Finite Group Databasebeta

← #35224458

Subgroups — #35224458

Order 20864.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#23782175C211yesmaximal, characteristic
2C2×C4⋊C4×D163C211yesmaximal, characteristic, complemented
3#23782001C211yesmaximal, characteristic, complemented
4#23782000C211yesmaximal, characteristic, complemented
5#23781997C212yesmaximal, complemented
6#23781981C211yesmaximal, characteristic, complemented
7#23781977C212yesmaximal, complemented
8#23781704C212yesmaximal, complemented
9#23781663C212yesmaximal, complemented
10#23781627C211yesmaximal, characteristic, complemented
11#481×C163C211yesmaximal, characteristic, complemented
12C2×#14579698C2211yescharacteristic
13C22×C4×D163—21no—
14C22×C4×D163C2211yescharacteristic
15C2×#14579682C2212yes—
16#19104717—21no—
17#19104715C2211yescharacteristic
18#19104714C2211yescharacteristic
19#19104706C2211yescharacteristic
20#19104703—21no—
21D4×Dic163—21no—
22(C22⋊D163)⋊C4—21no—
23#19104693C2211yescharacteristic
24C2×C4⋊Dic163C2211yescharacteristic
25C2×Dic163⋊C4—21no—
26C4⋊C4×D163—21no—
27C4⋊C4×D163—21no—
28C4⋊C4×D163C2212yes—
29C4⋊C4×D163C2212yes—
30#19104669C2212yescomplemented
31#19104668C2212yescomplemented
32#19104666C2212yescomplemented
33#19104660C2212yescomplemented
34#19104658C2212yescomplemented
35#19104656C2212yescomplemented
36C22⋊(C163⋊C8)C2211yescharacteristic
37(C2×D163)⋊C8C2212yes—
38#121×C163C2211yescharacteristic
39C2×C4⋊C4×C163C2211yescharacteristic
40#106×C163C2212yescomplemented
41#102×C163C2212yescomplemented
42C22⋊C8×C163C2211yescharacteristic
43C23×D163D411yescharacteristic
44#14579698—24no—
45C2×Dic326C2311yescharacteristic
46C2×C22⋊D163—21no—
47C2×C22⋊D163—21no—
48C22×Dic163—21no—
49C22×Dic163—21no—
50C22×Dic163D411yescharacteristic
51C2×C4×D163—24no—
52C2×C4×D163—22no—
53C2×C4×D163—22no—
54C2×C4×D163—44no—
55C2×C4×D163D411yescharacteristic
56C2×C4×D163D411yescharacteristic
57C2×C4×D163C2311yescharacteristic
58C2×C4×D163C2311yescharacteristic
59#14579682—24no—
60C2×C163⋊C8C2312yes—
61C4⋊Dic163—21no—
62C4⋊Dic163—21no—
63C4⋊Dic163C2312yes—
64C4⋊Dic163C2311yescharacteristic
65Dic163⋊C4—22no—
66Dic163⋊C4—22no—
67Dic163⋊C4—21no—
68C22⋊Dic163—21no—
69C22⋊Dic163—21no—
70(C2×D163)⋊C4—21no—
71(C2×D163)⋊C4—21no—
72C4×Dic163—21no—
73C2×D4×C163—21no—
74C2×D4×C163C2311yescharacteristic
75C22×C4×C163—21no—
76C22×C4×C163C2311yescharacteristic
77C2×C8×C163C2312yes—
78C4⋊C4×C163—21no—
79C4⋊C4×C163C2312yes—
80C4⋊C4×C163C2311yescharacteristic
81C22⋊C4×C163—21no—
82C22×D163—22no—
83C22×D163—22no—
84C22×D163—22no—
85C22×D163C2×D411yescharacteristic
86C22×D163C2×D411yescharacteristic
87Dic326—22no—
88C22⋊D163—42no—
89C22⋊D163—42no—
90C2×Dic163—22no—
91C2×Dic163—21no—
92C2×Dic163—21no—
93C2×Dic163—22no—
94C2×Dic163—42no—
95C2×Dic163—42no—
96C2×Dic163C2×D411yescharacteristic
97C2×Dic163C2×D411yescharacteristic
98C4×D163—22no—
99C4×D163—22no—
100C4×D163—22no—
101C4×D163—44no—
102C4×D163—44no—
103C4×D163#4114yes—
104C163⋊C8—22no—
105C23×C163—21no—
106C23×C163C2×D411yescharacteristic
107D4×C163—22no—
108D4×C163—42no—
109C2×C4×C163—22no—
110C2×C4×C163—21no—
111C2×C4×C163—22no—
112C2×C4×C163—42no—
113C2×C4×C163C2×D411yescharacteristic
114C2×C4×C163C2411yescharacteristic
115C8×C163—22no—
116C22×C163—21no—
117C22×C163—21no—
118C22×C163—41no—
119C22×C163—41no—
120C22×C163—41no—
121C22×C163#13712yes—
122C22×C163C22×D411yescharacteristic
123C2×D163—28no—
124C2×D163—21no—
125C2×D163—21no—
126C2×D163—21no—
127C2×D163—21no—
128C2×D163#12312yes—
129C2×D163AGL(1,ℤ/8)12yes—
130C4×C163—21no—
131C4×C163—42no—
132C4×C163—41no—
133C4×C163C2×#4112yes—
134Dic163—21no—
135Dic163—41no—
136Dic163—42no—
137Dic163—41no—
138Dic163#12312yes—
139C2×C163—22no—
140C2×C163—81no—
141C2×C163C2×#12311yescharacteristic
142C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C163C2×#13711yescharacteristic
144D163—22no—
145D163—22no—
146C163#309211yescharacteristic, complemented, Sylow
147#3092—1631nomaximal, Sylow
148#525—1631no—
149C22×C4⋊C4—1631no—
150#484—1631no—
151#484—1631no—
152#483—1632no—
153#481—1631no—
154#481D16311yescharacteristic, complemented
155#480—1632no—
156#428—1632no—
157#417—1632no—
158#407—1631no—
159C2×#41—1631no—
160C23×C4—3261no—
161C23×C4—1631no—
162C2×M16—1632no—
163#123—3261no—
164C22⋊Q8—1631no—
165C22⋊Q8—1631no—
166#121—1631no—
167#121C2×D16311yescharacteristic
168#120—3261no—
169C4×D4—3261no—
170C4×D4—3261no—
171#117—1631no—
172C2×C4⋊C4—3261no—
173C2×C4⋊C4—3261no—
174C2×C4⋊C4—3261no—
175C2×C4⋊C4—1631no—
176C2×C4⋊C4—1632no—
177C2×C4⋊C4—1632no—
178C2×C4⋊C4C2×D16311yescharacteristic
179#107—1632no—
180#107—1632no—
181#106—1632no—
182#106C2×D16312yescomplemented
183#103—1632no—
184#103—1632no—
185#102—1632no—
186#102C2×D16312yescomplemented
187C22⋊C8—1631no—
188C22⋊C8—1632no—
189C22⋊C8C2×D16311yescharacteristic
190C24—1631no—
191#41—3264no—
192C2×Q8—1631no—
193C2×D4—21no—
194C2×D4—3261no—
195C2×D4—3261no—
196C2×D4C22×D16311yescharacteristic
197C22×C4—21no—
198C22×C4—3264no—
199C22×C4—3262no—
200C22×C4—3262no—
201C22×C4—3261no—
202C22×C4—3261no—
203C22×C4—6524no—
204C22×C4—1631no—
205C22×C4—1631no—
206C22×C4—1631no—
207C22×C4—1631no—
208C22×C4—1631no—
209C22×C4C22×D16311yescharacteristic
210M16—3264no—
211C2×C8—1632no—
212C2×C8C22×D16312yes—
213C4⋊C4—21no—
214C4⋊C4—3262no—
215C4⋊C4—3262no—
216C4⋊C4—3261no—
217C4⋊C4—3261no—
218C4⋊C4—3261no—
219C4⋊C4—1632no—
220C4⋊C4—1631no—
221C4⋊C4C22×D16312yes—
222C4⋊C4C22×D16311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3261no—
225C22⋊C4—3261no—
226C22⋊C4—3261no—
227C22⋊C4—3261no—
228C42—3261no—
229C23—21no—
230C23—3262no—
231C23—3262no—
232C23—3262no—
233C23—1631no—
234C23—1631no—
235C23D4×D16311yescharacteristic
236Q8—3262no—
237D4—22no—
238D4—42no—
239D4—6522no—
240D4—6522no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—3262no—
246C2×C4—3262no—
247C2×C4—3262no—
248C2×C4—3262no—
249C2×C4—3261no—
250C2×C4—3261no—
251C2×C4—3262no—
252C2×C4—6524no—
253C2×C4—6524no—
254C2×C4—6522no—
255C2×C4—6522no—
256C2×C4—1631no—
257C2×C4—1631no—
258C2×C4—1634no—
259C2×C4D4×D16311yescharacteristic
260C2×C4C23×D16311yescharacteristic
261C8—22no—
262C8—3262no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—3268no—
269C22—3261no—
270C22—3261no—
271C22—3261no—
272C22—3261no—
273C22—1632no—
274C22—1632no—
275C22#1910476412yes—
276C22C2×D4×D16311yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—3261no—
281C4—6521no—
282C4—6522no—
283C4—6521no—
284C4—1632no—
285C4#41×D16312yes—
286C2—22no—
287C2—81no—
288C2—3262no—
289C2—3262no—
290C2#123×D16311yescharacteristic
291C2AGL(1,ℤ/8)×D16311yescharacteristic
292C2C2×#1910476411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#481×C163C2
socleC22×C163C22×D4
prime radical#23782175C2
hypercenter#481D163
hypocenterC163#3092
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D163C2semidirect
3. #23782001C2semidirect
4. #23782000C2semidirect
5. #23781997C2semidirect
6. #23781981C2semidirect
7. #23781977C2semidirect
8. #23781704C2semidirect
9. #23781663C2semidirect
10. #23781627C2semidirect
11. #481×C163C2semidirect
30. #19104669C22semidirect
31. #19104668C22semidirect
32. #19104666C22semidirect
33. #19104660C22semidirect
34. #19104658C22semidirect
35. #19104656C22semidirect
40. #106×C163C22semidirect
41. #102×C163C22semidirect
146. C163#3092semidirect
154. #481D163semidirect
182. #106C2×D163semidirect
186. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237821752
C2×C4⋊C4×D1632
#237820012
#237820002
#237819972
#237819812
#237819772
#237817042
#237816632
#237816272
#481×C1632
#3092163