Finite Group Databasebeta

← #35223842

Subgroups — #35223842

Order 20864.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19104781C211yesmaximal, characteristic, complemented
2C2×#19104763C211yesmaximal, characteristic, complemented
3C2×#19104755C211yesmaximal, characteristic, complemented
4#23782246C211yesmaximal, characteristic, complemented
5#23781830C212yesmaximal, complemented
6#23781817C212yesmaximal, complemented
7#23781813C212yesmaximal, complemented
8#23781800C212yesmaximal, complemented
9C2×#19104655C211yesmaximal, characteristic, complemented
10#23781633C211yesmaximal, characteristic, complemented
11#417×C163C211yesmaximal, characteristic, complemented
12#19104781—24no—
13#19104781—24no—
14C2×#14579700C2212yes—
15C2×#14579699—21no—
16C2×#14579699C2211yescharacteristic
17C2×#14579699C2211yescharacteristic, complemented
18C22×D652—21no—
19C22×D652C2211yescharacteristic
20C2×D4×D163—22no—
21C2×D4×D163C2212yes—
22#19104763—24no—
23#19104755—24no—
24C2×#14579689C2211yescharacteristic, complemented
25C2×#14579688C2212yescomplemented
26C2×#14579687C2211yescharacteristic, complemented
27C2×#14579686C2211yescharacteristic, complemented
28C2×D1304C2212yescomplemented
29C2×#14579684C2211yescharacteristic, complemented
30C22×C163⋊C8C2211yescharacteristic, complemented
31#19104715—21no—
32#19104715C2212yescomplemented
33#19104710—21no—
34#19104710C2212yescomplemented
35#19104709—22no—
36#19104655C2212yescomplemented
37#19104655C2212yescomplemented
38(C2×D163)⋊C8C2212yescomplemented
39(C2×D163)⋊C8C2212yescomplemented
40C2×#41×C163C2211yescharacteristic
41C2×M16×C163C2211yescharacteristic, complemented
42C2×C4⋊C4×C163C2211yescharacteristic, complemented
43#103×C163C2212yescomplemented
44#102×C163C2212yescomplemented
45C23×D163—22no—
46C23×D163—21no—
47#14579700—24no—
48#14579700—24no—
49#14579699—28no—
50#14579699—22no—
51#14579699—22no—
52#14579699—22no—
53#14579699—22no—
54C2×Dic326D411yescharacteristic
55C2×Dic326C2311yescharacteristic
56C2×D652—28no—
57C2×D652—22no—
58C2×D652—22no—
59C2×D652—21no—
60C2×D652—22no—
61C2×D652D411yescharacteristic
62C2×D652D414yescomplemented
63C2×D652C2311yescharacteristic
64C2×D652C2312yes—
65C2×C22⋊D163—22no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163D412yes—
68D4×D163—216no—
69D4×D163—24no—
70D4×D163—24no—
71C2×C4×D163—22no—
72C2×C4×D163—21no—
73C2×C4×D163C2312yes—
74#14579689—22no—
75#14579688—24no—
76#14579687—22no—
77#14579686—22no—
78D1304—24no—
79#14579684—22no—
80C2×C163⋊C8—22no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C4⋊Dic163—22no—
84(C2×D163)⋊C4—22no—
85(C2×D163)⋊C4—22no—
86#41×C163—22no—
87#41×C163D414yescomplemented
88C2×Q8×C163C2311yescharacteristic
89C2×D4×C163—21no—
90C2×D4×C163C2311yescharacteristic
91C22×C4×C163—21no—
92C22×C4×C163—21no—
93C22×C4×C163C2311yescharacteristic
94M16×C163—22no—
95C2×C8×C163C2312yes—
96C4⋊C4×C163—21no—
97C4⋊C4×C163C2312yes—
98C22×D163—22no—
99C22×D163—28no—
100C22×D163—24no—
101C22×D163—21no—
102C22×D163—82no—
103C22×D163—82no—
104C22×D163—81no—
105C22×D163—81no—
106C22×D163C2×D412yes—
107Dic326—22no—
108Dic326—21no—
109Dic326—21no—
110D652—28no—
111D652—22no—
112D652—24no—
113D652—22no—
114D652—21no—
115D652—21no—
116D652—82no—
117D652C2×D414yes—
118C22⋊D163—28no—
119C22⋊D163—24no—
120C22⋊D163—42no—
121C2×Dic163—21no—
122C2×Dic163C2×D412yes—
123C4×D163—28no—
124C4×D163—22no—
125C4×D163—22no—
126C4×D163—42no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163—21no—
130C23×C163C2×D411yescharacteristic
131Q8×C163—21no—
132Q8×C163C2×D412yes—
133D4×C163—24no—
134D4×C163—21no—
135D4×C163C2×D412yes—
136C2×C4×C163—21no—
137C2×C4×C163—24no—
138C2×C4×C163—22no—
139C2×C4×C163—81no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D412yes—
142C2×C4×C163C2×D412yes—
143C2×C4×C163C2411yescharacteristic
144C8×C163—22no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—22no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163#12012yes—
151C22×C163C22×D411yescharacteristic
152C2×D163—22no—
153C2×D163—24no—
154C2×D163—22no—
155C2×D163—22no—
156C2×D163—42no—
157C2×D163—42no—
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—84no—
162C2×D163—88no—
163C4×C163—22no—
164C4×C163—42no—
165C4×C163#12012yes—
166C4×C163C22×D411yescharacteristic
167C4×C163C22×D411yescharacteristic
168Dic163—22no—
169Dic163—41no—
170C2×C163—22no—
171C2×C163—42no—
172C2×C163C2×#12011yescharacteristic
173C2×C163#57611yescharacteristic
174C2×C163#57711yescharacteristic
175D163—42no—
176D163—44no—
177D163—81no—
178C163#301211yescharacteristic, complemented, Sylow
179#3012—1631nomaximal, Sylow
180C2×H(5,𝔽2)—1631no—
181C2×AGL(1,ℤ/8)—1631no—
182C2×#135—1631no—
183#537—1631no—
184#452—1632no—
185#450—1632no—
186#449—1632no—
187#447—1632no—
188#417D16311yescharacteristic, complemented
189C2×#102—1631no—
190#408—1631no—
191H(5,𝔽2)—3264no—
192H(5,𝔽2)—3264no—
193C2×#41—3261no—
194C2×#41—1632no—
195C2×#41—1631no—
196C2×#41—1631no—
197C2×#41C2×D16311yescharacteristic
198C22×D4—3261no—
199C22×D4—3262no—
200C22×D4—1632no—
201C22×D4—1631no—
202AGL(1,ℤ/8)—3264no—
203#135—3264no—
204C2×Q16—1631no—
205C2×SD16—1631no—
206C2×SD16—1631no—
207C2×SD16—1632no—
208C2×D8—1631no—
209C2×D8—1632no—
210C2×M16C2×D16311yescharacteristic, complemented
211C22×C8—1631no—
212C22⋊Q8—3261no—
213C22⋊Q8—1632no—
214#121—3261no—
215#121—3262no—
216#121—1632no—
217C2×C4⋊C4C2×D16311yescharacteristic, complemented
218#103C2×D16312yescomplemented
219#102—1632no—
220#102—1632no—
221#102C2×D16312yescomplemented
222C22⋊C8—1632no—
223C22⋊C8—1632no—
224C24—3262no—
225C24—3261no—
226#41—22no—
227#41—3268no—
228#41—3264no—
229#41—3264no—
230#41—3262no—
231#41—3262no—
232#41—3262no—
233#41—3262no—
234#41D65214yescomplemented
235C2×Q8—1631no—
236C2×Q8—1631no—
237C2×Q8C22×D16311yescharacteristic
238C2×D4—21no—
239C2×D4—3262no—
240C2×D4—3268no—
241C2×D4—3262no—
242C2×D4—32616no—
243C2×D4—3264no—
244C2×D4—3264no—
245C2×D4—3261no—
246C2×D4—3262no—
247C2×D4—3261no—
248C2×D4—3262no—
249C2×D4—1632no—
250C2×D4—1631no—
251C2×D4—1634no—
252C2×D4—1631no—
253C2×D4—1632no—
254C2×D4C22×D16311yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—3262no—
258C22×C4—3261no—
259C22×C4—1632no—
260C22×C4C22×D16311yescharacteristic
261Q16—3262no—
262SD16—3264no—
263SD16—3262no—
264SD16—3262no—
265D8—3264no—
266D8—3262no—
267M16—22no—
268C2×C8—3262no—
269C2×C8—1631no—
270C2×C8—1631no—
271C2×C8C22×D16312yes—
272C4⋊C4—21no—
273C4⋊C4—3262no—
274C4⋊C4C22×D16312yes—
275C22⋊C4—3262no—
276C22⋊C4—3262no—
277C23—21no—
278C23—13041no—
279C23—13041no—
280C23—13042no—
281C23—13042no—
282C23—3268no—
283C23—3264no—
284C23—3262no—
285C23—3261no—
286C23—1632no—
287C23D4×D16311yescharacteristic
288Q8—21no—
289Q8—3262no—
290Q8—3261no—
291Q8—3261no—
292Q8C2×D65212yes—
293D4—24no—
294D4—21no—
295D4—13042no—
296D4—3268no—
297D4—3268no—
298D4—3264no—
299D4—3261no—
300D4—3262no—
301D4—3261no—
302D4—3262no—
303D4—3264no—
304D4—6522no—
305D4—1634no—
306D4C2×D65212yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—3268no—
312C2×C4—3262no—
313C2×C4—3262no—
314C2×C4—3261no—
315C2×C4—6522no—
316C2×C4—1632no—
317C2×C4D4×D16312yes—
318C2×C4D4×D16311yescharacteristic
319C2×C4C2×D65212yes—
320C2×C4C23×D16311yescharacteristic
321C8—22no—
322C8—3261no—
323C8—3261no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—13044no—
330C22—13048no—
331C22—3262no—
332C22—3264no—
333C22—3262no—
334C22—3262no—
335C22—6521no—
336C22—6521no—
337C22—6521no—
338C22—6522no—
339C22—6522no—
340C22#1910470512yes—
341C22C2×D4×D16311yescharacteristic
342C4—22no—
343C4—42no—
344C4—3262no—
345C4—6521no—
346C4#1910470512yes—
347C4C2×D4×D16311yescharacteristic
348C4C22×D65211yescharacteristic
349C2—22no—
350C2—42no—
351C2—13041no—
352C2—6522no—
353C2—6524no—
354C2C2×#1910470511yescharacteristic
355C2#2378246911yescharacteristic
356C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#417×C163C2
socleC22×C163C22×D4
hypercenter#417D163
hypocenterC163#3012
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19104781C2semidirect
2. C2×#19104763C2semidirect
3. C2×#19104755C2semidirect
4. #23782246C2semidirect
5. #23781830C2semidirect
6. #23781817C2semidirect
7. #23781813C2semidirect
8. #23781800C2semidirect
9. C2×#19104655C2semidirect
10. #23781633C2semidirect
11. #417×C163C2semidirect
17. C2×#14579699C22semidirect
24. C2×#14579689C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579687C22semidirect
27. C2×#14579686C22semidirect
28. C2×D1304C22semidirect
29. C2×#14579684C22semidirect
30. C22×C163⋊C8C22semidirect
32. #19104715C22semidirect
34. #19104710C22semidirect
36. #19104655C22semidirect
37. #19104655C22semidirect
38. (C2×D163)⋊C8C22semidirect
39. (C2×D163)⋊C8C22semidirect
41. C2×M16×C163C22semidirect
42. C2×C4⋊C4×C163C22semidirect
43. #103×C163C22semidirect
44. #102×C163C22semidirect
62. C2×D652D4semidirect
87. #41×C163D4semidirect
178. C163#3012semidirect
188. #417D163semidirect
210. C2×M16C2×D163semidirect
217. C2×C4⋊C4C2×D163semidirect
218. #103C2×D163semidirect
221. #102C2×D163semidirect
234. #41D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047812
C2×#191047632
C2×#191047552
#237822462
#237818302
#237818172
#237818132
#237818002
C2×#191046552
#237816332
#417×C1632
#3012163