Finite Group Databasebeta

← #35218185

Subgroups — #35218185

Order 20864.

1088 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1088 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×#19104721C212yesmaximal, complemented
3C2×#19104712C211yesmaximal, characteristic, complemented
4#23781581C212yesmaximal, complemented
5#23781580C212yesmaximal, complemented
6#23781565C212yesmaximal, complemented
7#23781521C211yesmaximal, characteristic, complemented
8#23781520C211yesmaximal, characteristic, complemented
9#23781520C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic163C211yesmaximal, characteristic, complemented
11#379×C163C211yesmaximal, characteristic, complemented
12C24×D163C2211yescharacteristic
13C22×Dic326C2211yescharacteristic, complemented
14C22×C22⋊D163—28no—
15C22×C22⋊D163—24no—
16C22×C22⋊D163C2212yes—
17C22×C22⋊D163C2212yes—
18C23×Dic163C2211yescharacteristic
19#19104721—28no—
20#19104712—24no—
21C2×C4⋊Dic163C2211yescharacteristic, complemented
22C2×Dic163⋊C4C2211yescharacteristic, complemented
23C2×Dic163⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic163C2211yescharacteristic
25C2×C22⋊Dic163C2211yescharacteristic
26C2×C22⋊Dic163C2212yescomplemented
27C2×C22⋊Dic163C2212yescomplemented
28C2×(C2×D163)⋊C4C2212yescomplemented
29C2×(C2×D163)⋊C4C2212yescomplemented
30C2×(C2×D163)⋊C4C2212yescomplemented
31C2×(C2×D163)⋊C4C2212yescomplemented
32C2×C4×Dic163C2212yescomplemented
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—22no—
43C23×D163—42no—
44C23×D163C2312yes—
45C2×Dic326—22no—
46C2×Dic326—24no—
47C2×C22⋊D163—232no—
48C2×C22⋊D163—24no—
49C2×C22⋊D163—24no—
50C2×C22⋊D163—24no—
51C2×C22⋊D163—24no—
52C2×C22⋊D163—24no—
53C2×C22⋊D163D418yes—
54C22×Dic163—24no—
55C22×Dic163D414yes—
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2312yes—
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C4⋊Dic163—22no—
62Dic163⋊C4—22no—
63Dic163⋊C4—22no—
64C22⋊Dic163—22no—
65C22⋊Dic163—24no—
66C22⋊Dic163—22no—
67C22⋊Dic163—42no—
68(C2×D163)⋊C4—24no—
69(C2×D163)⋊C4—24no—
70(C2×D163)⋊C4—42no—
71(C2×D163)⋊C4—42no—
72C4×Dic163—24no—
73C24×C163—24no—
74C24×C163—22no—
75C24×C163—22no—
76C24×C163—22no—
77C24×C163—42no—
78C24×C163C2312yes—
79C24×C163C2311yescharacteristic
80C22×C4×C163C2311yescharacteristic
81C22×C4×C163C2312yes—
82C22×C4×C163C2311yescharacteristic
83C22⋊C4×C163—22no—
84C22⋊C4×C163—24no—
85C22⋊C4×C163—22no—
86C22⋊C4×C163—42no—
87C22×D163—28no—
88C22×D163—22no—
89C22×D163—22no—
90C22×D163—22no—
91C22×D163—22no—
92C22×D163—22no—
93C22×D163—22no—
94C22×D163—48no—
95C22×D163—48no—
96C22×D163—44no—
97C22×D163—44no—
98C22×D163C2×D414yes—
99Dic326—44no—
100C22⋊D163—216no—
101C22⋊D163—216no—
102C2×Dic163—22no—
103C2×Dic163—28no—
104C2×Dic163—22no—
105C2×Dic163—22no—
106C2×Dic163—21no—
107C2×Dic163—21no—
108C2×Dic163—22no—
109C2×Dic163—22no—
110C2×Dic163—22no—
111C2×Dic163C2×D412yes—
112C2×Dic163C2×D414yes—
113C2×Dic163C2×D412yes—
114C23×C163—22no—
115C23×C163—22no—
116C23×C163—28no—
117C23×C163—22no—
118C23×C163—22no—
119C23×C163—22no—
120C23×C163—22no—
121C23×C163—22no—
122C23×C163—22no—
123C23×C163—48no—
124C23×C163—48no—
125C23×C163—44no—
126C23×C163—44no—
127C23×C163C2×D414yes—
128C23×C163C2×D412yes—
129C23×C163#4112yes—
130C23×C163#4112yes—
131C23×C163C2411yescharacteristic
132C2×C4×C163—22no—
133C2×C4×C163—22no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—22no—
137C2×C4×C163—22no—
138C2×C4×C163—22no—
139C22×C163—22no—
140C22×C163—22no—
141C22×C163—24no—
142C22×C163—24no—
143C22×C163—24no—
144C22×C163—24no—
145C22×C163—24no—
146C22×C163—416no—
147C22×C163—42no—
148C22×C163—42no—
149C22×C163—42no—
150C22×C163—42no—
151C22×C163#12414yescomplemented
152C22×C163C22×D411yescharacteristic
153C22×C163C22×D412yes—
154C22×C163C2×#4111yescharacteristic
155C22×C163C2×#4111yescharacteristic
156C22×C163H(5,𝔽2)12yes—
157C2×D163—24no—
158C2×D163—24no—
159C2×D163—24no—
160C2×D163—416no—
161C2×D163—42no—
162C2×D163—42no—
163C2×D163—42no—
164C2×D163—42no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163—42no—
168Dic163—22no—
169Dic163—41no—
170Dic163—41no—
171Dic163—42no—
172Dic163#12014yes—
173C2×C163—24no—
174C2×C163—44no—
175C2×C163C2×#12011yescharacteristic
176C2×C163C2×#12411yescharacteristic
177C2×C163#53511yescharacteristic
178C2×C163#54612yes—
179C2×C163#54612yes—
180D163—44no—
181C163#243511yescharacteristic, complemented, Sylow
182#2435—1631nomaximal, Sylow
183C23×D4—1631no—
184C2×#124—1632no—
185C2×C22⋊Q8—1631no—
186#394—1632no—
187#394—1632no—
188#390—1632no—
189#379—1631no—
190#379—1631no—
191#379—1631no—
192#379D16311yescharacteristic, complemented
193C4×C22⋊C4—1631no—
194C25—1631no—
195C25C2×D16311yescharacteristic
196C22×Q8—1631no—
197C22×D4—3268no—
198C22×D4—3264no—
199C22×D4—1632no—
200C22×D4—1632no—
201C23×C4—1631no—
202#124—3268no—
203C22⋊Q8—3264no—
204C2×C4⋊C4—1631no—
205C2×C4⋊C4—1631no—
206C2×C4⋊C4—1631no—
207C2×C22⋊C4—1631no—
208C2×C22⋊C4—1632no—
209C2×C22⋊C4—1631no—
210C2×C22⋊C4—1632no—
211C2×C22⋊C4—1632no—
212C2×C22⋊C4—1632no—
213C2×C22⋊C4—1632no—
214C2×C22⋊C4—1632no—
215C2×C22⋊C4C2×D16311yescharacteristic
216C2×C22⋊C4C2×D16311yescharacteristic
217C2×C22⋊C4C2×D16312yescomplemented
218C2×C22⋊C4C2×D16312yescomplemented
219C2×C42—1632no—
220#95—1632no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—3264no—
227C24—3262no—
228C24—3262no—
229C24—3262no—
230C24—6522no—
231C24—1632no—
232C24C22×D16312yes—
233C24C22×D16311yescharacteristic
234C2×Q8—3262no—
235C2×Q8—3264no—
236C2×D4—32632no—
237C2×D4—3264no—
238C2×D4—3264no—
239C2×D4—3264no—
240C2×D4—3264no—
241C2×D4—3264no—
242C2×D4—1638no—
243C22×C4—3264no—
244C22×C4—1632no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1634no—
248C22×C4—1631no—
249C22×C4—1631no—
250C22×C4C22×D16312yes—
251C22×C4C22×D16311yescharacteristic
252C22×C4C22×D16311yescharacteristic
253C4⋊C4—3262no—
254C4⋊C4—3262no—
255C4⋊C4—3262no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—3264no—
261C22⋊C4—3264no—
262C22⋊C4—3262no—
263C22⋊C4—3264no—
264C22⋊C4—3262no—
265C22⋊C4—6522no—
266C22⋊C4—6522no—
267C22⋊C4—6522no—
268C42—3264no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—3268no—
283C23—3262no—
284C23—3262no—
285C23—3262no—
286C23—3262no—
287C23—3262no—
288C23—3262no—
289C23—6528no—
290C23—6528no—
291C23—6524no—
292C23—6524no—
293C23—1634no—
294C23D4×D16314yes—
295C23D4×D16312yes—
296C23#1457969812yes—
297C23#1457969912yes—
298C23C23×D16311yescharacteristic
299Q8—6524no—
300D4—32616no—
301D4—32616no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—3262no—
310C2×C4—3268no—
311C2×C4—3262no—
312C2×C4—3262no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3262no—
316C2×C4—3262no—
317C2×C4—3262no—
318C2×C4—1632no—
319C2×C4—1634no—
320C2×C4—1632no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—416no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—3264no—
334C22—3264no—
335C22—3264no—
336C22—65216no—
337C22—6522no—
338C22—6522no—
339C22—6522no—
340C22—6522no—
341C22#1910472114yescomplemented
342C22C2×D4×D16311yescharacteristic
343C22C2×D4×D16312yes—
344C22C2×#1457969811yescharacteristic
345C22C2×#1457969911yescharacteristic
346C22#1910478012yes—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—3262no—
351C4—6521no—
352C4—6521no—
353C4—6522no—
354C4—1634no—
355C2—24no—
356C2—44no—
357C2—6524no—
358C2#120×D16311yescharacteristic
359C2C2×#1910472111yescharacteristic
360C2#2378223011yescharacteristic
361C2#2378230212yes—
362C2#2378230512yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#379×C163C2
prime radicalC23×C22⋊D163C2
hypercenter#379D163
hypocenterC163#2435
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#19104721C2semidirect
3. C2×#19104712C2semidirect
4. #23781581C2semidirect
5. #23781580C2semidirect
6. #23781565C2semidirect
7. #23781521C2semidirect
8. #23781520C2semidirect
9. #23781520C2semidirect
10. C22⋊C4×Dic163C2semidirect
11. #379×C163C2semidirect
13. C22×Dic326C22semidirect
21. C2×C4⋊Dic163C22semidirect
22. C2×Dic163⋊C4C22semidirect
23. C2×Dic163⋊C4C22semidirect
26. C2×C22⋊Dic163C22semidirect
27. C2×C22⋊Dic163C22semidirect
28. C2×(C2×D163)⋊C4C22semidirect
29. C2×(C2×D163)⋊C4C22semidirect
30. C2×(C2×D163)⋊C4C22semidirect
31. C2×(C2×D163)⋊C4C22semidirect
32. C2×C4×Dic163C22semidirect
33. C163⋊#95C22semidirect
37. C2×C22⋊C4×C163C22semidirect
38. C2×C22⋊C4×C163C22semidirect
151. C22×C163#124semidirect
181. C163#2435semidirect
192. #379D163semidirect
217. C2×C22⋊C4C2×D163semidirect
218. C2×C22⋊C4C2×D163semidirect
341. C22#19104721semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1632
C2×#191047212
C2×#191047122
#237815812
#237815802
#237815652
#237815212
#237815202
C22⋊C4×Dic1632
#379×C1632
#2435163