Finite Group Databasebeta

← #35226050

Subgroups — #35226050

Order 20864.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104764C211yesmaximal, characteristic, complemented
2C22×#14579687C211yesmaximal, characteristic, complemented
3#23782328C211yesmaximal, characteristic, complemented
4#23782309C211yesmaximal, characteristic
5#23782056C211yesmaximal, characteristic, complemented
6#23781910C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781886C211yesmaximal, characteristic, complemented
9#23781868C211yesmaximal, characteristic, complemented
10#23781828C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12#23781802C211yesmaximal, characteristic, complemented
13#23781762C211yesmaximal, characteristic, complemented
14#23781733C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic, complemented
19#19104764—24no—
20C2×Dic652C2211yescharacteristic, complemented
21C2×Dic652C2211yescharacteristic, complemented
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579687—24no—
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic
26C2×#14579687C2211yescharacteristic
27C2×#14579687C2211yescharacteristic
28C2×#14579682C2211yescharacteristic, complemented
29C22×C163⋊C8C2211yescharacteristic
30#19104734—21no—
31#19104733C2211yescharacteristic, complemented
32#19104722C2211yescharacteristic
33#19104716—21no—
34#19104715C2211yescharacteristic, complemented
35#19104714C2211yescharacteristic
36#19104713—21no—
37#19104712—21no—
38#19104707—21no—
39#19104706C2211yescharacteristic
40#19104704—21no—
41#19104700C2211yescharacteristic, complemented
42D4×Dic163C2211yescharacteristic
43C4×C22⋊D163C2211yescharacteristic
44(C22⋊D163)⋊C4—21no—
45C2×C22⋊Dic163—21no—
46#19104668C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104659C2211yescharacteristic, complemented
51#19104658C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104645C2211yescharacteristic, complemented
54C22×D4×C163C2211yescharacteristic
55C2×SD16×C163C2211yescharacteristic
56C2×SD16×C163C2211yescharacteristic
57C22⋊Q8×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#102×C163C2211yescharacteristic
60C22⋊C8×C163C2211yescharacteristic
61#14579698—22no—
62#14579698—41no—
63C2×Dic326—21no—
64C2×Dic326—22no—
65C2×Dic326—21no—
66C2×Dic326—41no—
67C2×Dic326C2311yescharacteristic
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70Q8×D163—22no—
71Q8×D163—41no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163D411yescharacteristic
75C22×Dic163—21no—
76C22×Dic163—21no—
77C22×Dic163—21no—
78C2×C4×D163—21no—
79C2×C4×D163C2311yescharacteristic
80Dic652—22no—
81Dic652—41no—
82#14579689—22no—
83#14579687—22no—
84#14579687—22no—
85#14579687—22no—
86#14579687D414yescomplemented
87#14579682—22no—
88C2×C163⋊C8—22no—
89C2×C163⋊C8C2311yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C4⋊Dic163—21no—
92C4⋊Dic163C2311yescharacteristic
93Dic163⋊C4—21no—
94Dic163⋊C4—21no—
95Dic163⋊C4—21no—
96Dic163⋊C4—21no—
97Dic163⋊C4D411yescharacteristic
98C22⋊Dic163—21no—
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163D411yescharacteristic
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4D411yescharacteristic
106C4×Dic163—21no—
107C4×Dic163C2311yescharacteristic
108C24×C163—21no—
109C2×Q8×C163C2311yescharacteristic
110C2×D4×C163—22no—
111C2×D4×C163—21no—
112C2×D4×C163—41no—
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115C22×C4×C163C2311yescharacteristic
116SD16×C163—22no—
117SD16×C163—41no—
118C2×C8×C163C2311yescharacteristic
119C2×C8×C163C2311yescharacteristic
120C4⋊C4×C163—21no—
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C22×D163C2×D411yescharacteristic
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—21no—
129Dic326—42no—
130Dic326—42no—
131Dic326C2×D412yes—
132C22⋊D163—22no—
133C22⋊D163—42no—
134C22⋊D163—81no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—42no—
140C2×Dic163—42no—
141C2×Dic163—81no—
142C2×Dic163C2×D411yescharacteristic
143C2×Dic163C2×D411yescharacteristic
144C2×Dic163C2×D411yescharacteristic
145C4×D163—21no—
146C4×D163—21no—
147C4×D163—42no—
148C163⋊C8—21no—
149C163⋊C8C2×D412yes—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—42no—
153C23×C163—42no—
154C23×C163—81no—
155C23×C163C2×D411yescharacteristic
156Q8×C163—21no—
157Q8×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163—42no—
162D4×C163C2×D412yes—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163—21no—
171C22×C163—21no—
172C22×C163—21no—
173C22×C163—22no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—84no—
180C22×C163#13712yescomplemented
181C22×C163C22×D411yescharacteristic
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C4×C163—21no—
186C4×C163—41no—
187C4×C163—41no—
188C4×C163C22×D411yescharacteristic
189C4×C163H(5,𝔽2)11yescharacteristic
190Dic163—21no—
191Dic163—22no—
192Dic163—21no—
193Dic163—21no—
194Dic163—41no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—22no—
198C2×C163—42no—
199C2×C163—81no—
200C2×C163D4211yescharacteristic
201C2×C163C2×#13711yescharacteristic
202C2×C163#57711yescharacteristic
203D163—41no—
204C163#326611yescharacteristic, complemented, Sylow
205#3266—1631nomaximal, Sylow
206C2×#137—1631no—
207C22×SD16—1631no—
208D4×Q8—1631no—
209#546—1631no—
210#497—1631no—
211#468—1631no—
212#467—1631no—
213#464—1631no—
214#461—1631no—
215#452—1631no—
216#451—1631no—
217#450D16311yescharacteristic, complemented
218#448—1631no—
219#440—1631no—
220#435—1631no—
221C2×#41—1631no—
222C22×Q8—1631no—
223C22×Q8—1631no—
224C22×D4C2×D16311yescharacteristic
225#137—3264no—
226C2×Q16—1631no—
227C2×Q16—1631no—
228C2×Q16—1631no—
229C2×SD16—3264no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×SD16—1631no—
233C2×SD16—1631no—
234C2×SD16C2×D16311yescharacteristic
235C2×SD16C2×D16311yescharacteristic
236C2×M16—1631no—
237C22×C8—1631no—
238C4⋊Q8—3261no—
239C4⋊Q8—1631no—
240#124—1631no—
241#123—3261no—
242C22⋊Q8—3261no—
243C22⋊Q8—3261no—
244C22⋊Q8—1631no—
245C22⋊Q8—1631no—
246C22⋊Q8C2×D16311yescharacteristic
247#121—3261no—
248#121—1631no—
249#120—3261no—
250C4×Q8—1631no—
251C4×D4—3261no—
252C4×D4—1631no—
253C4×D4—1631no—
254C2×C22⋊C4—3261no—
255#107—1631no—
256C4⋊C8—1631no—
257#103—1631no—
258#103—1631no—
259#103—1631no—
260#103—1631no—
261#102C2×D16311yescharacteristic
262#102C2×D16311yescharacteristic
263C22⋊C8—1631no—
264C22⋊C8C2×D16311yescharacteristic
265#97—1631no—
266C24—21no—
267#41—3262no—
268#41—6521no—
269C2×Q8—3261no—
270C2×Q8—3262no—
271C2×Q8—3261no—
272C2×Q8—3262no—
273C2×Q8—6521no—
274C2×Q8—6521no—
275C2×Q8—1631no—
276C2×Q8—1631no—
277C2×Q8—1631no—
278C2×Q8C22×D16311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—1631no—
285C2×D4C22×D16311yescharacteristic
286C2×D4C22×D16311yescharacteristic
287C22×C4—3261no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4—1631no—
292C22×C4C22×D16311yescharacteristic
293Q16—3262no—
294Q16—3262no—
295Q16—6521no—
296SD16—22no—
297SD16—41no—
298SD16—3262no—
299SD16—3262no—
300SD16—3262no—
301SD16—1634no—
302M16—3262no—
303C2×C8—3262no—
304C2×C8—1631no—
305C2×C8—1631no—
306C2×C8C22×D16311yescharacteristic
307C2×C8C22×D16311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3261no—
310C4⋊C4—3261no—
311C4⋊C4—3261no—
312C4⋊C4—3261no—
313C4⋊C4—3261no—
314C4⋊C4—1631no—
315C4⋊C4—1631no—
316C4⋊C4C22×D16311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3261no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—1631no—
325C22⋊C4—1631no—
326C42—3261no—
327C42—1631no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1631no—
334C23D4×D16311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3261no—
338Q8—3261no—
339Q8—3261no—
340Q8—3261no—
341Q8—3261no—
342Q8—6522no—
343Q8—6522no—
344Q8—1632no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—13041no—
350D4—3262no—
351D4—6522no—
352D4D4×D16312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—13041no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—3261no—
361C2×C4—3261no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—6522no—
365C2×C4—6522no—
366C2×C4—6522no—
367C2×C4—1631no—
368C2×C4—1631no—
369C2×C4—1631no—
370C2×C4D4×D16311yescharacteristic
371C2×C4C23×D16311yescharacteristic
372C8—21no—
373C8—21no—
374C8—3261no—
375C8—1632no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3261no—
386C22—3261no—
387C22—3261no—
388C22#1910476412yescomplemented
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3261no—
394C4—3261no—
395C4—3261no—
396C4—3262no—
397C4—6521no—
398C4—6521no—
399C4—6521no—
400C4C2×D4×D16311yescharacteristic
401C4#1910478011yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—6521no—
406C2#2378229711yescharacteristic
407C2C2×#1910476411yescharacteristic
408C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
prime radical#23782309C2
hypercenter#450D163
hypocenterC163#3266
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19104764C2semidirect
2. C22×#14579687C2semidirect
3. #23782328C2semidirect
5. #23782056C2semidirect
6. #23781910C2semidirect
7. #23781904C2semidirect
8. #23781886C2semidirect
9. #23781868C2semidirect
10. #23781828C2semidirect
11. #23781823C2semidirect
12. #23781802C2semidirect
13. #23781762C2semidirect
14. #23781733C2semidirect
15. #450×C163C2semidirect
18. C2×Q8×D163C22semidirect
20. C2×Dic652C22semidirect
21. C2×Dic652C22semidirect
22. C2×#14579689C22semidirect
28. C2×#14579682C22semidirect
31. #19104733C22semidirect
34. #19104715C22semidirect
41. #19104700C22semidirect
46. #19104668C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
50. #19104659C22semidirect
51. #19104658C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104645C22semidirect
86. #14579687D4semidirect
180. C22×C163#137semidirect
204. C163#3266semidirect
217. #450D163semidirect
388. C22#19104764semidirect

Maximal subgroups

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

subgroupindex
C2×#191047642
C22×#145796872
#237823282
#237823092
#237820562
#237819102
#237819042
#237818862
#237818682
#237818282
#237818232
#237818022
#237817622
#237817332
#450×C1632
#3266163