Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
135 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 24 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 | #10634 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7504 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #7352 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#353 | — | 2 | 1 | no | — |
| 5 | #1961 | C22 | 1 | 1 | yes | characteristic |
| 6 | #1838 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #1545 | — | 2 | 1 | no | — |
| 8 | #1324 | — | 2 | 1 | no | — |
| 9 | #1286 | — | 2 | 1 | no | — |
| 10 | #1281 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#127 | D4 | 1 | 1 | yes | characteristic |
| 12 | #431 | — | 2 | 1 | no | — |
| 13 | #423 | — | 2 | 1 | no | — |
| 14 | C2×#102 | — | 4 | 1 | no | — |
| 15 | C2×#99 | — | 4 | 1 | no | — |
| 16 | C2×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | #382 | D4 | 1 | 1 | yes | characteristic |
| 18 | #353 | — | 2 | 2 | no | — |
| 19 | #353 | — | 4 | 1 | no | — |
| 20 | #341 | — | 4 | 1 | no | — |
| 21 | #331 | — | 4 | 1 | no | — |
| 22 | #331 | — | 4 | 1 | no | — |
| 23 | #324 | — | 4 | 1 | no | — |
| 24 | C22×D4 | — | 2 | 1 | no | — |
| 25 | C22×D4 | — | 4 | 1 | no | — |
| 26 | C2×M16 | — | 4 | 1 | no | — |
| 27 | C22×C8 | — | 2 | 1 | no | — |
| 28 | #127 | — | 2 | 1 | no | — |
| 29 | #127 | — | 4 | 1 | no | — |
| 30 | #127 | D8 | 1 | 1 | yes | characteristic |
| 31 | #127 | SD16 | 1 | 1 | yes | characteristic |
| 32 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 33 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 34 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 35 | C4⋊C8 | — | 4 | 1 | no | — |
| 36 | C4⋊C8 | — | 4 | 1 | no | — |
| 37 | C4⋊C8 | — | 4 | 1 | no | — |
| 38 | #102 | — | 8 | 2 | no | — |
| 39 | #99 | — | 4 | 2 | no | — |
| 40 | #99 | — | 8 | 1 | no | — |
| 41 | C22⋊C8 | — | 4 | 1 | no | — |
| 42 | C22⋊C8 | — | 8 | 1 | no | — |
| 43 | #97 | — | 4 | 1 | no | — |
| 44 | C4×C8 | — | 4 | 1 | no | — |
| 45 | #95 | — | 4 | 1 | no | — |
| 46 | C24 | — | 4 | 1 | no | — |
| 47 | C2×D4 | — | 2 | 1 | no | — |
| 48 | C2×D4 | — | 2 | 1 | no | — |
| 49 | C2×D4 | — | 4 | 1 | no | — |
| 50 | C2×D4 | — | 4 | 1 | no | — |
| 51 | C2×D4 | — | 4 | 1 | no | — |
| 52 | C2×D4 | — | 8 | 1 | no | — |
| 53 | C2×D4 | — | 8 | 1 | no | — |
| 54 | C2×D4 | — | 8 | 1 | no | — |
| 55 | C2×D4 | — | 16 | 2 | no | — |
| 56 | C22×C4 | — | 2 | 1 | no | — |
| 57 | C22×C4 | — | 4 | 1 | no | — |
| 58 | C22×C4 | — | 16 | 1 | no | — |
| 59 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 60 | M16 | — | 8 | 1 | no | — |
| 61 | M16 | — | 8 | 1 | no | — |
| 62 | C2×C8 | — | 4 | 1 | no | — |
| 63 | C2×C8 | — | 4 | 1 | no | — |
| 64 | C2×C8 | — | 4 | 1 | no | — |
| 65 | C2×C8 | — | 8 | 1 | no | — |
| 66 | C2×C8 | — | 8 | 1 | no | — |
| 67 | C2×C8 | — | 16 | 1 | no | — |
| 68 | C4⋊C4 | — | 8 | 1 | no | — |
| 69 | C22⋊C4 | — | 8 | 2 | no | — |
| 70 | C22⋊C4 | — | 16 | 1 | no | — |
| 71 | C42 | — | 2 | 1 | no | — |
| 72 | C42 | #102 | 1 | 1 | yes | characteristic |
| 73 | C42 | #104 | 1 | 1 | yes | characteristic |
| 74 | C23 | — | 4 | 1 | no | — |
| 75 | C23 | — | 4 | 1 | no | — |
| 76 | C23 | — | 8 | 1 | no | — |
| 77 | C23 | — | 8 | 1 | no | — |
| 78 | C23 | — | 16 | 2 | no | — |
| 79 | C23 | #354 | 1 | 1 | yes | characteristic |
| 80 | D4 | — | 8 | 1 | no | — |
| 81 | D4 | — | 8 | 1 | no | — |
| 82 | D4 | — | 16 | 1 | no | — |
| 83 | D4 | — | 16 | 1 | no | — |
| 84 | D4 | — | 16 | 1 | no | — |
| 85 | D4 | — | 16 | 2 | no | — |
| 86 | C2×C4 | — | 2 | 1 | no | — |
| 87 | C2×C4 | — | 2 | 1 | no | — |
| 88 | C2×C4 | — | 2 | 1 | no | — |
| 89 | C2×C4 | — | 2 | 1 | no | — |
| 90 | C2×C4 | — | 8 | 1 | no | — |
| 91 | C2×C4 | — | 8 | 1 | no | — |
| 92 | C2×C4 | — | 16 | 1 | no | — |
| 93 | C2×C4 | — | 16 | 2 | no | — |
| 94 | C2×C4 | — | 32 | 2 | no | — |
| 95 | C2×C4 | #327 | 1 | 1 | yes | characteristic |
| 96 | C2×C4 | #352 | 1 | 1 | yes | characteristic |
| 97 | C8 | — | 8 | 1 | no | — |
| 98 | C8 | — | 16 | 1 | no | — |
| 99 | C8 | — | 16 | 1 | no | — |
| 100 | C8 | — | 32 | 1 | no | — |
| 101 | C22 | — | 4 | 1 | no | — |
| 102 | C22 | — | 8 | 1 | no | — |
| 103 | C22 | — | 8 | 1 | no | — |
| 104 | C22 | — | 16 | 1 | no | — |
| 105 | C22 | — | 16 | 1 | no | — |
| 106 | C22 | — | 16 | 2 | no | — |
| 107 | C22 | — | 32 | 1 | no | — |
| 108 | C22 | #1344 | 1 | 1 | yes | characteristic |
| 109 | C22 | #1412 | 1 | 1 | yes | characteristic |
| 110 | C22 | #1414 | 1 | 1 | yes | characteristic |
| 111 | C4 | — | 2 | 1 | no | — |
| 112 | C4 | — | 2 | 1 | no | — |
| 113 | C4 | — | 4 | 1 | no | — |
| 114 | C4 | — | 4 | 1 | no | — |
| 115 | C4 | — | 16 | 1 | no | — |
| 116 | C4 | — | 32 | 2 | no | — |
| 117 | C2 | — | 2 | 1 | no | — |
| 118 | C2 | — | 2 | 1 | no | — |
| 119 | C2 | — | 2 | 1 | no | — |
| 120 | C2 | — | 16 | 1 | no | — |
| 121 | C2 | — | 16 | 1 | no | — |
| 122 | C2 | #7412 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7412 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1961 | C22 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1961 ◁ G | |
| LECS | G ▷ #1961 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #1838 | C4 | semidirect |
| 10. #1281 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10634 | 2 |
| #7504 | 2 |
| #7352 | 2 |