Finite Group Databasebeta

← #35226107

Subgroups — #35226107

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×#19104766C211yesmaximal, characteristic, complemented
2C22×#14579689C211yesmaximal, characteristic, complemented
3#23782324C211yesmaximal, characteristic
4#23782057C211yesmaximal, characteristic, complemented
5#23781921C211yesmaximal, characteristic, complemented
6#23781896C211yesmaximal, characteristic, complemented
7#23781887C211yesmaximal, characteristic, complemented
8#23781872C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781821C211yesmaximal, characteristic, complemented
11#23781820C211yesmaximal, characteristic, complemented
12#23781804C211yesmaximal, characteristic, complemented
13#23781770C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15D4×Q8×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18#19104766—24no—
19C2×#14579689—24no—
20C2×#14579689C2211yescharacteristic
21C2×#14579689C2211yescharacteristic
22C2×#14579689C2211yescharacteristic
23C2×#14579689C2211yescharacteristic
24C2×#14579689C2211yescharacteristic
25C2×#14579689C2211yescharacteristic
26C2×#14579686C2211yescharacteristic
27C2×#14579686C2211yescharacteristic
28C2×#14579686C2211yescharacteristic
29C2×#14579683C2211yescharacteristic
30C22×C163⋊C8C2211yescharacteristic
31#19104723C2211yescharacteristic
32#19104721—21no—
33#19104714C2211yescharacteristic
34#19104713—21no—
35#19104709C2211yescharacteristic
36#19104707—21no—
37C4×Dic326C2211yescharacteristic
38(C22⋊D163)⋊C4—21no—
39#19104666C2211yescharacteristic
40C4⋊(C163⋊C8)C2211yescharacteristic
41#19104659C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104655C2211yescharacteristic
47C22⋊(C163⋊C8)C2211yescharacteristic
48C22⋊(C163⋊C8)C2211yescharacteristic
49#19104646C2211yescharacteristic
50C22×Q8×C163C2211yescharacteristic
51C22×Q8×C163C2211yescharacteristic
52C4⋊Q8×C163—21no—
53C4⋊Q8×C163C2211yescharacteristic
54C22⋊Q8×C163—21no—
55C22⋊Q8×C163—21no—
56C22⋊Q8×C163C2211yescharacteristic
57C22⋊Q8×C163C2211yescharacteristic
58C4×Q8×C163C2211yescharacteristic
59C4×D4×C163—21no—
60C4×D4×C163C2211yescharacteristic
61#14579699—22no—
62#14579699—41no—
63C2×Dic326—22no—
64C2×Dic326—21no—
65C2×Dic326—41no—
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73#14579689—22no—
74#14579689—22no—
75#14579689—22no—
76#14579689—22no—
77#14579689—41no—
78#14579689D414yescomplemented
79#14579686—22no—
80#14579686—22no—
81#14579686—41no—
82#14579683—22no—
83C2×C163⋊C8—22no—
84C2×C163⋊C8C2311yescharacteristic
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C4⋊Dic163C2311yescharacteristic
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95C4×Dic163—21no—
96C2×Q8×C163—21no—
97C2×Q8×C163—21no—
98C2×Q8×C163—22no—
99C2×Q8×C163—22no—
100C2×Q8×C163—41no—
101C2×Q8×C163—41no—
102C2×Q8×C163C2311yescharacteristic
103C2×Q8×C163C2311yescharacteristic
104C2×Q8×C163C2311yescharacteristic
105C2×D4×C163D411yescharacteristic
106C22×C4×C163—21no—
107C22×C4×C163—21no—
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C4⋊C4×C163—21no—
111C4⋊C4×C163—21no—
112C4⋊C4×C163—21no—
113C4⋊C4×C163—21no—
114C4⋊C4×C163—21no—
115C4⋊C4×C163D411yescharacteristic
116C4⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163D411yescharacteristic
120C22⋊C4×C163D411yescharacteristic
121C42×C163—21no—
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—42no—
128Dic326C2×D412yes—
129D652—21no—
130D652—21no—
131C22⋊D163—42no—
132C22⋊D163—42no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—42no—
137C4×D163—42no—
138C163⋊C8—21no—
139C163⋊C8—21no—
140C163⋊C8—21no—
141C163⋊C8C2×D412yes—
142C23×C163C2×D411yescharacteristic
143C23×C163C2×D411yescharacteristic
144Q8×C163—21no—
145Q8×C163—21no—
146Q8×C163—21no—
147Q8×C163—21no—
148Q8×C163—21no—
149Q8×C163—42no—
150Q8×C163—42no—
151Q8×C163C2×D412yes—
152D4×C163—22no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—42no—
162C2×C4×C163—42no—
163C2×C4×C163C2×D411yescharacteristic
164C2×C4×C163C2×D411yescharacteristic
165C2×C4×C163C2×D411yescharacteristic
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2411yescharacteristic
168C22×C163—21no—
169C22×C163—21no—
170C22×C163—21no—
171C22×C163—21no—
172C22×C163—21no—
173C22×C163#13712yescomplemented
174C22×C163C22×D411yescharacteristic
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—41no—
178C4×C163—21no—
179C4×C163—21no—
180C4×C163—22no—
181C4×C163—21no—
182C4×C163—21no—
183C4×C163—41no—
184C4×C163—41no—
185C4×C163—41no—
186C4×C163C22×D411yescharacteristic
187C4×C163H(5,𝔽2)11yescharacteristic
188Dic163—22no—
189Dic163—41no—
190Dic163—41no—
191C2×C163—22no—
192C2×C163—41no—
193C2×C163D4211yescharacteristic
194C2×C163C2×#13711yescharacteristic
195C2×C163#57811yescharacteristic
196D163—81no—
197C163#327011yescharacteristic, complemented, Sylow
198#3270—1631nomaximal, Sylow
199C2×#137—1631no—
200C22×Q16—1631no—
201D4×Q8D16311yescharacteristic, complemented
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—3264no—
220C2×Q16—1631no—
221C2×Q16—1631no—
222C2×Q16—1631no—
223C2×Q16—1631no—
224C2×Q16—1631no—
225C2×Q16—1631no—
226C2×SD16—1631no—
227C2×SD16—1631no—
228C2×SD16—1631no—
229C2×M16—1631no—
230C22×C8—1631no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D16311yescharacteristic
233#124—3261no—
234#124—1631no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—3261no—
238C22⋊Q8—1631no—
239C22⋊Q8C2×D16311yescharacteristic
240C22⋊Q8C2×D16311yescharacteristic
241#121—3261no—
242#121—1631no—
243C4×Q8—1631no—
244C4×Q8C2×D16311yescharacteristic
245C4×D4—21no—
246C4×D4—3261no—
247C4×D4C2×D16311yescharacteristic
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—3262no—
268C2×Q8—3261no—
269C2×Q8—6521no—
270C2×Q8—1631no—
271C2×Q8—1631no—
272C2×Q8C22×D16311yescharacteristic
273C2×Q8C22×D16311yescharacteristic
274C2×Q8C22×D16311yescharacteristic
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—1631no—
278C2×D4C22⋊D16311yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3261no—
282C22×C4—3261no—
283C22×C4C22×D16311yescharacteristic
284C22×C4C22×D16311yescharacteristic
285Q16—3262no—
286Q16—3262no—
287Q16—3262no—
288Q16—3262no—
289Q16—6521no—
290Q16—1634no—
291SD16—3262no—
292SD16—3262no—
293SD16—6521no—
294M16—3262no—
295C2×C8—3262no—
296C2×C8—1631no—
297C2×C8—1631no—
298C2×C8—1631no—
299C2×C8—1631no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—3261no—
306C4⋊C4—3261no—
307C4⋊C4—1631no—
308C4⋊C4C22⋊D16311yescharacteristic
309C4⋊C4C22×D16311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3261no—
313C22⋊C4—3261no—
314C22⋊C4—3261no—
315C22⋊C4—3261no—
316C22⋊C4C22⋊D16311yescharacteristic
317C22⋊C4C22⋊D16311yescharacteristic
318C42—21no—
319C42—3261no—
320C42C22×D16311yescharacteristic
321C23—3261no—
322C23C2×C22⋊D16311yescharacteristic
323C23C2×C22⋊D16311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—3261no—
332Q8—3261no—
333Q8—3261no—
334Q8—6522no—
335Q8—1632no—
336Q8D4×D16312yes—
337D4—22no—
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—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3261no—
353C2×C4—3261no—
354C2×C4—3261no—
355C2×C4—6522no—
356C2×C4—6522no—
357C2×C4C2×C22⋊D16311yescharacteristic
358C2×C4C2×C22⋊D16311yescharacteristic
359C2×C4C2×C22⋊D16311yescharacteristic
360C2×C4C2×C22⋊D16311yescharacteristic
361C2×C4C23×D16311yescharacteristic
362C8—3261no—
363C8—3261no—
364C8—3261no—
365C8—1632no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—6521no—
372C22—6521no—
373C22—6521no—
374C22#1910476612yescomplemented
375C22C22×C22⋊D16311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3262no—
385C4—6521no—
386C4—6521no—
387C4C2×D4×D16311yescharacteristic
388C4#1910478011yescharacteristic
389C2—22no—
390C2—41no—
391C2—13041no—
392C2D4×C22⋊D16311yescharacteristic
393C2C2×#1910476611yescharacteristic
394C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingD4×Q8×C163C2
socleC22×C163C22×D4
prime radical#23782324C2
hypercenterD4×Q8D163
hypocenterC163#3270
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19104766C2semidirect
2. C22×#14579689C2semidirect
4. #23782057C2semidirect
5. #23781921C2semidirect
6. #23781896C2semidirect
7. #23781887C2semidirect
8. #23781872C2semidirect
9. #23781825C2semidirect
10. #23781821C2semidirect
11. #23781820C2semidirect
12. #23781804C2semidirect
13. #23781770C2semidirect
14. #23781737C2semidirect
15. D4×Q8×C163C2semidirect
78. #14579689D4semidirect
173. C22×C163#137semidirect
197. C163#3270semidirect
201. D4×Q8D163semidirect
374. C22#19104766semidirect

Maximal subgroups

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

subgroupindex
C2×#191047662
C22×#145796892
#237823242
#237820572
#237819212
#237818962
#237818872
#237818722
#237818252
#237818212
#237818202
#237818042
#237817702
#237817372
D4×Q8×C1632
#3270163