Finite Group Databasebeta

← #35227676

Subgroups — #35227676

Order 20864.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#23782467C211yesmaximal, characteristic, complemented
2#23782466C211yesmaximal, characteristic, complemented
3#23782103C211yesmaximal, characteristic, complemented
4#23782102C212yesmaximal, complemented
5#23782099C212yesmaximal, complemented
6#23782096C212yesmaximal, complemented
7#23782095C211yesmaximal, characteristic, complemented
8C2×#19104676C211yesmaximal, characteristic, complemented
9D16×D163C212yesmaximal, complemented
10#23782083C211yesmaximal, characteristic, complemented
11#509×C163C211yesmaximal, characteristic, complemented
12#19104781—21no—
13#19104780—21no—
14#19104762—22no—
15#19104761—21no—
16#19104760—21no—
17#19104759—22no—
18#19104758C2212yescomplemented
19#19104757C2211yescharacteristic, complemented
20#19104754C2212yescomplemented
21C2×D1304C2211yescharacteristic, complemented
22D8×D163C2212yescomplemented
23D8×D163C2212yescomplemented
24#19104741C2211yescharacteristic
25#19104681C2211yescharacteristic, complemented
26#19104680C2212yescomplemented
27#19104679C2211yescharacteristic, complemented
28#19104678C2211yescharacteristic, complemented
29D2608C2212yescomplemented
30#19104676C2211yescharacteristic, complemented
31#19104676C2212yescomplemented
32#19104676C2212yescomplemented
33#19104674C2212yescomplemented
34C2×C163⋊C16C2211yescharacteristic, complemented
35C16×D163C2212yescomplemented
36#135×C163C2211yescharacteristic, complemented
37C2×D8×C163C2211yescharacteristic, complemented
38SD32×C163C2212yescomplemented
39D16×C163C2212yescomplemented
40M32×C163C2211yescharacteristic, complemented
41#14579700—22no—
42#14579699—21no—
43#14579699—41no—
44#14579699D411yescharacteristic
45#14579698—22no—
46#14579698—41no—
47C2×D652—21no—
48C2×D652—41no—
49C2×C22⋊D163—42no—
50D4×D163—22no—
51D4×D163—22no—
52D4×D163—41no—
53D4×D163—42no—
54Dic652C2311yescharacteristic
55#14579688—21no—
56#14579687—22no—
57#14579686—22no—
58D1304—21no—
59D1304C2312yes—
60D1304C2311yescharacteristic
61#14579684—22no—
62#14579684—22no—
63#14579683D411yescharacteristic
64#14579682D412yes—
65C8×D163C2312yes—
66C163⋊C16C2311yescharacteristic
67C163⋊C16C2311yescharacteristic
68#41×C163—21no—
69C2×D4×C163—21no—
70Q16×C163C2311yescharacteristic
71SD16×C163—21no—
72D8×C163—21no—
73D8×C163C2311yescharacteristic
74D8×C163C2312yes—
75C2×C8×C163C2311yescharacteristic
76C16×C163C2312yes—
77C22×D163—41no—
78C22×D163—42no—
79C22×D163—42no—
80Dic326—21no—
81Dic326D811yescharacteristic
82D652—21no—
83D652—22no—
84D652—42no—
85D652—41no—
86D652D811yescharacteristic
87C22⋊D163—41no—
88C22⋊D163—42no—
89C22⋊D163—42no—
90C22⋊D163—42no—
91C22⋊D163D812yes—
92C2×Dic163—42no—
93C4×D163—41no—
94C4×D163—42no—
95C4×D163C2×D412yes—
96C163⋊C8C2×D412yes—
97C23×C163—41no—
98Q8×C163—21no—
99D4×C163—22no—
100D4×C163—21no—
101D4×C163—41no—
102D4×C163—41no—
103C2×C4×C163—41no—
104C2×C4×C163C2×D411yescharacteristic
105C8×C163C2×D411yescharacteristic
106C8×C163C2411yescharacteristic
107C22×C163—41no—
108C22×C163—42no—
109C22×C163—161no—
110C22×C163C2×D811yescharacteristic
111C2×D163—41no—
112C2×D163—42no—
113C2×D163—162no—
114C2×D163—162no—
115C2×D163—161no—
116C2×D163C2×D812yes—
117C4×C163—41no—
118C4×C163C2×D811yescharacteristic
119C4×C163C22×D411yescharacteristic
120Dic163—41no—
121Dic163C2×D812yes—
122C2×C163—21no—
123C2×C163—81no—
124C2×C163—82no—
125C2×C163C22×D811yescharacteristic
126D163—22no—
127D163—81no—
128D163—82no—
129C163#341511yescharacteristic, complemented, Sylow
130#3415—1631nomaximal, Sylow
131#576—1631no—
132#576—1631no—
133#509—1632no—
134#509—1632no—
135#509—1631no—
136#509D16311yescharacteristic, complemented
137#508—1631no—
138#508—1632no—
139C2×D16—1631no—
140C2×D16—1632no—
141#504—1631no—
142H(5,𝔽2)—3261no—
143H(5,𝔽2)—3261no—
144AGL(1,ℤ/8)—3262no—
145AGL(1,ℤ/8)—3262no—
146AGL(1,ℤ/8)—3261no—
147AGL(1,ℤ/8)—3261no—
148#135—1632no—
149#135—1632no—
150#135—1631no—
151#135C2×D16311yescharacteristic, complemented
152C2×D8—1632no—
153C2×D8—1632no—
154C2×D8—1631no—
155C2×D8C2×D16311yescharacteristic, complemented
156#131—1631no—
157Q32—1631no—
158SD32—1631no—
159SD32—1631no—
160SD32—1632no—
161SD32C2×D16312yescomplemented
162D16—1632no—
163D16—1631no—
164D16—1632no—
165D16—1632no—
166D16C2×D16312yescomplemented
167M32—1632no—
168M32C2×D16311yescharacteristic, complemented
169C2×C16—1631no—
170C2×C16—1632no—
171#41—21no—
172#41—3262no—
173#41—3262no—
174#41—3261no—
175#41—6521no—
176#41—6521no—
177#41—1631no—
178C2×D4—21no—
179C2×D4—3262no—
180C2×D4—3262no—
181C2×D4—3261no—
182C2×D4—6521no—
183C2×D4—6522no—
184C2×D4—6521no—
185C2×D4—6522no—
186Q16—1631no—
187Q16C22×D16311yescharacteristic
188SD16—21no—
189SD16—3262no—
190SD16—3262no—
191SD16—3261no—
192D8—21no—
193D8—3262no—
194D8—3262no—
195D8—3261no—
196D8—1632no—
197D8—1631no—
198D8C22×D16312yes—
199D8C22×D16311yescharacteristic
200M16—1631no—
201M16—1632no—
202C2×C8—1632no—
203C2×C8C22×D16311yescharacteristic
204C16—1631no—
205C16—1631no—
206C16C22×D16312yes—
207C23—41no—
208C23—6522no—
209C23—6522no—
210C23—6521no—
211Q8—21no—
212Q8—3261no—
213Q8—1631no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—3261no—
219D4—3262no—
220D4—6522no—
221D4—6522no—
222D4—6521no—
223D4—6522no—
224D4—6522no—
225D4—6521no—
226D4—1632no—
227D4—1631no—
228C2×C4—41no—
229C2×C4—6522no—
230C2×C4—6522no—
231C2×C4—6521no—
232C2×C4—1632no—
233C2×C4D4×D16311yescharacteristic
234C8—1632no—
235C8D4×D16311yescharacteristic
236C8C23×D16311yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—26081no—
241C22—26082no—
242C22—26082no—
243C22—6522no—
244C22—6521no—
245C22—1632no—
246C22D8×D16311yescharacteristic
247C4—41no—
248C4—6521no—
249C4—1632no—
250C4D8×D16311yescharacteristic
251C4C2×D4×D16311yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—13042no—
256C2—13041no—
257C2—3262no—
258C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#509×C163C2
socleC2×C163C22×D8
hypercenter#509D163
hypocenterC163#3415
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #23782467C2semidirect
2. #23782466C2semidirect
3. #23782103C2semidirect
4. #23782102C2semidirect
5. #23782099C2semidirect
6. #23782096C2semidirect
7. #23782095C2semidirect
8. C2×#19104676C2semidirect
9. D16×D163C2semidirect
10. #23782083C2semidirect
11. #509×C163C2semidirect
18. #19104758C22semidirect
19. #19104757C22semidirect
20. #19104754C22semidirect
21. C2×D1304C22semidirect
22. D8×D163C22semidirect
23. D8×D163C22semidirect
25. #19104681C22semidirect
26. #19104680C22semidirect
27. #19104679C22semidirect
28. #19104678C22semidirect
29. D2608C22semidirect
30. #19104676C22semidirect
31. #19104676C22semidirect
32. #19104676C22semidirect
33. #19104674C22semidirect
34. C2×C163⋊C16C22semidirect
35. C16×D163C22semidirect
36. #135×C163C22semidirect
37. C2×D8×C163C22semidirect
38. SD32×C163C22semidirect
39. D16×C163C22semidirect
40. M32×C163C22semidirect
129. C163#3415semidirect
136. #509D163semidirect
151. #135C2×D163semidirect
155. C2×D8C2×D163semidirect
161. SD32C2×D163semidirect
166. D16C2×D163semidirect
168. M32C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824672
#237824662
#237821032
#237821022
#237820992
#237820962
#237820952
C2×#191046762
D16×D1632
#237820832
#509×C1632
#3415163