — |
| 68 | #107 | — | 4 | 1 | no | — |
| 69 | #106 | — | 2 | 2 | no | — |
| 70 | C4⋊C8 | — | 8 | 2 | no | — |
| 71 | #103 | — | 4 | 2 | no | — |
| 72 | #103 | — | 4 | 2 | no | — |
| 73 | C4×C8 | — | 2 | 1 | no | — |
| 74 | C4×C8 | — | 2 | 1 | no | — |
| 75 | C32 | — | 4 | 2 | no | — |
| 76 | C2×Q8 | — | 4 | 2 | no | — |
| 77 | C2×Q8 | — | 8 | 1 | no | — |
| 78 | C22×C4 | — | 2 | 1 | no | — |
| 79 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 80 | Q16 | — | 4 | 2 | no | — |
| 81 | Q16 | — | 4 | 2 | no | — |
| 82 | Q16 | — | 8 | 2 | no | — |
| 83 | M16 | — | 8 | 1 | no | — |
| 84 | M16 | — | 8 | 1 | no | — |
| 85 | C2×C8 | — | 2 | 1 | no | — |
| 86 | C2×C8 | — | 2 | 1 | no | — |
| 87 | C2×C8 | — | 2 | 1 | no | — |
| 88 | C2×C8 | — | 4 | 1 | no | — |
| 89 | C2×C8 | — | 8 | 1 | no | — |
| 90 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 91 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 92 | C4⋊C4 | — | 4 | 2 | no | — |
| 93 | C4⋊C4 | — | 4 | 2 | no | — |
| 94 | C4⋊C4 | — | 8 | 1 | no | — |
| 95 | C4⋊C4 | — | 8 | 1 | no | — |
| 96 | C4⋊C4 | — | 8 | 2 | no | — |
| 97 | C4⋊C4 | — | 8 | 1 | no | — |
| 98 | C4⋊C4 | — | 8 | 2 | no | — |
| 99 | C22⋊C4 | — | 8 | 1 | no | — |
| 100 | C22⋊C4 | — | 8 | 2 | no | — |
| 101 | C42 | — | 2 | 1 | no | — |
| 102 | C42 | — | 8 | 1 | no | — |
| 103 | C42 | — | 8 | 2 | no | — |
| 104 | C42 | SD32 | 1 | 1 | yes | characteristic |
| 105 | C42 | SD32 | 1 | 1 | yes | characteristic |
| 106 | C16 | — | 2 | 2 | no | — |
| 107 | C16 | — | 4 | 1 | no | — |
| 108 | C23 | #509 | 1 | 1 | yes | characteristic |
| 109 | Q8 | — | 8 | 2 | no | — |
| 110 | Q8 | — | 8 | 2 | no | — |
| 111 | Q8 | — | 16 | 2 | no | — |
| 112 | C2×C4 | — | 2 | 1 | no | — |
| 113 | C2×C4 | — | 2 | 1 | no | — |
| 114 | C2×C4 | — | 2 | 1 | no | — |
| 115 | C2×C4 | — | 2 | 1 | no | — |
| 116 | C2×C4 | — | 4 | 2 | no | — |
| 117 | C2×C4 | — | 8 | 2 | no | — |
| 118 | C2×C4 | — | 8 | 2 | no | — |
| 119 | C2×C4 | — | 8 | 2 | no | — |
| 120 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 121 | C2×C4 | C2×SD32 | 1 | 1 | yes | characteristic |
| 122 | C8 | — | 2 | 1 | no | — |
| 123 | C8 | — | 4 | 1 | no | — |
| 124 | C8 | — | 16 | 1 | no | — |
| 125 | C8 | #456 | 1 | 1 | yes | characteristic |
| 126 | C8 | #458 | 1 | 1 | yes | characteristic |
| 127 | C22 | — | 2 | 1 | no | — |
| 128 | C22 | — | 4 | 1 | no | — |
| 129 | C22 | #2187 | 1 | 1 | yes | characteristic |
| 130 | C4 | — | 2 | 2 | no | — |
| 131 | C4 | — | 2 | 1 | no | — |
| 132 | C4 | — | 4 | 1 | no | — |
| 133 | C4 | — | 16 | 2 | no | — |
| 134 | C4 | — | 16 | 2 | no | — |
| 135 | C4 | — | 16 | 2 | no | — |
| 136 | C4 | #1610 | 1 | 1 | yes | characteristic |
| 137 | C4 | #2190 | 1 | 1 | yes | characteristic |
| 138 | C2 | — | 2 | 1 | no | — |
| 139 | C2 | — | 4 | 1 | no | — |
| 140 | C2 | #12291 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12291 |
| commutator, Frattini, cosocle | C2×M32 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33821 | 2 |
| #13718 | 2 |
| #13519 | 2 |
| #7423 | 2 |
| #7364 | 2 |