Finite Group Databasebeta

← #35217861

Subgroups — #35217861

Order 20864.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D163C211yesmaximal, characteristic
2C2×#19104717C212yesmaximal, complemented
3C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
4#23781581C212yesmaximal, complemented
5#23781580C212yesmaximal, complemented
6#23781541C212yesmaximal, complemented
7#23781540C211yesmaximal, characteristic, complemented
8#23781537C211yesmaximal, characteristic, complemented
9#23781521C211yesmaximal, characteristic, complemented
10#23781520C211yesmaximal, characteristic, complemented
11#379×C163C211yesmaximal, characteristic, complemented
12C24×D163C2211yescharacteristic
13C22×C22⋊D163—24no—
14C22×C22⋊D163—28no—
15C22×C22⋊D163C2212yes—
16C22×C22⋊D163C2212yes—
17C23×Dic163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic, complemented
19C22×C4×D163C2211yescharacteristic, complemented
20#19104717—28no—
21C2×C4⋊Dic163C2212yescomplemented
22C2×Dic163⋊C4C2212yescomplemented
23C2×C22⋊Dic163C2211yescharacteristic
24C2×C22⋊Dic163C2211yescharacteristic
25C2×C22⋊Dic163C2212yescomplemented
26C2×C22⋊Dic163C2212yescomplemented
27C2×(C2×D163)⋊C4C2212yescomplemented
28C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D163)⋊C4C2212yescomplemented
31C22⋊C4×D163—24no—
32C163⋊#95C2212yescomplemented
33C163⋊#95C2212yescomplemented
34C25×C163C2211yescharacteristic
35C2×C22⋊C4×C163C2211yescharacteristic
36C2×C22⋊C4×C163C2211yescharacteristic
37C2×C22⋊C4×C163C2212yescomplemented
38C2×C22⋊C4×C163C2212yescomplemented
39C23×D163—24no—
40C23×D163—22no—
41C23×D163—22no—
42C23×D163—42no—
43C23×D163D414yes—
44C23×D163C2311yescharacteristic
45C23×D163C2311yescharacteristic
46C2×C22⋊D163—232no—
47C2×C22⋊D163—24no—
48C2×C22⋊D163—24no—
49C2×C22⋊D163—24no—
50C2×C22⋊D163—24no—
51C2×C22⋊D163—24no—
52C2×C22⋊D163D418yes—
53C22×Dic163—24no—
54C22×Dic163—22no—
55C22×Dic163C2311yescharacteristic
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2312yes—
58C22×Dic163C2312yes—
59C2×C4×D163—22no—
60C2×C4×D163—22no—
61C2×C4×D163—24no—
62C2×C4×D163—42no—
63C4⋊Dic163—24no—
64Dic163⋊C4—24no—
65C22⋊Dic163—22no—
66C22⋊Dic163—24no—
67C22⋊Dic163—22no—
68C22⋊Dic163—42no—
69(C2×D163)⋊C4—22no—
70(C2×D163)⋊C4—22no—
71(C2×D163)⋊C4—24no—
72(C2×D163)⋊C4—42no—
73C24×C163—24no—
74C24×C163—22no—
75C24×C163—22no—
76C24×C163—22no—
77C24×C163—42no—
78C24×C163C2312yes—
79C24×C163C2311yescharacteristic
80C22×C4×C163C2311yescharacteristic
81C22×C4×C163C2311yescharacteristic
82C22×C4×C163C2312yes—
83C22⋊C4×C163—22no—
84C22⋊C4×C163—24no—
85C22⋊C4×C163—22no—
86C22⋊C4×C163—42no—
87C22×D163—21no—
88C22×D163—21no—
89C22×D163—22no—
90C22×D163—28no—
91C22×D163—22no—
92C22×D163—21no—
93C22×D163—21no—
94C22×D163—22no—
95C22×D163—28no—
96C22×D163—44no—
97C22×D163—48no—
98C22×D163—48no—
99C22×D163C2×D412yes—
100C22×D163C2×D414yes—
101C22×D163C2×D412yes—
102C22⋊D163—216no—
103C22⋊D163—216no—
104C2×Dic163—28no—
105C2×Dic163—22no—
106C2×Dic163—22no—
107C2×Dic163—22no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—22no—
111C2×Dic163—22no—
112C2×Dic163—22no—
113C2×Dic163—22no—
114C2×Dic163C2×D414yes—
115C4×D163—44no—
116C4×D163—44no—
117C23×C163—22no—
118C23×C163—22no—
119C23×C163—28no—
120C23×C163—22no—
121C23×C163—22no—
122C23×C163—22no—
123C23×C163—22no—
124C23×C163—22no—
125C23×C163—22no—
126C23×C163—48no—
127C23×C163—48no—
128C23×C163—44no—
129C23×C163—44no—
130C23×C163C2×D414yes—
131C23×C163C2×D412yes—
132C23×C163#4112yes—
133C23×C163#4112yes—
134C23×C163C2411yescharacteristic
135C2×C4×C163—22no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—22no—
139C2×C4×C163—22no—
140C2×C4×C163—22no—
141C2×C4×C163—22no—
142C22×C163—22no—
143C22×C163—22no—
144C22×C163—24no—
145C22×C163—24no—
146C22×C163—24no—
147C22×C163—24no—
148C22×C163—24no—
149C22×C163—416no—
150C22×C163—42no—
151C22×C163—42no—
152C22×C163—42no—
153C22×C163—42no—
154C22×C163#12314yescomplemented
155C22×C163C22×D411yescharacteristic
156C22×C163C22×D412yes—
157C22×C163C2×#4111yescharacteristic
158C22×C163C2×#4111yescharacteristic
159C22×C163H(5,𝔽2)12yes—
160C2×D163—22no—
161C2×D163—22no—
162C2×D163—24no—
163C2×D163—24no—
164C2×D163—22no—
165C2×D163—22no—
166C2×D163—22no—
167C2×D163—416no—
168C2×D163—42no—
169C2×D163—42no—
170C2×D163#12014yes—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163—42no—
174Dic163—24no—
175Dic163—41no—
176Dic163—41no—
177Dic163—42no—
178C2×C163—24no—
179C2×C163—44no—
180C2×C163C2×#12011yescharacteristic
181C2×C163C2×#12311yescharacteristic
182C2×C163#53411yescharacteristic
183C2×C163#54612yes—
184C2×C163#54612yes—
185D163—24no—
186D163—42no—
187C163#240511yescharacteristic, complemented, Sylow
188#2405—1631nomaximal, Sylow
189C23×D4—1631no—
190C2×#123—1632no—
191C22×C22⋊C4—1631no—
192#394—1632no—
193#394—1632no—
194#386—1632no—
195#385—1631no—
196#385—1631no—
197#379—1631no—
198#379—1631no—
199#379D16311yescharacteristic, complemented
200C25—1631no—
201C25C2×D16311yescharacteristic
202C22×D4—3264no—
203C22×D4—3268no—
204C22×D4—1632no—
205C22×D4—1632no—
206C23×C4—1631no—
207C23×C4—1631no—
208C23×C4—1631no—
209#123—3268no—
210C2×C4⋊C4—1632no—
211C2×C4⋊C4—1632no—
212C2×C22⋊C4—3264no—
213C2×C22⋊C4—1631no—
214C2×C22⋊C4—1632no—
215C2×C22⋊C4—1631no—
216C2×C22⋊C4—1631no—
217C2×C22⋊C4—1631no—
218C2×C22⋊C4—1632no—
219C2×C22⋊C4—1632no—
220C2×C22⋊C4—1632no—
221C2×C22⋊C4C2×D16311yescharacteristic
222C2×C22⋊C4C2×D16311yescharacteristic
223C2×C22⋊C4C2×D16312yescomplemented
224C2×C22⋊C4C2×D16312yescomplemented
225#95—1632no—
226#95—1632no—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—3264no—
233C24—3262no—
234C24—3262no—
235C24—6522no—
236C24—1631no—
237C24—1634no—
238C24—1631no—
239C24C22×D16312yes—
240C24C22×D16311yescharacteristic
241C2×D4—32632no—
242C2×D4—3264no—
243C2×D4—3264no—
244C2×D4—3264no—
245C2×D4—3264no—
246C2×D4—3264no—
247C2×D4—1638no—
248C22×C4—3262no—
249C22×C4—3262no—
250C22×C4—3264no—
251C22×C4—3262no—
252C22×C4—3264no—
253C22×C4—6522no—
254C22×C4—1631no—
255C22×C4—1632no—
256C22×C4—1631no—
257C22×C4—1632no—
258C22×C4C22×D16311yescharacteristic
259C22×C4C22×D16312yes—
260C22×C4C22×D16311yescharacteristic
261C4⋊C4—3264no—
262C4⋊C4—3264no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—3262no—
268C22⋊C4—3262no—
269C22⋊C4—3264no—
270C22⋊C4—3262no—
271C22⋊C4—3264no—
272C22⋊C4—3262no—
273C22⋊C4—6522no—
274C22⋊C4—6522no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—3268no—
289C23—3261no—
290C23—3261no—
291C23—3262no—
292C23—3262no—
293C23—3261no—
294C23—3261no—
295C23—3262no—
296C23—3268no—
297C23—6524no—
298C23—6528no—
299C23—6528no—
300C23—1632no—
301C23—1634no—
302C23—1632no—
303C23D4×D16314yes—
304C23D4×D16312yes—
305C23#1457969812yes—
306C23#1457969912yes—
307C23C23×D16311yescharacteristic
308D4—32616no—
309D4—32616no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—3268no—
318C2×C4—3262no—
319C2×C4—3262no—
320C2×C4—3262no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3262no—
324C2×C4—3262no—
325C2×C4—3262no—
326C2×C4—3262no—
327C2×C4—6524no—
328C2×C4—6524no—
329C2×C4—1634no—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—3262no—
343C22—3262no—
344C22—3264no—
345C22—3264no—
346C22—3262no—
347C22—3262no—
348C22—3262no—
349C22—65216no—
350C22—6522no—
351C22—6522no—
352C22—1634no—
353C22#1910471714yescomplemented
354C22C2×D4×D16311yescharacteristic
355C22C2×D4×D16312yes—
356C22C2×#1457969811yescharacteristic
357C22C2×#1457969911yescharacteristic
358C22#1910478012yes—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—3264no—
363C4—6521no—
364C4—6521no—
365C4—6522no—
366C2—24no—
367C2—44no—
368C2—3264no—
369C2—6522no—
370C2#120×D16311yescharacteristic
371C2C2×#1910471711yescharacteristic
372C2#2378222511yescharacteristic
373C2#2378230212yes—
374C2#2378230512yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#379×C163C2
prime radicalC23×C22⋊D163C2
hypercenter#379D163
hypocenterC163#2405
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#19104717C2semidirect
3. C2×C22⋊C4×D163C2semidirect
4. #23781581C2semidirect
5. #23781580C2semidirect
6. #23781541C2semidirect
7. #23781540C2semidirect
8. #23781537C2semidirect
9. #23781521C2semidirect
10. #23781520C2semidirect
11. #379×C163C2semidirect
18. C22×C4×D163C22semidirect
19. C22×C4×D163C22semidirect
21. C2×C4⋊Dic163C22semidirect
22. C2×Dic163⋊C4C22semidirect
25. C2×C22⋊Dic163C22semidirect
26. C2×C22⋊Dic163C22semidirect
27. C2×(C2×D163)⋊C4C22semidirect
28. C2×(C2×D163)⋊C4C22semidirect
29. C2×(C2×D163)⋊C4C22semidirect
30. C2×(C2×D163)⋊C4C22semidirect
32. C163⋊#95C22semidirect
33. C163⋊#95C22semidirect
37. C2×C22⋊C4×C163C22semidirect
38. C2×C22⋊C4×C163C22semidirect
154. C22×C163#123semidirect
187. C163#2405semidirect
199. #379D163semidirect
223. C2×C22⋊C4C2×D163semidirect
224. C2×C22⋊C4C2×D163semidirect
353. C22#19104717semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1632
C2×#191047172
C2×C22⋊C4×D1632
#237815812
#237815802
#237815412
#237815402
#237815372
#237815212
#237815202
#379×C1632
#2405163