Finite Group Databasebeta

← #11574

Subgroups — #11574

Order 256.

187 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 53 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#2567C211yesmaximal, characteristic
2#1467C211yesmaximal, characteristic
3#1461C211yesmaximal, characteristic
4#1325C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7#432—21no—
8#431—21no—
9#395—21no—
10#395—21no—
11#391C412yes—
12#389—21no—
13#389—21no—
14#384C2212yes—
15C4×C4⋊C4C411yescharacteristic
16#324—21no—
17#324—21no—
18#324C2212yes—
19#324C2212yes—
20C22×Q8—21no—
21C22×Q8C2×C411yescharacteristic
22C4⋊Q8—41no—
23C4⋊Q8—41no—
24C4×Q8—41no—
25C4×Q8—41no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4C2×C412yes—
31C2×C4⋊C4D411yescharacteristic
32C2×C4⋊C4D411yescharacteristic
33C2×C4⋊C4C2311yescharacteristic
34C2×C42C2×C412yes—
35C2×C42C2×C411yescharacteristic
36C2×C42D411yescharacteristic
37C2×C42D411yescharacteristic
38C4⋊C8—41no—
39C4⋊C8—41no—
40C22⋊C8—22no—
41C22⋊C8—22no—
42#97—41no—
43C4×C8—41no—
44#95—21no—
45#95—22no—
46#95—22no—
47C2×Q8—22no—
48C2×Q8—22no—
49C2×Q8—41no—
50C2×Q8—42no—
51C2×Q8—82no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4C22⋊C411yescharacteristic
55C22×C4C22⋊C411yescharacteristic
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22×C411yescharacteristic
59C22×C4C2×D411yescharacteristic
60C22×C4C2×D411yescharacteristic
61C2×C8—42no—
62C2×C8—42no—
63C4⋊C4—24no—
64C4⋊C4—41no—
65C4⋊C4—41no—
66C4⋊C4—41no—
67C4⋊C4—42no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—82no—
71C4⋊C4—82no—
72C42—21no—
73C42—21no—
74C42—21no—
75C42—21no—
76C42—21no—
77C42—21no—
78C42—24no—
79C23C2×C22⋊C411yescharacteristic
80Q8—44no—
81Q8—84no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—24no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4#9911yescharacteristic
103C2×C4#9911yescharacteristic
104C2×C4#10112yes—
105C2×C4#13112yes—
106C8—82no—
107C8—82no—
108C22—21no—
109C22—21no—
110C22#40811yescharacteristic
111C22C2×#9911yescharacteristic
112C22C2×#10111yescharacteristic
113C4—21no—
114C4—22no—
115C4—22no—
116C4—44no—
117C4—41no—
118C4—41no—
119C4—41no—
120C4—41no—
121C4—82no—
122C4#35612yes—
123C2—21no—
124C2—21no—
125C2#147211yescharacteristic
126C2#212811yescharacteristic
127C2C2×#35611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#25672
#14672
#14612
#13252