Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1196 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 266 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×#2250 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C2×#2249 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#2246 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #33308 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #12539 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×Q32 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×SD32 | C22 | 1 | 6 | yes | — |
| 8 | C22×D16 | C22 | 1 | 3 | yes | — |
| 9 | C22×D16 | C22 | 1 | 6 | yes | — |
| 10 | C2×#496 | — | 2 | 6 | no | — |
| 11 | C2×#495 | C22 | 1 | 6 | yes | — |
| 12 | C2×#493 | C22 | 1 | 6 | yes | — |
| 13 | #2657 | — | 2 | 2 | no | — |
| 14 | #2250 | — | 2 | 24 | no | — |
| 15 | #2249 | — | 2 | 12 | no | — |
| 16 | #2246 | — | 2 | 12 | no | — |
| 17 | C2×#346 | C22 | 1 | 3 | yes | — |
| 18 | C2×C4×C16 | C22 | 1 | 3 | yes | — |
| 19 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 21 | C22×SD16 | — | 2 | 6 | no | — |
| 22 | C22×D8 | — | 2 | 6 | no | — |
| 23 | C22×D8 | C23 | 1 | 6 | yes | — |
| 24 | C2×#127 | — | 2 | 6 | no | — |
| 25 | C2×#124 | — | 2 | 6 | no | — |
| 26 | C2×Q32 | — | 2 | 6 | no | — |
| 27 | C2×SD32 | — | 2 | 12 | no | — |
| 28 | C2×SD32 | — | 2 | 24 | no | — |
| 29 | C2×D16 | — | 2 | 12 | no | — |
| 30 | C2×D16 | — | 2 | 12 | no | — |
| 31 | C2×D16 | — | 2 | 6 | no | — |
| 32 | C2×D16 | — | 2 | 24 | no | — |
| 33 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 34 | C22×C16 | C23 | 1 | 3 | yes | — |
| 35 | #496 | — | 4 | 24 | no | — |
| 36 | #495 | — | 2 | 12 | no | — |
| 37 | #495 | — | 2 | 12 | no | — |
| 38 | #493 | — | 2 | 12 | no | — |
| 39 | #493 | — | 2 | 12 | no | — |
| 40 | C2×#97 | D4 | 1 | 3 | yes | — |
| 41 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 42 | #392 | — | 4 | 8 | no | — |
| 43 | #376 | D4 | 1 | 1 | yes | characteristic |
| 44 | #346 | D4 | 1 | 12 | yes | complemented |
| 45 | C4×C16 | D4 | 1 | 12 | yes | complemented |
| 46 | C22×Q8 | — | 2 | 2 | no | — |
| 47 | C22×D4 | — | 2 | 6 | no | — |
| 48 | C22×D4 | — | 4 | 12 | no | — |
| 49 | C2×Q16 | — | 2 | 6 | no | — |
| 50 | C2×Q16 | — | 2 | 6 | no | — |
| 51 | C2×SD16 | — | 4 | 12 | no | — |
| 52 | C2×SD16 | — | 4 | 24 | no | — |
| 53 | C2×D8 | — | 2 | 12 | no | — |
| 54 | C2×D8 | — | 2 | 12 | no | — |
| 55 | C2×D8 | — | 2 | 6 | no | — |
| 56 | C2×D8 | — | 2 | 6 | no | — |
| 57 | C2×D8 | — | 4 | 12 | no | — |
| 58 | C2×D8 | — | 4 | 24 | no | — |
| 59 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 60 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 61 | #127 | — | 4 | 12 | no | — |
| 62 | #127 | — | 4 | 12 | no | — |
| 63 | #124 | — | 4 | 12 | no | — |
| 64 | #124 | — | 4 | 12 | no | — |
| 65 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 66 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 67 | Q32 | — | 4 | 4 | no | — |
| 68 | SD32 | — | 4 | 24 | no | — |
| 69 | D16 | — | 4 | 12 | no | — |
| 70 | D16 | — | 4 | 24 | no | — |
| 71 | C2×C16 | C2×D4 | 1 | 6 | yes | — |
| 72 | C2×C16 | C2×D4 | 1 | 6 | yes | — |
| 73 | C2×C16 | C2×D4 | 1 | 12 | yes | — |
| 74 | #97 | D8 | 1 | 12 | yes | — |
| 75 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 76 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 77 | #95 | D8 | 1 | 4 | yes | — |
| 78 | C24 | — | 4 | 6 | no | — |
| 79 | C2×Q8 | — | 4 | 6 | no | — |
| 80 | C2×Q8 | — | 4 | 6 | no | — |
| 81 | C2×D4 | — | 4 | 6 | no | — |
| 82 | C2×D4 | — | 4 | 12 | no | — |
| 83 | C2×D4 | — | 4 | 6 | no | — |
| 84 | C2×D4 | — | 4 | 12 | no | — |
| 85 | C2×D4 | — | 8 | 24 | no | — |
| 86 | C2×D4 | — | 16 | 24 | no | — |
| 87 | C22×C4 | — | 4 | 2 | no | — |
| 88 | C22×C4 | C2×D8 | 1 | 6 | yes | — |
| 89 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 90 | Q16 | — | 4 | 6 | no | — |
| 91 | Q16 | — | 4 | 2 | no | — |
| 92 | SD16 | — | 8 | 24 | no | — |
| 93 | D8 | — | 4 | 6 | no | — |
| 94 | D8 | — | 4 | 12 | no | — |
| 95 | D8 | — | 4 | 6 | no | — |
| 96 | D8 | — | 8 | 24 | no | — |
| 97 | C2×C8 | #127 | 1 | 6 | yes | — |
| 98 | C2×C8 | C2×D8 | 1 | 12 | yes | — |
| 99 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 100 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 101 | C22⋊C4 | — | 8 | 24 | no | — |
| 102 | C42 | #127 | 1 | 6 | yes | — |
| 103 | C42 | C2×D8 | 1 | 6 | yes | — |
| 104 | C16 | #127 | 1 | 4 | yes | — |
| 105 | C16 | #127 | 1 | 12 | yes | — |
| 106 | C23 | — | 8 | 6 | no | — |
| 107 | C23 | — | 8 | 12 | no | — |
| 108 | C23 | — | 8 | 24 | no | — |
| 109 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 110 | Q8 | — | 8 | 2 | no | — |
| 111 | Q8 | — | 8 | 6 | no | — |
| 112 | D4 | — | 8 | 6 | no | — |
| 113 | D4 | — | 8 | 12 | no | — |
| 114 | D4 | — | 8 | 6 | no | — |
| 115 | D4 | — | 16 | 48 | no | — |
| 116 | C2×C4 | — | 2 | 12 | no | — |
| 117 | C2×C4 | — | 8 | 6 | no | — |
| 118 | C2×C4 | #493 | 1 | 12 | yes | — |
| 119 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 120 | C2×C4 | C22×D8 | 1 | 3 | yes | — |
| 121 | C8 | #493 | 1 | 12 | yes | — |
| 122 | C8 | C2×#127 | 1 | 1 | yes | characteristic |
| 123 | C8 | C2×#127 | 1 | 3 | yes | — |
| 124 | C22 | — | 16 | 6 | no | — |
| 125 | C22 | — | 16 | 24 | no | — |
| 126 | C22 | — | 16 | 12 | no | — |
| 127 | C22 | C2×#493 | 1 | 3 | yes | — |
| 128 | C22 | #3415 | 1 | 4 | yes | — |
| 129 | C4 | — | 2 | 12 | no | — |
| 130 | C4 | — | 16 | 2 | no | — |
| 131 | C4 | #2867 | 1 | 1 | yes | characteristic |
| 132 | C4 | C2×#493 | 1 | 3 | yes | — |
| 133 | C2 | — | 32 | 6 | no | — |
| 134 | C2 | #33301 | 1 | 1 | yes | characteristic |
| 135 | C2 | #33720 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #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. C2×#2250 | C2 | semidirect |
| 2. C2×#2249 | C2 | semidirect |
| 3. C2×#2246 | C2 | semidirect |
| 4. #33308 | C2 | semidirect |
| 5. #12539 | C2 | semidirect |
| 44. #346 | D4 | semidirect |
| 45. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2250 | 2 |
| C2×#2249 | 2 |
| C2×#2246 | 2 |
| #33308 | 2 |
| #12539 | 2 |