Finite Group Databasebeta

← #35226326

Subgroups — #35226326

Order 20864.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782474C212yesmaximal, complemented
2#23782473C212yesmaximal, complemented
3#23782039C211yesmaximal, characteristic, complemented
4#23781931C212yesmaximal, complemented
5#23781930C212yesmaximal, complemented
6#23781847C211yesmaximal, characteristic, complemented
7#23781834C212yesmaximal, complemented
8#23781779C212yesmaximal, complemented
9#456×C163C211yesmaximal, characteristic, complemented
10#19104783C2211yescharacteristic
11#19104781C2212yescomplemented
12#19104767—22no—
13#19104766—22no—
14#19104766C2212yescomplemented
15#19104763C2212yes—
16#19104762—22no—
17#19104760—22no—
18#19104760C2212yes—
19#19104758—22no—
20#19104755—22no—
21#19104755C2212yes—
22C2×#14579686—22no—
23C2×#14579686C2212yescomplemented
24C2×#14579686C2212yescomplemented
25SD16×D163—22no—
26#19104743C2212yescomplemented
27#19104742C2212yescomplemented
28#19104732C2211yescharacteristic
29#19104670C2212yescomplemented
30#19104663C2212yescomplemented
31#19104654C2211yescharacteristic, complemented
32#19104651C2212yescomplemented
33C4×C163⋊C8C2211yescharacteristic, complemented
342−1+4×C163C2211yescharacteristic
35#137×C163C2212yescomplemented
36C4⋊Q8×C163C2211yescharacteristic, complemented
37#104×C163C2212yescomplemented
38#101×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579700—22no—
41#14579699—22no—
42#14579699—41no—
43#14579699C2312yes—
44Q8×D163—22no—
45C2×D652—22no—
46C2×D652—41no—
47C2×D652C2312yes—
48D4×D163—22no—
49D4×D163—42no—
50#14579689—22no—
51#14579689D412yescomplemented
52#14579688—22no—
53#14579687D412yescomplemented
54#14579686—22no—
55#14579686—22no—
56#14579686—22no—
57#14579686—22no—
58#14579686—22no—
59#14579686—22no—
60#14579686D412yescomplemented
61D1304—22no—
62#14579684—22no—
63#14579684D412yescomplemented
64#14579683—22no—
65#14579683C2312yes—
66#14579682—22no—
67C2×C163⋊C8—22no—
68C2×C163⋊C8C2312yes—
69C8×D163—22no—
70#41×C163—22no—
71#41×C163—41no—
72#41×C163C2312yes—
73C2×Q8×C163—22no—
74C2×Q8×C163C2311yescharacteristic
75C2×Q8×C163C2311yescharacteristic
76Q16×C163—22no—
77SD16×C163—22no—
78M16×C163C2312yes—
79C4⋊C4×C163—41no—
80C42×C163C2311yescharacteristic
81C22×D163—22no—
82C22×D163—42no—
83Dic326—41no—
84Dic326C2×D412yes—
85D652—22no—
86D652—22no—
87D652—42no—
88D652—41no—
89D652—82no—
90D652C2×D412yes—
91C22⋊D163—22no—
92C22⋊D163—42no—
93C4×D163—22no—
94C4×D163—42no—
95C4×D163—42no—
96C163⋊C8—22no—
97C163⋊C8C2×D412yes—
98C163⋊C8C2×D412yes—
99Q8×C163—22no—
100Q8×C163—22no—
101Q8×C163—41no—
102Q8×C163C2×D412yes—
103D4×C163—22no—
104D4×C163—41no—
105D4×C163C2×D412yes—
106C2×C4×C163—21no—
107C2×C4×C163—22no—
108C2×C4×C163—21no—
109C2×C4×C163—21no—
110C2×C4×C163—42no—
111C2×C4×C163C2411yescharacteristic
112C8×C163—22no—
113C22×C163—22no—
114C22×C163H(5,𝔽2)11yescharacteristic
115C2×D163—22no—
116C2×D163—42no—
117C2×D163—82no—
118C2×D163—162no—
119C4×C163—22no—
120C4×C163—22no—
121C4×C163—41no—
122C4×C163—41no—
123C4×C163C22×D412yes—
124Dic163—22no—
125C2×C163—21no—
126C2×C163—42no—
127C2×C163D4211yescharacteristic
128D163—42no—
129D163—82no—
130C163#329111yescharacteristic, complemented, Sylow
131#3291—1631nomaximal, Sylow
132#577—1632no—
133#577—1632no—
134#492—1631no—
135#471—1632no—
136#471—1632no—
137#456—1631no—
138#456D16311yescharacteristic, complemented
139#453—1632no—
140#443—1632no—
1412−1+4—1631no—
1422−1+4C2×D16311yescharacteristic
143H(5,𝔽2)—1632no—
144#137—3262no—
145#137—3262no—
146#137—1632no—
147#137C2×D16312yescomplemented
148AGL(1,ℤ/8)—3262no—
149AGL(1,ℤ/8)—3262no—
150AGL(1,ℤ/8)—1632no—
151AGL(1,ℤ/8)—1632no—
152#135—3262no—
153#135—3262no—
154#135—1632no—
155C2×SD16—3262no—
156C2×SD16—3262no—
157C2×SD16—1632no—
158C2×SD16—1632no—
159#131—1632no—
160#131—1632no—
161C4⋊Q8C2×D16311yescharacteristic, complemented
162#127—1631no—
163#108—1632no—
164#104—1632no—
165#104C2×D16312yescomplemented
166#101—1631no—
167#101C2×D16311yescharacteristic, complemented
168#100—1632no—
169C4×C8—1631no—
170#41—22no—
171#41—41no—
172#41—3262no—
173#41—3262no—
174#41—3262no—
175#41—6521no—
176#41—1632no—
177#41C22×D16312yes—
178C2×Q8—22no—
179C2×Q8—3262no—
180C2×Q8C22×D16311yescharacteristic
181C2×Q8C22×D16311yescharacteristic
182C2×D4—3262no—
183C2×D4—3262no—
184C2×D4—6521no—
185C2×D4—6522no—
186C2×D4—1632no—
187Q16—22no—
188Q16—3262no—
189Q16—1632no—
190SD16—22no—
191SD16—3262no—
192SD16—3262no—
193SD16—3262no—
194SD16—3262no—
195SD16—3262no—
196SD16—3262no—
197SD16—3262no—
198SD16—1632no—
199SD16—1632no—
200D8—3262no—
201D8—3262no—
202D8—1632no—
203M16—3262no—
204M16—3262no—
205M16—1632no—
206M16C22×D16312yes—
207C2×C8—3262no—
208C2×C8—3262no—
209C2×C8—1632no—
210C4⋊C4—41no—
211C42C22×D16311yescharacteristic
212C23—3262no—
213C23—6522no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—6521no—
218Q8—1632no—
219Q8D4×D16312yes—
220D4—22no—
221D4—41no—
222D4—13042no—
223D4—3262no—
224D4—3262no—
225D4—3262no—
226D4—6522no—
227D4—6521no—
228D4—6522no—
229D4—1632no—
230D4D4×D16312yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3262no—
237C2×C4—6522no—
238C2×C4—6522no—
239C2×C4C23×D16311yescharacteristic
240C8—22no—
241C8—3262no—
242C8—1632no—
243C8—1632no—
244C22—22no—
245C22—13042no—
246C22—26082no—
247C22—3262no—
248C22—6522no—
249C22#1910478011yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—3262no—
255C4C2×D4×D16312yes—
256C2—21no—
257C2—42no—
258C2—13042no—
259C2—6522no—
260C2#2378229711yescharacteristic
subgroupG/H
centerC2#23782297
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#456×C163C2
socleC2×C163D42
hypercenter#456D163
hypocenterC163#3291
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23782474C2semidirect
2. #23782473C2semidirect
3. #23782039C2semidirect
4. #23781931C2semidirect
5. #23781930C2semidirect
6. #23781847C2semidirect
7. #23781834C2semidirect
8. #23781779C2semidirect
9. #456×C163C2semidirect
11. #19104781C22semidirect
14. #19104766C22semidirect
23. C2×#14579686C22semidirect
24. C2×#14579686C22semidirect
26. #19104743C22semidirect
27. #19104742C22semidirect
29. #19104670C22semidirect
30. #19104663C22semidirect
31. #19104654C22semidirect
32. #19104651C22semidirect
33. C4×C163⋊C8C22semidirect
35. #137×C163C22semidirect
36. C4⋊Q8×C163C22semidirect
37. #104×C163C22semidirect
38. #101×C163C22semidirect
51. #14579689D4semidirect
53. #14579687D4semidirect
60. #14579686D4semidirect
63. #14579684D4semidirect
130. C163#3291semidirect
138. #456D163semidirect
147. #137C2×D163semidirect
161. C4⋊Q8C2×D163semidirect
165. #104C2×D163semidirect
167. #101C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824742
#237824732
#237820392
#237819312
#237819302
#237818472
#237818342
#237817792
#456×C1632
#3291163