Finite Group Databasebeta

← #35226106

Subgroups — #35226106

Order 20864.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104767C211yesmaximal, characteristic, complemented
2Q8×C22⋊D163C211yesmaximal, characteristic, complemented
3#23782326C211yesmaximal, characteristic
4#23782058C211yesmaximal, characteristic, complemented
5#23781920C211yesmaximal, characteristic, complemented
6#23781894C211yesmaximal, characteristic, complemented
7#23781889C211yesmaximal, characteristic, complemented
8#23781873C211yesmaximal, characteristic, complemented
9#23781826C211yesmaximal, characteristic, complemented
10#23781821C211yesmaximal, characteristic, complemented
11#23781819C211yesmaximal, characteristic, complemented
12#23781804C211yesmaximal, characteristic, complemented
13#23781772C211yesmaximal, characteristic, complemented
14#23781736C211yesmaximal, characteristic, complemented
15C22×Q16×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18#19104767—24no—
19C2×#14579689C2211yescharacteristic
20C2×#14579689C2211yescharacteristic
21C2×#14579688C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23C2×#14579686C2211yescharacteristic
24C2×#14579682C2211yescharacteristic
25#19104733—21no—
26#19104733C2211yescharacteristic
27#19104724—21no—
28#19104724C2211yescharacteristic
29#19104714—21no—
30#19104714C2211yescharacteristic
31#19104711—21no—
32#19104711—21no—
33#19104711C2211yescharacteristic
34#19104711C2211yescharacteristic
35#19104709—21no—
36#19104709C2211yescharacteristic
37Q8×Dic163C2211yescharacteristic
38Q8×Dic163C2211yescharacteristic
39C4×C22⋊D163—21no—
40C4×C22⋊D163—21no—
41C4×C22⋊D163C2211yescharacteristic
42#19104667C2211yescharacteristic
43Dic163⋊C8C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104659C2211yescharacteristic
48#19104659C2211yescharacteristic
49#19104657C2211yescharacteristic
50C22⋊(C163⋊C8)C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52#19104645C2211yescharacteristic
53C22×Q8×C163C2211yescharacteristic
54C22×Q8×C163C2211yescharacteristic
55C2×Q16×C163—24no—
56C2×Q16×C163C2211yescharacteristic
57C2×Q16×C163C2211yescharacteristic
58C2×Q16×C163C2211yescharacteristic
59C2×Q16×C163C2211yescharacteristic
60C22×C8×C163C2211yescharacteristic
61#14579700—22no—
62#14579700—41no—
63C2×Dic326—21no—
64C2×Dic326C2311yescharacteristic
65Q8×D163—22no—
66Q8×D163—41no—
67C2×D652—21no—
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163D411yescharacteristic
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579689—22no—
75#14579689—41no—
76#14579688—22no—
77#14579686—22no—
78#14579686—41no—
79#14579682—22no—
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C4⋊Dic163—21no—
83C4⋊Dic163—21no—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91Dic163⋊C4D411yescharacteristic
92C22⋊Dic163—21no—
93C22⋊Dic163D411yescharacteristic
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4D411yescharacteristic
100C4×Dic163—21no—
101C4×Dic163—21no—
102C4×Dic163C2311yescharacteristic
103C2×Q8×C163—21no—
104C2×Q8×C163—21no—
105C2×Q8×C163—22no—
106C2×Q8×C163—22no—
107C2×Q8×C163—41no—
108C2×Q8×C163—41no—
109C2×Q8×C163C2311yescharacteristic
110C2×Q8×C163C2311yescharacteristic
111C2×Q8×C163C2311yescharacteristic
112C2×Q8×C163C2311yescharacteristic
113C22×C4×C163—21no—
114C22×C4×C163—21no—
115C22×C4×C163C2311yescharacteristic
116Q16×C163—22no—
117Q16×C163—22no—
118Q16×C163—22no—
119Q16×C163D414yescomplemented
120C2×C8×C163—22no—
121C2×C8×C163C2311yescharacteristic
122C2×C8×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163C2×D411yescharacteristic
125Dic326—21no—
126Dic326—21no—
127Dic326—42no—
128D652—21no—
129D652—21no—
130D652—42no—
131C22⋊D163—22no—
132C22⋊D163—42no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163C2×D411yescharacteristic
137C2×Dic163C2×D411yescharacteristic
138C2×Dic163C2×D411yescharacteristic
139C4×D163—21no—
140C4×D163—21no—
141C4×D163—42no—
142C4×D163—42no—
143C163⋊C8—21no—
144C163⋊C8—21no—
145C23×C163C2×D411yescharacteristic
146Q8×C163—21no—
147Q8×C163—21no—
148Q8×C163—21no—
149Q8×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152Q8×C163—42no—
153Q8×C163—42no—
154Q8×C163C2×D412yes—
155Q8×C163C2×D412yes—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—42no—
163C2×C4×C163—42no—
164C2×C4×C163C2×D411yescharacteristic
165C2×C4×C163C2411yescharacteristic
166C8×C163—21no—
167C8×C163C2×D412yes—
168C22×C163—21no—
169C22×C163—21no—
170C22×C163#13712yescomplemented
171C22×C163C22×D411yescharacteristic
172C2×D163—21no—
173C2×D163—21no—
174C2×D163—21no—
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—41no—
178C4×C163—21no—
179C4×C163—22no—
180C4×C163—22no—
181C4×C163—41no—
182C4×C163—41no—
183C4×C163C22×D411yescharacteristic
184C4×C163H(5,𝔽2)11yescharacteristic
185Dic163—21no—
186Dic163—21no—
187Dic163—21no—
188Dic163—41no—
189Dic163—41no—
190Dic163—41no—
191C2×C163—22no—
192C2×C163D4211yescharacteristic
193C2×C163C2×#13711yescharacteristic
194C2×C163#57811yescharacteristic
195D163—41no—
196D163—81no—
197C163#327011yescharacteristic, complemented, Sylow
198#3270—1631nomaximal, Sylow
199C2×#137—1631no—
200C22×Q16D16311yescharacteristic, complemented
201D4×Q8—1631no—
202#548—1631no—
203#497—1631no—
204#470—1631no—
205#465—1631no—
206#464—1631no—
207#462—1631no—
208#452—1631no—
209#451—1631no—
210#451—1631no—
211#448—1631no—
212#441—1631no—
213#435—1631no—
214C2×#41—1631no—
215C22×Q8—1631no—
216C22×Q8C2×D16311yescharacteristic
217C22×Q8C2×D16311yescharacteristic
218#137—3264no—
219C2×Q16—24no—
220C2×Q16—1631no—
221C2×Q16—1631no—
222C2×Q16C2×D16311yescharacteristic
223C2×Q16C2×D16311yescharacteristic
224C2×Q16C2×D16311yescharacteristic
225C2×Q16C2×D16311yescharacteristic
226C2×SD16—1631no—
227C2×SD16—1631no—
228C2×SD16—1631no—
229C2×M16—1631no—
230C22×C8C2×D16311yescharacteristic
231C4⋊Q8—3261no—
232C4⋊Q8—1631no—
233#124—3261no—
234#124—1631no—
235C22⋊Q8—3261no—
236C22⋊Q8—3261no—
237C22⋊Q8—3261no—
238C22⋊Q8—1631no—
239C22⋊Q8—1631no—
240C22⋊Q8—1631no—
241#121—3261no—
242#121—1631no—
243C4×Q8—1631no—
244C4×Q8—1631no—
245C4×D4—3261no—
246C4×D4—3261no—
247C4×D4—1631no—
248#106—1631no—
249C4⋊C8—1631no—
250#103—1631no—
251#103—1631no—
252#103—1631no—
253#103—1631no—
254#103—1631no—
255#102—1631no—
256C22⋊C8—1631no—
257C22⋊C8—1631no—
258#97—1631no—
259#41—3262no—
260#41—6521no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—3261no—
268C2×Q8—3262no—
269C2×Q8—6521no—
270C2×Q8—1631no—
271C2×Q8C22×D16311yescharacteristic
272C2×Q8C22×D16311yescharacteristic
273C2×Q8C22×D16311yescharacteristic
274C2×Q8C22×D16311yescharacteristic
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—1631no—
278C2×D4—1631no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3261no—
282C22×C4—3261no—
283C22×C4—1631no—
284C22×C4C22×D16311yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—3262no—
289Q16—6521no—
290Q16C22⋊D16314yescomplemented
291SD16—3262no—
292SD16—3262no—
293SD16—6521no—
294M16—3262no—
295C2×C8—22no—
296C2×C8—1631no—
297C2×C8—1631no—
298C2×C8C22×D16311yescharacteristic
299C2×C8C22×D16311yescharacteristic
300C4⋊C4—3261no—
301C4⋊C4—3261no—
302C4⋊C4—3261no—
303C4⋊C4—3261no—
304C4⋊C4—3261no—
305C4⋊C4—3261no—
306C4⋊C4—3261no—
307C4⋊C4—1631no—
308C4⋊C4—1631no—
309C4⋊C4—1631no—
310C22⋊C4—3261no—
311C22⋊C4—3261no—
312C22⋊C4—3261no—
313C22⋊C4—3261no—
314C22⋊C4—3261no—
315C22⋊C4—3261no—
316C22⋊C4—1631no—
317C22⋊C4—1631no—
318C42—3261no—
319C42—3261no—
320C42—1631no—
321C23—3261no—
322C23—1631no—
323C23D4×D16311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—3261no—
333Q8—3261no—
334Q8—6522no—
335Q8C2×C22⋊D16312yes—
336Q8C2×C22⋊D16312yes—
337D4—3262no—
338D4—3261no—
339D4—3261no—
340D4—6522no—
341D4—6522no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—3261no—
353C2×C4—3261no—
354C2×C4—3261no—
355C2×C4—6522no—
356C2×C4—6522no—
357C2×C4—1631no—
358C2×C4—1631no—
359C2×C4—1631no—
360C2×C4D4×D16311yescharacteristic
361C2×C4C23×D16311yescharacteristic
362C8—21no—
363C8—3261no—
364C8—3261no—
365C8C2×C22⋊D16312yes—
366C22—21no—
367C22—21no—
368C22—3261no—
369C22—3261no—
370C22—3261no—
371C22—6521no—
372C22—6521no—
373C22—6521no—
374C22#1910476712yescomplemented
375C22C2×D4×D16311yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—3261no—
382C4—3261no—
383C4—3261no—
384C4—6521no—
385C4—6521no—
386C4—6521no—
387C4C22×C22⋊D16311yescharacteristic
388C4#1910478011yescharacteristic
389C2—22no—
390C2—13041no—
391C2—6521no—
392C2D4×C22⋊D16311yescharacteristic
393C2C2×#1910476711yescharacteristic
394C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC22×Q16×C163C2
socleC22×C163C22×D4
prime radical#23782326C2
hypercenterC22×Q16D163
hypocenterC163#3270
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#19104767C2semidirect
2. Q8×C22⋊D163C2semidirect
4. #23782058C2semidirect
5. #23781920C2semidirect
6. #23781894C2semidirect
7. #23781889C2semidirect
8. #23781873C2semidirect
9. #23781826C2semidirect
10. #23781821C2semidirect
11. #23781819C2semidirect
12. #23781804C2semidirect
13. #23781772C2semidirect
14. #23781736C2semidirect
15. C22×Q16×C163C2semidirect
119. Q16×C163D4semidirect
170. C22×C163#137semidirect
197. C163#3270semidirect
200. C22×Q16D163semidirect
290. Q16C22⋊D163semidirect
374. C22#19104767semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
Q8×C22⋊D1632
#237823262
#237820582
#237819202
#237818942
#237818892
#237818732
#237818262
#237818212
#237818192
#237818042
#237817722
#237817362
C22×Q16×C1632
#3270163