Finite Group Databasebeta

← #35223052

Subgroups — #35223052

Order 20864.

772 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D163C211yesmaximal, characteristic
2C2×M16×D163C211yesmaximal, characteristic, complemented
3C2×#19104693C211yesmaximal, characteristic, complemented
4#23781672C212yesmaximal, complemented
5#23781668C212yesmaximal, complemented
6#23781667C211yesmaximal, characteristic, complemented
7#23781666C211yesmaximal, characteristic, complemented
8#23781664C211yesmaximal, characteristic, complemented
9#23781660C212yesmaximal, complemented
10#23781658C212yesmaximal, complemented
11#417×C163C211yesmaximal, characteristic, complemented
12C2×#14579700C2211yescharacteristic
13C2×#14579699—21no—
14C2×#14579699C2211yescharacteristic
15C2×#14579698—21no—
16C2×#14579698C2211yescharacteristic
17#41×D163—24no—
18#41×D163C418yescomplemented
19C2×Q8×D163C2211yescharacteristic
20C2×D4×D163—21no—
21C2×D4×D163C2211yescharacteristic
22C22×C4×D163—21no—
23C22×C4×D163C2211yescharacteristic
24C2×#14579683C2211yescharacteristic, complemented
25C2×#14579682C2212yes—
26M16×D163—24no—
27C2×C8×D163C2212yescomplemented
28#19104693—21no—
29#19104693—21no—
30#19104693C2212yes—
31#19104693C2212yescomplemented
32C2×C4⋊Dic163C2211yescharacteristic, complemented
33C2×(C2×D163)⋊C4—21no—
34C2×C4×Dic163—21no—
35#19104660C2212yescomplemented
36#19104659C2212yescomplemented
37#19104658C2212yescomplemented
38#19104657C2212yescomplemented
39#19104656C2212yescomplemented
40#19104655C2212yescomplemented
41C2×#41×C163C2211yescharacteristic
42C2×M16×C163C2211yescharacteristic, complemented
43C2×C4⋊C4×C163C2211yescharacteristic, complemented
44#103×C163C2212yescomplemented
45#102×C163C2212yescomplemented
46C23×D163—21no—
47C23×D163D411yescharacteristic
48#14579700—22no—
49#14579700C2×C414yescomplemented
50#14579699—28no—
51#14579699—22no—
52#14579699C2×C414yes—
53#14579698—28no—
54#14579698—22no—
55#14579698C2×C414yescomplemented
56C2×Dic326—21no—
57C2×Dic326C2311yescharacteristic
58Q8×D163—22no—
59Q8×D163C2×C414yescomplemented
60C2×D652—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65D4×D163—28no—
66D4×D163—22no—
67D4×D163C2×C414yescomplemented
68C22×Dic163—21no—
69C22×Dic163—21no—
70C22×Dic163D411yescharacteristic
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—28no—
74C2×C4×D163—21no—
75C2×C4×D163C2×C414yes—
76C2×C4×D163D411yescharacteristic
77C2×C4×D163D411yescharacteristic
78C2×C4×D163D414yes—
79C2×C4×D163C2311yescharacteristic
80C2×C4×D163C2311yescharacteristic
81#14579683—22no—
82#14579682—24no—
83C2×C163⋊C8C2312yes—
84C8×D163—24no—
85C4⋊Dic163—21no—
86C4⋊Dic163C2312yes—
87(C2×D163)⋊C4—22no—
88(C2×D163)⋊C4—22no—
89C4×Dic163—22no—
90C4×Dic163—22no—
91#41×C163—22no—
92#41×C163C2×C414yes—
93C2×Q8×C163C2311yescharacteristic
94C2×D4×C163—21no—
95C2×D4×C163C2311yescharacteristic
96C22×C4×C163—21no—
97C22×C4×C163—21no—
98C22×C4×C163C2311yescharacteristic
99M16×C163—22no—
100C2×C8×C163C2312yes—
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2312yes—
103C22×D163—21no—
104C22×D163—24no—
105C22×D163—41no—
106C22×D163—41no—
107C22×D163—42no—
108C22×D163—42no—
109C22×D163C22⋊C414yes—
110C22×D163C2×D411yescharacteristic
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—24no—
114Dic326C22×C412yes—
115D652—24no—
116D652—21no—
117D652C22×C412yes—
118C22⋊D163—24no—
119C22⋊D163—24no—
120C22⋊D163—24no—
121C2×Dic163—21no—
122C2×Dic163—24no—
123C2×Dic163—22no—
124C2×Dic163—42no—
125C2×Dic163C22⋊C414yes—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C4×D163—24no—
129C4×D163—24no—
130C4×D163—24no—
131C4×D163C22⋊C414yes—
132C4×D163C22⋊C414yes—
133C4×D163C22×C412yes—
134C4×D163C22×C412yes—
135C4×D163C2×D412yes—
136C4×D163C2×D412yes—
137C163⋊C8—22no—
138C23×C163—21no—
139C23×C163C2×D411yescharacteristic
140Q8×C163—21no—
141Q8×C163C22×C412yes—
142D4×C163—24no—
143D4×C163—21no—
144D4×C163C22×C412yes—
145C2×C4×C163—21no—
146C2×C4×C163—24no—
147C2×C4×C163—22no—
148C2×C4×C163—42no—
149C2×C4×C163C22×C412yes—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D412yes—
152C2×C4×C163C2411yescharacteristic
153C8×C163—22no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—22no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163C2×C22⋊C412yes—
160C22×C163C22×D411yescharacteristic
161C2×D163—21no—
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—21no—
165C2×D163—22no—
166C2×D163—44no—
167C2×D163—48no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163C2×C22⋊C412yes—
171C2×D163C2×C22⋊C412yes—
172C4×C163—22no—
173C4×C163—42no—
174C4×C163C2×C22⋊C412yes—
175C4×C163C23×C411yescharacteristic
176C4×C163C22×D411yescharacteristic
177Dic163—22no—
178Dic163—42no—
179Dic163C2×C22⋊C412yes—
180Dic163C2×C22⋊C412yes—
181C2×C163—22no—
182C2×C163—42no—
183C2×C163C22×C22⋊C411yescharacteristic
184C2×C163#57511yescharacteristic
185C2×C163#57511yescharacteristic
186D163—22no—
187D163—22no—
188D163—42no—
189C163#289611yescharacteristic, complemented, Sylow
190#2896—1631nomaximal, Sylow
191C22×#41—1631no—
192C22×M16—1631no—
193C2×#117—1631no—
194#419—1632no—
195#418—1632no—
196#417—1631no—
197#417—1631no—
198#417—1631no—
199#417D16311yescharacteristic, complemented
200#416—1632no—
201#416—1632no—
202C2×#41—3261no—
203C2×#41—3261no—
204C2×#41—3264no—
205C2×#41—1638no—
206C2×#41—1631no—
207C2×#41—1631no—
208C2×#41—1631no—
209C2×#41C2×D16311yescharacteristic
210C22×Q8—1631no—
211C22×D4—3261no—
212C22×D4—1631no—
213C23×C4—3261no—
214C23×C4—1631no—
215C2×M16—3264no—
216C2×M16—1631no—
217C2×M16—1632no—
218C2×M16C2×D16311yescharacteristic, complemented
219C22×C8—1632no—
220#117—3261no—
221#117—3261no—
222#117—1632no—
223#117—1632no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4C2×D16311yescharacteristic, complemented
226C2×C22⋊C4—3261no—
227C2×C42—3261no—
228#103—1632no—
229#103—1632no—
230#103—1632no—
231#103C2×D16312yescomplemented
232#102—1632no—
233#102—1632no—
234#102—1632no—
235#102C2×D16312yescomplemented
236C24—3261no—
237C24—1631no—
238#41—22no—
239#41—3268no—
240#41—3268no—
241#41—3262no—
242#41—3262no—
243#41—3262no—
244#41—1634no—
245#41—1634no—
246#41—1634no—
247#41C4×D16314yes—
248C2×Q8—3261no—
249C2×Q8—3262no—
250C2×Q8—1634no—
251C2×Q8—1631no—
252C2×Q8C22×D16311yescharacteristic
253C2×D4—21no—
254C2×D4—3261no—
255C2×D4—3268no—
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—3262no—
259C2×D4—3261no—
260C2×D4—1634no—
261C2×D4—1631no—
262C2×D4C22×D16311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3261no—
266C22×C4—3261no—
267C22×C4—3268no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4—3261no—
271C22×C4—1631no—
272C22×C4—1634no—
273C22×C4—1631no—
274C22×C4—1631no—
275C22×C4—1634no—
276C22×C4—1631no—
277C22×C4—1631no—
278C22×C4C22×D16311yescharacteristic
279M16—22no—
280M16—3264no—
281M16—3262no—
282C2×C8—3264no—
283C2×C8—1632no—
284C2×C8C22×D16312yes—
285C4⋊C4—21no—
286C4⋊C4—3261no—
287C4⋊C4—1632no—
288C4⋊C4C22×D16312yes—
289C22⋊C4—3262no—
290C22⋊C4—3262no—
291C42—3262no—
292C42—3262no—
293C23—21no—
294C23—3264no—
295C23—3261no—
296C23—6521no—
297C23—6521no—
298C23—6522no—
299C23—6522no—
300C23—1631no—
301C23—1631no—
302C23—1634no—
303C23D4×D16311yescharacteristic
304Q8—21no—
305Q8—3264no—
306Q8—3261no—
307Q8—1632no—
308Q8C2×C4×D16312yes—
309D4—24no—
310D4—21no—
311D4—3264no—
312D4—3264no—
313D4—3264no—
314D4—3264no—
315D4—3261no—
316D4—1632no—
317D4C2×C4×D16312yes—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—3264no—
323C2×C4—3264no—
324C2×C4—3261no—
325C2×C4—3264no—
326C2×C4—3264no—
327C2×C4—3262no—
328C2×C4—6522no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4—1634no—
332C2×C4—1632no—
333C2×C4—1632no—
334C2×C4—1632no—
335C2×C4—1632no—
336C2×C4—1634no—
337C2×C4—1634no—
338C2×C4C2×C4×D16312yes—
339C2×C4D4×D16312yes—
340C2×C4D4×D16311yescharacteristic
341C2×C4C23×D16311yescharacteristic
342C8—22no—
343C8—3262no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—3261no—
350C22—3261no—
351C22—3261no—
352C22—3261no—
353C22—3262no—
354C22—6524no—
355C22—6528no—
356C22—6521no—
357C22—6521no—
358C22—1632no—
359C22—1632no—
360C22C22⋊C4×D16312yes—
361C22C2×D4×D16311yescharacteristic
362C4—22no—
363C4—42no—
364C4—3262no—
365C4—6522no—
366C4—1632no—
367C4—1632no—
368C4C22⋊C4×D16312yes—
369C4C22×C4×D16311yescharacteristic
370C4C2×D4×D16311yescharacteristic
371C2—22no—
372C2—42no—
373C2—3262no—
374C2—3262no—
375C2—6522no—
376C2C2×C22⋊C4×D16311yescharacteristic
377C2#2378246111yescharacteristic
378C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutatorC4×C163C23×C4
FrattiniC2×C4C23×D163
Fitting#417×C163C2
socleC22×C163C22×D4
prime radicalC2×#41×D163C2
cosocleC2×C4×C163C24
hypercenter#417D163
hypocenterC163#2896
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×M16×D163C2semidirect
3. C2×#19104693C2semidirect
4. #23781672C2semidirect
5. #23781668C2semidirect
6. #23781667C2semidirect
7. #23781666C2semidirect
8. #23781664C2semidirect
9. #23781660C2semidirect
10. #23781658C2semidirect
11. #417×C163C2semidirect
18. #41×D163C4semidirect
24. C2×#14579683C22semidirect
27. C2×C8×D163C22semidirect
31. #19104693C22semidirect
32. C2×C4⋊Dic163C22semidirect
35. #19104660C22semidirect
36. #19104659C22semidirect
37. #19104658C22semidirect
38. #19104657C22semidirect
39. #19104656C22semidirect
40. #19104655C22semidirect
42. C2×M16×C163C22semidirect
43. C2×C4⋊C4×C163C22semidirect
44. #103×C163C22semidirect
45. #102×C163C22semidirect
49. #14579700C2×C4semidirect
55. #14579698C2×C4semidirect
59. Q8×D163C2×C4semidirect
67. D4×D163C2×C4semidirect
189. C163#2896semidirect
199. #417D163semidirect
218. C2×M16C2×D163semidirect
225. C2×C4⋊C4C2×D163semidirect
231. #103C2×D163semidirect
235. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1632
C2×M16×D1632
C2×#191046932
#237816722
#237816682
#237816672
#237816662
#237816642
#237816602
#237816582
#417×C1632
#2896163