Finite Group Databasebeta

← #12273

Subgroups — #12273

Order 256.

182 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 35 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#2586C211yesmaximal, characteristic
2#2028C211yesmaximal, characteristic
3#1611C211yesmaximal, characteristic
4#1605C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic
6#1346C211yesmaximal, characteristic
7#1319C211yesmaximal, characteristic
8#544C2211yescharacteristic
9#483—21no—
10#451—21no—
11#419—21no—
12#398C2211yescharacteristic
13#395—21no—
14#395C2211yescharacteristic
15#391—21no—
16#391C2211yescharacteristic
17#389—21no—
18#387—21no—
19#387—21no—
20C4×C4⋊C4—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×Q8D411yescharacteristic
29C2×Q16—41no—
30C2×M16—21no—
31C4⋊Q8—41no—
32#125—41no—
33C22⋊Q8—21no—
34C22⋊Q8—21no—
35#117—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4C2311yescharacteristic
43C2×C42—21no—
44C2×C42—21no—
45C2×C42D411yescharacteristic
46#106—41no—
47#103—41no—
48#103—41no—
49#103—41no—
50#103—41no—
51C22⋊C8—21no—
52C22⋊C8—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×Q8—42no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—41no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79Q16—82no—
80M16—41no—
81M16—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C4⋊C4—24no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—81no—
98C4⋊C4—81no—
99C22⋊C4—41no—
100C42—41no—
101C42—42no—
102C42—41no—
103C42—81no—
104C23#12011yescharacteristic
105Q8—42no—
106Q8—42no—
107Q8—81no—
108Q8—81no—
109Q8—81no—
110Q8—81no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4#13511yescharacteristic
138C2×C4#13711yescharacteristic
139C8—81no—
140C8—81no—
141C8—81no—
142C22—21no—
143C22—21no—
144C22#45211yescharacteristic
145C22#45511yescharacteristic
146C22B(4,𝔽2)11yescharacteristic
147C4—21no—
148C4—21no—
149C4—42no—
150C4—42no—
151C4—42no—
152C4—41no—
153C4—41no—
154C4—42no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4—81no—
159C2—21no—
160C2—21no—
161C2#160611yescharacteristic
162C2#219711yescharacteristic
163C2#220511yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC23#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ 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
#25862
#20282
#16112
#16052
#13542
#13462
#13192