Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
139 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 139 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 | #10530 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7373 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #7273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #3253 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×#364 | — | 2 | 1 | no | — |
| 6 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 7 | #1560 | — | 2 | 1 | no | — |
| 8 | #1492 | — | 2 | 1 | no | — |
| 9 | #1327 | — | 2 | 1 | no | — |
| 10 | #1309 | C4 | 1 | 1 | yes | characteristic |
| 11 | #532 | — | 2 | 1 | no | — |
| 12 | C2×M32 | — | 4 | 1 | no | — |
| 13 | #476 | — | 4 | 1 | no | — |
| 14 | #469 | — | 4 | 1 | no | — |
| 15 | #459 | — | 4 | 1 | no | — |
| 16 | C2×#108 | — | 4 | 1 | no | — |
| 17 | #423 | — | 2 | 1 | no | — |
| 18 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | #423 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 21 | #364 | — | 2 | 2 | no | — |
| 22 | #364 | — | 4 | 1 | no | — |
| 23 | #349 | — | 4 | 1 | no | — |
| 24 | #333 | — | 4 | 1 | no | — |
| 25 | #331 | — | 4 | 1 | no | — |
| 26 | D4⋊C8 | — | 4 | 1 | no | — |
| 27 | D4⋊C8 | — | 4 | 1 | no | — |
| 28 | C2×#41 | — | 4 | 1 | no | — |
| 29 | C2×D8 | — | 4 | 1 | no | — |
| 30 | C2×M16 | — | 2 | 1 | no | — |
| 31 | C2×M16 | — | 2 | 1 | no | — |
| 32 | C2×M16 | — | 4 | 1 | no | — |
| 33 | C22×C8 | — | 2 | 1 | no | — |
| 34 | C4⋊Q8 | — | 2 | 1 | no | — |
| 35 | #127 | — | 2 | 1 | no | — |
| 36 | #124 | — | 4 | 1 | no | — |
| 37 | #121 | — | 4 | 1 | no | — |
| 38 | #121 | — | 4 | 1 | no | — |
| 39 | C4×D4 | — | 4 | 1 | no | — |
| 40 | C4×D4 | — | 4 | 1 | no | — |
| 41 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 42 | M32 | — | 4 | 2 | no | — |
| 43 | M32 | — | 8 | 1 | no | — |
| 44 | M32 | — | 8 | 1 | no | — |
| 45 | C2×C16 | — | 8 | 1 | no | — |
| 46 | #108 | — | 8 | 1 | no | — |
| 47 | #108 | — | 8 | 1 | no | — |
| 48 | #106 | — | 4 | 1 | no | — |
| 49 | C4⋊C8 | — | 4 | 1 | no | — |
| 50 | C4⋊C8 | — | 4 | 1 | no | — |
| 51 | C4⋊C8 | — | 4 | 1 | no | — |
| 52 | C4⋊C8 | — | 4 | 1 | no | — |
| 53 | #102 | — | 4 | 2 | no | — |
| 54 | C4×C8 | — | 2 | 1 | no | — |
| 55 | C4×C8 | D8 | 1 | 1 | yes | characteristic |
| 56 | C4×C8 | SD16 | 1 | 1 | yes | characteristic |
| 57 | #41 | — | 16 | 1 | no | — |
| 58 | #41 | — | 16 | 1 | no | — |
| 59 | C2×Q8 | — | 4 | 1 | no | — |
| 60 | C2×D4 | — | 4 | 1 | no | — |
| 61 | C2×D4 | — | 4 | 1 | no | — |
| 62 | C2×D4 | — | 4 | 1 | no | — |
| 63 | C2×D4 | — | 8 | 1 | no | — |
| 64 | C22×C4 | — | 2 | 1 | no | — |
| 65 | C22×C4 | — | 8 | 1 | no | — |
| 66 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 67 | D8 | — | 16 | 1 | no | — |
| 68 | M16 | — | 8 | 1 | no | — |
| 69 | M16 | — | 8 | 1 | no | — |
| 70 | M16 | — | 16 | 1 | no | — |
| 71 | C2×C8 | — | 2 | 1 | no | — |
| 72 | C2×C8 | — | 2 | 1 | no | — |
| 73 | C2×C8 | — | 2 | 1 | no | — |
| 74 | C2×C8 | — | 2 | 1 | no | — |
| 75 | C2×C8 | — | 2 | 1 | no | — |
| 76 | C2×C8 | — | 2 | 1 | no | — |
| 77 | C2×C8 | — | 4 | 1 | no | — |
| 78 | C2×C8 | — | 4 | 1 | no | — |
| 79 | C2×C8 | — | 8 | 1 | no | — |
| 80 | C4⋊C4 | — | 4 | 1 | no | — |
| 81 | C4⋊C4 | — | 4 | 1 | no | — |
| 82 | C22⋊C4 | — | 8 | 1 | no | — |
| 83 | C22⋊C4 | — | 8 | 1 | no | — |
| 84 | C42 | — | 2 | 1 | no | — |
| 85 | C42 | #102 | 1 | 1 | yes | characteristic |
| 86 | C42 | #104 | 1 | 1 | yes | characteristic |
| 87 | C16 | — | 8 | 2 | no | — |
| 88 | C16 | — | 16 | 1 | no | — |
| 89 | C23 | — | 8 | 1 | no | — |
| 90 | C23 | #355 | 1 | 1 | yes | characteristic |
| 91 | Q8 | — | 16 | 1 | no | — |
| 92 | D4 | — | 8 | 1 | no | — |
| 93 | D4 | — | 8 | 1 | no | — |
| 94 | D4 | — | 8 | 1 | no | — |
| 95 | D4 | — | 8 | 1 | no | — |
| 96 | D4 | — | 16 | 1 | no | — |
| 97 | D4 | — | 16 | 2 | no | — |
| 98 | C2×C4 | — | 2 | 1 | no | — |
| 99 | C2×C4 | — | 2 | 1 | no | — |
| 100 | C2×C4 | — | 2 | 1 | no | — |
| 101 | C2×C4 | — | 2 | 1 | no | — |
| 102 | C2×C4 | — | 4 | 1 | no | — |
| 103 | C2×C4 | — | 4 | 1 | no | — |
| 104 | C2×C4 | — | 8 | 1 | no | — |
| 105 | C2×C4 | — | 16 | 1 | no | — |
| 106 | C2×C4 | — | 16 | 1 | no | — |
| 107 | C2×C4 | #327 | 1 | 1 | yes | characteristic |
| 108 | C2×C4 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 109 | C8 | — | 2 | 1 | no | — |
| 110 | C8 | — | 2 | 1 | no | — |
| 111 | C8 | — | 2 | 1 | no | — |
| 112 | C8 | — | 2 | 1 | no | — |
| 113 | C8 | — | 8 | 1 | no | — |
| 114 | C8 | — | 8 | 1 | no | — |
| 115 | C8 | — | 16 | 1 | no | — |
| 116 | C22 | — | 2 | 1 | no | — |
| 117 | C22 | — | 4 | 1 | no | — |
| 118 | C22 | — | 16 | 1 | no | — |
| 119 | C22 | — | 16 | 1 | no | — |
| 120 | C22 | — | 16 | 1 | no | — |
| 121 | C22 | #1345 | 1 | 1 | yes | characteristic |
| 122 | C4 | — | 2 | 1 | no | — |
| 123 | C4 | — | 2 | 1 | no | — |
| 124 | C4 | — | 2 | 1 | no | — |
| 125 | C4 | — | 4 | 1 | no | — |
| 126 | C4 | — | 16 | 1 | no | — |
| 127 | C4 | #1402 | 1 | 1 | yes | characteristic |
| 128 | C4 | #1405 | 1 | 1 | yes | characteristic |
| 129 | C2 | — | 2 | 1 | no | — |
| 130 | C2 | — | 4 | 1 | no | — |
| 131 | C2 | — | 32 | 1 | no | — |
| 132 | C2 | #7417 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7417 |
| commutator | #423 | C2×C4 |
| Frattini, cosocle | #1848 | C22 |
| prime radical | #3253 | C4 |
| LCS | G ▷ #423 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1848 ◁ G | |
| LECS | G ▷ #1848 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #7373 | C2 | semidirect |
| 3. #7273 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10530 | 2 |
| #7373 | 2 |
| #7273 | 2 |