Finite Group Databasebeta

← #35226230

Subgroups — #35226230

Order 20864.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579687C211yesmaximal, characteristic, complemented
2C2×SD16×D163C211yesmaximal, characteristic, complemented
3#23782328C211yesmaximal, characteristic, complemented
4D4×C22⋊D163C211yesmaximal, characteristic
5#23782036C211yesmaximal, characteristic, complemented
6#23781896C211yesmaximal, characteristic, complemented
7#23781893C211yesmaximal, characteristic, complemented
8#23781868C211yesmaximal, characteristic, complemented
9#23781864C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781806C211yesmaximal, characteristic, complemented
12#23781802C211yesmaximal, characteristic, complemented
13#23781756C211yesmaximal, characteristic, complemented
14(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic, complemented
17C2×Q8×D163C2211yescharacteristic, complemented
18C22×C22⋊D163—21no—
19C2×D4×D163C2211yescharacteristic
20C2×#14579688C2211yescharacteristic, complemented
21C2×#14579688C2211yescharacteristic, complemented
22C2×#14579687—24no—
23C2×#14579687C2211yescharacteristic
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic, complemented
26C2×#14579687C2211yescharacteristic, complemented
27C2×#14579686C2211yescharacteristic, complemented
28SD16×D163—24no—
29C22×C163⋊C8C2211yescharacteristic, complemented
30C2×C8×D163C2211yescharacteristic, complemented
31#19104734—21no—
32#19104733C2211yescharacteristic, complemented
33#19104731C2211yescharacteristic
34#19104715C2211yescharacteristic, complemented
35#19104713—21no—
36#19104712—21no—
37#19104709C2211yescharacteristic
38#19104707—21no—
39#19104706C2211yescharacteristic
40#19104704—21no—
41#19104703—21no—
42#19104700C2211yescharacteristic, complemented
43D4×Dic163C2211yescharacteristic
44C4×C22⋊D163C2211yescharacteristic
45(C22⋊D163)⋊C4—21no—
46#19104666C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104659C2211yescharacteristic, complemented
51#19104655C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C22×D4×C163C2211yescharacteristic
55C2×SD16×C163C2211yescharacteristic
56C2×SD16×C163C2211yescharacteristic, complemented
57C22⋊Q8×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59#102×C163C2211yescharacteristic, complemented
60C22⋊C8×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C2×Dic326—21no—
63C2×Dic326—22no—
64C2×Dic326—21no—
65C2×Dic326—41no—
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68Q8×D163—22no—
69Q8×D163—41no—
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163—44no—
77C2×C22⋊D163D411yescharacteristic
78D4×D163—22no—
79D4×D163—41no—
80C22×Dic163—21no—
81C22×Dic163—21no—
82C2×C4×D163—21no—
83C2×C4×D163C2311yescharacteristic
84#14579688—22no—
85#14579688—41no—
86#14579687—22no—
87#14579687—22no—
88#14579687—22no—
89#14579687D414yescomplemented
90#14579686—22no—
91C2×C163⋊C8—22no—
92C2×C163⋊C8C2311yescharacteristic
93C2×C163⋊C8C2311yescharacteristic
94C8×D163—22no—
95C4⋊Dic163—21no—
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4—21no—
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100Dic163⋊C4D411yescharacteristic
101C22⋊Dic163—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163D411yescharacteristic
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4D411yescharacteristic
107C4×Dic163—21no—
108C4×Dic163C2311yescharacteristic
109C24×C163—21no—
110C2×Q8×C163C2311yescharacteristic
111C2×D4×C163—22no—
112C2×D4×C163—21no—
113C2×D4×C163—41no—
114C2×D4×C163C2311yescharacteristic
115C2×D4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117SD16×C163—22no—
118SD16×C163—41no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163—21no—
122C4⋊C4×C163C2311yescharacteristic
123C22⋊C4×C163—21no—
124C22×D163—21no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163—81no—
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—42no—
133Dic326—42no—
134Dic326C2×D412yescomplemented
135D652—21no—
136D652—21no—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C22⋊D163—42no—
140C22⋊D163—42no—
141C22⋊D163—81no—
142C22⋊D163SD1614yescomplemented
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—42no—
147C2×Dic163—42no—
148C2×Dic163C2×D411yescharacteristic
149C2×Dic163C2×D411yescharacteristic
150C2×Dic163C2×D411yescharacteristic
151C4×D163—21no—
152C4×D163—21no—
153C4×D163—42no—
154C163⋊C8—21no—
155C163⋊C8C2×D412yescomplemented
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—42no—
159C23×C163—42no—
160C23×C163—81no—
161C23×C163C2×D411yescharacteristic
162Q8×C163—21no—
163Q8×C163—21no—
164D4×C163—21no—
165D4×C163—21no—
166D4×C163—21no—
167D4×C163—42no—
168D4×C163C2×D412yes—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163—21no—
173C2×C4×C163C2×D411yescharacteristic
174C2×C4×C163C2411yescharacteristic
175C8×C163—21no—
176C8×C163—21no—
177C22×C163—21no—
178C22×C163—21no—
179C22×C163—22no—
180C22×C163—41no—
181C22×C163—41no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—84no—
186C22×C163C2×SD1612yescomplemented
187C22×C163C22×D411yescharacteristic
188C2×D163—21no—
189C2×D163—21no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—84no—
194C2×D163C2×SD1612yescomplemented
195C4×C163—21no—
196C4×C163—41no—
197C4×C163—41no—
198C4×C163C22×D411yescharacteristic
199C4×C163H(5,𝔽2)11yescharacteristic
200Dic163—21no—
201Dic163—22no—
202Dic163—21no—
203Dic163—41no—
204Dic163—41no—
205Dic163C2×SD1612yescomplemented
206C2×C163—22no—
207C2×C163—42no—
208C2×C163—81no—
209C2×C163D4211yescharacteristic
210C2×C163C22×SD1611yescharacteristic
211C2×C163#57711yescharacteristic
212D163—22no—
213D163—81no—
214C163D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—1631nomaximal, Sylow
216C22×SD16—1631no—
217C22×SD16—1631no—
218D4×Q8—1631no—
219D42—1631no—
220#492—1631no—
221#465—1631no—
222#465—1631no—
223#461—1631no—
224#460—1631no—
225#450—1631no—
226#450D16311yescharacteristic, complemented
227#448—1631no—
228#448—1631no—
229C4×SD16—1631no—
230C8×D4—1631no—
231C22×Q8—1631no—
232C22×Q8—1631no—
233C22×D4—3261no—
234C22×D4—1631no—
235C22×D4C2×D16311yescharacteristic
236C2×SD16—3264no—
237C2×SD16—3264no—
238C2×SD16—1631no—
239C2×SD16—1631no—
240C2×SD16—1631no—
241C2×SD16—1631no—
242C2×SD16—1631no—
243C2×SD16—1631no—
244C2×SD16—1631no—
245C2×SD16C2×D16311yescharacteristic
246C2×SD16C2×D16311yescharacteristic, complemented
247C22×C8—1631no—
248C22×C8—1631no—
249C4⋊Q8—3261no—
250C4⋊Q8—1631no—
251#127—1631no—
252C22⋊Q8—3261no—
253C22⋊Q8—3261no—
254C22⋊Q8—1631no—
255C22⋊Q8C2×D16311yescharacteristic, complemented
256#121—3261no—
257#121—1631no—
258#121—1631no—
259#120—3261no—
260#120—3261no—
261C4×Q8—1631no—
262C4×D4—3261no—
263C4×D4—1631no—
264C4×D4—1631no—
265#106—1631no—
266C4⋊C8—1631no—
267#103—1631no—
268#103—1631no—
269#103—1631no—
270#102—1631no—
271#102C2×D16311yescharacteristic
272#102C2×D16311yescharacteristic, complemented
273C22⋊C8—1631no—
274C22⋊C8C2×D16311yescharacteristic, complemented
275C4×C8—1631no—
276C24—21no—
277C24—3261no—
278C2×Q8—3261no—
279C2×Q8—3262no—
280C2×Q8—3261no—
281C2×Q8—3262no—
282C2×Q8—6521no—
283C2×Q8—6521no—
284C2×Q8—1631no—
285C2×Q8—1631no—
286C2×Q8C22×D16311yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—3262no—
291C2×D4—3261no—
292C2×D4—3261no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—3261no—
296C2×D4—6521no—
297C2×D4—6524no—
298C2×D4—1631no—
299C2×D4—1631no—
300C2×D4C22×D16311yescharacteristic
301C2×D4C22×D16311yescharacteristic
302C22×C4—3261no—
303C22×C4—3261no—
304C22×C4—3261no—
305C22×C4—1631no—
306C22×C4C22×D16311yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—3262no—
310SD16—3262no—
311SD16—3262no—
312SD16—3262no—
313SD16—3262no—
314SD16—6521no—
315SD16—1634no—
316C2×C8—3262no—
317C2×C8—3262no—
318C2×C8—1631no—
319C2×C8—1631no—
320C2×C8C22×D16311yescharacteristic
321C2×C8C22×D16311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—3261no—
324C4⋊C4—3261no—
325C4⋊C4—3261no—
326C4⋊C4—3261no—
327C4⋊C4—1631no—
328C4⋊C4—1631no—
329C4⋊C4C22×D16311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—3261no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4—3261no—
335C22⋊C4—1631no—
336C22⋊C4—1631no—
337C42—3261no—
338C42—1631no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—13041no—
345C23—3261no—
346C23—6522no—
347C23—6522no—
348C23—1631no—
349C23D4×D16311yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—3261no—
353Q8—3261no—
354Q8—3261no—
355Q8—6522no—
356Q8—6522no—
357Q8—1632no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—13041no—
363D4—3261no—
364D4—3261no—
365D4—6522no—
366D4—6522no—
367D4—6522no—
368D4—6522no—
369D4—1634no—
370D4D4×D16312yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3261no—
376C2×C4—3261no—
377C2×C4—3261no—
378C2×C4—3261no—
379C2×C4—3261no—
380C2×C4—6522no—
381C2×C4—6522no—
382C2×C4—6522no—
383C2×C4—1631no—
384C2×C4—1631no—
385C2×C4—1631no—
386C2×C4D4×D16311yescharacteristic
387C2×C4C23×D16311yescharacteristic
388C8—21no—
389C8—21no—
390C8—3261no—
391C8—1632no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—13044no—
402C22—3261no—
403C22—3261no—
404C22—6521no—
405C22—6521no—
406C22—6521no—
407C22—1632no—
408C22SD16×D16312yescomplemented
409C22C2×D4×D16311yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—3261no—
414C4—3261no—
415C4—3262no—
416C4—6521no—
417C4—6521no—
418C4—1632no—
419C4C2×D4×D16311yescharacteristic
420C4#1910478011yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—13041no—
425C2—3262no—
426C2#2378229711yescharacteristic
427C2C2×SD16×D16311yescharacteristic
428C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
prime radicalD4×C22⋊D163C2
hypercenter#450D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14579687C2semidirect
2. C2×SD16×D163C2semidirect
3. #23782328C2semidirect
5. #23782036C2semidirect
6. #23781896C2semidirect
7. #23781893C2semidirect
8. #23781868C2semidirect
9. #23781864C2semidirect
10. #23781817C2semidirect
11. #23781806C2semidirect
12. #23781802C2semidirect
13. #23781756C2semidirect
14. (C22⋊D163)⋊C8C2semidirect
15. #450×C163C2semidirect
16. C22×Dic326C22semidirect
17. C2×Q8×D163C22semidirect
20. C2×#14579688C22semidirect
21. C2×#14579688C22semidirect
25. C2×#14579687C22semidirect
26. C2×#14579687C22semidirect
27. C2×#14579686C22semidirect
29. C22×C163⋊C8C22semidirect
30. C2×C8×D163C22semidirect
32. #19104733C22semidirect
34. #19104715C22semidirect
42. #19104700C22semidirect
46. #19104666C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
50. #19104659C22semidirect
51. #19104655C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. C8×Dic163C22semidirect
56. C2×SD16×C163C22semidirect
57. C22⋊Q8×C163C22semidirect
59. #102×C163C22semidirect
60. C22⋊C8×C163C22semidirect
89. #14579687D4semidirect
134. Dic326C2×D4semidirect
142. C22⋊D163SD16semidirect
155. C163⋊C8C2×D4semidirect
186. C22×C163C2×SD16semidirect
194. C2×D163C2×SD16semidirect
205. Dic163C2×SD16semidirect
214. C163D4×SD16semidirect
226. #450D163semidirect
246. C2×SD16C2×D163semidirect
255. C22⋊Q8C2×D163semidirect
272. #102C2×D163semidirect
274. C22⋊C8C2×D163semidirect
408. C22SD16×D163semidirect

Maximal subgroups

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

subgroupindex
C22×#145796872
C2×SD16×D1632
#237823282
D4×C22⋊D1632
#237820362
#237818962
#237818932
#237818682
#237818642
#237818172
#237818062
#237818022
#237817562
(C22⋊D163)⋊C82
#450×C1632
D4×SD16163