Finite Group Databasebeta

← #35225011

Subgroups — #35225011

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×#19104767C211yesmaximal, characteristic, complemented
2C2×#19104762C211yesmaximal, characteristic, complemented
3C2×#19104758C211yesmaximal, characteristic, complemented
4C2×SD16×D163C211yesmaximal, characteristic, complemented
5#23782240C211yesmaximal, characteristic, complemented
6#23782233C211yesmaximal, characteristic, complemented
7#23782058C211yesmaximal, characteristic, complemented
8#23782054C211yesmaximal, characteristic, complemented
9#23782051C211yesmaximal, characteristic, complemented
10#23782050C211yesmaximal, characteristic, complemented
11#23782047C211yesmaximal, characteristic, complemented
12#23782039C211yesmaximal, characteristic, complemented
13#23782036C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#497×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579700C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic
20#19104767—24no—
21#19104762—24no—
22#19104758—24no—
23C2×#14579689C2211yescharacteristic, complemented
24C2×#14579688C2211yescharacteristic
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579687C2211yescharacteristic
27C2×#14579686C2211yescharacteristic
28C2×#14579686C2211yescharacteristic
29C2×#14579686C2211yescharacteristic, complemented
30C2×#14579686C2211yescharacteristic, complemented
31C2×#14579686C2211yescharacteristic, complemented
32SD16×D163—24no—
33C2×D1304C2211yescharacteristic
34C2×D1304C2211yescharacteristic, complemented
35C2×#14579684C2211yescharacteristic
36C2×#14579682C2211yescharacteristic
37C2×C8×D163C2211yescharacteristic
38#19104733C2211yescharacteristic, complemented
39#19104732C2211yescharacteristic
40#19104731C2211yescharacteristic
41#19104724C2211yescharacteristic
42#19104724C2211yescharacteristic, complemented
43#19104723C2211yescharacteristic, complemented
44#19104720—22no—
45#19104713—22no—
46#19104708—22no—
47#19104705—22no—
48#19104692C2211yescharacteristic
49#19104645C2211yescharacteristic, complemented
50C4×C163⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×Q16×C163C2211yescharacteristic, complemented
53C2×Q16×C163C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic
55C2×SD16×C163C2211yescharacteristic
56C4⋊Q8×C163C2211yescharacteristic, complemented
57#124×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579700—22no—
61#14579700—22no—
62#14579700—22no—
63#14579700—22no—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66Q8×D163—22no—
67Q8×D163—22no—
68C2×D652—21no—
69C2×D652—21no—
70C2×D652—22no—
71C2×D652C2311yescharacteristic
72C2×D652C2311yescharacteristic
73C2×D652C2311yescharacteristic
74C2×C22⋊D163—21no—
75C2×C22⋊D163—22no—
76D4×D163—22no—
77D4×D163—22no—
78C2×C4×D163—21no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163C2311yescharacteristic
82#14579689—22no—
83#14579688—22no—
84#14579688—22no—
85#14579687—22no—
86#14579686—22no—
87#14579686—22no—
88#14579686—22no—
89#14579686—22no—
90#14579686—22no—
91D1304—22no—
92D1304—22no—
93#14579684—22no—
94#14579682D414yescomplemented
95C2×C163⋊C8C2311yescharacteristic
96C2×C163⋊C8C2311yescharacteristic
97C8×D163D414yescomplemented
98Dic163⋊C4—22no—
99Dic163⋊C4D412yes—
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×Q8×C163C2311yescharacteristic
108C2×D4×C163C2311yescharacteristic
109Q16×C163—22no—
110Q16×C163—22no—
111SD16×C163—22no—
112SD16×C163—22no—
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C4⋊C4×C163—22no—
116C22⋊C4×C163—22no—
117C42×C163C2311yescharacteristic
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—42no—
122C22×D163—82no—
123C22×D163C2×D411yescharacteristic
124Dic326—21no—
125Dic326—21no—
126Dic326—42no—
127D652—21no—
128D652—21no—
129D652—21no—
130D652—21no—
131D652—21no—
132D652—21no—
133D652—42no—
134D652—42no—
135D652—82no—
136C22⋊D163—42no—
137C22⋊D163—82no—
138C2×Dic163—21no—
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—
149Q8×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152Q8×C163—21no—
153Q8×C163—21no—
154Q8×C163—21no—
155D4×C163—21no—
156D4×C163—21no—
157C2×C4×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—42no—
166C22×C163C22×D411yescharacteristic
167C2×D163—22no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—42no—
172C2×D163—42no—
173C2×D163—42no—
174C2×D163—84no—
175C2×D163#12712yes—
176C4×C163—22no—
177C4×C163—41no—
178C4×C163—41no—
179C4×C163—41no—
180C4×C163C22×D411yescharacteristic
181C4×C163C22×D411yescharacteristic
182Dic163—22no—
183Dic163—41no—
184Dic163#12712yes—
185C2×C163—81no—
186C2×C163C2×#12711yescharacteristic
187C2×C163#57712yes—
188D163—22no—
189D163—81no—
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#497D16311yescharacteristic, complemented
202#496—1631no—
203#496—1631no—
204#495—1631no—
205#495—1631no—
206#492—1631no—
207#492—1631no—
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×Q16C2×D16311yescharacteristic, complemented
218C2×Q16C2×D16311yescharacteristic, complemented
219C2×SD16—3264no—
220C2×SD16—1631no—
221C2×SD16—1631no—
222C2×SD16—1631no—
223C2×SD16—1631no—
224C2×SD16—1631no—
225C2×SD16—1631no—
226C2×SD16—1631no—
227C2×SD16—1631no—
228C2×SD16C2×D16311yescharacteristic
229C2×SD16C2×D16311yescharacteristic
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#127—1631no—
239#124—1631no—
240#124—1631no—
241#124—1631no—
242#124C2×D16311yescharacteristic
243#123—3262no—
244C22⋊Q8—3262no—
245#121—3262no—
246#120—3262no—
247#117—1631no—
248#97—1631no—
249#97C2×D16311yescharacteristic, complemented
250C4×C8—1631no—
251C4×C8—1631no—
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×Q8C22×D16311yescharacteristic
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—3262no—
271C2×D4—1631no—
272C2×D4—1631no—
273C2×D4—1631no—
274C2×D4C22×D16311yescharacteristic
275C22×C4—3261no—
276C22×C4—3261no—
277C22×C4—3261no—
278C22×C4—1631no—
279Q16—22no—
280Q16—22no—
281Q16—3262no—
282SD16—22no—
283SD16—22no—
284SD16—3262no—
285SD16—3262no—
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—22no—
305C22⋊C4—3262no—
306C22⋊C4—3262no—
307C22⋊C4—3262no—
308C22⋊C4—1632no—
309C42—1631no—
310C42C22×D16311yescharacteristic
311C23—21no—
312C23—13042no—
313C23—3261no—
314C23—3261no—
315C23—3261no—
316C23—6522no—
317C23—1631no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—3261no—
325Q8—3261no—
326Q8—6522no—
327D4—21no—
328D4—21no—
329D4—13042no—
330D4—13042no—
331D4—3261no—
332D4—3261no—
333D4—3261no—
334D4—3261no—
335D4—3261no—
336D4—3261no—
337D4—6522no—
338D4—6522no—
339D4—6522no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
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—42no—
359C22—13044no—
360C22—3262no—
361C22—6521no—
362C22—6521no—
363C22—6521no—
364C22—6522no—
365C22—6522no—
366C22—6522no—
367C22—1632no—
368C22C2×D4×D16311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3262no—
374C4—6521no—
375C4—1632no—
376C4C2×D4×D16311yescharacteristic
377C4C2×D4×D16311yescharacteristic
378C2—81no—
379C2—13041no—
380C2—13041no—
381C2—13041no—
382C2—3262no—
383C2#127×D16311yescharacteristic
384C2#2378247412yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#497×C163C2
socleC22×C163C22×D4
hypercenter#497D163
hypocenterC163#3154
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19104767C2semidirect
2. C2×#19104762C2semidirect
3. C2×#19104758C2semidirect
4. C2×SD16×D163C2semidirect
5. #23782240C2semidirect
6. #23782233C2semidirect
7. #23782058C2semidirect
8. #23782054C2semidirect
9. #23782051C2semidirect
10. #23782050C2semidirect
11. #23782047C2semidirect
12. #23782039C2semidirect
13. #23782036C2semidirect
14. #23781622C2semidirect
15. #497×C163C2semidirect
23. C2×#14579689C22semidirect
25. C2×#14579688C22semidirect
29. C2×#14579686C22semidirect
30. C2×#14579686C22semidirect
31. C2×#14579686C22semidirect
34. C2×D1304C22semidirect
38. #19104733C22semidirect
42. #19104724C22semidirect
43. #19104723C22semidirect
49. #19104645C22semidirect
50. C4×C163⋊C8C22semidirect
51. C8×Dic163C22semidirect
52. C2×Q16×C163C22semidirect
53. C2×Q16×C163C22semidirect
56. C4⋊Q8×C163C22semidirect
58. #97×C163C22semidirect
94. #14579682D4semidirect
97. C8×D163D4semidirect
192. C163#3154semidirect
201. #497D163semidirect
217. C2×Q16C2×D163semidirect
218. C2×Q16C2×D163semidirect
236. C4⋊Q8C2×D163semidirect
249. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
C2×#191047622
C2×#191047582
C2×SD16×D1632
#237822402
#237822332
#237820582
#237820542
#237820512
#237820502
#237820472
#237820392
#237820362
#237816222
#497×C1632
#3154163