Finite Group Databasebeta

← #309577

Subgroups — #309577

Order 512.

546 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 146 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#24994C212yesmaximal
2#22965C212yesmaximal
3#10676C211yesmaximal, characteristic
4#10482C212yesmaximal
5#10237C211yesmaximal, characteristic
6#9613C212yesmaximal
7#9543C212yesmaximal
8#8928C211yesmaximal, characteristic
9#8884C212yesmaximal
10C2×#477C2212yes—
11#2582C2211yescharacteristic
12#2369C2211yescharacteristic
13Q8×C4⋊C4C2212yes—
14#2099—22no—
15#2072—22no—
16#1990C2212yes—
17#1984C2212yes—
18#1956C2211yescharacteristic
19#1948C2212yes—
20#1947C2212yes—
21#1942—22no—
22#1927—22no—
23#1921—22no—
24#1920—22no—
25#1863—22no—
26#1845C2211yescharacteristic
27#1834C2212yes—
28#1803C2212yes—
29C8×C4⋊C4C2212yes—
30C4×#103—22no—
31C4×#103C2212yes—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×Q8⋊C8C2212yes—
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2311yescharacteristic
40C2×C4×Q8—22no—
41C2×C4×Q8C2312yes—
42#477—24no—
43#477—24no—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107—41no—
47C2×#106—22no—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×#103—22no—
51C2×#103—22no—
52C2×#103—41no—
53C2×#103D412yes—
54C2×#103D412yes—
55C2×C4×C8—22no—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2312yes—
58#398—41no—
59#391—22no—
60#391—41no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—42no—
65#384—22no—
66#384C2312yes—
67#384C2311yescharacteristic
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2312yes—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340D412yes—
79#340D412yes—
80#336—22no—
81#336—22no—
82#336—22no—
83#334—24no—
84#333—22no—
85Q8⋊C8—24no—
86C22×Q8—21no—
87C22×Q8C2×D412yes—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C4⋊Q8—22no—
94C4⋊Q8—21no—
95C4⋊Q8#4112yes—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—44no—
101C4×Q8—82no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42C2411yescharacteristic
121#107—28no—
122#107—82no—
123#107—82no—
124#106—44no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8#4114yes—
129#103—28no—
130#103—24no—
131#103—24no—
132#103—44no—
133#103—82no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—82no—
138#95—22no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—41no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—41no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8#13514yes—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4C22×D412yes—
161C22×C4H(5,𝔽2)11yescharacteristic
162C2×C8—22no—
163C2×C8—24no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—84no—
193C4⋊C4—81no—
194C4⋊C4—82no—
195C4⋊C4—161no—
196C4⋊C4—162no—
197C4⋊C4—161no—
198C4⋊C4#13714yes—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—84no—
204C42—161no—
205C42C2×#4111yescharacteristic
206C42C2×#4112yes—
207C422−1+411yescharacteristic
208C23D4211yescharacteristic
209Q8—24no—
210Q8—24no—
211Q8—42no—
212Q8—42no—
213Q8—82no—
214Q8—161no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4#52512yes—
238C2×C4#53911yescharacteristic
239C2×C4#54712yes—
240C2×C4#55911yescharacteristic
241C2×C4C2×#13512yes—
242C2×C4C2×#13712yes—
243C2×C4#57712yes—
244C2×C4#57812yes—
245C8—42no—
246C8—42no—
247C8—42no—
248C8—44no—
249C8—82no—
250C22#245511yescharacteristic
251C22#327012yes—
252C22#328212yes—
253C22#328911yescharacteristic
254C22#329211yescharacteristic
255C4—21no—
256C4—21no—
257C4—44no—
258C4—44no—
259C4—82no—
260C4—82no—
261C4—82no—
262C4—162no—
263C4—161no—
264C4#310812yes—
265C4#312012yes—
266C4#333612yes—
267C4#335612yes—
268C2#1979712yes—
269C2#2000311yescharacteristic
270C2#2011711yescharacteristic
271C2#2655812yes—
272C2#2657811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ 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
#249942
#229652
#106762
#104822
#102372
#96132
#95432
#89282
#88842