Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
220 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #7317 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7315 | C2 | 1 | 3 | yes | maximal |
| 3 | #7308 | C2 | 1 | 3 | yes | maximal |
| 4 | #1784 | C22 | 1 | 3 | yes | — |
| 5 | #1784 | C22 | 1 | 3 | yes | — |
| 6 | #1446 | C22 | 1 | 1 | yes | characteristic |
| 7 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 8 | C2×#101 | — | 2 | 3 | no | — |
| 9 | C2×#101 | — | 2 | 3 | no | — |
| 10 | C2×#101 | — | 2 | 3 | no | — |
| 11 | C2×#101 | — | 2 | 3 | no | — |
| 12 | #407 | — | 2 | 3 | no | — |
| 13 | #407 | — | 2 | 3 | no | — |
| 14 | C2×#95 | — | 2 | 3 | no | — |
| 15 | C22×Q8 | — | 8 | 3 | no | — |
| 16 | C22×Q8 | #41 | 1 | 3 | yes | — |
| 17 | C22×Q8 | #41 | 1 | 1 | yes | characteristic |
| 18 | C23×C4 | — | 4 | 1 | no | — |
| 19 | C23×C4 | #41 | 1 | 3 | yes | — |
| 20 | C2×M16 | — | 4 | 3 | no | — |
| 21 | C2×M16 | — | 4 | 3 | no | — |
| 22 | #101 | — | 8 | 3 | no | — |
| 23 | #101 | — | 8 | 3 | no | — |
| 24 | #101 | — | 8 | 3 | no | — |
| 25 | #101 | — | 8 | 3 | no | — |
| 26 | C22⋊C8 | — | 8 | 3 | no | — |
| 27 | C22⋊C8 | — | 8 | 3 | no | — |
| 28 | #95 | — | 8 | 3 | no | — |
| 29 | C24 | #126 | 1 | 1 | yes | characteristic |
| 30 | C2×Q8 | — | 4 | 3 | no | — |
| 31 | C2×Q8 | — | 4 | 3 | no | — |
| 32 | C2×Q8 | — | 4 | 3 | no | — |
| 33 | C2×Q8 | — | 4 | 3 | no | — |
| 34 | C2×Q8 | — | 8 | 8 | no | — |
| 35 | C22×C4 | — | 4 | 3 | no | — |
| 36 | C22×C4 | — | 4 | 3 | no | — |
| 37 | C22×C4 | — | 4 | 3 | no | — |
| 38 | C22×C4 | — | 4 | 2 | no | — |
| 39 | C22×C4 | — | 8 | 3 | no | — |
| 40 | C22×C4 | — | 8 | 3 | no | — |
| 41 | C22×C4 | #126 | 1 | 3 | yes | — |
| 42 | C22×C4 | #126 | 1 | 3 | yes | — |
| 43 | M16 | — | 8 | 3 | no | — |
| 44 | M16 | — | 8 | 3 | no | — |
| 45 | M16 | — | 8 | 3 | no | — |
| 46 | M16 | — | 8 | 3 | no | — |
| 47 | C2×C8 | — | 8 | 3 | no | — |
| 48 | C2×C8 | — | 8 | 3 | no | — |
| 49 | C23 | — | 2 | 3 | no | — |
| 50 | C23 | — | 4 | 1 | no | — |
| 51 | C23 | — | 4 | 1 | no | — |
| 52 | C23 | #401 | 1 | 1 | yes | characteristic |
| 53 | Q8 | — | 8 | 6 | no | — |
| 54 | Q8 | — | 8 | 2 | no | — |
| 55 | C2×C4 | — | 2 | 3 | no | — |
| 56 | C2×C4 | — | 2 | 3 | no | — |
| 57 | C2×C4 | — | 2 | 3 | no | — |
| 58 | C2×C4 | — | 2 | 3 | no | — |
| 59 | C2×C4 | — | 2 | 3 | no | — |
| 60 | C2×C4 | — | 2 | 3 | no | — |
| 61 | C2×C4 | — | 8 | 6 | no | — |
| 62 | C2×C4 | — | 8 | 6 | no | — |
| 63 | C2×C4 | — | 16 | 3 | no | — |
| 64 | C2×C4 | — | 16 | 1 | no | — |
| 65 | C8 | — | 16 | 3 | no | — |
| 66 | C8 | — | 16 | 3 | no | — |
| 67 | C22 | — | 2 | 3 | no | — |
| 68 | C22 | — | 2 | 3 | no | — |
| 69 | C22 | — | 4 | 1 | no | — |
| 70 | C22 | — | 4 | 3 | no | — |
| 71 | C22 | — | 4 | 1 | no | — |
| 72 | C22 | #2104 | 1 | 3 | yes | — |
| 73 | C4 | — | 4 | 3 | no | — |
| 74 | C4 | — | 4 | 3 | no | — |
| 75 | C4 | — | 16 | 2 | no | — |
| 76 | C2 | — | 2 | 3 | no | — |
| 77 | C2 | — | 4 | 1 | no | — |
| 78 | C2 | — | 4 | 1 | no | — |
| 79 | C2 | #10690 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10690 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | #126 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #7317 | 2 |
| #7315 | 2 |
| #7308 | 2 |