Finite Group Databasebeta

← #7086

Subgroups — #7086

Order 256.

132 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 25 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#2001C211yesmaximal, characteristic
2#1314C211yesmaximal, characteristic
3#1293C211yesmaximal, characteristic
4C22×Q16C411yescharacteristic
5C2×C22⋊Q8—21no—
6C2×#103—21no—
7#407—21no—
8C2×C22⋊C8C2211yescharacteristic
9C22⋊C16—21no—
10#340C411yescharacteristic
11C22×Q8—21no—
12C23×C4D411yescharacteristic
13C2×Q16—41no—
14C2×Q16—42no—
15C2×M16—41no—
16C22×C8C2×C411yescharacteristic
17C22×C8D411yescharacteristic
18C22⋊Q8—42no—
19C22⋊Q8—42no—
20C2×C4⋊C4—21no—
21C2×C4⋊C4—41no—
22C2×C22⋊C4—41no—
23C2×C16—41no—
24#103—42no—
25C22⋊C8—22no—
26C22⋊C8—41no—
27C22⋊C8—41no—
28C24SD1611yescharacteristic
29C2×Q8—22no—
30C2×Q8—41no—
31C2×Q8—41no—
32C2×Q8—41no—
33C2×Q8—41no—
34C2×Q8—41no—
35C22×C4—21no—
36C22×C4—21no—
37C22×C4—41no—
38C22×C4—41no—
39C22×C4—41no—
40C22×C4—41no—
41C22×C4C22⋊C411yescharacteristic
42C22×C4D811yescharacteristic
43Q16—42no—
44Q16—42no—
45M16—81no—
46M16—81no—
47C2×C8—21no—
48C2×C8—21no—
49C2×C8—41no—
50C2×C8—41no—
51C2×C8—42no—
52C4⋊C4—22no—
53C4⋊C4—41no—
54C4⋊C4—44no—
55C22⋊C4—44no—
56C16—81no—
57C23—21no—
58C23—21no—
59C23—21no—
60C23—41no—
61C23—41no—
62C23#10211yescharacteristic
63Q8—42no—
64Q8—42no—
65Q8—81no—
66Q8—81no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—21no—
70C2×C4—41no—
71C2×C4—41no—
72C2×C4—41no—
73C2×C4—42no—
74C2×C4—42no—
75C2×C4—81no—
76C2×C4—81no—
77C2×C4—81no—
78C2×C4—81no—
79C2×C4—81no—
80C2×C4#9911yescharacteristic
81C2×C4#10411yescharacteristic
82C8—41no—
83C8—41no—
84C8—82no—
85C22—21no—
86C22—21no—
87C22—42no—
88C22—41no—
89C22—41no—
90C22—41no—
91C22—81no—
92C22#32711yescharacteristic
93C22#35911yescharacteristic
94C22#36211yescharacteristic
95C4—21no—
96C4—41no—
97C4—82no—
98C4—82no—
99C4AΣL(1,𝔽16)11yescharacteristic
100C4#35311yescharacteristic
101C2—21no—
102C2—21no—
103C2—81no—
104C2#134211yescharacteristic
105C2#134311yescharacteristic
106C2#135011yescharacteristic
subgroupG/H
center, socleC22#327
commutatorC22×C8C2×C4
Frattini, cosocleC2×C22⋊C8C22
prime radicalC24SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#20012
#13142
#12932