Finite Group Databasebeta

← #11993

Subgroups — #11993

Order 256.

180 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#3152C211yesmaximal, characteristic
2#2165C212yesmaximal, complemented
3#1359C212yesmaximal, complemented
4#1358C212yesmaximal, complemented
5C2×#137—22no—
6#532C411yescharacteristic
7C2×C4⋊Q8C411yescharacteristic
8#497—22no—
9#494C411yescharacteristic
10#494C2211yescharacteristic
11#492C411yescharacteristic
12#492C2211yescharacteristic
13C4×M16C2211yescharacteristic
14#364C2214yes—
15#349—22no—
16#347—22no—
17C2×#41—21no—
18C22×Q8—21no—
19#137—48no—
20C2×Q16—22no—
21C2×Q16—22no—
22C2×SD16—22no—
23C2×SD16—22no—
24C2×M16D412yes—
25C4⋊Q8—21no—
26C4⋊Q8—41no—
27C4⋊Q8C2×C411yescharacteristic
28C4⋊Q8C2×C411yescharacteristic
29C4⋊Q8C2×C411yescharacteristic
30#127C2×C411yescharacteristic
31#124—21no—
32#121—41no—
33C4×D4—41no—
34C2×C4⋊C4—41no—
35C2×C42C2×C411yescharacteristic
36M32—24no—
37#97D412yes—
38C4×C8C2×C411yescharacteristic
39C4×C8C2311yescharacteristic
40#41—44no—
41C2×Q8—21no—
42C2×Q8—21no—
43C2×Q8—21no—
44C2×Q8—44no—
45C2×Q8—41no—
46C2×D4—21no—
47C2×D4—41no—
48C2×D4—41no—
49C22×C4—21no—
50C22×C4—41no—
51C22×C4—41no—
52C22×C4C22⋊C411yescharacteristic
53Q16—44no—
54Q16—44no—
55SD16—44no—
56SD16—44no—
57M16—24no—
58C2×C8C22⋊C412yes—
59C2×C8C2×D412yes—
60C4⋊C4—41no—
61C4⋊C4—41no—
62C4⋊C4—41no—
63C4⋊C4—42no—
64C22⋊C4—42no—
65C42—21no—
66C42C22⋊C411yescharacteristic
67C42C22×C411yescharacteristic
68C16—44no—
69C23—41no—
70C23#10011yescharacteristic
71Q8—42no—
72Q8—42no—
73Q8—42no—
74Q8—82no—
75D4—42no—
76D4—82no—
77D4—82no—
78C2×C4—21no—
79C2×C4—22no—
80C2×C4—21no—
81C2×C4—42no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—82no—
86C2×C4—82no—
87C2×C4#10011yescharacteristic
88C2×C4C2×C22⋊C411yescharacteristic
89C8—22no—
90C8AGL(1,ℤ/8)14yes—
91C22—21no—
92C22—41no—
93C22—82no—
94C22—81no—
95C22C2×#10011yescharacteristic
96C4—22no—
97C4—21no—
98C4—41no—
99C4—81no—
100C4—81no—
101C4—81no—
102C4#41812yes—
103C2—21no—
104C2—41no—
105C2—161no—
106C2#155511yescharacteristic
subgroupG/H
center, socleC2#1555
commutatorC42C22×C4
Frattini, cosocleC4×C8C23
prime radical#532C4
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2165C2semidirect
3. #1359C2semidirect
4. #1358C2semidirect

Maximal subgroups

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

subgroupindex
#31522
#21652
#13592
#13582