Finite Group Databasebeta

← #35224043

Subgroups — #35224043

Order 20864.

494 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic
2C2×SD16×D163C211yesmaximal, characteristic, complemented
3#23782240C211yesmaximal, characteristic, complemented
4#23782137C211yesmaximal, characteristic, complemented
5#23781883C211yesmaximal, characteristic, complemented
6#23781882C211yesmaximal, characteristic, complemented
7#23781881C211yesmaximal, characteristic, complemented
8#23781880C211yesmaximal, characteristic, complemented
9#23781869C211yesmaximal, characteristic, complemented
10#23781868C211yesmaximal, characteristic, complemented
11#23781867C211yesmaximal, characteristic, complemented
12#23781690C211yesmaximal, characteristic, complemented
13#23781672C211yesmaximal, characteristic, complemented
14#23781660C211yesmaximal, characteristic, complemented
15#461×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104762—24no—
20C2×#14579688C2211yescharacteristic
21C2×#14579687C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23C2×#14579686C2211yescharacteristic
24SD16×D163—24no—
25C2×D1304C2211yescharacteristic
26C2×#14579684C2211yescharacteristic
27C2×#14579682C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic
29#19104733C2211yescharacteristic, complemented
30#19104724C2211yescharacteristic, complemented
31#19104723C2211yescharacteristic, complemented
32#19104720—21no—
33#19104720—21no—
34#19104713—21no—
35#19104713—21no—
36C4×D652C2211yescharacteristic
37D4×Dic163C2211yescharacteristic
38#19104697C2211yescharacteristic
39C4×C22⋊D163—21no—
40(C22⋊D163)⋊C4—21no—
41#19104693C2211yescharacteristic
42#19104692C2211yescharacteristic
43C2×(C2×D163)⋊C4—21no—
44C22⋊C4×D163—21no—
45C4⋊(C163⋊C8)C2211yescharacteristic, complemented
46Dic163⋊C8C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104659C2211yescharacteristic, complemented
51#19104658C2211yescharacteristic, complemented
52#19104657C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic
55C2×SD16×C163C2211yescharacteristic
56C4⋊Q8×C163C2211yescharacteristic, complemented
57C4×D4×C163C2211yescharacteristic
58C4⋊C8×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#103×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579700—22no—
63#14579700—22no—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66Q8×D163—22no—
67Q8×D163—22no—
68C2×D652—21no—
69C2×D652C2311yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72D4×D163—22no—
73D4×D163D414yes—
74C22×Dic163—21no—
75C2×C4×D163—21no—
76C2×C4×D163—21no—
77C2×C4×D163—21no—
78C2×C4×D163—21no—
79C2×C4×D163C2311yescharacteristic
80#14579688—22no—
81#14579687—22no—
82#14579686—22no—
83#14579686—22no—
84D1304—22no—
85#14579684—22no—
86#14579682—22no—
87C2×C163⋊C8C2311yescharacteristic
88C2×C163⋊C8C2311yescharacteristic
89C8×D163—22no—
90C4⋊Dic163C2311yescharacteristic
91Dic163⋊C4—21no—
92Dic163⋊C4—21no—
93Dic163⋊C4D411yescharacteristic
94Dic163⋊C4D411yescharacteristic
95C22⋊Dic163—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4D411yescharacteristic
104(C2×D163)⋊C4D411yescharacteristic
105C4×Dic163—21no—
106C4×Dic163C2311yescharacteristic
107C2×Q8×C163C2311yescharacteristic
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110C22×C4×C163—21no—
111SD16×C163—22no—
112SD16×C163—22no—
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C4⋊C4×C163—21no—
116C4⋊C4×C163—21no—
117C4⋊C4×C163C2311yescharacteristic
118C22⋊C4×C163—21no—
119C42×C163C2311yescharacteristic
120C22×D163—21no—
121C22×D163—24no—
122C22×D163—21no—
123C22×D163—81no—
124C22×D163—81no—
125C22×D163C2×D411yescharacteristic
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129D652—21no—
130D652—21no—
131D652—21no—
132D652—42no—
133D652C2×D412yes—
134C22⋊D163—24no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—81no—
138C2×Dic163C2×D411yescharacteristic
139C2×Dic163C2×D411yescharacteristic
140C2×Dic163C2×D411yescharacteristic
141C4×D163—42no—
142C4×D163—42no—
143C4×D163—81no—
144C4×D163—81no—
145C4×D163C2×D412yes—
146C4×D163#4112yes—
147C163⋊C8—21no—
148C163⋊C8—21no—
149C23×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152Q8×C163—21no—
153Q8×C163—21no—
154D4×C163—21no—
155D4×C163C2×D412yes—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—81no—
160C2×C4×C163C2×D411yescharacteristic
161C2×C4×C163C2×D411yescharacteristic
162C2×C4×C163C2411yescharacteristic
163C8×C163—21no—
164C8×C163—21no—
165C22×C163—22no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163C22×D411yescharacteristic
169C2×D163—21no—
170C2×D163—21no—
171C2×D163—22no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—84no—
178C2×D163#12112yes—
179C4×C163—21no—
180C4×C163—21no—
181C4×C163—41no—
182C4×C163—41no—
183C4×C163—41no—
184C4×C163C22×D411yescharacteristic
185C4×C163C2×#4111yescharacteristic
186Dic163—21no—
187Dic163—21no—
188Dic163—41no—
189Dic163—41no—
190Dic163#12112yes—
191C2×C163—42no—
192C2×C163C2×#12111yescharacteristic
193C2×C163#57711yescharacteristic
194C2×C163#57711yescharacteristic
195D163—22no—
196D163—42no—
197D163—81no—
198C163#304211yescharacteristic, complemented, Sylow
199#3042—1631nomaximal, Sylow
200C2×AGL(1,ℤ/8)—1631no—
201C22×SD16—1631no—
202#536—1631no—
203#518—1631no—
204#463—1631no—
205#463—1631no—
206#463—1631no—
207#463—1631no—
208#461—1631no—
209#461—1631no—
210#461—1631no—
211#461D16311yescharacteristic, complemented
212#424—1631no—
213#419—1631no—
214#416—1631no—
215C2×#41—1631no—
216C22×Q8—1631no—
217C22×D4—1631no—
218AGL(1,ℤ/8)—3264no—
219C2×SD16—3264no—
220C2×SD16—1631no—
221C2×SD16—1631no—
222C2×SD16—1631no—
223C2×SD16—1631no—
224C2×SD16C2×D16311yescharacteristic
225C2×SD16C2×D16311yescharacteristic
226C2×D8—1631no—
227C2×D8—1631no—
228C2×M16—1631no—
229C22×C8—1631no—
230C4⋊Q8—1631no—
231C4⋊Q8C2×D16311yescharacteristic, complemented
232#124—1631no—
233#124—1631no—
234#123—3261no—
235#123—3261no—
236C22⋊Q8—3261no—
237C22⋊Q8—3261no—
238C4×D4—3261no—
239C4×D4—3261no—
240C4×D4—1631no—
241C4×D4—1631no—
242C4×D4—1631no—
243C4×D4C2×D16311yescharacteristic
244#117—1631no—
245#117—1631no—
246C2×C22⋊C4—3261no—
247C2×C22⋊C4—3261no—
248C4⋊C8—1631no—
249C4⋊C8—1631no—
250C4⋊C8—1631no—
251C4⋊C8C2×D16311yescharacteristic, complemented
252#103—1631no—
253#103—1631no—
254#103—1631no—
255#103—1631no—
256#103C2×D16311yescharacteristic, complemented
257#103C2×D16311yescharacteristic, complemented
258#102—1631no—
259#102—1631no—
260C24—3261no—
261#41—3262no—
262#41—3262no—
263C2×Q8—3261no—
264C2×Q8—3262no—
265C2×Q8—3262no—
266C2×Q8—1631no—
267C2×Q8C22×D16311yescharacteristic
268C2×Q8C22×D16311yescharacteristic
269C2×D4—3261no—
270C2×D4—3262no—
271C2×D4—3261no—
272C2×D4—1634no—
273C2×D4—1631no—
274C2×D4—1631no—
275C2×D4C22×D16311yescharacteristic
276C22×C4—21no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—3261no—
280C22×C4—3261no—
281C22×C4—3261no—
282C22×C4—1631no—
283SD16—22no—
284SD16—22no—
285SD16—3262no—
286SD16—3262no—
287SD16—3262no—
288SD16—3262no—
289D8—3262no—
290D8—3262no—
291M16—3262no—
292C2×C8—3262no—
293C2×C8—1631no—
294C2×C8—1631no—
295C2×C8C22×D16311yescharacteristic
296C2×C8C22×D16311yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—3261no—
300C4⋊C4—3261no—
301C4⋊C4—1631no—
302C4⋊C4—1631no—
303C4⋊C4—1631no—
304C4⋊C4C22×D16311yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—3261no—
307C22⋊C4—3261no—
308C22⋊C4—3261no—
309C22⋊C4—3261no—
310C22⋊C4—3261no—
311C22⋊C4—3261no—
312C22⋊C4—3261no—
313C22⋊C4—3261no—
314C22⋊C4—1631no—
315C22⋊C4—1631no—
316C42—3261no—
317C42—1631no—
318C42C22×D16311yescharacteristic
319C23—21no—
320C23—13041no—
321C23—13041no—
322C23—3264no—
323C23—3261no—
324C23—3261no—
325C23—1631no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—3261no—
331Q8—3261no—
332Q8—6522no—
333D4—21no—
334D4—3264no—
335D4—3261no—
336D4—3261no—
337D4—3261no—
338D4—6522no—
339D4—1632no—
340D4D4×D16312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—13041no—
346C2×C4—13041no—
347C2×C4—13041no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—6522no—
351C2×C4—6522no—
352C2×C4—1631no—
353C2×C4—1631no—
354C2×C4—1631no—
355C2×C4—1632no—
356C2×C4—1632no—
357C2×C4D4×D16311yescharacteristic
358C2×C4D4×D16311yescharacteristic
359C2×C4C23×D16311yescharacteristic
360C8—21no—
361C8—21no—
362C8—3261no—
363C8—3261no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—13044no—
368C22—3261no—
369C22—3261no—
370C22—3262no—
371C22—6521no—
372C22—6521no—
373C22—6521no—
374C22—6521no—
375C22—6521no—
376C22—1632no—
377C22C2×D4×D16311yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3261no—
384C4—3261no—
385C4—6521no—
386C4—6521no—
387C4—1632no—
388C4C2×D4×D16311yescharacteristic
389C4#41×D16311yescharacteristic
390C2—42no—
391C2—13041no—
392C2—3262no—
393C2—6522no—
394C2#121×D16311yescharacteristic
395C2#2378247111yescharacteristic
396C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#461×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104762C2
hypercenter#461D163
hypocenterC163#3042
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D163C2semidirect
3. #23782240C2semidirect
4. #23782137C2semidirect
5. #23781883C2semidirect
6. #23781882C2semidirect
7. #23781881C2semidirect
8. #23781880C2semidirect
9. #23781869C2semidirect
10. #23781868C2semidirect
11. #23781867C2semidirect
12. #23781690C2semidirect
13. #23781672C2semidirect
14. #23781660C2semidirect
15. #461×C163C2semidirect
29. #19104733C22semidirect
30. #19104724C22semidirect
31. #19104723C22semidirect
45. C4⋊(C163⋊C8)C22semidirect
46. Dic163⋊C8C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104659C22semidirect
51. #19104658C22semidirect
52. #19104657C22semidirect
53. #19104655C22semidirect
56. C4⋊Q8×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #103×C163C22semidirect
60. #103×C163C22semidirect
198. C163#3042semidirect
211. #461D163semidirect
231. C4⋊Q8C2×D163semidirect
251. C4⋊C8C2×D163semidirect
256. #103C2×D163semidirect
257. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C2×SD16×D1632
#237822402
#237821372
#237818832
#237818822
#237818812
#237818802
#237818692
#237818682
#237818672
#237816902
#237816722
#237816602
#461×C1632
#3042163