Finite Group Databasebeta

← #11568

Subgroups — #11568

Order 256.

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

normal subgroups only

This group has 197 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#2546C211yesmaximal, characteristic
2#1467C211yesmaximal, characteristic
3#1461C211yesmaximal, characteristic
4#1319C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic
6C2×C4⋊Q8C412yes—
7C2×C4×Q8C2211yescharacteristic
8#432—21no—
9#431—21no—
10#398—21no—
11#398—21no—
12#391C2212yes—
13#389—21no—
14#389—21no—
15C4×C4⋊C4C411yescharacteristic
16#324—21no—
17#324—21no—
18#324C2212yes—
19#324C2212yes—
20C22×Q8C2×C412yes—
21C22×Q8C2×C411yescharacteristic
22C4⋊Q8—41no—
23C4⋊Q8—41no—
24C4⋊Q8—44no—
25C4×Q8—41no—
26C4×Q8—41no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4D411yescharacteristic
33C2×C4⋊C4D411yescharacteristic
34C2×C4⋊C4C2311yescharacteristic
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C412yes—
37C2×C42D411yescharacteristic
38C2×C42D411yescharacteristic
39C4⋊C8—41no—
40C4⋊C8—41no—
41C22⋊C8—22no—
42C22⋊C8—22no—
43#97—41no—
44C4×C8—41no—
45#95—21no—
46#95—22no—
47#95—22no—
48C2×Q8—22no—
49C2×Q8—22no—
50C2×Q8—24no—
51C2×Q8—41no—
52C2×Q8—42no—
53C2×Q8—42no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22×C411yescharacteristic
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C2×C8—42no—
64C2×C8—42no—
65C4⋊C4—24no—
66C4⋊C4—41no—
67C4⋊C4—41no—
68C4⋊C4—41no—
69C4⋊C4—42no—
70C4⋊C4—44no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—44no—
74C42—21no—
75C42—21no—
76C42—21no—
77C42—21no—
78C42—21no—
79C42—21no—
80C42—42no—
81C23C2×C22⋊C411yescharacteristic
82Q8—44no—
83Q8—84no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—24no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4#9911yescharacteristic
105C2×C4#9911yescharacteristic
106C2×C4#10012yes—
107C2×C4#13112yes—
108C8—82no—
109C8—82no—
110C22—21no—
111C22—21no—
112C22#40811yescharacteristic
113C22C2×#9911yescharacteristic
114C22C2×#10011yescharacteristic
115C4—21no—
116C4—22no—
117C4—22no—
118C4—44no—
119C4—41no—
120C4—41no—
121C4—41no—
122C4—41no—
123C4—82no—
124C4#35612yes—
125C2—21no—
126C2—21no—
127C2#147111yescharacteristic
128C2#212911yescharacteristic
129C2C2×#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
#25462
#14672
#14612
#13192