Finite Group Databasebeta

← #94385

Subgroups — #94385

Order 512.

222 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 46 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#11478C211yesmaximal, characteristic
2#8962C211yesmaximal, characteristic
3#7571C211yesmaximal, characteristic
4#2983C411yescharacteristic
5C4×C4⋊Q8C2211yescharacteristic
6C4×C4⋊C8—21no—
7#1479—21no—
8#1461—21no—
9#1461C411yescharacteristic
10#1325C412yes—
11#1319—21no—
12#1276—21no—
13#1272—21no—
14#1272C412yes—
15C2×C4⋊Q8Q811yescharacteristic
16C2×C4×Q8C2×C411yescharacteristic
17C2×C4×Q8D411yescharacteristic
18#446—42no—
19#432—41no—
20#431—21no—
21#431—21no—
22#431—41no—
23C2×C4⋊C8—41no—
24C2×C4×C8—41no—
25#391C2×C411yescharacteristic
26#391D411yescharacteristic
27#389—41no—
28#384C2×C411yescharacteristic
29C4×C4⋊C4—41no—
30C43D411yescharacteristic
31#336—22no—
32Q8⋊C8—42no—
33#324—22no—
34#324—41no—
35#324—41no—
36C22×Q8C4⋊C411yescharacteristic
37C22×Q8C4⋊C411yescharacteristic
38C22×C8—42no—
39C4⋊Q8—22no—
40C4⋊Q8—22no—
41C4×Q8—41no—
42C4×Q8—41no—
43C4×Q8—41no—
44C4×Q8—41no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—41no—
48C2×C4⋊C4—41no—
49C2×C4⋊C4C4211yescharacteristic
50C2×C4⋊C4C22⋊C411yescharacteristic
51C2×C42—21no—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42C22⋊C411yescharacteristic
56C2×C42C22⋊C411yescharacteristic
57C2×C42C4⋊C411yescharacteristic
58C4⋊C8—22no—
59C4⋊C8—41no—
60C4⋊C8—42no—
61C4⋊C8—44no—
62C4⋊C8—81no—
63C22⋊C8—41no—
64C22⋊C8—42no—
65C22⋊C8—41no—
66#97—42no—
67C4×C8—41no—
68C4×C8—41no—
69C4×C8—82no—
70C4×C8—81no—
71#95—41no—
72#95—41no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—21no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×Q8#10812yes—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4#9511yescharacteristic
88C22×C4#9911yescharacteristic
89C22×C4#9911yescharacteristic
90C2×C8—44no—
91C2×C8—42no—
92C2×C8—82no—
93C2×C8—82no—
94C2×C8—81no—
95C2×C8—82no—
96C4⋊C4—44no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—42no—
100C4⋊C4—81no—
101C4⋊C4—82no—
102C42—22no—
103C42—22no—
104C42—21no—
105C42—21no—
106C42—21no—
107C42—41no—
108C42—42no—
109C42—41no—
110C42—41no—
111C42—41no—
112C42—41no—
113C42#10812yes—
114C23#34211yescharacteristic
115Q8—42no—
116Q8—81no—
117Q8—81no—
118Q8—81no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—22no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—21no—
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—41no—
138C2×C4—41no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4#34111yescharacteristic
142C2×C4#34111yescharacteristic
143C2×C4AΣL(1,𝔽16)11yescharacteristic
144C2×C4#35211yescharacteristic
145C8—84no—
146C8—82no—
147C8—161no—
148C8—162no—
149C22—21no—
150C22—21no—
151C22#129811yescharacteristic
152C22#139011yescharacteristic
153C22#139511yescharacteristic
154C4—24no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—41no—
159C4—41no—
160C4—41no—
161C4—42no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4#140512yes—
166C2—21no—
167C2—21no—
168C2#728511yescharacteristic
169C2#748611yescharacteristic
170C2#749111yescharacteristic
subgroupG/H
centerC2×C4AΣL(1,𝔽16)
commutatorC2×C4⋊C4C42
Frattini, cosocleC4×C4⋊Q8C22
socleC22#1390
prime radicalC23#342
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C4×C4⋊Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#114782
#89622
#75712