Finite Group Databasebeta

← #326584

Subgroups — #326584

Order 512.

398 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 126 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#22300C211yesmaximal, characteristic
2#22258C211yesmaximal, characteristic
3#10670C211yesmaximal, characteristic
4#10668C211yesmaximal, characteristic
5#10373C211yesmaximal, characteristic
6#10373C211yesmaximal, characteristic
7#9749C212yesmaximal
8#9629C212yesmaximal
9#8881C212yesmaximal
10#8879C212yesmaximal
11#8783C211yesmaximal, characteristic
12#3160C2211yescharacteristic
13#2961C2211yescharacteristic
14#2961C2211yescharacteristic
15#2402C2211yescharacteristic
16#1995C2211yescharacteristic
17#1995C2211yescharacteristic
18#1940C2212yes—
19#1939—22no—
20#1922—22no—
21#1922—22no—
22#1916—22no—
23#1843C2211yescharacteristic
24#1837C2211yescharacteristic
25C8×C4⋊C4C2212yes—
26#1746C2211yescharacteristic
27#1571C2212yes—
28#1570C2212yes—
29#1564C2212yes—
30#1503C2212yes—
31#1503C2212yes—
32#1481C2212yes—
33#1481C2212yes—
34#1303C2212yes—
35#1303C2212yes—
36#1281C2212yes—
37#1274C2211yescharacteristic
38#1274C2211yescharacteristic
39#533C2311yescharacteristic
40#533C2311yescharacteristic
41#500Q812yes—
42#499—22no—
43#498Q812yes—
44#446—42no—
45#446—42no—
46#431C2312yes—
47#431C2312yes—
48#427—22no—
49#424—21no—
50#424—21no—
51#423C2311yescharacteristic
52#423C2311yescharacteristic
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56#405—21no—
57#405—21no—
58C2×#97C2311yescharacteristic
59C2×C4×C8—22no—
60C2×C4×C8C2311yescharacteristic
61#400—22no—
62#398—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4C2312yes—
65#376C2311yescharacteristic
66#341—22no—
67#339—22no—
68#339—22no—
69#336—22no—
70#336—22no—
71#336D412yes—
72#336D412yes—
73#335—22no—
74#335—42no—
75#334—22no—
76#334—42no—
77#328—22no—
78#328—22no—
79#328—22no—
80#328—22no—
81Q8⋊C8—42no—
82Q8⋊C8—42no—
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—22no—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D412yes—
89C4⋊Q8C2×Q812yes—
90C4⋊Q8C2×Q812yes—
91#125—21no—
92#125—21no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97#117—21no—
98#117—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C42—22no—
103C2×C42C2×D412yes—
104C2×C42C2411yescharacteristic
105#107—24no—
106#106—24no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—82no—
116C4⋊C8—82no—
117C22⋊C8—22no—
118C22⋊C8—22no—
119#97—22no—
120#97—41no—
121#97—41no—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—41no—
125C4×C8—41no—
126C4×C8—82no—
127C4×C8C2×Q811yescharacteristic
128C4×C8C2×Q811yescharacteristic
129C4×C8#4112yes—
130#95—22no—
131#95—22no—
132#95—22no—
133#95#4112yes—
134#95#4112yes—
135C2×Q8—22no—
136C2×Q8—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4C22×D411yescharacteristic
141C22×C4C2×#4112yes—
142M16—42no—
143M16—42no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8C22⋊Q812yes—
158C2×C8C22⋊Q812yes—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C42—22no—
171C42—22no—
172C42—41no—
173C42—41no—
174C42—42no—
175C42—42no—
176C42C22×Q811yescharacteristic
177C42C2×#4111yescharacteristic
178C42H(5,𝔽2)12yes—
179C23#52511yescharacteristic
180Q8—82no—
181Q8—82no—
182C2×C4—24no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4C2×C22⋊Q811yescharacteristic
195C2×C4#53511yescharacteristic
196C2×C4#55412yes—
197C2×C4#55912yes—
198C2×C4#57712yes—
199C2×C4#57812yes—
200C8—22no—
201C8—22no—
202C8—41no—
203C8—41no—
204C8—82no—
205C8—82no—
206C8—82no—
207C8—81no—
208C8—81no—
209C22—22no—
210C22#258011yescharacteristic
211C22#312012yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4—82no—
219C4#308212yes—
220C4#322111yescharacteristic
221C4#322311yescharacteristic
222C2—22no—
223C2#2926611yescharacteristic
224C2#2927012yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radicalC23#525
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#223002
#222582
#106702
#106682
#103732
#97492
#96292
#88812
#88792
#87832