Finite Group Databasebeta

← #35226052

Subgroups — #35226052

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
2C2×SD16×D163C211yesmaximal, characteristic, complemented
3#23782330C211yesmaximal, characteristic, complemented
4#23782310C211yesmaximal, characteristic
5#23782057C211yesmaximal, characteristic, complemented
6#23781914C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781824C211yesmaximal, characteristic, complemented
10#23781822C211yesmaximal, characteristic, complemented
11#23781814C211yesmaximal, characteristic, complemented
12#23781805C211yesmaximal, characteristic, complemented
13#23781765C211yesmaximal, characteristic, complemented
14#23781735C211yesmaximal, characteristic, complemented
15#461×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×Q8×D163C2211yescharacteristic, complemented
19C2×D4×D163C2211yescharacteristic
20#19104764—24no—
21C2×Dic652C2211yescharacteristic, complemented
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579689C2211yescharacteristic, complemented
24C2×#14579688C2211yescharacteristic
25C2×#14579687C2211yescharacteristic
26C2×#14579686C2211yescharacteristic
27C2×#14579686C2211yescharacteristic
28SD16×D163—24no—
29C2×#14579682C2211yescharacteristic, complemented
30C2×C8×D163C2211yescharacteristic
31#19104734—21no—
32#19104723C2211yescharacteristic
33#19104719—21no—
34#19104715C2211yescharacteristic
35#19104713—21no—
36#19104713—21no—
37#19104711C2211yescharacteristic
38#19104711C2211yescharacteristic, complemented
39#19104710C2211yescharacteristic
40#19104709—21no—
41#19104705—21no—
42C4×Dic326C2211yescharacteristic, complemented
43C4×D652C2211yescharacteristic
44#19104697—21no—
45C2×(C2×D163)⋊C4—21no—
46#19104668C2211yescharacteristic, complemented
47#19104660C2211yescharacteristic, complemented
48#19104658C2211yescharacteristic, complemented
49#19104657C2211yescharacteristic
50#19104656C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104646C2211yescharacteristic, 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#14579698—22no—
63#14579698—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—
79C22×Dic163—21no—
80C2×C4×D163—21no—
81C2×C4×D163—21no—
82C2×C4×D163C2311yescharacteristic
83C2×C4×D163C2311yescharacteristic
84Dic652—22no—
85#14579689—22no—
86#14579689—41no—
87#14579688—22no—
88#14579687D414yescomplemented
89#14579686—22no—
90#14579686—41no—
91#14579682—22no—
92C2×C163⋊C8C2311yescharacteristic
93C2×C163⋊C8C2311yescharacteristic
94C8×D163—22no—
95C4⋊Dic163—21no—
96C4⋊Dic163D411yescharacteristic
97C4⋊Dic163C2311yescharacteristic
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100Dic163⋊C4—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4D411yescharacteristic
108(C2×D163)⋊C4D411yescharacteristic
109C4×Dic163—21no—
110C2×Q8×C163C2311yescharacteristic
111C2×Q8×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C22×C4×C163—21no—
114SD16×C163—22no—
115SD16×C163—22no—
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C4⋊C4×C163—21no—
119C4⋊C4×C163—21no—
120C4⋊C4×C163C2311yescharacteristic
121C22⋊C4×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—42no—
133Dic326—42no—
134Dic326C2×D412yes—
135D652—22no—
136D652—21no—
137D652—21no—
138D652—81no—
139C22⋊D163—42no—
140C22⋊D163—42no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—42no—
145C2×Dic163C2×D411yescharacteristic
146C2×Dic163C2×D411yescharacteristic
147C4×D163—21no—
148C4×D163—21no—
149C4×D163—21no—
150C4×D163—21no—
151C4×D163—42no—
152C4×D163—42no—
153C163⋊C8—21no—
154C163⋊C8C2×D412yes—
155C23×C163—21no—
156Q8×C163—21no—
157Q8×C163—21no—
158Q8×C163—21no—
159Q8×C163—21no—
160D4×C163—21no—
161D4×C163C2×D412yes—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—81no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163—21no—
171C22×C163—22no—
172C22×C163—41no—
173C22×C163—41no—
174C22×C163C22×D411yescharacteristic
175C2×D163—21no—
176C2×D163—21no—
177C2×D163—21no—
178C2×D163—21no—
179C2×D163—21no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—84no—
184C2×D163#13712yescomplemented
185C4×C163—21no—
186C4×C163—21no—
187C4×C163—41no—
188C4×C163—41no—
189C4×C163—41no—
190C4×C163C22×D411yescharacteristic
191C4×C163H(5,𝔽2)11yescharacteristic
192Dic163—21no—
193Dic163—21no—
194Dic163—22no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—42no—
198C2×C163D4211yescharacteristic
199C2×C163C2×#13711yescharacteristic
200C2×C163#57711yescharacteristic
201D163—22no—
202D163—41no—
203D163—81no—
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#461D16311yescharacteristic, complemented
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×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⋊Q8C2×D16311yescharacteristic, complemented
240#124—1631no—
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×Q8—1631no—
251C4×D4—3261no—
252C4×D4—1631no—
253C4×D4C2×D16311yescharacteristic
254C2×C22⋊C4—3261no—
255#107—1631no—
256C4⋊C8C2×D16311yescharacteristic, complemented
257#103—1631no—
258#103—1631no—
259#103C2×D16311yescharacteristic, complemented
260#103C2×D16311yescharacteristic, complemented
261#102—1631no—
262#102—1631no—
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—21no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4—1631no—
292C22×C4—1631no—
293Q16—3262no—
294Q16—3262no—
295Q16—6521no—
296SD16—22no—
297SD16—22no—
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—21no—
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—
327C42C22×D16311yescharacteristic
328C23—21no—
329C23—13041no—
330C23—3261no—
331C23—6522no—
332C23—6522no—
333C23—1631no—
334C23—1631no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3261no—
340Q8—3261no—
341Q8—3261no—
342Q8—6522no—
343Q8—6522no—
344Q8—1632no—
345D4—21no—
346D4—13041no—
347D4—3262no—
348D4—3261no—
349D4—3261no—
350D4—6522no—
351D4—6522no—
352D4D4×D16312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
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—22no—
377C22—41no—
378C22—41no—
379C22—13044no—
380C22—3261no—
381C22—3261no—
382C22—3261no—
383C22—3261no—
384C22—3261no—
385C22—6521no—
386C22—6521no—
387C22—6521no—
388C22—1632no—
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3261no—
396C4—3261no—
397C4—3262no—
398C4—6521no—
399C4—6521no—
400C4C2×D4×D16311yescharacteristic
401C4#1910478111yescharacteristic
402C2—42no—
403C2—13041no—
404C2—3262no—
405C2—6521no—
406C2#2378229911yescharacteristic
407C2#137×D16311yescharacteristic
408C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#461×C163C2
socleC22×C163C22×D4
prime radical#23782310C2
hypercenter#461D163
hypocenterC163#3266
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
1. C2×#19104764C2semidirect
2. C2×SD16×D163C2semidirect
3. #23782330C2semidirect
5. #23782057C2semidirect
6. #23781914C2semidirect
7. #23781905C2semidirect
8. #23781888C2semidirect
9. #23781824C2semidirect
10. #23781822C2semidirect
11. #23781814C2semidirect
12. #23781805C2semidirect
13. #23781765C2semidirect
14. #23781735C2semidirect
15. #461×C163C2semidirect
18. C2×Q8×D163C22semidirect
21. C2×Dic652C22semidirect
22. C2×#14579689C22semidirect
23. C2×#14579689C22semidirect
29. C2×#14579682C22semidirect
38. #19104711C22semidirect
42. C4×Dic326C22semidirect
46. #19104668C22semidirect
47. #19104660C22semidirect
48. #19104658C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104646C22semidirect
56. C4⋊Q8×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #103×C163C22semidirect
60. #103×C163C22semidirect
88. #14579687D4semidirect
184. C2×D163#137semidirect
204. C163#3266semidirect
214. #461D163semidirect
239. C4⋊Q8C2×D163semidirect
256. C4⋊C8C2×D163semidirect
259. #103C2×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×#191047642
C2×SD16×D1632
#237823302
#237823102
#237820572
#237819142
#237819052
#237818882
#237818242
#237818222
#237818142
#237818052
#237817652
#237817352
#461×C1632
#3266163