Finite Group Databasebeta

← #508×D163 #35227650

Subgroups — #508×D163 #35227650

Order 20864.

368 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D163C212yesmaximal, complemented
2#23782098C211yesmaximal, characteristic, complemented
3#23782097C211yesmaximal, characteristic, complemented
4#23782096C212yesmaximal, complemented
5#23782095C212yesmaximal, complemented
6#23782094C211yesmaximal, characteristic, complemented
7Q32×D163C211yesmaximal, characteristic, complemented
8SD32×D163C212yesmaximal, complemented
9D16×D163C211yesmaximal, characteristic, complemented
10C2×C16×D163C211yesmaximal, characteristic, complemented
11#508×C163C211yesmaximal, characteristic, complemented
12#41×D163—22no—
13#19104758C2212yescomplemented
14#19104757C2212yescomplemented
15#19104756—22no—
16#19104755—22no—
17#19104754C2212yescomplemented
18Q16×D163C2212yes—
19SD16×D163—22no—
20D8×D163C2212yes—
21C2×C8×D163C2211yescharacteristic
22Dic1304C2211yescharacteristic, complemented
23#19104681C2212yescomplemented
24#19104680C2212yescomplemented
25#19104679C2212yescomplemented
26#19104678C2212yescomplemented
27D2608C2211yescharacteristic, complemented
28#19104676C2212yescomplemented
29C2×C163⋊C16C2211yescharacteristic, complemented
30C16×D163C2211yescharacteristic
31C16×D163C2211yescharacteristic
32C16×D163C2211yescharacteristic, complemented
33C16×D163C2211yescharacteristic, complemented
34#135×C163C2212yescomplemented
35Q32×C163C2211yescharacteristic, complemented
36SD32×C163C2212yescomplemented
37D16×C163C2211yescharacteristic, complemented
38C2×C16×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579699—22no—
41#14579699—42no—
42#14579698—22no—
43#14579698—42no—
44Q8×D163—22no—
45D4×D163—22no—
46D4×D163—42no—
47C2×C4×D163—42no—
48C2×C4×D163D411yescharacteristic
49Dic652C2312yes—
50#14579689—22no—
51#14579688—22no—
52#14579687—22no—
53#14579686—22no—
54D1304C2312yes—
55#14579684—22no—
56C2×C163⋊C8D411yescharacteristic
57C8×D163D411yescharacteristic
58C8×D163D411yescharacteristic
59C8×D163C2311yescharacteristic
60C8×D163C2311yescharacteristic
61C163⋊C16C2311yescharacteristic
62C163⋊C16C2311yescharacteristic
63#41×C163—22no—
64Q16×C163C2312yes—
65SD16×C163—22no—
66D8×C163C2312yes—
67C2×C8×C163C2311yescharacteristic
68C16×C163C2311yescharacteristic
69C16×C163C2311yescharacteristic
70C22×D163—42no—
71C22×D163D811yescharacteristic
72Dic326—22no—
73Dic326—42no—
74D652—22no—
75D652—42no—
76C22⋊D163—42no—
77C22⋊D163—42no—
78C22⋊D163—42no—
79C2×Dic163—42no—
80C2×Dic163D811yescharacteristic
81C4×D163—42no—
82C4×D163—42no—
83C4×D163—42no—
84C4×D163D811yescharacteristic
85C4×D163D811yescharacteristic
86C4×D163C2×D411yescharacteristic
87C4×D163C2×D411yescharacteristic
88C163⋊C8C2×D411yescharacteristic
89C163⋊C8C2×D411yescharacteristic
90Q8×C163—22no—
91D4×C163—22no—
92D4×C163—42no—
93C2×C4×C163—42no—
94C2×C4×C163C2×D411yescharacteristic
95C8×C163C2×D411yescharacteristic
96C8×C163C2411yescharacteristic
97C22×C163—42no—
98C22×C163C2×D811yescharacteristic
99C2×D163—22no—
100C2×D163—42no—
101C2×D163—84no—
102C2×D163C2×D811yescharacteristic
103C2×D163C2×D811yescharacteristic
104C4×C163—42no—
105C4×C163C2×D811yescharacteristic
106C4×C163C22×D411yescharacteristic
107Dic163—42no—
108Dic163C2×D811yescharacteristic
109Dic163C2×D811yescharacteristic
110C2×C163—21no—
111C2×C163—82no—
112C2×C163C22×D811yescharacteristic
113D163—21no—
114D163—82no—
115D163#50812yesdirect product factor
116C163C2×#50811yescharacteristic, complemented, Sylow
117C2×#508—1631nomaximal, Sylow
118C2×#135—1632no—
119#508—1632no—
120#508—1631no—
121#508—1632no—
122#508—1631no—
123#508—1631no—
124#508D16311yescharacteristic, direct product factor
125C2×Q32—1631no—
126C2×SD32—1632no—
127C2×D16—1631no—
128C22×C16—1631no—
129C2×#41—3262no—
130#135—3262no—
131#135—3262no—
132#135—1632no—
133#135—1632no—
134#135—1632no—
135#135C2×D16312yescomplemented
136C2×Q16—1632no—
137C2×SD16—3262no—
138C2×D8—1632no—
139C22×C8—1631no—
140Q32—1632no—
141Q32—1631no—
142Q32C2×D16311yescharacteristic, complemented
143SD32—1632no—
144SD32—1632no—
145SD32—1632no—
146SD32C2×D16312yescomplemented
147D16—1632no—
148D16—1631no—
149D16C2×D16311yescharacteristic, complemented
150C2×C16—1631no—
151C2×C16—1631no—
152C2×C16—1631no—
153C2×C16—1631no—
154C2×C16—1631no—
155C2×C16C2×D16311yescharacteristic, complemented
156#41—22no—
157#41—3262no—
158#41—3262no—
159#41—3262no—
160#41—6522no—
161#41—6522no—
162C2×Q8—3262no—
163C2×D4—3262no—
164C2×D4—6522no—
165C22×C4—6522no—
166C22×C4—1631no—
167Q16—3262no—
168Q16—1632no—
169Q16C22×D16312yes—
170SD16—22no—
171SD16—3262no—
172SD16—3262no—
173SD16—3262no—
174D8—3262no—
175D8—1632no—
176D8C22×D16312yes—
177C2×C8—1631no—
178C2×C8—1631no—
179C2×C8—1631no—
180C2×C8—1631no—
181C2×C8—1631no—
182C2×C8C22×D16311yescharacteristic
183C16—1631no—
184C16—1631no—
185C16C22×D16311yescharacteristic
186C16C22×D16311yescharacteristic
187C23—6522no—
188C23—1631no—
189Q8—22no—
190Q8—3262no—
191Q8—6522no—
192D4—22no—
193D4—42no—
194D4—3262no—
195D4—6522no—
196D4—6522no—
197D4—6522no—
198D4—6522no—
199C2×C4—42no—
200C2×C4—6522no—
201C2×C4—6522no—
202C2×C4—6522no—
203C2×C4—6522no—
204C2×C4—1631no—
205C2×C4—1631no—
206C2×C4—1631no—
207C2×C4—1631no—
208C2×C4—1631no—
209C2×C4D4×D16311yescharacteristic
210C8—1631no—
211C8—1631no—
212C8D4×D16311yescharacteristic
213C8C23×D16311yescharacteristic
214C22—42no—
215C22—13044no—
216C22—3262no—
217C22—6522no—
218C22—1631no—
219C22—1631no—
220C22D8×D16311yescharacteristic
221C4—42no—
222C4—6522no—
223C4—1631no—
224C4—1631no—
225C4D8×D16311yescharacteristic
226C4C2×D4×D16311yescharacteristic
227C2—21no—
228C2—82no—
229C2—13042no—
230C2—3261no—
231C2—1632no—
232C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC4D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#508×C163C2
socleC2×C163C22×D8
hypercenter#508D163
hypocenterC163C2×#508
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D163C2semidirect
2. #23782098C2semidirect
3. #23782097C2semidirect
4. #23782096C2semidirect
5. #23782095C2semidirect
6. #23782094C2semidirect
7. Q32×D163C2semidirect
8. SD32×D163C2semidirect
9. D16×D163C2semidirect
10. C2×C16×D163C2semidirect
11. #508×C163C2semidirect
13. #19104758C22semidirect
14. #19104757C22semidirect
17. #19104754C22semidirect
22. Dic1304C22semidirect
23. #19104681C22semidirect
24. #19104680C22semidirect
25. #19104679C22semidirect
26. #19104678C22semidirect
27. D2608C22semidirect
28. #19104676C22semidirect
29. C2×C163⋊C16C22semidirect
32. C16×D163C22semidirect
33. C16×D163C22semidirect
34. #135×C163C22semidirect
35. Q32×C163C22semidirect
36. SD32×C163C22semidirect
37. D16×C163C22semidirect
38. C2×C16×C163C22semidirect
115. D163#508direct (N × H)
116. C163C2×#508semidirect
124. #508D163direct (N × H)
135. #135C2×D163semidirect
142. Q32C2×D163semidirect
146. SD32C2×D163semidirect
149. D16C2×D163semidirect
155. C2×C16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#135×D1632
#237820982
#237820972
#237820962
#237820952
#237820942
Q32×D1632
SD32×D1632
D16×D1632
C2×C16×D1632
#508×C1632
C2×#508163