Finite Group Databasebeta

← #35226229

Subgroups — #35226229

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×#14579688C211yesmaximal, characteristic, complemented
2C2×SD16×D163C211yesmaximal, characteristic, complemented
3#23782328C211yesmaximal, characteristic, complemented
4#23782299C211yesmaximal, characteristic
5#23782036C211yesmaximal, characteristic, complemented
6#23781895C211yesmaximal, characteristic, complemented
7#23781869C211yesmaximal, characteristic, complemented
8#23781864C211yesmaximal, characteristic, complemented
9#23781817C211yesmaximal, characteristic, complemented
10#23781815C211yesmaximal, characteristic, complemented
11#23781805C211yesmaximal, characteristic, complemented
12#23781803C211yesmaximal, characteristic, complemented
13#23781752C211yesmaximal, characteristic, complemented
14C8×C22⋊D163C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic, complemented
17C2×Q8×D163C2211yescharacteristic, complemented
18C22×D652C2211yescharacteristic
19C2×D4×D163—21no—
20C2×D4×D163C2211yescharacteristic
21C2×#14579688—24no—
22C2×#14579688C2211yescharacteristic
23C2×#14579688C2211yescharacteristic
24C2×#14579688C2211yescharacteristic, complemented
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579687C2211yescharacteristic, complemented
27C2×#14579687C2211yescharacteristic, complemented
28C2×#14579686C2211yescharacteristic
29C2×#14579686C2211yescharacteristic, complemented
30SD16×D163—24no—
31C2×C8×D163C2211yescharacteristic, complemented
32#19104734—21no—
33#19104733C2211yescharacteristic, complemented
34#19104731C2211yescharacteristic
35#19104715C2211yescharacteristic, complemented
36#19104713—21no—
37#19104712—21no—
38#19104710C2211yescharacteristic
39#19104708—21no—
40#19104705—21no—
41#19104703—21no—
42#19104700C2211yescharacteristic, complemented
43#19104697C2211yescharacteristic
44C4×C22⋊D163C2211yescharacteristic
45(C22⋊D163)⋊C4—21no—
46Dic163⋊C8C2211yescharacteristic, complemented
47#19104658C2211yescharacteristic, complemented
48#19104658C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic
50#19104655C2211yescharacteristic, complemented
51C22⋊(C163⋊C8)C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic, complemented
56C22⋊Q8×C163C2211yescharacteristic, complemented
57#121×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63C2×Dic326—21no—
64C2×Dic326—22no—
65C2×Dic326—21no—
66C2×Dic326—41no—
67C2×Dic326C2311yescharacteristic
68C2×Dic326C2311yescharacteristic
69Q8×D163—22no—
70Q8×D163—41no—
71C2×D652—22no—
72C2×D652—21no—
73C2×D652—21no—
74C2×D652—41no—
75C2×D652C2311yescharacteristic
76C2×D652C2311yescharacteristic
77C2×C22⋊D163—21no—
78C2×C22⋊D163—21no—
79C2×C22⋊D163—21no—
80C2×C22⋊D163D411yescharacteristic
81D4×D163—22no—
82D4×D163—41no—
83D4×D163—44no—
84C22×Dic163—21no—
85C2×C4×D163—21no—
86C2×C4×D163—21no—
87C2×C4×D163C2311yescharacteristic
88#14579688—22no—
89#14579688—22no—
90#14579688—22no—
91#14579688D414yescomplemented
92#14579687—22no—
93#14579687—41no—
94#14579686—22no—
95#14579686—41no—
96C2×C163⋊C8C2311yescharacteristic
97C2×C163⋊C8C2311yescharacteristic
98C8×D163—22no—
99C4⋊Dic163—21no—
100Dic163⋊C4—21no—
101Dic163⋊C4—21no—
102Dic163⋊C4—21no—
103Dic163⋊C4D411yescharacteristic
104C22⋊Dic163D411yescharacteristic
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4D411yescharacteristic
109C4×Dic163—21no—
110C4×Dic163C2311yescharacteristic
111C2×Q8×C163C2311yescharacteristic
112C2×D4×C163—21no—
113C2×D4×C163C2311yescharacteristic
114C22×C4×C163C2311yescharacteristic
115SD16×C163—22no—
116C2×C8×C163—22no—
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C4⋊C4×C163—21no—
120C4⋊C4×C163C2311yescharacteristic
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C22⋊C4×C163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—81no—
131C22×D163—81no—
132C22×D163C2×D411yescharacteristic
133Dic326—21no—
134Dic326—21no—
135Dic326—21no—
136Dic326—42no—
137Dic326—42no—
138Dic326C2×D412yescomplemented
139D652—21no—
140D652—21no—
141D652—21no—
142D652—42no—
143D652—42no—
144D652C2×D412yes—
145C22⋊D163—42no—
146C22⋊D163—42no—
147C22⋊D163—81no—
148C22⋊D163SD1614yescomplemented
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163—42no—
152C2×Dic163C2×D411yescharacteristic
153C2×Dic163C2×D411yescharacteristic
154C2×Dic163C2×D411yescharacteristic
155C4×D163—21no—
156C4×D163—21no—
157C4×D163—42no—
158C4×D163—42no—
159C163⋊C8—21no—
160C163⋊C8—21no—
161C23×C163—21no—
162C23×C163C2×D411yescharacteristic
163Q8×C163—21no—
164Q8×C163—21no—
165D4×C163—21no—
166D4×C163—21no—
167D4×C163—42no—
168C2×C4×C163—21no—
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×C163C2×D412yescomplemented
177C22×C163—21no—
178C22×C163—21no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163—41no—
182C22×C163C2×SD1612yescomplemented
183C22×C163C22×D411yescharacteristic
184C2×D163—21no—
185C2×D163—21no—
186C2×D163—22no—
187C2×D163—41no—
188C2×D163—41no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—84no—
193C2×D163—84no—
194C2×D163C2×SD1612yescomplemented
195C4×C163—21no—
196C4×C163—41no—
197C4×C163—41no—
198C4×C163—41no—
199C4×C163C22×D411yescharacteristic
200C4×C163H(5,𝔽2)11yescharacteristic
201Dic163—21no—
202Dic163—22no—
203Dic163—21no—
204Dic163—41no—
205Dic163C2×SD1612yescomplemented
206C2×C163—22no—
207C2×C163—81no—
208C2×C163D4211yescharacteristic
209C2×C163C22×SD1611yescharacteristic
210C2×C163#57711yescharacteristic
211D163—22no—
212D163—42no—
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#465D16311yescharacteristic, complemented
223#461—1631no—
224#460—1631no—
225#450—1631no—
226#450—1631no—
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×D4—1631no—
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×SD16—1631no—
246C2×SD16C2×D16311yescharacteristic, complemented
247C22×C8—1631no—
248C22×C8C2×D16311yescharacteristic, complemented
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#121C2×D16311yescharacteristic
259#120—3261no—
260#120—3261no—
261C4×Q8—1631no—
262C4×D4—3261no—
263C4×D4—1631no—
264C4×D4—1631no—
265#106C2×D16311yescharacteristic, complemented
266C4⋊C8—1631no—
267#103—1631no—
268#103—1631no—
269#103C2×D16311yescharacteristic, complemented
270#102—1631no—
271#102—1631no—
272#102C2×D16311yescharacteristic, complemented
273C22⋊C8—1631no—
274C22⋊C8—1631no—
275C4×C8—1631no—
276C24—3261no—
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—21no—
288C2×D4—3262no—
289C2×D4—3261no—
290C2×D4—3262no—
291C2×D4—3261no—
292C2×D4—3261no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—6521no—
296C2×D4—6521no—
297C2×D4—6524no—
298C2×D4—1631no—
299C2×D4—1631no—
300C2×D4—1631no—
301C2×D4C22×D16311yescharacteristic
302C22×C4—3261no—
303C22×C4—3261no—
304C22×C4—3261no—
305C22×C4—1631no—
306C22×C4C22×D16311yescharacteristic
307SD16—22no—
308SD16—3262no—
309SD16—3262no—
310SD16—3262no—
311SD16—3262no—
312SD16—3262no—
313SD16—6521no—
314SD16—6521no—
315SD16—1634no—
316C2×C8—22no—
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⋊C4C22×D16311yescharacteristic
329C4⋊C4C22×D16311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4—3261no—
335C22⋊C4—1631no—
336C22⋊C4—1631no—
337C42—3261no—
338C42—1631no—
339C23—21no—
340C23—13041no—
341C23—13041no—
342C23—3261no—
343C23—3261no—
344C23—6522no—
345C23—6522no—
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—42no—
361D4—13041no—
362D4—3261no—
363D4—3261no—
364D4—3261no—
365D4—6522no—
366D4—6522no—
367D4—6522no—
368D4—6522no—
369D4—1634no—
370D4—1632no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
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—3261no—
390C8—3261no—
391C8D4×D16312yescomplemented
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—13044no—
398C22—13044no—
399C22—3261no—
400C22—3261no—
401C22—3262no—
402C22—6521no—
403C22—6521no—
404C22—6521no—
405C22—6521no—
406C22—6521no—
407C22—1632no—
408C22SD16×D16312yescomplemented
409C22C2×D4×D16311yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—3261no—
415C4—3261no—
416C4—3262no—
417C4—6521no—
418C4—1632no—
419C4C2×D4×D16311yescharacteristic
420C4#1910478111yescharacteristic
421C2—22no—
422C2—81no—
423C2—13041no—
424C2—3262no—
425C2—6522no—
426C2#2378229911yescharacteristic
427C2C2×SD16×D16311yescharacteristic
428C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#23782299C2
hypercenter#465D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C22×#14579688C2semidirect
2. C2×SD16×D163C2semidirect
3. #23782328C2semidirect
5. #23782036C2semidirect
6. #23781895C2semidirect
7. #23781869C2semidirect
8. #23781864C2semidirect
9. #23781817C2semidirect
10. #23781815C2semidirect
11. #23781805C2semidirect
12. #23781803C2semidirect
13. #23781752C2semidirect
14. C8×C22⋊D163C2semidirect
15. #465×C163C2semidirect
16. C22×Dic326C22semidirect
17. C2×Q8×D163C22semidirect
24. C2×#14579688C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579687C22semidirect
27. C2×#14579687C22semidirect
29. C2×#14579686C22semidirect
31. C2×C8×D163C22semidirect
33. #19104733C22semidirect
35. #19104715C22semidirect
42. #19104700C22semidirect
46. Dic163⋊C8C22semidirect
47. #19104658C22semidirect
48. #19104658C22semidirect
50. #19104655C22semidirect
51. C22⋊(C163⋊C8)C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. C8×Dic163C22semidirect
54. C2×SD16×C163C22semidirect
55. C22×C8×C163C22semidirect
56. C22⋊Q8×C163C22semidirect
58. #106×C163C22semidirect
59. #103×C163C22semidirect
60. #102×C163C22semidirect
91. #14579688D4semidirect
138. Dic326C2×D4semidirect
148. C22⋊D163SD16semidirect
176. C8×C163C2×D4semidirect
182. C22×C163C2×SD16semidirect
194. C2×D163C2×SD16semidirect
205. Dic163C2×SD16semidirect
214. C163D4×SD16semidirect
222. #465D163semidirect
246. C2×SD16C2×D163semidirect
248. C22×C8C2×D163semidirect
255. C22⋊Q8C2×D163semidirect
265. #106C2×D163semidirect
269. #103C2×D163semidirect
272. #102C2×D163semidirect
391. C8D4×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×#145796882
C2×SD16×D1632
#237823282
#237822992
#237820362
#237818952
#237818692
#237818642
#237818172
#237818152
#237818052
#237818032
#237817522
C8×C22⋊D1632
#465×C1632
D4×SD16163