Finite Group Databasebeta

← #35226041

Subgroups — #35226041

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×#19104767C211yesmaximal, characteristic, complemented
2C2×SD16×D163C211yesmaximal, characteristic, complemented
3Q8×D652C211yesmaximal, characteristic, complemented
4#23782308C211yesmaximal, characteristic
5#23782060C211yesmaximal, characteristic, complemented
6#23781910C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781871C211yesmaximal, characteristic, complemented
9#23781830C211yesmaximal, characteristic, complemented
10#23781819C211yesmaximal, characteristic, complemented
11#23781817C211yesmaximal, characteristic, complemented
12#23781802C211yesmaximal, characteristic, complemented
13#23781763C211yesmaximal, characteristic, complemented
14#23781735C211yesmaximal, characteristic, complemented
15#464×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic, complemented
19C2×D4×D163C2211yescharacteristic
20#19104767—24no—
21C2×#14579689C2211yescharacteristic, complemented
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579688C2211yescharacteristic
24C2×#14579688C2211yescharacteristic
25C2×#14579687C2211yescharacteristic
26C2×#14579687C2211yescharacteristic
27C2×#14579686C2211yescharacteristic
28SD16×D163—24no—
29C2×#14579682C2211yescharacteristic, complemented
30C2×C8×D163C2211yescharacteristic
31#19104735—21no—
32#19104735C2211yescharacteristic, complemented
33#19104717—21no—
34#19104715—21no—
35#19104715—21no—
36#19104715C2211yescharacteristic
37#19104715C2211yescharacteristic, complemented
38#19104711C2211yescharacteristic
39#19104708—21no—
40#19104706C2211yescharacteristic
41#19104705—21no—
42C4×D652—21no—
43C4×D652C2211yescharacteristic
44C4×D652C2211yescharacteristic
45C22⋊C4×D163—21no—
46#19104668C2211yescharacteristic, complemented
47#19104660C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104655C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104646C2211yescharacteristic, complemented
54C2×Q16×C163C2211yescharacteristic, complemented
55C2×SD16×C163C2211yescharacteristic
56#124×C163C2211yescharacteristic
57C4×Q8×C163C2211yescharacteristic, complemented
58C4⋊C8×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic
61C23×D163—21no—
62#14579700—22no—
63#14579700—41no—
64C2×Dic326—21no—
65C2×Dic326—21no—
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68Q8×D163—22no—
69Q8×D163—22no—
70Q8×D163—41no—
71Q8×D163—41no—
72C2×D652—21no—
73C2×D652D411yescharacteristic
74C2×D652C2311yescharacteristic
75C2×C22⋊D163—21no—
76C2×C22⋊D163—21no—
77D4×D163—22no—
78D4×D163—41no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163—21no—
82C2×C4×D163—21no—
83C2×C4×D163C2311yescharacteristic
84C2×C4×D163C2311yescharacteristic
85#14579689—22no—
86#14579689—41no—
87#14579688—22no—
88#14579688—22no—
89#14579687—22no—
90#14579687—41no—
91#14579686D414yescomplemented
92#14579682—22no—
93C2×C163⋊C8C2311yescharacteristic
94C2×C163⋊C8C2311yescharacteristic
95C8×D163—22no—
96C4⋊Dic163—21no—
97C4⋊Dic163—21no—
98C4⋊Dic163—21no—
99C4⋊Dic163—21no—
100C4⋊Dic163D411yescharacteristic
101C4⋊Dic163C2311yescharacteristic
102Dic163⋊C4—21no—
103C22⋊Dic163—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4D411yescharacteristic
109(C2×D163)⋊C4D411yescharacteristic
110C2×Q8×C163C2311yescharacteristic
111C2×Q8×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113Q16×C163—22no—
114SD16×C163—22no—
115C2×C8×C163C2311yescharacteristic
116C2×C8×C163C2311yescharacteristic
117C4⋊C4×C163—21no—
118C4⋊C4×C163C2311yescharacteristic
119C22⋊C4×C163—21no—
120C22⋊C4×C163—21no—
121C42×C163—21no—
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163—81no—
127C22×D163C2×D411yescharacteristic
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—21no—
133Dic326—42no—
134Dic326—42no—
135D652—22no—
136D652—21no—
137D652—42no—
138D652C2×D412yes—
139C22⋊D163—42no—
140C22⋊D163—81no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163C2×D411yescharacteristic
145C2×Dic163C2×D411yescharacteristic
146C4×D163—21no—
147C4×D163—21no—
148C4×D163—21no—
149C4×D163—21no—
150C4×D163—42no—
151C4×D163—42no—
152C4×D163—42no—
153C4×D163—81no—
154C163⋊C8—21no—
155C163⋊C8C2×D412yes—
156C23×C163—21no—
157Q8×C163—21no—
158Q8×C163—21no—
159Q8×C163—21no—
160Q8×C163C2×D412yes—
161D4×C163—21no—
162D4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163—21no—
171C22×C163—41no—
172C22×C163—41no—
173C22×C163—41no—
174C22×C163C22×D411yescharacteristic
175C2×D163—21no—
176C2×D163—21no—
177C2×D163—21no—
178C2×D163—22no—
179C2×D163—21no—
180C2×D163—21no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—84no—
184C2×D163#13712yescomplemented
185C4×C163—22no—
186C4×C163—21no—
187C4×C163—21no—
188C4×C163—41no—
189C4×C163—41no—
190C4×C163C22×D411yescharacteristic
191C4×C163H(5,𝔽2)11yescharacteristic
192Dic163—21no—
193Dic163—21no—
194Dic163—41no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—81no—
198C2×C163D4211yescharacteristic
199C2×C163C2×#13711yescharacteristic
200C2×C163#57711yescharacteristic
201D163—22no—
202D163—41no—
203D163—42no—
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#464D16311yescharacteristic, complemented
214#461—1631no—
215#452—1631no—
216#451—1631no—
217#450—1631no—
218#448—1631no—
219#440—1631no—
220#435—1631no—
221C2×#41—1631no—
222C22×Q8—1631no—
223C22×Q8—1631no—
224C22×D4—1631no—
225#137—3264no—
226C2×Q16—1631no—
227C2×Q16—1631no—
228C2×Q16C2×D16311yescharacteristic, complemented
229C2×SD16—3264no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×SD16—1631no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16C2×D16311yescharacteristic
236C2×M16—1631no—
237C22×C8—1631no—
238C4⋊Q8—3261no—
239C4⋊Q8—1631no—
240#124C2×D16311yescharacteristic
241#123—3261no—
242C22⋊Q8—3261no—
243C22⋊Q8—3261no—
244C22⋊Q8—1631no—
245C22⋊Q8—1631no—
246C22⋊Q8—1631no—
247#121—3261no—
248#121—1631no—
249#120—3261no—
250C4×Q8C2×D16311yescharacteristic, complemented
251C4×D4—3261no—
252C4×D4—1631no—
253C4×D4—1631no—
254C2×C22⋊C4—3261no—
255#107—1631no—
256C4⋊C8C2×D16311yescharacteristic, complemented
257#103—1631no—
258#103—1631no—
259#103—1631no—
260#103C2×D16311yescharacteristic, complemented
261#102—1631no—
262#102C2×D16311yescharacteristic
263C22⋊C8—1631no—
264C22⋊C8—1631no—
265#97—1631no—
266C24—3261no—
267#41—3262no—
268#41—6521no—
269C2×Q8—3261no—
270C2×Q8—3261no—
271C2×Q8—3262no—
272C2×Q8—3262no—
273C2×Q8—6521no—
274C2×Q8—6521no—
275C2×Q8—1631no—
276C2×Q8—1631no—
277C2×Q8C22×D16311yescharacteristic
278C2×Q8C22×D16311yescharacteristic
279C2×D4—3261no—
280C2×D4—3262no—
281C2×D4—3261no—
282C2×D4—3261no—
283C2×D4—6521no—
284C2×D4—1631no—
285C2×D4—1631no—
286C2×D4C22×D16311yescharacteristic
287C22×C4—3261no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4—1631no—
292C22×C4—1631no—
293Q16—22no—
294Q16—3262no—
295Q16—6521no—
296SD16—22no—
297SD16—3262no—
298SD16—3262no—
299SD16—3262no—
300SD16—6521no—
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—21no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—1631no—
325C22⋊C4—1631no—
326C42—21no—
327C42C22×D16311yescharacteristic
328C23—21no—
329C23—13041no—
330C23—3261no—
331C23—6522no—
332C23—6522no—
333C23—1631no—
334C23—1631no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—3261no—
339Q8—3261no—
340Q8—3261no—
341Q8—3261no—
342Q8—6522no—
343Q8—6522no—
344Q8D4×D16312yes—
345D4—21no—
346D4—21no—
347D4—13041no—
348D4—3262no—
349D4—3261no—
350D4—6522no—
351D4—6522no—
352D4—1632no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—13041no—
357C2×C4—3261no—
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×C4D4×D16311yescharacteristic
370C2×C4D4×D16311yescharacteristic
371C2×C4C23×D16311yescharacteristic
372C8—21no—
373C8—21no—
374C8—3261no—
375C8—1632no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—13044no—
380C22—3261no—
381C22—3261no—
382C22—3261no—
383C22—3261no—
384C22—3261no—
385C22—3262no—
386C22—6521no—
387C22—6521no—
388C22—1632no—
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—3261no—
396C4—3261no—
397C4—6521no—
398C4—6521no—
399C4—6521no—
400C4C2×D4×D16311yescharacteristic
401C4#1910478111yescharacteristic
402C2—81no—
403C2—3262no—
404C2—6522no—
405C2—6521no—
406C2#2378229911yescharacteristic
407C2#137×D16311yescharacteristic
408C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#464×C163C2
socleC22×C163C22×D4
prime radical#23782308C2
hypercenter#464D163
hypocenterC163#3266
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19104767C2semidirect
2. C2×SD16×D163C2semidirect
3. Q8×D652C2semidirect
5. #23782060C2semidirect
6. #23781910C2semidirect
7. #23781905C2semidirect
8. #23781871C2semidirect
9. #23781830C2semidirect
10. #23781819C2semidirect
11. #23781817C2semidirect
12. #23781802C2semidirect
13. #23781763C2semidirect
14. #23781735C2semidirect
15. #464×C163C2semidirect
18. C2×Q8×D163C22semidirect
21. C2×#14579689C22semidirect
22. C2×#14579689C22semidirect
29. C2×#14579682C22semidirect
32. #19104735C22semidirect
37. #19104715C22semidirect
46. #19104668C22semidirect
47. #19104660C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104646C22semidirect
54. C2×Q16×C163C22semidirect
57. C4×Q8×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #103×C163C22semidirect
91. #14579686D4semidirect
184. C2×D163#137semidirect
204. C163#3266semidirect
213. #464D163semidirect
228. C2×Q16C2×D163semidirect
250. C4×Q8C2×D163semidirect
256. C4⋊C8C2×D163semidirect
260. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
C2×SD16×D1632
Q8×D6522
#237823082
#237820602
#237819102
#237819052
#237818712
#237818302
#237818192
#237818172
#237818022
#237817632
#237817352
#464×C1632
#3266163