Finite Group Databasebeta

← #13620

Subgroups — #13620

Order 256.

145 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#508C211yesmaximal, characteristic
2C2×M64C211yesmaximal, characteristic, complemented
3C2×#368C211yesmaximal, characteristic, complemented
4#1417C214yesmaximal, complemented
5C2×#135—21no—
6#508—21no—
7#508—21no—
8#508C412yes—
9#508C2212yes—
10C2×Q32C411yescharacteristic
11C2×SD32—21no—
12C2×D16C411yescharacteristic
13C22×C16C2211yescharacteristic
14C2×#108—21no—
15M64—21no—
16M64—21no—
17C2×C32C2212yescomplemented
18#368—21no—
19#368C2212yescomplemented
20C2×#41—41no—
21#135—22no—
22#135—22no—
23#135—41no—
24#135—41no—
25C2×Q16—21no—
26C2×SD16—41no—
27C2×D8—21no—
28C2×M16—41no—
29C22×C8D411yescharacteristic
30Q32—21no—
31Q32C2×C412yes—
32SD32—22no—
33SD32—22no—
34D16—21no—
35D16C2×C412yes—
36C2×C16C2×C411yescharacteristic
37C2×C16C2×C411yescharacteristic
38C2×C16D411yescharacteristic
39C2×C16D411yescharacteristic
40C2×C16D411yescharacteristic
41C2×C16C2311yescharacteristic
42#108—22no—
43#108—41no—
44C32—22no—
45#41—42no—
46#41—42no—
47#41—81no—
48#41—81no—
49C2×Q8—41no—
50C2×D4—41no—
51C2×D4—81no—
52C22×C4—81no—
53C22×C4D811yescharacteristic
54Q16—22no—
55Q16—41no—
56SD16—42no—
57SD16—42no—
58D8—22no—
59D8—41no—
60M16—42no—
61M16—81no—
62C2×C8—81no—
63C2×C8C22⋊C411yescharacteristic
64C2×C8C22⋊C411yescharacteristic
65C2×C8D811yescharacteristic
66C2×C8SD1611yescharacteristic
67C2×C8SD1611yescharacteristic
68C2×C8C2×D411yescharacteristic
69C16C22⋊C411yescharacteristic
70C16C22⋊C411yescharacteristic
71C16C22×C411yescharacteristic
72C16C2×D411yescharacteristic
73C23—81no—
74C23SD3211yescharacteristic
75Q8—42no—
76Q8—81no—
77D4—42no—
78D4—82no—
79D4—82no—
80D4—81no—
81C2×C4—81no—
82C2×C4—82no—
83C2×C4—82no—
84C2×C4#10211yescharacteristic
85C2×C4#10211yescharacteristic
86C2×C4D1611yescharacteristic
87C2×C4D1611yescharacteristic
88C2×C4SD3211yescharacteristic
89C2×C4C2×D811yescharacteristic
90C8—82no—
91C8#10211yescharacteristic
92C8#10211yescharacteristic
93C8C2×C22⋊C411yescharacteristic
94C8C2×SD1611yescharacteristic
95C22—41no—
96C22—82no—
97C22—162no—
98C22#35711yescharacteristic
99C22#35711yescharacteristic
100C22C2×SD3211yescharacteristic
101C4—82no—
102C4#35711yescharacteristic
103C4#35711yescharacteristic
104C4C2×#10211yescharacteristic
105C4C2×D1611yescharacteristic
106C2—21no—
107C2—21no—
108C2—21no—
109C2—162no—
110C2C2×#35711yescharacteristic
subgroupG/H
centerC4C2×D16
commutatorC16C22×C4
Frattini, cosocleC2×C16C23
socleC2C2×#357
prime radicalC2×#508C2
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×M64C2semidirect
3. C2×#368C2semidirect
4. #1417C2semidirect
17. C2×C32C22semidirect
19. #368C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5082
C2×M642
C2×#3682
#14172