Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
198 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 198 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 | #13584 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11992 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #9423 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #3151 | C22 | 1 | 1 | yes | characteristic |
| 5 | #2980 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #2165 | — | 2 | 1 | no | — |
| 7 | C2×#353 | — | 2 | 1 | no | — |
| 8 | #1555 | — | 2 | 1 | no | — |
| 9 | #1536 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #1409 | — | 2 | 1 | no | — |
| 11 | #1408 | C4 | 1 | 2 | yes | — |
| 12 | #1359 | — | 2 | 1 | no | — |
| 13 | #1357 | C4 | 1 | 2 | yes | complemented |
| 14 | #1291 | — | 2 | 1 | no | — |
| 15 | C2×AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 16 | #532 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×#127 | D4 | 1 | 1 | yes | characteristic |
| 18 | #496 | — | 4 | 1 | no | — |
| 19 | #493 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | #493 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | #492 | D4 | 1 | 1 | yes | characteristic |
| 22 | #492 | Q8 | 1 | 1 | yes | characteristic |
| 23 | #435 | — | 4 | 2 | no | — |
| 24 | #432 | — | 2 | 1 | no | — |
| 25 | #432 | — | 2 | 1 | no | — |
| 26 | C2×#99 | — | 4 | 1 | no | — |
| 27 | #408 | — | 4 | 1 | no | — |
| 28 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 29 | #364 | — | 2 | 2 | no | — |
| 30 | #353 | — | 2 | 2 | no | — |
| 31 | #353 | — | 4 | 1 | no | — |
| 32 | #349 | — | 4 | 1 | no | — |
| 33 | #347 | — | 4 | 1 | no | — |
| 34 | #331 | — | 4 | 2 | no | — |
| 35 | #329 | — | 4 | 1 | no | — |
| 36 | C2×#41 | — | 2 | 1 | no | — |
| 37 | C22×D4 | — | 2 | 1 | no | — |
| 38 | C22×D4 | — | 4 | 1 | no | — |
| 39 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 40 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 41 | C2×SD16 | — | 4 | 1 | no | — |
| 42 | C2×SD16 | — | 4 | 1 | no | — |
| 43 | C2×D8 | — | 4 | 1 | no | — |
| 44 | C2×D8 | — | 4 | 1 | no | — |
| 45 | C2×M16 | — | 2 | 1 | no | — |
| 46 | C2×M16 | — | 4 | 1 | no | — |
| 47 | C22×C8 | — | 4 | 1 | no | — |
| 48 | C4⋊Q8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 49 | #127 | — | 2 | 1 | no | — |
| 50 | #127 | — | 4 | 1 | no | — |
| 51 | #127 | C42 | 1 | 1 | yes | characteristic |
| 52 | #127 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | #127 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 54 | #124 | — | 2 | 1 | no | — |
| 55 | #121 | — | 4 | 1 | no | — |
| 56 | C4×D4 | — | 4 | 1 | no | — |
| 57 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 58 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 59 | M32 | — | 4 | 2 | no | — |
| 60 | C4⋊C8 | — | 2 | 2 | no | — |
| 61 | C4⋊C8 | — | 4 | 1 | no | — |
| 62 | #99 | — | 4 | 2 | no | — |
| 63 | #99 | — | 8 | 1 | no | — |
| 64 | C22⋊C8 | — | 4 | 1 | no | — |
| 65 | C22⋊C8 | — | 4 | 2 | no | — |
| 66 | C22⋊C8 | — | 4 | 1 | no | — |
| 67 | #97 | — | 2 | 1 | no | — |
| 68 | #97 | — | 4 | 1 | no | — |
| 69 | #97 | — | 4 | 1 | no | — |
| 70 | C4×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C4×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 72 | C24 | — | 4 | 1 | no | — |
| 73 | #41 | — | 8 | 1 | no | — |
| 74 | #41 | — | 8 | 1 | no | — |
| 75 | C2×Q8 | — | 2 | 1 | no | — |
| 76 | C2×D4 | — | 2 | 1 | no | — |
| 77 | C2×D4 | — | 2 | 1 | no | — |
| 78 | C2×D4 | — | 2 | 1 | no | — |
| 79 | C2×D4 | — | 4 | 1 | no | — |
| 80 | C2×D4 | — | 4 | 1 | no | — |
| 81 | C2×D4 | — | 4 | 1 | no | — |
| 82 | C2×D4 | — | 4 | 1 | no | — |
| 83 | C2×D4 | — | 4 | 1 | no | — |
| 84 | C2×D4 | — | 8 | 1 | no | — |
| 85 | C2×D4 | — | 8 | 1 | no | — |
| 86 | C2×D4 | — | 8 | 1 | no | — |
| 87 | C2×D4 | — | 16 | 2 | no | — |
| 88 | C22×C4 | — | 2 | 1 | no | — |
| 89 | C22×C4 | — | 4 | 1 | no | — |
| 90 | C22×C4 | — | 16 | 1 | no | — |
| 91 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 92 | SD16 | — | 8 | 2 | no | — |
| 93 | SD16 | — | 8 | 2 | no | — |
| 94 | D8 | — | 8 | 2 | no | — |
| 95 | D8 | — | 8 | 2 | no | — |
| 96 | M16 | — | 4 | 2 | no | — |
| 97 | M16 | — | 16 | 1 | no | — |
| 98 | C2×C8 | — | 2 | 1 | no | — |
| 99 | C2×C8 | — | 2 | 1 | no | — |
| 100 | C2×C8 | — | 4 | 2 | no | — |
| 101 | C2×C8 | — | 8 | 1 | no | — |
| 102 | C2×C8 | — | 16 | 1 | no | — |
| 103 | C4⋊C4 | — | 4 | 1 | no | — |
| 104 | C22⋊C4 | — | 4 | 2 | no | — |
| 105 | C22⋊C4 | — | 8 | 2 | no | — |
| 106 | C22⋊C4 | — | 16 | 1 | no | — |
| 107 | C42 | — | 2 | 1 | no | — |
| 108 | C42 | #95 | 1 | 1 | yes | characteristic |
| 109 | C42 | #99 | 1 | 1 | yes | characteristic |
| 110 | C16 | — | 8 | 2 | no | — |
| 111 | C23 | — | 4 | 1 | no | — |
| 112 | C23 | — | 4 | 1 | no | — |
| 113 | C23 | — | 4 | 1 | no | — |
| 114 | C23 | — | 8 | 1 | no | — |
| 115 | C23 | — | 8 | 1 | no | — |
| 116 | C23 | — | 16 | 2 | no | — |
| 117 | C23 | #352 | 1 | 1 | yes | characteristic |
| 118 | Q8 | — | 8 | 1 | no | — |
| 119 | D4 | — | 8 | 1 | no | — |
| 120 | D4 | — | 8 | 1 | no | — |
| 121 | D4 | — | 8 | 1 | no | — |
| 122 | D4 | — | 16 | 1 | no | — |
| 123 | D4 | — | 16 | 1 | no | — |
| 124 | D4 | — | 16 | 1 | no | — |
| 125 | D4 | — | 16 | 1 | no | — |
| 126 | D4 | — | 16 | 2 | no | — |
| 127 | D4 | — | 16 | 1 | no | — |
| 128 | C2×C4 | — | 2 | 1 | no | — |
| 129 | C2×C4 | — | 2 | 1 | no | — |
| 130 | C2×C4 | — | 4 | 1 | no | — |
| 131 | C2×C4 | — | 4 | 1 | no | — |
| 132 | C2×C4 | — | 8 | 1 | no | — |
| 133 | C2×C4 | — | 8 | 1 | no | — |
| 134 | C2×C4 | — | 8 | 1 | no | — |
| 135 | C2×C4 | — | 16 | 2 | no | — |
| 136 | C2×C4 | — | 32 | 2 | no | — |
| 137 | C2×C4 | #342 | 1 | 1 | yes | characteristic |
| 138 | C2×C4 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 139 | C8 | — | 2 | 2 | no | — |
| 140 | C8 | — | 4 | 1 | no | — |
| 141 | C8 | — | 8 | 2 | no | — |
| 142 | C8 | — | 16 | 1 | no | — |
| 143 | C22 | — | 2 | 1 | no | — |
| 144 | C22 | — | 4 | 1 | no | — |
| 145 | C22 | — | 8 | 2 | no | — |
| 146 | C22 | — | 8 | 1 | no | — |
| 147 | C22 | — | 8 | 1 | no | — |
| 148 | C22 | — | 8 | 1 | no | — |
| 149 | C22 | — | 16 | 1 | no | — |
| 150 | C22 | — | 16 | 1 | no | — |
| 151 | C22 | — | 16 | 2 | no | — |
| 152 | C22 | — | 32 | 1 | no | — |
| 153 | C22 | #1390 | 1 | 1 | yes | characteristic |
| 154 | C4 | — | 2 | 1 | no | — |
| 155 | C4 | — | 2 | 1 | no | — |
| 156 | C4 | — | 4 | 1 | no | — |
| 157 | C4 | — | 4 | 1 | no | — |
| 158 | C4 | — | 8 | 1 | no | — |
| 159 | C4 | — | 32 | 2 | no | — |
| 160 | C2 | — | 2 | 1 | no | — |
| 161 | C2 | — | 4 | 1 | no | — |
| 162 | C2 | — | 16 | 1 | no | — |
| 163 | C2 | — | 16 | 1 | no | — |
| 164 | C2 | — | 16 | 1 | no | — |
| 165 | C2 | #7493 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7493 |
| commutator | #127 | C42 |
| Frattini, prime radical, cosocle | #3151 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #2980 | C4 | semidirect |
| 9. #1536 | C4 | semidirect |
| 13. #1357 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13584 | 2 |
| #11992 | 2 |
| #9423 | 2 |