Finite Group Databasebeta

← #35225009

Subgroups — #35225009

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×#19104759C211yesmaximal, characteristic, complemented
3C2×#19104754C211yesmaximal, characteristic, complemented
4C2×SD16×D163C211yesmaximal, characteristic, complemented
5#23782235C211yesmaximal, characteristic, complemented
6#23782228C211yesmaximal, characteristic, complemented
7#23782059C211yesmaximal, characteristic, complemented
8#23782056C211yesmaximal, characteristic, complemented
9#23782053C211yesmaximal, characteristic, complemented
10#23782048C211yesmaximal, characteristic, complemented
11#23782047C211yesmaximal, characteristic, complemented
12#23782037C211yesmaximal, characteristic, complemented
13#23782036C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#496×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic
20#19104764—24no—
21#19104759—24no—
22#19104754—24no—
23C2×Dic652C2211yescharacteristic, complemented
24C2×Dic652C2211yescharacteristic, complemented
25C2×#14579689C2211yescharacteristic, complemented
26C2×#14579688C2211yescharacteristic
27C2×#14579688C2211yescharacteristic
28C2×#14579687C2211yescharacteristic
29C2×#14579687C2211yescharacteristic
30C2×#14579687C2211yescharacteristic, complemented
31C2×#14579687C2211yescharacteristic, complemented
32C2×#14579687C2211yescharacteristic, complemented
33C2×#14579686C2211yescharacteristic
34SD16×D163—24no—
35C2×#14579684C2211yescharacteristic
36C2×#14579682C2211yescharacteristic
37C2×C8×D163C2211yescharacteristic
38#19104735C2211yescharacteristic, complemented
39#19104733C2211yescharacteristic, complemented
40#19104731C2211yescharacteristic
41#19104723C2211yescharacteristic
42#19104722C2211yescharacteristic
43#19104722C2211yescharacteristic, complemented
44#19104717—22no—
45#19104712—22no—
46#19104707—22no—
47#19104703—22no—
48#19104692C2211yescharacteristic
49#19104645C2211yescharacteristic, complemented
50C4×C163⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
55C2×D8×C163C2211yescharacteristic, complemented
56#127×C163C2211yescharacteristic
57#124×C163C2211yescharacteristic, complemented
58#97×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579698—22no—
61#14579698—22no—
62#14579698—22no—
63#14579698—22no—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68Q8×D163—22no—
69Q8×D163—22no—
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—22no—
75D4×D163—22no—
76D4×D163—22no—
77C22×Dic163—21no—
78C22×Dic163—21no—
79C2×C4×D163—21no—
80C2×C4×D163C2311yescharacteristic
81Dic652—22no—
82Dic652—22no—
83#14579689—22no—
84#14579688—22no—
85#14579688—22no—
86#14579687—22no—
87#14579687—22no—
88#14579687—22no—
89#14579687—22no—
90#14579687—22no—
91#14579686—22no—
92#14579684—22no—
93#14579682D414yescomplemented
94C2×C163⋊C8C2311yescharacteristic
95C2×C163⋊C8C2311yescharacteristic
96C8×D163D414yescomplemented
97C4⋊Dic163—22no—
98Dic163⋊C4—22no—
99Dic163⋊C4D412yes—
100C22⋊Dic163—22no—
101C22⋊Dic163—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4D412yes—
104C4×Dic163C2311yescharacteristic
105C2×Q8×C163C2311yescharacteristic
106C2×D4×C163—22no—
107C2×D4×C163C2311yescharacteristic
108C2×D4×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110SD16×C163—22no—
111SD16×C163—22no—
112D8×C163—22no—
113D8×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C22⋊C4×C163—22no—
117C42×C163C2311yescharacteristic
118C22×D163—21no—
119C22×D163—42no—
120C22×D163—82no—
121C22×D163C2×D411yescharacteristic
122Dic326—21no—
123Dic326—21no—
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129D652—21no—
130D652—21no—
131C22⋊D163—42no—
132C22⋊D163—42no—
133C22⋊D163—42no—
134C22⋊D163—82no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—42no—
139C2×Dic163—42no—
140C2×Dic163C2×D412yes—
141C2×Dic163C2×D411yescharacteristic
142C4×D163—42no—
143C4×D163C2×D412yes—
144C4×D163C2×D412yes—
145C163⋊C8C2×D412yes—
146C163⋊C8C2×D412yes—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152D4×C163—21no—
153D4×C163—21no—
154D4×C163—21no—
155D4×C163—21no—
156D4×C163—21no—
157D4×C163—21no—
158D4×C163—82no—
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—41no—
167C22×C163—42no—
168C22×C163—42no—
169C22×C163—42no—
170C22×C163C22×D411yescharacteristic
171C2×D163—22no—
172C2×D163—41no—
173C2×D163—42no—
174C2×D163—84no—
175C2×D163#12712yes—
176C4×C163—22no—
177C4×C163—41no—
178C4×C163C22×D411yescharacteristic
179C4×C163C22×D411yescharacteristic
180Dic163—22no—
181Dic163—41no—
182Dic163—41no—
183Dic163—41no—
184Dic163#12712yes—
185C2×C163—81no—
186C2×C163—81no—
187C2×C163—81no—
188C2×C163C2×#12711yescharacteristic
189C2×C163#57712yes—
190D163—22no—
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#496D16311yescharacteristic, complemented
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×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×SD16—1631no—
227C2×SD16—1631no—
228C2×SD16C2×D16311yescharacteristic
229C2×SD16C2×D16311yescharacteristic, complemented
230C2×D8—1631no—
231C2×D8C2×D16311yescharacteristic
232C2×D8C2×D16311yescharacteristic, complemented
233C2×M16—1631no—
234C22×C8—1631no—
235C4⋊Q8—1631no—
236C4⋊Q8—1631no—
237#127—1631no—
238#127C2×D16311yescharacteristic
239#124—1631no—
240#124—1631no—
241#124—1631no—
242#124C2×D16311yescharacteristic, complemented
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×Q8—1631no—
262C2×Q8—1631no—
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×D4C22×D16311yescharacteristic
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—3262no—
285SD16—3262no—
286SD16—3262no—
287SD16—3262no—
288SD16—3262no—
289SD16—3262no—
290SD16—3262no—
291SD16—3262no—
292D8—22no—
293D8—22no—
294D8—3262no—
295M16—1634no—
296C2×C8—1631no—
297C2×C8—1631no—
298C2×C8—1634no—
299C2×C8C22×D16311yescharacteristic
300C2×C8C22×D16311yescharacteristic
301C4⋊C4—3262no—
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—21no—
313C23—21no—
314C23—13042no—
315C23—3261no—
316C23—6522no—
317C23—1631no—
318Q8—21no—
319Q8—21no—
320Q8—3261no—
321Q8—3261no—
322Q8—3261no—
323Q8—3261no—
324Q8—3261no—
325Q8—3261no—
326Q8—6522no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—82no—
334D4—13042no—
335D4—3261no—
336D4—3261no—
337D4—6522no—
338D4—6522no—
339D4—6522no—
340C2×C4—21no—
341C2×C4—3261no—
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—41no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—13044no—
364C22—3262no—
365C22—6521no—
366C22—6522no—
367C22—1632no—
368C22C2×D4×D16311yescharacteristic
369C4—22no—
370C4—41no—
371C4—3262no—
372C4—6521no—
373C4—6521no—
374C4—6521no—
375C4—1632no—
376C4C2×D4×D16311yescharacteristic
377C4C2×D4×D16311yescharacteristic
378C2—81no—
379C2—81no—
380C2—81no—
381C2—13041no—
382C2—3262no—
383C2#127×D16311yescharacteristic
384C2#2378247512yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#496×C163C2
socleC22×C163C22×D4
hypercenter#496D163
hypocenterC163#3154
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19104764C2semidirect
2. C2×#19104759C2semidirect
3. C2×#19104754C2semidirect
4. C2×SD16×D163C2semidirect
5. #23782235C2semidirect
6. #23782228C2semidirect
7. #23782059C2semidirect
8. #23782056C2semidirect
9. #23782053C2semidirect
10. #23782048C2semidirect
11. #23782047C2semidirect
12. #23782037C2semidirect
13. #23782036C2semidirect
14. #23781622C2semidirect
15. #496×C163C2semidirect
23. C2×Dic652C22semidirect
24. C2×Dic652C22semidirect
25. C2×#14579689C22semidirect
30. C2×#14579687C22semidirect
31. C2×#14579687C22semidirect
32. C2×#14579687C22semidirect
38. #19104735C22semidirect
39. #19104733C22semidirect
43. #19104722C22semidirect
49. #19104645C22semidirect
50. C4×C163⋊C8C22semidirect
51. C8×Dic163C22semidirect
53. C2×SD16×C163C22semidirect
55. C2×D8×C163C22semidirect
57. #124×C163C22semidirect
58. #97×C163C22semidirect
93. #14579682D4semidirect
96. C8×D163D4semidirect
192. C163#3154semidirect
203. #496D163semidirect
229. C2×SD16C2×D163semidirect
232. C2×D8C2×D163semidirect
242. #124C2×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×#191047642
C2×#191047592
C2×#191047542
C2×SD16×D1632
#237822352
#237822282
#237820592
#237820562
#237820532
#237820482
#237820472
#237820372
#237820362
#237816222
#496×C1632
#3154163