Finite Group Databasebeta

← #232161

Subgroups — #232161

Order 512.

443 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 136 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#19750C211yesmaximal, characteristic
2#9430C212yesmaximal, complemented
3#9429C211yesmaximal, characteristic, complemented
4#9421C211yesmaximal, characteristic, complemented
5#9419C212yesmaximal, complemented
6#9417C212yesmaximal, complemented
7#9417C212yesmaximal, complemented
8#9417C212yesmaximal, complemented
9#8803C212yesmaximal, complemented
10#3184C2211yescharacteristic
11#3184C2212yes—
12#3152C2211yescharacteristic
13#3141C2212yes—
14#2980C2212yes—
15#2980C2211yescharacteristic
16#2381C2211yescharacteristic
17#1721C411yescharacteristic, complemented
18#1720C412yescomplemented
19#1717C412yescomplemented
20#1716C411yescharacteristic, complemented
21#1709C411yescharacteristic, complemented
22#1708C411yescharacteristic, complemented
23#1558C2211yescharacteristic
24#1556C2212yes—
25#1536C2212yes—
26#1536C2211yescharacteristic
27#1514C2212yes—
28#1514C2212yes—
29#1514C2212yes—
30#1455C2211yescharacteristic
31#1291C2212yes—
32#1291C2212yes—
33#1291C2212yes—
34#1288C2212yes—
35#1288C2212yes—
36#1288C2212yes—
37C2×#137—41no—
38#532C2312yes—
39#532C2311yescharacteristic
40C2×C4⋊Q8C2311yescharacteristic
41#498C2×C411yescharacteristic
42#498C2×C412yes—
43#497—41no—
44#494C2×C411yescharacteristic
45#494C2×C411yescharacteristic
46#493C2×C411yescharacteristic
47#493C2×C412yes—
48#492C2×C411yescharacteristic
49#492C2×C411yescharacteristic
50#489—42no—
51#487C2×C412yes—
52#487C2×C412yes—
53#486C2×C412yes—
54#486C2×C412yes—
55#486C2×C412yes—
56#486C2×C412yes—
57#486C2×C412yes—
58#469—41no—
59#469—42no—
60#442—42no—
61#442—41no—
62#435—42no—
63#435—42no—
64#435—42no—
65#432C2312yes—
66#432C2312yes—
67#432C2312yes—
68#417—42no—
69#408—41no—
70#408—42no—
71C4×M16C2312yes—
72C4×M16C2311yescharacteristic
73#391—41no—
74#387—42no—
75#383C2312yes—
76#333—41no—
77#332—42no—
78#329—42no—
79#329—41no—
80#327—42no—
81#327—42no—
82#327—42no—
83#322C2×C411yescharacteristic
84#322C2×C412yes—
85C82C2×C411yescharacteristic
86C2×#41—21no—
87C2×#41—22no—
88C22×Q8—21no—
89#137—82no—
90#137—82no—
91C2×Q16—41no—
92C2×Q16—41no—
93C2×SD16—41no—
94C2×SD16—41no—
95C2×D8—41no—
96C2×D8—42no—
97C2×M16—21no—
98C2×M16—22no—
99C2×M16—42no—
100C2×M16—41no—
101C22×C8—42no—
102C22×C8—41no—
103C4⋊Q8—22no—
104C4⋊Q8—21no—
105C4⋊Q8C22×C411yescharacteristic
106C4⋊Q8C22×C411yescharacteristic
107C4⋊Q8C22×C412yes—
108C4⋊Q8C22×C411yescharacteristic
109#127C22×C412yes—
110#127C22×C411yescharacteristic
111#124—22no—
112#124—21no—
113#121—22no—
114#121—22no—
115#121—22no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—41no—
121C2×C42—41no—
122C2×C42C2411yescharacteristic
123#106—41no—
124#106—42no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128#103—42no—
129#103—42no—
130#102—42no—
131#102—42no—
132#102—42no—
133#102—42no—
134#102—42no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—44no—
139C22⋊C8—42no—
140#97—22no—
141#97—21no—
142#97—22no—
143#97—22no—
144#97—22no—
145C4×C8C22×C412yes—
146C4×C8C22×C411yescharacteristic
147C4×C8C22×C411yescharacteristic
148C4×C8C22×C412yes—
149#95—22no—
150#95—22no—
151#95—22no—
152#95—42no—
153#41—81no—
154#41—81no—
155#41—82no—
156#41—82no—
157C2×Q8—22no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—41no—
162C2×Q8—81no—
163C2×Q8—81no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—41no—
170C2×D4—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4H(5,𝔽2)12yes—
176C22×C4H(5,𝔽2)11yescharacteristic
177Q16—82no—
178Q16—82no—
179SD16—82no—
180SD16—82no—
181D8—82no—
182D8—84no—
183M16—42no—
184M16—82no—
185M16—161no—
186M16—162no—
187C2×C8—21no—
188C2×C8—22no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—161no—
195C2×C8—162no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—161no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C42—161no—
207C42C23×C411yescharacteristic
208C42H(5,𝔽2)12yes—
209C42H(5,𝔽2)11yescharacteristic
210C23—41no—
211C23—42no—
212C23#56011yescharacteristic
213Q8—81no—
214Q8—81no—
215Q8—81no—
216Q8—82no—
217Q8—161no—
218D4—82no—
219D4—82no—
220D4—82no—
221D4—81no—
222D4—82no—
223D4—161no—
224D4—162no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—84no—
239C2×C4#51812yes—
240C2×C4#51811yescharacteristic
241C2×C4#56012yes—
242C2×C4#56011yescharacteristic
243C8—22no—
244C8—41no—
245C8—42no—
246C8—42no—
247C8—82no—
248C8—82no—
249C8—82no—
250C22—21no—
251C22—22no—
252C22—81no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—82no—
257C22#238011yescharacteristic
258C4—21no—
259C4—21no—
260C4—22no—
261C4—22no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4—82no—
266C4—162no—
267C2—41no—
268C2—161no—
269C2—162no—
270C2#2336111yescharacteristic
271C2#2336212yes—
subgroupG/H
center, socleC22#2380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19750C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #9430C2semidirect
3. #9429C2semidirect
4. #9421C2semidirect
5. #9419C2semidirect
6. #9417C2semidirect
7. #9417C2semidirect
8. #9417C2semidirect
9. #8803C2semidirect
17. #1721C4semidirect
18. #1720C4semidirect
19. #1717C4semidirect
20. #1716C4semidirect
21. #1709C4semidirect
22. #1708C4semidirect

Maximal subgroups

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

subgroupindex
#197502
#94302
#94292
#94212
#94192
#94172
#88032