Finite Group Databasebeta

← #35224997

Subgroups — #35224997

Order 20864.

572 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104767C211yesmaximal, characteristic, complemented
2C2×#19104764C211yesmaximal, characteristic, complemented
3C2×#19104762C211yesmaximal, characteristic, complemented
4C2×#19104759C211yesmaximal, characteristic, complemented
5#23782214C211yesmaximal, characteristic, complemented
6#124×D163C211yesmaximal, characteristic, complemented
7#23782058C211yesmaximal, characteristic, complemented
8#23782056C211yesmaximal, characteristic, complemented
9#23782053C211yesmaximal, characteristic, complemented
10#23782051C211yesmaximal, characteristic, complemented
11#23782049C211yesmaximal, characteristic, complemented
12#23782047C211yesmaximal, characteristic, complemented
13#23782047C211yesmaximal, characteristic, complemented
14C4×#14579682C211yesmaximal, characteristic, complemented
15#495×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic
20#19104767—24no—
21#19104764—24no—
22#19104762—24no—
23#19104759—24no—
24C2×Dic652C2211yescharacteristic, complemented
25C2×#14579689C2211yescharacteristic, complemented
26C2×#14579689C2211yescharacteristic, complemented
27C2×#14579688C2211yescharacteristic
28C2×#14579688C2211yescharacteristic, complemented
29C2×#14579687C2211yescharacteristic
30C2×#14579687C2211yescharacteristic, complemented
31C2×#14579686C2211yescharacteristic
32C2×#14579686C2211yescharacteristic, complemented
33C2×D1304C2211yescharacteristic
34C2×#14579684C2211yescharacteristic
35C2×#14579684C2211yescharacteristic
36C2×#14579682C2211yescharacteristic
37C2×#14579682C2211yescharacteristic
38#19104733C2211yescharacteristic, complemented
39#19104731C2211yescharacteristic, complemented
40#19104724C2211yescharacteristic
41#19104723C2211yescharacteristic
42#19104723C2211yescharacteristic, complemented
43#19104722C2211yescharacteristic
44#19104721—22no—
45#19104708—22no—
46(C22⋊D163)⋊C4—22no—
47C22⋊C4×D163—22no—
48C42×D163C2211yescharacteristic
49#19104645C2211yescharacteristic, complemented
50#19104645C2211yescharacteristic, complemented
51C4×C163⋊C8C2211yescharacteristic, complemented
52C2×Q16×C163C2211yescharacteristic, complemented
53C2×SD16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×D8×C163C2211yescharacteristic
56#124×C163C2211yescharacteristic
57#124×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579700—22no—
61#14579700—22no—
62#14579698—22no—
63#14579698—22no—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67Q8×D163—22no—
68Q8×D163—22no—
69C2×D652—21no—
70C2×D652C2311yescharacteristic
71C2×D652C2311yescharacteristic
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—22no—
75D4×D163—22no—
76D4×D163—22no—
77C22×Dic163—21no—
78C2×C4×D163—21no—
79C2×C4×D163—21no—
80C2×C4×D163D412yes—
81C2×C4×D163C2311yescharacteristic
82Dic652—22no—
83#14579689—22no—
84#14579689—22no—
85#14579688—22no—
86#14579688—22no—
87#14579687—22no—
88#14579687—22no—
89#14579686—22no—
90#14579686—22no—
91D1304—22no—
92#14579684—22no—
93#14579684—22no—
94#14579682D414yescomplemented
95#14579682D414yescomplemented
96C2×C163⋊C8C2311yescharacteristic
97C2×C163⋊C8C2311yescharacteristic
98Dic163⋊C4—22no—
99C22⋊Dic163—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103C4×Dic163D412yes—
104C4×Dic163C2311yescharacteristic
105C2×Q8×C163C2311yescharacteristic
106C2×Q8×C163C2311yescharacteristic
107C2×D4×C163C2311yescharacteristic
108C2×D4×C163C2311yescharacteristic
109Q16×C163—22no—
110SD16×C163—22no—
111SD16×C163—22no—
112D8×C163—22no—
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C22⋊C4×C163—22no—
116C22⋊C4×C163—22no—
117C42×C163C2311yescharacteristic
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—44no—
121C22×D163—42no—
122C22×D163C2×D411yescharacteristic
123Dic326—21no—
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—42no—
128D652—21no—
129D652—21no—
130D652—21no—
131D652—21no—
132D652—42no—
133C22⋊D163—44no—
134C22⋊D163—42no—
135C22⋊D163—42no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—42no—
139C2×Dic163C2×D412yes—
140C2×Dic163C2×D411yescharacteristic
141C4×D163—42no—
142C4×D163—42no—
143C4×D163—42no—
144C4×D163C2×D412yes—
145C4×D163C2×D412yes—
146C163⋊C8C2×D412yes—
147C163⋊C8C2×D412yes—
148C23×C163—21no—
149C23×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152Q8×C163—21no—
153Q8×C163—21no—
154D4×C163—21no—
155D4×C163—21no—
156D4×C163—21no—
157D4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163C2×D412yes—
161C2×C4×C163C2411yescharacteristic
162C8×C163C2×D412yes—
163C8×C163C2×D412yes—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—42no—
167C22×C163—42no—
168C22×C163C22×D411yescharacteristic
169C2×D163—22no—
170C2×D163—41no—
171C2×D163—41no—
172C2×D163—42no—
173C2×D163—42no—
174C2×D163—84no—
175C2×D163#12712yes—
176C4×C163—22no—
177C4×C163—41no—
178C4×C163—41no—
179C4×C163C22×D411yescharacteristic
180C4×C163C22×D411yescharacteristic
181Dic163—22no—
182Dic163—41no—
183Dic163—41no—
184Dic163#12712yes—
185C2×C163—81no—
186C2×C163—81no—
187C2×C163C2×#12711yescharacteristic
188C2×C163#57512yes—
189D163—22no—
190D163—81no—
191D163—81no—
192C163#315311yescharacteristic, complemented, Sylow
193#3153—1631nomaximal, Sylow
194C2×#137—1631no—
195C2×#137—1631no—
196C2×AGL(1,ℤ/8)—1631no—
197C2×AGL(1,ℤ/8)—1631no—
198#532—1631no—
199C2×#124—1631no—
200#497—1631no—
201#497—1631no—
202#496—1631no—
203#496—1631no—
204#495—1631no—
205#495—1631no—
206#495—1631no—
207#495D16311yescharacteristic, complemented
208C4×M16—1631no—
209C2×#41—1631no—
210C2×#41—1631no—
211C22×Q8—1631no—
212C22×D4—1631no—
213#137—3264no—
214#137—3264no—
215AGL(1,ℤ/8)—3264no—
216AGL(1,ℤ/8)—3264no—
217C2×Q16—1631no—
218C2×Q16—1631no—
219C2×Q16—1631no—
220C2×Q16C2×D16311yescharacteristic, complemented
221C2×SD16—1631no—
222C2×SD16—1631no—
223C2×SD16—1631no—
224C2×SD16—1631no—
225C2×SD16—1631no—
226C2×SD16—1631no—
227C2×SD16C2×D16311yescharacteristic
228C2×SD16C2×D16311yescharacteristic, complemented
229C2×D8—1631no—
230C2×D8—1631no—
231C2×D8—1631no—
232C2×D8C2×D16311yescharacteristic
233C2×M16—1631no—
234C2×M16—1631no—
235C4⋊Q8—1631no—
236#127—1631no—
237#124—3262no—
238#124—1631no—
239#124—1631no—
240#124—1631no—
241#124—1631no—
242#124C2×D16311yescharacteristic
243#124C2×D16311yescharacteristic, complemented
244#121—3262no—
245C4×D4—3262no—
246C2×C22⋊C4—3262no—
247C2×C42—1631no—
248#97—1631no—
249#97—1631no—
250C4×C8—1631no—
251C4×C8C2×D16311yescharacteristic, complemented
252C24—3261no—
253#41—3262no—
254#41—3262no—
255#41—3262no—
256#41—3262no—
257C2×Q8—3261no—
258C2×Q8—3262no—
259C2×Q8—3262no—
260C2×Q8—1631no—
261C2×Q8—1631no—
262C2×Q8C22×D16311yescharacteristic
263C2×Q8C22×D16311yescharacteristic
264C2×D4—3261no—
265C2×D4—3261no—
266C2×D4—3262no—
267C2×D4—3262no—
268C2×D4—3262no—
269C2×D4—3261no—
270C2×D4—1631no—
271C2×D4—1631no—
272C2×D4C22×D16311yescharacteristic
273C2×D4C22×D16311yescharacteristic
274C22×C4—3261no—
275C22×C4—3261no—
276C22×C4—3261no—
277C22×C4—1632no—
278C22×C4—1631no—
279Q16—22no—
280Q16—3262no—
281Q16—3262no—
282Q16—3262no—
283SD16—22no—
284SD16—22no—
285SD16—3262no—
286SD16—3262no—
287SD16—3262no—
288SD16—3262no—
289SD16—3262no—
290SD16—3262no—
291D8—22no—
292D8—3262no—
293D8—3262no—
294D8—3262no—
295M16—1634no—
296M16—1634no—
297C2×C8—1631no—
298C2×C8—1631no—
299C2×C8C22×D16311yescharacteristic
300C2×C8C22×D16311yescharacteristic
301C4⋊C4—3262no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—3262no—
305C22⋊C4—3262no—
306C22⋊C4—3262no—
307C22⋊C4—3262no—
308C42—1632no—
309C42—1631no—
310C42C22×D16311yescharacteristic
311C23—21no—
312C23—21no—
313C23—3261no—
314C23—3261no—
315C23—6524no—
316C23—6522no—
317C23—1631no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—3261no—
323Q8—3261no—
324Q8—3261no—
325Q8—3261no—
326Q8—6522no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—3261no—
332D4—3261no—
333D4—3261no—
334D4—3261no—
335D4—6522no—
336D4—6524no—
337D4—6522no—
338D4—6522no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—3261no—
342C2×C4—3261no—
343C2×C4—6522no—
344C2×C4—6522no—
345C2×C4—6522no—
346C2×C4—6522no—
347C2×C4—1632no—
348C2×C4—1631no—
349C2×C4—1632no—
350C2×C4—1632no—
351C2×C4D4×D16312yes—
352C2×C4C23×D16311yescharacteristic
353C8—1632no—
354C8—1632no—
355C8D4×D16312yes—
356C8D4×D16312yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—13044no—
362C22—3262no—
363C22—6521no—
364C22—6521no—
365C22—6522no—
366C22—6522no—
367C22—1632no—
368C22C2×D4×D16311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—3262no—
373C4—6521no—
374C4—6521no—
375C4—1632no—
376C4C2×D4×D16311yescharacteristic
377C4C2×D4×D16311yescharacteristic
378C2—81no—
379C2—81no—
380C2—13041no—
381C2—13041no—
382C2—3262no—
383C2#127×D16311yescharacteristic
384C2#2378246212yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#495×C163C2
socleC22×C163C22×D4
hypercenter#495D163
hypocenterC163#3153
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19104767C2semidirect
2. C2×#19104764C2semidirect
3. C2×#19104762C2semidirect
4. C2×#19104759C2semidirect
5. #23782214C2semidirect
6. #124×D163C2semidirect
7. #23782058C2semidirect
8. #23782056C2semidirect
9. #23782053C2semidirect
10. #23782051C2semidirect
11. #23782049C2semidirect
12. #23782047C2semidirect
13. #23782047C2semidirect
14. C4×#14579682C2semidirect
15. #495×C163C2semidirect
24. C2×Dic652C22semidirect
25. C2×#14579689C22semidirect
26. C2×#14579689C22semidirect
28. C2×#14579688C22semidirect
30. C2×#14579687C22semidirect
32. C2×#14579686C22semidirect
38. #19104733C22semidirect
39. #19104731C22semidirect
42. #19104723C22semidirect
49. #19104645C22semidirect
50. #19104645C22semidirect
51. C4×C163⋊C8C22semidirect
52. C2×Q16×C163C22semidirect
54. C2×SD16×C163C22semidirect
57. #124×C163C22semidirect
58. C4×C8×C163C22semidirect
94. #14579682D4semidirect
95. #14579682D4semidirect
192. C163#3153semidirect
207. #495D163semidirect
220. C2×Q16C2×D163semidirect
228. C2×SD16C2×D163semidirect
243. #124C2×D163semidirect
251. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
C2×#191047642
C2×#191047622
C2×#191047592
#237822142
#124×D1632
#237820582
#237820562
#237820532
#237820512
#237820492
#237820472
C4×#145796822
#495×C1632
#3153163