Finite Group Databasebeta

← #35227682

Subgroups — #35227682

Order 20864.

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

normal subgroups only

This group has 360 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#23782479C211yesmaximal, characteristic, complemented
2#23782466C211yesmaximal, characteristic, complemented
3#23782108C211yesmaximal, characteristic, complemented
4#23782104C212yesmaximal, complemented
5#23782099C212yesmaximal, complemented
6#23782095C211yesmaximal, characteristic, complemented
7#23782094C212yesmaximal, complemented
8C2×#19104679C211yesmaximal, characteristic, complemented
9SD32×D163C212yesmaximal, complemented
10#23782083C211yesmaximal, characteristic, complemented
11#509×C163C211yesmaximal, characteristic, complemented
12#19104782—21no—
13#19104780—21no—
14#19104765—21no—
15#19104764—22no—
16#19104761—21no—
17#19104759—22no—
18#19104757C2211yescharacteristic
19#19104754C2212yes—
20#19104754C2212yescomplemented
21C2×Dic652C2211yescharacteristic, complemented
22Q16×D163C2212yescomplemented
23D8×D163C2212yes—
24#19104741C2211yescharacteristic
25Dic1304C2212yescomplemented
26#19104681C2211yescharacteristic, complemented
27#19104680C2212yescomplemented
28#19104679C2212yescomplemented
29#19104679C2212yescomplemented
30#19104679C2211yescharacteristic, complemented
31#19104678C2211yescharacteristic, complemented
32#19104676C2211yescharacteristic, complemented
33#19104674C2212yescomplemented
34C2×C163⋊C16C2211yescharacteristic, complemented
35C16×D163C2212yescomplemented
36#135×C163C2211yescharacteristic, complemented
37C2×D8×C163C2211yescharacteristic
38SD32×C163C2212yescomplemented
39D16×C163C2212yescomplemented
40M32×C163C2211yescharacteristic, complemented
41#14579699—21no—
42#14579699—41no—
43#14579699D411yescharacteristic
44#14579698—22no—
45#14579698—22no—
46#14579698—41no—
47#14579698—42no—
48C2×Dic326—21no—
49C2×Dic326—41no—
50Q8×D163—22no—
51C2×C22⋊D163—42no—
52D4×D163—22no—
53D4×D163—41no—
54Dic652—21no—
55Dic652C2312yes—
56Dic652C2311yescharacteristic
57#14579689—22no—
58#14579688—21no—
59#14579687—22no—
60#14579687—22no—
61D1304C2311yescharacteristic
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—42no—
78Dic326—21no—
79Dic326—22no—
80Dic326—41no—
81Dic326—42no—
82Dic326D811yescharacteristic
83D652—21no—
84D652D811yescharacteristic
85C22⋊D163—41no—
86C22⋊D163—42no—
87C22⋊D163—42no—
88C22⋊D163—42no—
89C22⋊D163D812yes—
90C2×Dic163—41no—
91C2×Dic163—42no—
92C2×Dic163—42no—
93C4×D163—42no—
94C4×D163—41no—
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—162no—
113C2×D163C2×D812yes—
114C4×C163—41no—
115C4×C163C2×D811yescharacteristic
116C4×C163C22×D411yescharacteristic
117Dic163—41no—
118Dic163—42no—
119Dic163C2×D812yes—
120C2×C163—21no—
121C2×C163—82no—
122C2×C163—81no—
123C2×C163C22×D811yescharacteristic
124D163—22no—
125D163—81no—
126C163#341611yescharacteristic, complemented, Sylow
127#3416—1631nomaximal, Sylow
128#578—1631no—
129#576—1631no—
130#510—1632no—
131#510—1631no—
132#509—1632no—
133#509D16311yescharacteristic, complemented
134#508—1631no—
135#508—1632no—
136C2×SD32—1631no—
137C2×SD32—1632no—
138#504—1631no—
1392−1+4—3261no—
140H(5,𝔽2)—3261no—
141#137—3262no—
142#137—3261no—
143AGL(1,ℤ/8)—3262no—
144AGL(1,ℤ/8)—3261no—
145#135—1632no—
146#135—1632no—
147#135—1631no—
148#135C2×D16311yescharacteristic, complemented
149C2×Q16—1632no—
150C2×Q16—1631no—
151C2×D8—1632no—
152C2×D8C2×D16311yescharacteristic
153#131—1631no—
154Q32—1631no—
155Q32—1632no—
156SD32—1631no—
157SD32—1632no—
158SD32—1632no—
159SD32—1631no—
160SD32—1632no—
161SD32C2×D16312yescomplemented
162D16—1631no—
163D16C2×D16312yescomplemented
164M32—1632no—
165M32C2×D16311yescharacteristic, complemented
166C2×C16—1631no—
167C2×C16—1632no—
168#41—21no—
169#41—3262no—
170#41—3262no—
171#41—3261no—
172#41—6521no—
173#41—6521no—
174#41—6522no—
175#41—1631no—
176C2×Q8—3262no—
177C2×Q8—3261no—
178C2×Q8—6521no—
179C2×D4—21no—
180C2×D4—3262no—
181C2×D4—6521no—
182C2×D4—6522no—
183Q16—3262no—
184Q16—3261no—
185Q16—1632no—
186Q16—1631no—
187Q16C22×D16311yescharacteristic
188SD16—21no—
189SD16—3262no—
190SD16—3262no—
191SD16—3261no—
192D8—21no—
193D8—3262no—
194D8—1631no—
195D8C22×D16311yescharacteristic
196D8C22×D16312yes—
197M16—1631no—
198M16—1632no—
199C2×C8—1632no—
200C2×C8C22×D16311yescharacteristic
201C16—1631no—
202C16—1631no—
203C16C22×D16312yes—
204C23—41no—
205C23—6522no—
206Q8—21no—
207Q8—3261no—
208Q8—3262no—
209Q8—6522no—
210Q8—6521no—
211Q8—1631no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3261no—
217D4—6522no—
218D4—6522no—
219D4—6521no—
220D4—6522no—
221D4—1632no—
222D4—1631no—
223C2×C4—41no—
224C2×C4—6522no—
225C2×C4—6521no—
226C2×C4—6522no—
227C2×C4—6522no—
228C2×C4—6521no—
229C2×C4—1632no—
230C2×C4D4×D16311yescharacteristic
231C8—1632no—
232C8D4×D16311yescharacteristic
233C8C23×D16311yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—26082no—
238C22—6521no—
239C22—1632no—
240C22D8×D16311yescharacteristic
241C4—41no—
242C4—6521no—
243C4—6522no—
244C4—1632no—
245C4D8×D16311yescharacteristic
246C4C2×D4×D16311yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—13041no—
251C2—3262no—
252C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#509×C163C2
socleC2×C163C22×D8
hypercenter#509D163
hypocenterC163#3416
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. #23782479C2semidirect
2. #23782466C2semidirect
3. #23782108C2semidirect
4. #23782104C2semidirect
5. #23782099C2semidirect
6. #23782095C2semidirect
7. #23782094C2semidirect
8. C2×#19104679C2semidirect
9. SD32×D163C2semidirect
10. #23782083C2semidirect
11. #509×C163C2semidirect
20. #19104754C22semidirect
21. C2×Dic652C22semidirect
22. Q16×D163C22semidirect
25. Dic1304C22semidirect
26. #19104681C22semidirect
27. #19104680C22semidirect
28. #19104679C22semidirect
29. #19104679C22semidirect
30. #19104679C22semidirect
31. #19104678C22semidirect
32. #19104676C22semidirect
33. #19104674C22semidirect
34. C2×C163⋊C16C22semidirect
35. C16×D163C22semidirect
36. #135×C163C22semidirect
38. SD32×C163C22semidirect
39. D16×C163C22semidirect
40. M32×C163C22semidirect
126. C163#3416semidirect
133. #509D163semidirect
148. #135C2×D163semidirect
161. SD32C2×D163semidirect
163. D16C2×D163semidirect
165. M32C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824792
#237824662
#237821082
#237821042
#237820992
#237820952
#237820942
C2×#191046792
SD32×D1632
#237820832
#509×C1632
#3416163