Finite Group Databasebeta

← #12240

Subgroups — #12240

Order 256.

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

normal subgroups only

This group has 175 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#2792C211yesmaximal, characteristic
2#2060C211yesmaximal, characteristic
3#1611C211yesmaximal, characteristic
4#1605C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic
6#1346C211yesmaximal, characteristic
7#1325C211yesmaximal, characteristic
8#544C2211yescharacteristic
9#483—21no—
10#451—21no—
11#428—21no—
12#398—21no—
13#398C2211yescharacteristic
14#395—21no—
15#395C2211yescharacteristic
16#389—21no—
17#389—21no—
18#387—21no—
19#387—21no—
20#384C2211yescharacteristic
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C2×Q16—41no—
29C2×M16—21no—
30C4⋊Q8—41no—
31#125—41no—
32C22⋊Q8—21no—
33C22⋊Q8—21no—
34#117—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4C2311yescharacteristic
45C2×C42—21no—
46C2×C42D411yescharacteristic
47#107—41no—
48#106—41no—
49#106—41no—
50#103—41no—
51#103—41no—
52C22⋊C8—21no—
53C22⋊C8—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—41no—
65C2×Q8—81no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75Q16—82no—
76M16—22no—
77M16—41no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C4⋊C4—24no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C4⋊C4—81no—
95C4⋊C4—81no—
96C4⋊C4—81no—
97C4⋊C4—81no—
98C22⋊C4—41no—
99C42—21no—
100C42—21no—
101C42—41no—
102C42—81no—
103C23#12011yescharacteristic
104Q8—42no—
105Q8—81no—
106Q8—81no—
107Q8—81no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
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#13511yescharacteristic
135C2×C4#13711yescharacteristic
136C8—42no—
137C8—81no—
138C8—81no—
139C22—21no—
140C22—21no—
141C22#45211yescharacteristic
142C22#45611yescharacteristic
143C22B(4,𝔽2)11yescharacteristic
144C4—21no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—42no—
149C4—41no—
150C4—41no—
151C4—81no—
152C4—81no—
153C4—81no—
154C4—81no—
155C4—81no—
156C2—21no—
157C2—21no—
158C2#160311yescharacteristic
159C2#220011yescharacteristic
160C2#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
#27922
#20602
#16112
#16052
#13542
#13462
#13252