Finite Group Databasebeta

← #309576

Subgroups — #309576

Order 512.

554 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 158 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#25062C212yesmaximal
2#22965C212yesmaximal
3#10674C211yesmaximal, characteristic
4#10483C212yesmaximal
5#10237C211yesmaximal, characteristic
6#9614C212yesmaximal
7#9544C212yesmaximal
8#8929C211yesmaximal, characteristic
9#8884C212yesmaximal
10C2×#477C2212yes—
11#2582C2211yescharacteristic
12#2369C2211yescharacteristic
13Q8×C4⋊C4C2212yes—
14#2095—22no—
15#2083—22no—
16#1990C2212yes—
17#1984C2212yes—
18#1956C2211yescharacteristic
19#1947C2212yes—
20#1947C2212yes—
21#1942—22no—
22#1928—22no—
23#1921—22no—
24#1921—22no—
25#1863—22no—
26#1845C2212yes—
27#1845C2211yescharacteristic
28#1803C2212yes—
29C8×C4⋊C4C2212yes—
30C4×#103—22no—
31C4×#103C2212yes—
32C2×#335C2212yes—
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—22no—
47C2×#107—22no—
48C2×#107—22no—
49C2×#107—41no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#103—22no—
53C2×#103—22no—
54C2×#103—41no—
55C2×#103D412yes—
56C2×#103D412yes—
57C2×C4×C8—22no—
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2312yes—
60#398—41no—
61#391—22no—
62#391—41no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—42no—
67#384—22no—
68#384C2312yes—
69#384C2311yescharacteristic
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2312yes—
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#335—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—44no—
123#107—82no—
124#107—82no—
125#107—82no—
126#107—82no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8#4114yes—
131#103—28no—
132#103—42no—
133#103—82no—
134#103—82no—
135#103Q1618yes—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—82no—
140#95—22no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—41no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—41no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8C2×Q1614yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4C22×D412yes—
163C22×C4H(5,𝔽2)11yescharacteristic
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—24no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8C2×Q1614yes—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—82no—
196C4⋊C4—84no—
197C4⋊C4—161no—
198C4⋊C4—162no—
199C4⋊C4—161no—
200C4⋊C4C2×Q1614yes—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—84no—
206C42—161no—
207C42C2×#4111yescharacteristic
208C42C2×#4112yes—
209C422−1+411yescharacteristic
210C23D4211yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—42no—
214Q8—42no—
215Q8—82no—
216Q8—161no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—24no—
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—42no—
231C2×C4—42no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—82no—
239C2×C4#52512yes—
240C2×C4#53911yescharacteristic
241C2×C4#54712yes—
242C2×C4#55911yescharacteristic
243C2×C4C22×Q1612yes—
244C2×C4#57512yes—
245C2×C4#57612yes—
246C2×C4#57812yes—
247C8—24no—
248C8—42no—
249C8—42no—
250C8—44no—
251C8—82no—
252C22#245511yescharacteristic
253C22#326812yes—
254C22D4×Q1612yes—
255C22#328911yescharacteristic
256C22#329211yescharacteristic
257C4—21no—
258C4—21no—
259C4—44no—
260C4—44no—
261C4—82no—
262C4—82no—
263C4—82no—
264C4—162no—
265C4—161no—
266C4#310412yes—
267C4#312412yes—
268C4#333812yes—
269C4#335212yes—
270C2#1981512yes—
271C2Q16211yescharacteristic
272C2#2013811yescharacteristic
273C2#2655912yes—
274C2#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
#250622
#229652
#106742
#104832
#102372
#96142
#95442
#89292
#88842