Finite Group Databasebeta

← #35226042

Subgroups — #35226042

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×#19104765C211yesmaximal, characteristic, complemented
2C22×#14579687C211yesmaximal, characteristic, complemented
3D4×Dic326C211yesmaximal, characteristic, complemented
4#23782060C211yesmaximal, characteristic, complemented
5#23781909C211yesmaximal, characteristic, complemented
6#23781904C211yesmaximal, characteristic, complemented
7#23781884C211yesmaximal, characteristic, complemented
8#23781870C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781820C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781803C211yesmaximal, characteristic, complemented
13#23781765C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15#546×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C22×Dic326C2211yescharacteristic, complemented
18#19104765—24no—
19C2×#14579689C2211yescharacteristic, complemented
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579689C2211yescharacteristic, complemented
22C2×#14579687—24no—
23C2×#14579687C2211yescharacteristic
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic
26C2×#14579687C2211yescharacteristic
27C2×#14579687C2211yescharacteristic
28C2×#14579687C2211yescharacteristic
29C2×#14579683C2211yescharacteristic, complemented
30C22×C163⋊C8C2211yescharacteristic
31#19104735C2211yescharacteristic, complemented
32#19104734—21no—
33#19104714C2211yescharacteristic
34#19104714C2211yescharacteristic, complemented
35#19104712—21no—
36#19104712—21no—
37C4×Dic326C2211yescharacteristic, complemented
38D4×Dic163—21no—
39#19104668C2211yescharacteristic, complemented
40C4⋊(C163⋊C8)C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47C22⋊(C163⋊C8)C2211yescharacteristic
48C22⋊(C163⋊C8)C2211yescharacteristic, complemented
49#19104646C2211yescharacteristic, complemented
50C2×#41×C163C2211yescharacteristic
51C22×D4×C163C2211yescharacteristic
52#124×C163C2211yescharacteristic
53#123×C163—21no—
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163—21no—
56#121×C163C2211yescharacteristic
57#120×C163—21no—
58C4×D4×C163C2211yescharacteristic
59C4×D4×C163C2211yescharacteristic
60C2×C22⋊C4×C163—21no—
61C2×Dic326—22no—
62C2×Dic326—22no—
63C2×Dic326—21no—
64C2×Dic326—21no—
65C2×Dic326—41no—
66C2×Dic326—41no—
67C2×Dic326C2311yescharacteristic
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70C22×Dic163—21no—
71C22×Dic163—21no—
72#14579689—22no—
73#14579689—22no—
74#14579689—41no—
75#14579687—22no—
76#14579687—22no—
77#14579687—22no—
78#14579687—22no—
79#14579687—41no—
80#14579687D414yescomplemented
81#14579683—22no—
82C2×C163⋊C8—22no—
83C2×C163⋊C8C2311yescharacteristic
84C2×C163⋊C8C2311yescharacteristic
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C4⋊Dic163—21no—
88C4⋊Dic163—21no—
89C4⋊Dic163C2311yescharacteristic
90Dic163⋊C4—21no—
91Dic163⋊C4—21no—
92Dic163⋊C4—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C4×Dic163—21no—
96C24×C163—21no—
97#41×C163—22no—
98#41×C163—41no—
99C2×Q8×C163C2311yescharacteristic
100C2×D4×C163—22no—
101C2×D4×C163—21no—
102C2×D4×C163—21no—
103C2×D4×C163—21no—
104C2×D4×C163—41no—
105C2×D4×C163D411yescharacteristic
106C2×D4×C163C2311yescharacteristic
107C2×D4×C163C2311yescharacteristic
108C22×C4×C163—21no—
109C22×C4×C163—21no—
110C22×C4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C4⋊C4×C163—21no—
113C4⋊C4×C163D411yescharacteristic
114C4⋊C4×C163C2311yescharacteristic
115C22⋊C4×C163—21no—
116C22⋊C4×C163—21no—
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163C2311yescharacteristic
123Dic326—21no—
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129Dic326—42no—
130Dic326C2×D412yes—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—42no—
136C2×Dic163—42no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C163⋊C8—21no—
140C163⋊C8C2×D412yes—
141C23×C163—21no—
142C23×C163—21no—
143C23×C163—42no—
144C23×C163—42no—
145C23×C163—81no—
146C23×C163C2×D411yescharacteristic
147C23×C163C2×D411yescharacteristic
148Q8×C163—21no—
149Q8×C163—21no—
150D4×C163—22no—
151D4×C163—21no—
152D4×C163—21no—
153D4×C163—21no—
154D4×C163—42no—
155D4×C163—42no—
156D4×C163—81no—
157D4×C163C2×D412yes—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—42no—
165C2×C4×C163—81no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2×D411yescharacteristic
170C2×C4×C163C2411yescharacteristic
171C22×C163—21no—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—22no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163—41no—
182C22×C163—84no—
183C22×C163#13712yescomplemented
184C22×C163C22×D411yescharacteristic
185C4×C163—21no—
186C4×C163—21no—
187C4×C163—21no—
188C4×C163—21no—
189C4×C163—41no—
190C4×C163—41no—
191C4×C163C22×D411yescharacteristic
192C4×C163H(5,𝔽2)11yescharacteristic
193Dic163—22no—
194Dic163—41no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—22no—
198C2×C163—41no—
199C2×C163—42no—
200C2×C163—81no—
201C2×C163D4211yescharacteristic
202C2×C163C2×#13711yescharacteristic
203C2×C163#57711yescharacteristic
204C163#326611yescharacteristic, complemented, Sylow
205#3266—1631nomaximal, Sylow
206C2×#137—1631no—
207C22×SD16—1631no—
208D4×Q8—1631no—
209#546D16311yescharacteristic
210#497—1631no—
211#468—1631no—
212#467—1631no—
213#464—1631no—
214#461—1631no—
215#452—1631no—
216#451—1631no—
217#450—1631no—
218#448—1631no—
219#440—1631no—
220#435—1631no—
221C2×#41C2×D16311yescharacteristic
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×SD16—1631no—
235C2×SD16—1631no—
236C2×M16—1631no—
237C22×C8—1631no—
238C4⋊Q8—3261no—
239C4⋊Q8—1631no—
240#124C2×D16311yescharacteristic
241#123—21no—
242C22⋊Q8—3261no—
243C22⋊Q8—3261no—
244C22⋊Q8—1631no—
245C22⋊Q8—1631no—
246C22⋊Q8C2×D16311yescharacteristic
247#121—21no—
248#121C2×D16311yescharacteristic
249#120—21no—
250C4×Q8—1631no—
251C4×D4—3261no—
252C4×D4C2×D16311yescharacteristic
253C4×D4C2×D16311yescharacteristic
254C2×C22⋊C4—21no—
255#107—1631no—
256C4⋊C8—1631no—
257#103—1631no—
258#103—1631no—
259#103—1631no—
260#103—1631no—
261#102—1631no—
262#102—1631no—
263C22⋊C8—1631no—
264C22⋊C8—1631no—
265#97—1631no—
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—3262no—
270C2×Q8—3262no—
271C2×Q8—3261no—
272C2×Q8—3261no—
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—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D16311yescharacteristic
285C2×D4C22×D16311yescharacteristic
286C2×D4C22×D16311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4C22×D16311yescharacteristic
292C22×C4C22×D16311yescharacteristic
293Q16—3262no—
294Q16—3262no—
295Q16—6521no—
296SD16—3262no—
297SD16—3262no—
298SD16—3262no—
299SD16—3262no—
300SD16—6521no—
301SD16—1634no—
302M16—3262no—
303C2×C8—3262no—
304C2×C8—1631no—
305C2×C8—1631no—
306C2×C8—1631no—
307C2×C8—1631no—
308C4⋊C4—21no—
309C4⋊C4—3261no—
310C4⋊C4—3261no—
311C4⋊C4—3261no—
312C4⋊C4—3261no—
313C4⋊C4—3261no—
314C4⋊C4—1631no—
315C4⋊C4C22⋊D16311yescharacteristic
316C4⋊C4C22×D16311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4C22⋊D16311yescharacteristic
325C22⋊C4C22⋊D16311yescharacteristic
326C42—3261no—
327C42C22×D16311yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D16311yescharacteristic
334C23C2×C22⋊D16311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3261no—
338Q8—3261no—
339Q8—3261no—
340Q8—3261no—
341Q8—3261no—
342Q8—6522no—
343Q8—6522no—
344Q8—1632no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D16312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—3261no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—3261no—
365C2×C4—6522no—
366C2×C4—6522no—
367C2×C4C2×C22⋊D16311yescharacteristic
368C2×C4C2×C22⋊D16311yescharacteristic
369C2×C4C2×C22⋊D16311yescharacteristic
370C2×C4C2×C22⋊D16311yescharacteristic
371C2×C4C23×D16311yescharacteristic
372C8—3261no—
373C8—3261no—
374C8—3261no—
375C8—1632no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1910476512yescomplemented
389C22C22×C22⋊D16311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—3262no—
397C4—6521no—
398C4—6521no—
399C4—6521no—
400C4C2×D4×D16311yescharacteristic
401C4#1910478011yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D16311yescharacteristic
407C2C2×#1910476511yescharacteristic
408C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting, prime radical#546×C163C2
socleC22×C163C22×D4
hypercenter#546D163
hypocenterC163#3266
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19104765C2semidirect
2. C22×#14579687C2semidirect
3. D4×Dic326C2semidirect
4. #23782060C2semidirect
5. #23781909C2semidirect
6. #23781904C2semidirect
7. #23781884C2semidirect
8. #23781870C2semidirect
9. #23781827C2semidirect
10. #23781820C2semidirect
11. #23781816C2semidirect
12. #23781803C2semidirect
13. #23781765C2semidirect
14. #23781737C2semidirect
17. C22×Dic326C22semidirect
19. C2×#14579689C22semidirect
20. C2×#14579689C22semidirect
21. C2×#14579689C22semidirect
29. C2×#14579683C22semidirect
31. #19104735C22semidirect
34. #19104714C22semidirect
37. C4×Dic326C22semidirect
39. #19104668C22semidirect
40. C4⋊(C163⋊C8)C22semidirect
41. #19104659C22semidirect
42. #19104658C22semidirect
43. #19104658C22semidirect
44. #19104658C22semidirect
48. C22⋊(C163⋊C8)C22semidirect
49. #19104646C22semidirect
80. #14579687D4semidirect
183. C22×C163#137semidirect
204. C163#3266semidirect
388. C22#19104765semidirect

Maximal subgroups

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

subgroupindex
C2×#191047652
C22×#145796872
D4×Dic3262
#237820602
#237819092
#237819042
#237818842
#237818702
#237818272
#237818202
#237818162
#237818032
#237817652
#237817372
#546×C1632
#3266163