Finite Group Databasebeta

← #35227679

Subgroups — #35227679

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#23782467C211yesmaximal, characteristic, complemented
3#23782107C212yesmaximal, complemented
4#23782105C211yesmaximal, characteristic, complemented
5#23782100C211yesmaximal, characteristic, complemented
6#23782099C212yesmaximal, complemented
7#23782096C212yesmaximal, complemented
8C2×#19104680C211yesmaximal, characteristic, complemented
9SD32×D163C212yesmaximal, complemented
10#23782081C211yesmaximal, characteristic, complemented
11#508×C163C211yesmaximal, characteristic, complemented
12#19104782—21no—
13#19104781—21no—
14#19104765—21no—
15#19104764—22no—
16#19104762—22no—
17#19104760—21no—
18#19104758C2212yes—
19#19104754C2212yescomplemented
20C2×Dic652C2211yescharacteristic, complemented
21Q16×D163C2212yescomplemented
22C2×D1304C2211yescharacteristic
23D8×D163C2212yes—
24#19104741C2211yescharacteristic
25#19104681C2212yescomplemented
26#19104680C2212yescomplemented
27#19104680C2212yescomplemented
28#19104679C2212yescomplemented
29#19104678C2212yescomplemented
30#19104676C2212yescomplemented
31#19104675C2211yescharacteristic, complemented
32#19104674C2212yescomplemented
33C16×D163C2212yescomplemented
34#135×C163C2211yescharacteristic
35#135×C163C2211yescharacteristic, complemented
36Q32×C163C2211yescharacteristic, complemented
37SD32×C163C2211yescharacteristic, complemented
38SD32×C163C2211yescharacteristic, complemented
39D16×C163C2211yescharacteristic, complemented
40C2×C16×C163C2211yescharacteristic, complemented
41#14579700—22no—
42#14579699—41no—
43#14579699—41no—
44#14579699D411yescharacteristic
45#14579698—22no—
46#14579698—42no—
47C2×Dic326—21no—
48C2×Dic326—41no—
49Q8×D163—22no—
50C2×D652—21no—
51C2×D652—41no—
52D4×D163—22no—
53D4×D163—42no—
54Dic652—21no—
55Dic652C2312yes—
56#14579689—22no—
57#14579687—22no—
58#14579686—22no—
59D1304—21no—
60D1304C2312yes—
61#14579684—22no—
62#14579683D411yescharacteristic
63#14579682D412yes—
64C8×D163C2312yes—
65C163⋊C16C2312yes—
66#41×C163—21no—
67#41×C163—21no—
68Q16×C163C2311yescharacteristic
69Q16×C163C2311yescharacteristic
70SD16×C163—21no—
71SD16×C163—21no—
72D8×C163C2311yescharacteristic
73D8×C163C2311yescharacteristic
74C2×C8×C163C2311yescharacteristic
75C16×C163C2311yescharacteristic
76C16×C163C2311yescharacteristic
77C22×D163—41no—
78C22×D163—42no—
79Dic326—22no—
80Dic326—41no—
81Dic326—42no—
82Dic326D811yescharacteristic
83D652—22no—
84D652—42no—
85D652—41no—
86D652D811yescharacteristic
87C22⋊D163—42no—
88C22⋊D163—42no—
89C22⋊D163D812yes—
90C2×Dic163—41no—
91C2×Dic163—42no—
92C4×D163—42no—
93C4×D163—42no—
94C4×D163C2×D412yes—
95C163⋊C8C2×D412yes—
96Q8×C163—21no—
97Q8×C163—21no—
98D4×C163—21no—
99D4×C163—21no—
100D4×C163—41no—
101D4×C163—41no—
102C2×C4×C163—41no—
103C2×C4×C163—41no—
104C2×C4×C163C2×D411yescharacteristic
105C8×C163C2×D411yescharacteristic
106C8×C163C2411yescharacteristic
107C22×C163—41no—
108C22×C163—41no—
109C22×C163C2×D811yescharacteristic
110C2×D163—42no—
111C2×D163—162no—
112C2×D163—161no—
113C2×D163C2×D812yes—
114C4×C163—41no—
115C4×C163—41no—
116C4×C163C2×D811yescharacteristic
117C4×C163C22×D411yescharacteristic
118Dic163—42no—
119Dic163C2×D812yes—
120C2×C163—21no—
121C2×C163—81no—
122C2×C163—81no—
123C2×C163C22×D811yescharacteristic
124D163—22no—
125D163—82no—
126C163#341611yescharacteristic, complemented, Sylow
127#3416—1631nomaximal, Sylow
128#578—1631no—
129#576—1631no—
130#510—1631no—
131#510—1632no—
132#509—1631no—
133#509—1632no—
134#508—1632no—
135#508D16311yescharacteristic, complemented
136C2×SD32—1632no—
137C2×SD32—1631no—
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#135C2×D16311yescharacteristic
148#135C2×D16311yescharacteristic, complemented
149C2×Q16—1632no—
150C2×Q16—1631no—
151C2×D8—1632no—
152C2×D8—1631no—
153#131—1631no—
154Q32—1632no—
155Q32C2×D16311yescharacteristic, complemented
156SD32—1632no—
157SD32—1632no—
158SD32—1632no—
159SD32—1632no—
160SD32C2×D16311yescharacteristic, complemented
161SD32C2×D16311yescharacteristic, complemented
162D16—1632no—
163D16C2×D16311yescharacteristic, complemented
164M32—1631no—
165M32—1632no—
166C2×C16—1632no—
167C2×C16C2×D16311yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—3262no—
171#41—3262no—
172#41—6521no—
173#41—6521no—
174#41—6522no—
175#41—1631no—
176C2×Q8—3262no—
177C2×Q8—3261no—
178C2×Q8—6521no—
179C2×D4—3262no—
180C2×D4—3261no—
181C2×D4—6521no—
182C2×D4—6522no—
183Q16—3262no—
184Q16—3261no—
185Q16—1632no—
186Q16C22×D16311yescharacteristic
187Q16C22×D16311yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—3262no—
191SD16—3262no—
192D8—3262no—
193D8—3261no—
194D8—1632no—
195D8C22×D16311yescharacteristic
196D8C22×D16311yescharacteristic
197M16—1631no—
198M16—1632no—
199C2×C8—1632no—
200C2×C8C22×D16311yescharacteristic
201C16—1632no—
202C16C22×D16311yescharacteristic
203C16C22×D16311yescharacteristic
204C23—6522no—
205C23—6521no—
206Q8—21no—
207Q8—21no—
208Q8—3262no—
209Q8—6522no—
210Q8—6521no—
211Q8—1631no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3262no—
217D4—6522no—
218D4—6522no—
219D4—6522no—
220D4—6521no—
221D4—1632no—
222D4—1631no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—6522no—
226C2×C4—6522no—
227C2×C4—6521no—
228C2×C4—6522no—
229C2×C4—1632no—
230C2×C4D4×D16311yescharacteristic
231C8—1632no—
232C8D4×D16311yescharacteristic
233C8C23×D16311yescharacteristic
234C22—41no—
235C22—41no—
236C22—26081no—
237C22—26082no—
238C22—6522no—
239C22—1632no—
240C22D8×D16311yescharacteristic
241C4—41no—
242C4—41no—
243C4—6522no—
244C4—1632no—
245C4D8×D16311yescharacteristic
246C4C2×D4×D16311yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—13042no—
251C2—3262no—
252C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#508×C163C2
socleC2×C163C22×D8
hypercenter#508D163
hypocenterC163#3416
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #23782479C2semidirect
2. #23782467C2semidirect
3. #23782107C2semidirect
4. #23782105C2semidirect
5. #23782100C2semidirect
6. #23782099C2semidirect
7. #23782096C2semidirect
8. C2×#19104680C2semidirect
9. SD32×D163C2semidirect
10. #23782081C2semidirect
11. #508×C163C2semidirect
19. #19104754C22semidirect
20. C2×Dic652C22semidirect
21. Q16×D163C22semidirect
25. #19104681C22semidirect
26. #19104680C22semidirect
27. #19104680C22semidirect
28. #19104679C22semidirect
29. #19104678C22semidirect
30. #19104676C22semidirect
31. #19104675C22semidirect
32. #19104674C22semidirect
33. C16×D163C22semidirect
35. #135×C163C22semidirect
36. Q32×C163C22semidirect
37. SD32×C163C22semidirect
38. SD32×C163C22semidirect
39. D16×C163C22semidirect
40. C2×C16×C163C22semidirect
126. C163#3416semidirect
135. #508D163semidirect
148. #135C2×D163semidirect
155. Q32C2×D163semidirect
160. SD32C2×D163semidirect
161. SD32C2×D163semidirect
163. D16C2×D163semidirect
167. C2×C16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824792
#237824672
#237821072
#237821052
#237821002
#237820992
#237820962
C2×#191046802
SD32×D1632
#237820812
#508×C1632
#3416163