Finite Group Databasebeta

← #11571

Subgroups — #11571

Order 256.

175 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 175 conjugacy classes of subgroups — too many to draw. Only its 51 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#2565C211yesmaximal, characteristic
2#1467C211yesmaximal, characteristic
3#1461C211yesmaximal, characteristic
4#1325C212yesmaximal
5#1324C212yesmaximal
6C2×#125C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#432—21no—
9#431—21no—
10#398—21no—
11#398—21no—
12#391C411yescharacteristic
13#391C411yescharacteristic
14#387—21no—
15#387—21no—
16#384C2211yescharacteristic
17#382C2211yescharacteristic
18C4×C4⋊C4C411yescharacteristic
19#324—21no—
20#324—21no—
21#324C2212yes—
22#324C2212yes—
23C22×Q8C2×C411yescharacteristic
24#125—41no—
25#125—41no—
26C4×Q8—41no—
27C4×Q8—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4C2311yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C4⋊C8—41no—
44C4⋊C8—41no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#97—41no—
48C4×C8—41no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56C2×Q8—22no—
57C2×Q8—22no—
58C2×Q8—41no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22×C411yescharacteristic
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C2×C8—42no—
71C2×C8—42no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C42—21no—
84C42—21no—
85C42—21no—
86C42—21no—
87C42—21no—
88C42—21no—
89C42—22no—
90C42—22no—
91C23C2×C22⋊C411yescharacteristic
92Q8—44no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4#9911yescharacteristic
119C2×C4#9911yescharacteristic
120C2×C4#10112yes—
121C2×C4#13112yes—
122C8—82no—
123C8—82no—
124C22—21no—
125C22—21no—
126C22#40811yescharacteristic
127C22C2×#9911yescharacteristic
128C22C2×#10111yescharacteristic
129C4—21no—
130C4—21no—
131C4—22no—
132C4—22no—
133C4—42no—
134C4—41no—
135C4—41no—
136C4—41no—
137C4—41no—
138C4—81no—
139C4—81no—
140C4—81no—
141C2—21no—
142C2—21no—
143C2#147211yescharacteristic
144C2#212611yescharacteristic
145C2#213311yescharacteristic
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
#25652
#14672
#14612
#13252
#13242