Finite Group Databasebeta

← #35218186

Subgroups — #35218186

Order 20864.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×D4×D163C211yesmaximal, characteristic
2C2×#19104723C211yesmaximal, characteristic, complemented
3C2×#19104722C211yesmaximal, characteristic, complemented
4C2×#19104715C211yesmaximal, characteristic, complemented
5#23781581C214yesmaximal, complemented
6#23781566C211yesmaximal, characteristic, complemented
7#23781520C214yesmaximal, complemented
8C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
9#390×C163C211yesmaximal, characteristic, complemented
10C24×D163C2212yes—
11C22×Dic326C2211yescharacteristic, complemented
12C22×D652C2211yescharacteristic
13C22×C22⋊D163C2212yes—
14C2×D4×D163—24no—
15C2×D4×D163—24no—
16C2×D4×D163—24no—
17C22×C4×D163C2211yescharacteristic
18#19104723—24no—
19#19104722—24no—
20#19104715—24no—
21C2×C4⋊Dic163C2212yescomplemented
22C2×C22⋊Dic163C2214yescomplemented
23C2×(C2×D163)⋊C4C2212yes—
24C2×(C2×D163)⋊C4C2212yes—
25C2×(C2×D163)⋊C4C2214yescomplemented
26C2×(C2×D163)⋊C4C2214yescomplemented
27C2×C4×Dic163C2211yescharacteristic, complemented
28C163⋊#95C2212yescomplemented
29C22×D4×C163C2211yescharacteristic
30C2×C4⋊C4×C163C2211yescharacteristic, complemented
31C2×C22⋊C4×C163C2214yescomplemented
32C2×C42×C163C2211yescharacteristic, complemented
33C23×D163—24no—
34C23×D163—24no—
35C23×D163—28no—
36C23×D163—24no—
37C23×D163—44no—
38C23×D163C2312yes—
39C23×D163C2311yescharacteristic
40C2×Dic326—22no—
41C2×Dic326—22no—
42C2×Dic326—22no—
43C2×D652—22no—
44C2×D652—22no—
45C2×D652D414yes—
46C2×C22⋊D163—24no—
47C2×C22⋊D163—24no—
48C2×C22⋊D163—24no—
49D4×D163—232no—
50C22×Dic163C2312yes—
51C22×Dic163C2312yes—
52C22×Dic163C2311yescharacteristic
53C2×C4×D163—22no—
54C2×C4×D163—22no—
55C2×C4×D163D414yes—
56C4⋊Dic163—24no—
57C22⋊Dic163—28no—
58(C2×D163)⋊C4—24no—
59(C2×D163)⋊C4—28no—
60(C2×D163)⋊C4—24no—
61(C2×D163)⋊C4—44no—
62C4×Dic163—22no—
63C24×C163C2312yes—
64C2×D4×C163—22no—
65C2×D4×C163—22no—
66C2×D4×C163D414yes—
67C22×C4×C163C2312yes—
68C22×C4×C163C2312yes—
69C22×C4×C163C2311yescharacteristic
70C4⋊C4×C163—22no—
71C22⋊C4×C163—44no—
72C42×C163—22no—
73C22×D163—21no—
74C22×D163—21no—
75C22×D163—22no—
76C22×D163—216no—
77C22×D163—22no—
78C22×D163—22no—
79C22×D163—24no—
80C22×D163—24no—
81C22×D163—416no—
82C22×D163—416no—
83C22×D163—48no—
84C22×D163—48no—
85C22×D163C2×D414yes—
86C22×D163C2×D412yes—
87C22×D163#4114yes—
88Dic326—44no—
89D652—28no—
90C22⋊D163—216no—
91C2×Dic163—21no—
92C2×Dic163—21no—
93C2×Dic163—22no—
94C2×Dic163—22no—
95C2×Dic163—22no—
96C2×Dic163—22no—
97C2×Dic163—22no—
98C2×Dic163—22no—
99C2×Dic163C2×D412yes—
100C4×D163—28no—
101C23×C163—22no—
102C23×C163—22no—
103C23×C163—24no—
104C23×C163—24no—
105C23×C163C2×D414yes—
106C23×C163C2411yescharacteristic
107D4×C163—28no—
108C2×C4×C163—22no—
109C2×C4×C163—22no—
110C2×C4×C163—22no—
111C2×C4×C163—22no—
112C2×C4×C163—24no—
113C2×C4×C163C2×D412yes—
114C2×C4×C163C2×D412yes—
115C2×C4×C163C2×D412yes—
116C22×C163—24no—
117C22×C163—28no—
118C22×C163—42no—
119C22×C163—42no—
120C22×C163—44no—
121C22×C163C22×D411yescharacteristic
122C22×C163C22×D411yescharacteristic
123C22×C163C22×D411yescharacteristic
124C22×C163C2×#4112yes—
125C22×C163H(5,𝔽2)12yes—
126C2×D163—22no—
127C2×D163—22no—
128C2×D163—24no—
129C2×D163—24no—
130C2×D163—24no—
131C2×D163—28no—
132C2×D163—432no—
133C2×D163—42no—
134C2×D163—42no—
135C2×D163—44no—
136C2×D163#12414yescomplemented
137C4×C163—42no—
138C4×C163—42no—
139C4×C163#12014yes—
140Dic163—22no—
141Dic163—42no—
142Dic163—42no—
143C2×C163—44no—
144C2×C163C2×#12011yescharacteristic
145C2×C163C2×#12411yescharacteristic
146C2×C163#53511yescharacteristic
147C2×C163#54612yes—
148C2×C163#54612yes—
149D163—24no—
150D163—44no—
151C163#243511yescharacteristic, complemented, Sylow
152#2435—1631nomaximal, Sylow
153C23×D4—1631no—
154C2×#124—1631no—
155C2×#124—1631no—
156C2×C22⋊Q8—1631no—
157#394—1634no—
158#390—1631no—
159#390D16311yescharacteristic, complemented
160#379—1634no—
161C4×C22⋊C4—1631no—
162C25—1632no—
163C22×Q8—1631no—
164C22×D4—3264no—
165C22×D4—3264no—
166C22×D4—3264no—
167C22×D4—1632no—
168C22×D4—1631no—
169C22×D4C2×D16311yescharacteristic
170C23×C4—1631no—
171#124—3264no—
172#124—3264no—
173C22⋊Q8—3264no—
174C2×C4⋊C4—1632no—
175C2×C4⋊C4C2×D16311yescharacteristic, complemented
176C2×C22⋊C4—1634no—
177C2×C22⋊C4—1632no—
178C2×C22⋊C4—1634no—
179C2×C22⋊C4—1632no—
180C2×C22⋊C4—1634no—
181C2×C22⋊C4C2×D16314yescomplemented
182C2×C42—1631no—
183C2×C42C2×D16311yescharacteristic, complemented
184#95—1632no—
185C24—3264no—
186C24—3264no—
187C24—3268no—
188C24—3264no—
189C24—6524no—
190C24—1632no—
191C24—1631no—
192C24C22×D16312yes—
193C2×Q8—3262no—
194C2×Q8—3262no—
195C2×Q8—3262no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—32632no—
199C2×D4—3264no—
200C2×D4—3264no—
201C2×D4—3264no—
202C2×D4—3262no—
203C2×D4—3262no—
204C2×D4—1634no—
205C2×D4D65214yes—
206C22×C4—3262no—
207C22×C4—3262no—
208C22×C4—1632no—
209C22×C4—1632no—
210C22×C4—1631no—
211C22×C4—1634no—
212C22×C4C22×D16312yes—
213C22×C4C22×D16312yes—
214C22×C4C22×D16311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—3264no—
217C22⋊C4—44no—
218C22⋊C4—3264no—
219C22⋊C4—3268no—
220C22⋊C4—3264no—
221C22⋊C4—3268no—
222C22⋊C4—6524no—
223C42—22no—
224C42—3262no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—32616no—
230C23—3261no—
231C23—3261no—
232C23—3262no—
233C23—3262no—
234C23—3262no—
235C23—3264no—
236C23—3264no—
237C23—65216no—
238C23—65216no—
239C23—6528no—
240C23—6528no—
241C23—1634no—
242C23—1632no—
243C23—1634no—
244C23C2×D65214yes—
245C23C23×D16311yescharacteristic
246Q8—6524no—
247D4—28no—
248D4—32616no—
249D4—3268no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—3261no—
256C2×C4—3261no—
257C2×C4—3268no—
258C2×C4—3262no—
259C2×C4—3262no—
260C2×C4—3262no—
261C2×C4—3262no—
262C2×C4—3262no—
263C2×C4—3262no—
264C2×C4—1632no—
265C2×C4D4×D16312yes—
266C2×C4D4×D16312yes—
267C2×C4C2×D65212yes—
268C22—24no—
269C22—28no—
270C22—42no—
271C22—42no—
272C22—44no—
273C22—3262no—
274C22—3262no—
275C22—3264no—
276C22—3264no—
277C22—3264no—
278C22—3268no—
279C22—65232no—
280C22—6522no—
281C22—6522no—
282C22—6524no—
283C22—1634no—
284C22C2×D4×D16311yescharacteristic
285C22C2×D4×D16311yescharacteristic
286C22C22×D65211yescharacteristic
287C22#41×D16312yes—
288C22#1910478012yes—
289C4—42no—
290C4—42no—
291C4—3262no—
292C4—6522no—
293C4—6522no—
294C4#1910470514yes—
295C2—44no—
296C2—3264no—
297C2—6524no—
298C2C2×#1910470511yescharacteristic
299C2#124×D16311yescharacteristic
300C2#2378222811yescharacteristic
301C2#2378231012yes—
302C2#2378231212yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#390×C163C2
prime radicalC22×D4×D163C2
hypercenter#390D163
hypocenterC163#2435
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#19104723C2semidirect
3. C2×#19104722C2semidirect
4. C2×#19104715C2semidirect
5. #23781581C2semidirect
6. #23781566C2semidirect
7. #23781520C2semidirect
8. C4×(C2×D163)⋊C4C2semidirect
9. #390×C163C2semidirect
11. C22×Dic326C22semidirect
21. C2×C4⋊Dic163C22semidirect
22. C2×C22⋊Dic163C22semidirect
25. C2×(C2×D163)⋊C4C22semidirect
26. C2×(C2×D163)⋊C4C22semidirect
27. C2×C4×Dic163C22semidirect
28. C163⋊#95C22semidirect
30. C2×C4⋊C4×C163C22semidirect
31. C2×C22⋊C4×C163C22semidirect
32. C2×C42×C163C22semidirect
136. C2×D163#124semidirect
151. C163#2435semidirect
159. #390D163semidirect
175. C2×C4⋊C4C2×D163semidirect
181. C2×C22⋊C4C2×D163semidirect
183. C2×C42C2×D163semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1632
C2×#191047232
C2×#191047222
C2×#191047152
#237815812
#237815662
#237815202
C4×(C2×D163)⋊C42
#390×C1632
#2435163