Finite Group Databasebeta

← #98911

Subgroups — #98911

Order 512.

314 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 102 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#21341C211yesmaximal, characteristic
2#8785C211yesmaximal, characteristic
3C4×#334C211yesmaximal, characteristic
4#7577C212yesmaximal
5#7022C212yesmaximal
6#2295C2211yescharacteristic
7#1984C411yescharacteristic
8#1984C411yescharacteristic
9#1939C411yescharacteristic
10#1939C411yescharacteristic
11#1916—21no—
12#1916—21no—
13#1831C2212yes—
14#1768—21no—
15#1767C2211yescharacteristic
16C4×C4⋊C8C2212yes—
17#1746—21no—
18C42×C8C2211yescharacteristic
19C2×#334—21no—
20#1277—21no—
21#1277—21no—
22#1277—21no—
23#1272—22no—
24#1271—22no—
25C2×C4×Q8C2×C411yescharacteristic
26C2×C4⋊C8—22no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×#97—21no—
33C2×C4×C8—22no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#391D411yescharacteristic
41#391Q811yescharacteristic
42#389D411yescharacteristic
43#389Q811yescharacteristic
44#387—21no—
45#387—21no—
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2×C411yescharacteristic
48#376—21no—
49C43C2311yescharacteristic
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—22no—
58#336—22no—
59#336—22no—
60#334—24no—
61C22×Q8C4⋊C411yescharacteristic
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C4×Q8—22no—
71C4×Q8—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C4⋊C411yescharacteristic
76C2×C4⋊C4C4⋊C411yescharacteristic
77C2×C4⋊C4C4⋊C411yescharacteristic
78C2×C42—21no—
79C2×C42—21no—
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×Q811yescharacteristic
85C2×C42#4111yescharacteristic
86C2×C42#4111yescharacteristic
87C4⋊C8—24no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95#97—42no—
96C4×C8—22no—
97C4×C8—22no—
98C4×C8—22no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—41no—
102C4×C8—82no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110C2×Q8—22no—
111C2×Q8—41no—
112C2×Q8—41no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C2×C4⋊C411yescharacteristic
119C22×C4#11711yescharacteristic
120C22×C4#12111yescharacteristic
121C22×C4#12111yescharacteristic
122C22×C4C22⋊Q811yescharacteristic
123C22×C4C22⋊Q811yescharacteristic
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—44no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8#10412yes—
148C4⋊C4—22no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C42—21no—
155C42—21no—
156C42—22no—
157C42—41no—
158C42—41no—
159C42—41no—
160C42—41no—
161C42—41no—
162C42—41no—
163C42—41no—
164C42#13112yes—
165C42#13112yes—
166C42#13512yes—
167C42#13712yes—
168C23#38011yescharacteristic
169Q8—82no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4#40811yescharacteristic
196C2×C4#40811yescharacteristic
197C2×C4#41611yescharacteristic
198C2×C4#41911yescharacteristic
199C2×C4C2×#10411yescharacteristic
200C2×C4#42111yescharacteristic
201C2×C4#42412yes—
202C2×C4#43312yes—
203C2×C4#46412yes—
204C2×C4#47112yes—
205C2×C4#47912yes—
206C8—22no—
207C8—41no—
208C8—41no—
209C8—41no—
210C8—84no—
211C8—81no—
212C8—81no—
213C8—162no—
214C22#179811yescharacteristic
215C22#180511yescharacteristic
216C22#180611yescharacteristic
217C22#181211yescharacteristic
218C22#181311yescharacteristic
219C22#181411yescharacteristic
220C22#181511yescharacteristic
221C4—22no—
222C4—22no—
223C4—22no—
224C4—22no—
225C4—22no—
226C4—41no—
227C4—81no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4#148211yescharacteristic
232C4#148511yescharacteristic
233C4#149411yescharacteristic
234C4#149711yescharacteristic
235C2#857011yescharacteristic
236C2#860311yescharacteristic
237C2#860711yescharacteristic
238C2#868111yescharacteristic
239C2#868711yescharacteristic
240C2#869011yescharacteristic
241C2#869611yescharacteristic
subgroupG/H
center, socle, prime radicalC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#213412
#87852
C4×#3342
#75772
#70222