Finite Group Databasebeta

← #35225005

Subgroups — #35225005

Order 20864.

564 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×#19104764C211yesmaximal, characteristic, complemented
2C2×#19104762C211yesmaximal, characteristic, complemented
3C2×#19104756C211yesmaximal, characteristic, complemented
4C2×SD16×D163C211yesmaximal, characteristic, complemented
5#23782240C211yesmaximal, characteristic, complemented
6#23782228C211yesmaximal, characteristic, complemented
7#23782057C211yesmaximal, characteristic, complemented
8#23782056C211yesmaximal, characteristic, complemented
9#23782052C211yesmaximal, characteristic, complemented
10#23782051C211yesmaximal, characteristic, complemented
11#23782049C211yesmaximal, characteristic, complemented
12#23782046C211yesmaximal, characteristic, complemented
13#23782036C211yesmaximal, characteristic, complemented
14#23781619C211yesmaximal, characteristic, complemented
15#492×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic
20#19104764—24no—
21#19104762—24no—
22#19104756—24no—
23C2×Dic652C2211yescharacteristic, complemented
24C2×#14579689C2211yescharacteristic, complemented
25C2×#14579689C2211yescharacteristic, complemented
26C2×#14579688C2211yescharacteristic
27C2×#14579688C2211yescharacteristic, complemented
28C2×#14579687C2211yescharacteristic
29C2×#14579687C2211yescharacteristic, complemented
30C2×#14579686C2211yescharacteristic
31C2×#14579686C2211yescharacteristic
32SD16×D163—24no—
33C2×D1304C2211yescharacteristic
34C2×#14579684C2211yescharacteristic
35C2×#14579684C2211yescharacteristic, complemented
36C2×#14579682C2211yescharacteristic
37C2×C8×D163C2211yescharacteristic
38#19104733C2211yescharacteristic, complemented
39#19104731C2211yescharacteristic
40#19104724C2211yescharacteristic, complemented
41#19104723C2211yescharacteristic
42#19104723C2211yescharacteristic, complemented
43#19104722C2211yescharacteristic
44#19104720—22no—
45#19104713—22no—
46#19104707—22no—
47#19104703—22no—
48#19104692C2211yescharacteristic
49#19104646C2211yescharacteristic, complemented
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×SD16×C163C2211yescharacteristic, complemented
56C4⋊Q8×C163C2211yescharacteristic, complemented
57#127×C163C2211yescharacteristic
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×D163C2311yescharacteristic
81Dic652—22no—
82#14579689—22no—
83#14579689—22no—
84#14579688—22no—
85#14579688—22no—
86#14579687—22no—
87#14579687—22no—
88#14579686—22no—
89#14579686—22no—
90D1304—22no—
91#14579684—22no—
92#14579684—22no—
93#14579682D414yescomplemented
94C2×C163⋊C8C2311yescharacteristic
95C2×C163⋊C8C2311yescharacteristic
96C8×D163D414yescomplemented
97Dic163⋊C4—22no—
98Dic163⋊C4D412yes—
99C22⋊Dic163—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4D412yes—
104C4×Dic163C2311yescharacteristic
105C2×Q8×C163C2311yescharacteristic
106C2×Q8×C163C2311yescharacteristic
107C2×D4×C163—22no—
108C2×D4×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110SD16×C163—22no—
111SD16×C163—22no—
112SD16×C163—22no—
113SD16×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C4⋊C4×C163—22no—
117C42×C163C2311yescharacteristic
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—42no—
121C22×D163—82no—
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—42no—
134C22⋊D163—42no—
135C22⋊D163—82no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—42no—
139C2×Dic163C2×D412yes—
140C2×Dic163C2×D411yescharacteristic
141C4×D163—42no—
142C4×D163—42no—
143C4×D163C2×D412yes—
144C4×D163C2×D412yes—
145C163⋊C8C2×D412yes—
146C163⋊C8C2×D412yes—
147C23×C163—21no—
148C23×C163—21no—
149Q8×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152Q8×C163—21no—
153D4×C163—21no—
154D4×C163—21no—
155D4×C163—21no—
156D4×C163—21no—
157D4×C163—82no—
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#57712yes—
189D163—22no—
190D163—81no—
191D163—81no—
192C163#315411yescharacteristic, complemented, Sylow
193#3154—1631nomaximal, Sylow
194C2×#137—1631no—
195C2×AGL(1,ℤ/8)—1631no—
196C2×#135—1631no—
197C22×SD16—1631no—
198#536—1631no—
199#535—1631no—
200#497—1631no—
201#497—1631no—
202#496—1631no—
203#496—1631no—
204#495—1631no—
205#495—1631no—
206#492—1631no—
207#492D16311yescharacteristic, complemented
208#405—1631no—
209C2×#41—1631no—
210C2×#41—1631no—
211C22×Q8—1631no—
212C22×D4—1631no—
213#137—3264no—
214AGL(1,ℤ/8)—3264no—
215#135—3264no—
216C2×Q16—1631no—
217C2×Q16—1631no—
218C2×Q16—1631no—
219C2×SD16—3264no—
220C2×SD16—1631no—
221C2×SD16—1631no—
222C2×SD16—1631no—
223C2×SD16—1631no—
224C2×SD16—1631no—
225C2×SD16—1631no—
226C2×SD16C2×D16311yescharacteristic
227C2×SD16C2×D16311yescharacteristic
228C2×SD16C2×D16311yescharacteristic, complemented
229C2×SD16C2×D16311yescharacteristic, complemented
230C2×D8—1631no—
231C2×D8—1631no—
232C2×D8—1631no—
233C2×M16—1631no—
234C22×C8—1631no—
235C4⋊Q8—1631no—
236C4⋊Q8C2×D16311yescharacteristic, complemented
237#127—1631no—
238#127C2×D16311yescharacteristic
239#124—1631no—
240#124—1631no—
241#124—1631no—
242#124—1631no—
243#123—3262no—
244C22⋊Q8—3262no—
245#121—3262no—
246#120—3262no—
247#117—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—22no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—3262no—
268C2×D4—3262no—
269C2×D4—3262no—
270C2×D4—3261no—
271C2×D4—1631no—
272C2×D4—1631no—
273C2×D4C22×D16311yescharacteristic
274C2×D4C22×D16311yescharacteristic
275C22×C4—3261no—
276C22×C4—3261no—
277C22×C4—3261no—
278C22×C4—1631no—
279Q16—3262no—
280Q16—3262no—
281Q16—3262no—
282SD16—22no—
283SD16—22no—
284SD16—22no—
285SD16—22no—
286SD16—3262no—
287SD16—3262no—
288SD16—3262no—
289SD16—3262no—
290SD16—3262no—
291SD16—3262no—
292D8—3262no—
293D8—3262no—
294D8—3262no—
295M16—1634no—
296C2×C8—1631no—
297C2×C8—1631no—
298C2×C8—1634no—
299C2×C8C22×D16311yescharacteristic
300C2×C8C22×D16311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—3262no—
303C4⋊C4—1632no—
304C22⋊C4—3262no—
305C22⋊C4—3262no—
306C22⋊C4—3262no—
307C22⋊C4—3262no—
308C22⋊C4—1632no—
309C42—1631no—
310C42C22×D16311yescharacteristic
311C23—21no—
312C23—21no—
313C23—13042no—
314C23—3261no—
315C23—3261no—
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—82no—
332D4—13042no—
333D4—3261no—
334D4—3261no—
335D4—3261no—
336D4—3261no—
337D4—6522no—
338D4—6522no—
339D4—6522no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3261no—
343C2×C4—3261no—
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#2378247112yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#492×C163C2
socleC22×C163C22×D4
hypercenter#492D163
hypocenterC163#3154
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#19104764C2semidirect
2. C2×#19104762C2semidirect
3. C2×#19104756C2semidirect
4. C2×SD16×D163C2semidirect
5. #23782240C2semidirect
6. #23782228C2semidirect
7. #23782057C2semidirect
8. #23782056C2semidirect
9. #23782052C2semidirect
10. #23782051C2semidirect
11. #23782049C2semidirect
12. #23782046C2semidirect
13. #23782036C2semidirect
14. #23781619C2semidirect
15. #492×C163C2semidirect
23. C2×Dic652C22semidirect
24. C2×#14579689C22semidirect
25. C2×#14579689C22semidirect
27. C2×#14579688C22semidirect
29. C2×#14579687C22semidirect
35. C2×#14579684C22semidirect
38. #19104733C22semidirect
40. #19104724C22semidirect
42. #19104723C22semidirect
49. #19104646C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
54. C2×SD16×C163C22semidirect
55. C2×SD16×C163C22semidirect
56. C4⋊Q8×C163C22semidirect
58. C4×C8×C163C22semidirect
93. #14579682D4semidirect
96. C8×D163D4semidirect
192. C163#3154semidirect
207. #492D163semidirect
228. C2×SD16C2×D163semidirect
229. C2×SD16C2×D163semidirect
236. C4⋊Q8C2×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×#191047642
C2×#191047622
C2×#191047562
C2×SD16×D1632
#237822402
#237822282
#237820572
#237820562
#237820522
#237820512
#237820492
#237820462
#237820362
#237816192
#492×C1632
#3154163