Finite Group Databasebeta

← #35226025

Subgroups — #35226025

Order 20864.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D163C211yesmaximal, characteristic, complemented
2#23782322C211yesmaximal, characteristic, complemented
3D4×C22⋊D163C211yesmaximal, characteristic, complemented
4#23782053C211yesmaximal, characteristic, complemented
5#23781911C211yesmaximal, characteristic, complemented
6#23781898C211yesmaximal, characteristic, complemented
7#23781878C211yesmaximal, characteristic, complemented
8#23781863C211yesmaximal, characteristic, complemented
9#23781816C211yesmaximal, characteristic, complemented
10#23781810C211yesmaximal, characteristic, complemented
11#23781802C211yesmaximal, characteristic, complemented
12#23781797C211yesmaximal, characteristic, complemented
13#23781766C211yesmaximal, characteristic, complemented
14#23781734C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic, complemented
17C22×C22⋊D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×#14579688C2211yescharacteristic, complemented
20C2×#14579687C2211yescharacteristic
21C2×#14579687C2211yescharacteristic, complemented
22C2×#14579686C2211yescharacteristic, complemented
23SD16×D163—24no—
24C2×#14579684C2211yescharacteristic, complemented
25C2×C8×D163C2211yescharacteristic, complemented
26#19104731C2211yescharacteristic
27#19104722—21no—
28#19104722C2211yescharacteristic, complemented
29#19104714—21no—
30#19104714C2211yescharacteristic, complemented
31#19104709C2211yescharacteristic
32#19104707—21no—
33#19104706—21no—
34#19104706C2211yescharacteristic
35#19104706C2211yescharacteristic, complemented
36#19104704—21no—
37#19104703—21no—
38Q8×Dic163C2211yescharacteristic, complemented
39D4×Dic163C2211yescharacteristic
40C4×C22⋊D163—21no—
41C4×C22⋊D163C2211yescharacteristic
42#19104669C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104659C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic, complemented
49#19104656C2211yescharacteristic, complemented
50C22⋊(C163⋊C8)C2211yescharacteristic, complemented
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52#19104645C2211yescharacteristic, complemented
53C2×#41×C163C2211yescharacteristic, complemented
54C22×D4×C163C2211yescharacteristic
55AGL(1,ℤ/8)×C163—24no—
56C2×SD16×C163C2211yescharacteristic
57C2×SD16×C163C2211yescharacteristic, complemented
58C2×D8×C163C2211yescharacteristic
59C2×D8×C163C2211yescharacteristic, complemented
60C2×M16×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64Q8×D163—22no—
65Q8×D163—41no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—44no—
74C2×C22⋊D163D411yescharacteristic
75D4×D163—22no—
76D4×D163—41no—
77C22×Dic163—21no—
78C2×C4×D163—21no—
79C2×C4×D163C2311yescharacteristic
80#14579688—22no—
81#14579687—22no—
82#14579687—41no—
83#14579686—22no—
84#14579684—41no—
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C8×D163—22no—
88C4⋊Dic163—21no—
89C4⋊Dic163C2311yescharacteristic
90C4⋊Dic163C2311yescharacteristic
91Dic163⋊C4—21no—
92Dic163⋊C4D411yescharacteristic
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C22⋊Dic163—21no—
96C22⋊Dic163—21no—
97C22⋊Dic163—21no—
98C22⋊Dic163—21no—
99C22⋊Dic163D411yescharacteristic
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4D411yescharacteristic
102C4×Dic163—21no—
103C4×Dic163C2311yescharacteristic
104C24×C163—21no—
105#41×C163—22no—
106#41×C163—41no—
107C2×Q8×C163C2311yescharacteristic
108C2×D4×C163—22no—
109C2×D4×C163—21no—
110C2×D4×C163—21no—
111C2×D4×C163—41no—
112C2×D4×C163C2311yescharacteristic
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115C22×C4×C163—21no—
116C22×C4×C163C2311yescharacteristic
117SD16×C163—22no—
118SD16×C163D414yescomplemented
119D8×C163—22no—
120D8×C163—22no—
121M16×C163—22no—
122C2×C8×C163C2311yescharacteristic
123C2×C8×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163—81no—
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131Dic326—42no—
132D652—21no—
133D652—21no—
134C22⋊D163—22no—
135C22⋊D163—42no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C22⋊D163—42no—
140C22⋊D163—81no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—42no—
145C2×Dic163C2×D411yescharacteristic
146C2×Dic163C2×D411yescharacteristic
147C2×Dic163C2×D411yescharacteristic
148C4×D163—21no—
149C4×D163—21no—
150C4×D163—42no—
151C163⋊C8—21no—
152C163⋊C8—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163—42no—
157C23×C163—42no—
158C23×C163—81no—
159C23×C163C2×D411yescharacteristic
160Q8×C163—21no—
161Q8×C163C2×D412yescomplemented
162D4×C163—21no—
163D4×C163—21no—
164D4×C163—21no—
165D4×C163—21no—
166D4×C163—21no—
167D4×C163—42no—
168D4×C163—42no—
169D4×C163C2×D412yes—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163—21no—
173C2×C4×C163—42no—
174C2×C4×C163C2×D411yescharacteristic
175C2×C4×C163C2411yescharacteristic
176C8×C163—21no—
177C8×C163C2×D412yescomplemented
178C22×C163—21no—
179C22×C163—22no—
180C22×C163—21no—
181C22×C163—21no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163—84no—
191C22×C163C22×D411yescharacteristic
192C2×D163—21no—
193C2×D163—21no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—84no—
198C2×D163AGL(1,ℤ/8)12yescomplemented
199C4×C163—21no—
200C4×C163—22no—
201C4×C163C22×D411yescharacteristic
202C4×C163H(5,𝔽2)11yescharacteristic
203Dic163—21no—
204Dic163—21no—
205Dic163—21no—
206Dic163—41no—
207Dic163—41no—
208Dic163—41no—
209C2×C163—42no—
210C2×C163—41no—
211C2×C163—81no—
212C2×C163—81no—
213C2×C163D4211yescharacteristic
214C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C163#57711yescharacteristic
216D163—22no—
217D163—81no—
218C163#326511yescharacteristic, complemented, Sylow
219#3265—1631nomaximal, Sylow
220C2×AGL(1,ℤ/8)D16311yescharacteristic, complemented
221C22×SD16—1631no—
222#548—1631no—
223D42—1631no—
224#496—1631no—
225#468—1631no—
226#466—1631no—
227#463—1631no—
228#460—1631no—
229#450—1631no—
230#449—1631no—
231#448—1631no—
232#447—1631no—
233#440—1631no—
234#435—1631no—
235C2×#41C2×D16311yescharacteristic, complemented
236C22×Q8—1631no—
237C22×D4—3261no—
238C22×D4—1631no—
239C22×D4C2×D16311yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—3264no—
242C2×SD16—1631no—
243C2×SD16—1631no—
244C2×SD16—1631no—
245C2×SD16—1631no—
246C2×SD16C2×D16311yescharacteristic
247C2×SD16C2×D16311yescharacteristic, complemented
248C2×D8—1631no—
249C2×D8C2×D16311yescharacteristic
250C2×D8C2×D16311yescharacteristic, complemented
251C2×M16C2×D16311yescharacteristic, complemented
252C22×C8—1631no—
253#127—1631no—
254#124—3261no—
255#124—1631no—
256C22⋊Q8—3261no—
257C22⋊Q8—1631no—
258#121—3261no—
259#121—3261no—
260#121—1631no—
261#121—1631no—
262#121—1631no—
263#120—3261no—
264#120—3261no—
265C4×Q8—1631no—
266C4×D4—3261no—
267C4×D4—1631no—
268C4×D4—1631no—
269#107—1631no—
270C4⋊C8—1631no—
271#103—1631no—
272#103—1631no—
273#102—1631no—
274#102—1631no—
275#102—1631no—
276#102—1631no—
277C22⋊C8—1631no—
278C22⋊C8—1631no—
279#97—1631no—
280C24—21no—
281C24—3261no—
282#41—22no—
283#41—41no—
284C2×Q8—3261no—
285C2×Q8—3262no—
286C2×Q8—6521no—
287C2×Q8—1631no—
288C2×Q8C22×D16311yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—3262no—
294C2×D4—3261no—
295C2×D4—3261no—
296C2×D4—3261no—
297C2×D4—3261no—
298C2×D4—3261no—
299C2×D4—3261no—
300C2×D4—6521no—
301C2×D4—6524no—
302C2×D4—1631no—
303C2×D4—1631no—
304C2×D4C22×D16311yescharacteristic
305C2×D4C22×D16311yescharacteristic
306C2×D4C22×D16311yescharacteristic
307C22×C4—21no—
308C22×C4—3261no—
309C22×C4—3261no—
310C22×C4—1631no—
311C22×C4C22×D16311yescharacteristic
312SD16—22no—
313SD16—3262no—
314SD16—3262no—
315SD16—3262no—
316SD16—6521no—
317SD16C22⋊D16314yescomplemented
318D8—22no—
319D8—22no—
320D8—6521no—
321M16—22no—
322C2×C8—3262no—
323C2×C8—1631no—
324C2×C8—1631no—
325C2×C8C22×D16311yescharacteristic
326C2×C8C22×D16311yescharacteristic
327C4⋊C4—3261no—
328C4⋊C4—3261no—
329C4⋊C4—1631no—
330C4⋊C4—1631no—
331C4⋊C4—1631no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4—3261no—
335C22⋊C4—3261no—
336C22⋊C4—3261no—
337C22⋊C4—3261no—
338C22⋊C4—3261no—
339C22⋊C4—1631no—
340C22⋊C4—1631no—
341C42—3261no—
342C42—1631no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—13041no—
350C23—3261no—
351C23—6522no—
352C23—6522no—
353C23—1631no—
354C23D4×D16311yescharacteristic
355Q8—21no—
356Q8—3261no—
357Q8—3261no—
358Q8—6522no—
359Q8C2×C22⋊D16312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—13041no—
368D4—3262no—
369D4—3261no—
370D4—3261no—
371D4—6522no—
372D4—6522no—
373D4—6522no—
374D4—6522no—
375D4—6522no—
376D4C2×C22⋊D16312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—3261no—
382C2×C4—3261no—
383C2×C4—3261no—
384C2×C4—3261no—
385C2×C4—3261no—
386C2×C4—6522no—
387C2×C4—6522no—
388C2×C4—1631no—
389C2×C4—1631no—
390C2×C4—1631no—
391C2×C4D4×D16311yescharacteristic
392C2×C4C23×D16311yescharacteristic
393C8—21no—
394C8—3261no—
395C8—3261no—
396C8C2×C22⋊D16312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—13044no—
411C22—3261no—
412C22—3261no—
413C22—6521no—
414C22—6521no—
415C22—6521no—
416C22—1632no—
417C22C2×D4×D16311yescharacteristic
418C4—21no—
419C4—22no—
420C4—3261no—
421C4—3261no—
422C4—3261no—
423C4—6521no—
424C4—6521no—
425C4—6521no—
426C4C22×C22⋊D16311yescharacteristic
427C4#1910478011yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—13041no—
433C2—3262no—
434C2D4×C22⋊D16311yescharacteristic
435C2AGL(1,ℤ/8)×D16311yescharacteristic
436C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×AGL(1,ℤ/8)×C163C2
socleC22×C163C22×D4
hypercenterC2×AGL(1,ℤ/8)D163
hypocenterC163#3265
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. #23782322C2semidirect
3. D4×C22⋊D163C2semidirect
4. #23782053C2semidirect
5. #23781911C2semidirect
6. #23781898C2semidirect
7. #23781878C2semidirect
8. #23781863C2semidirect
9. #23781816C2semidirect
10. #23781810C2semidirect
11. #23781802C2semidirect
12. #23781797C2semidirect
13. #23781766C2semidirect
14. #23781734C2semidirect
15. C2×AGL(1,ℤ/8)×C163C2semidirect
16. C2×Q8×D163C22semidirect
19. C2×#14579688C22semidirect
21. C2×#14579687C22semidirect
22. C2×#14579686C22semidirect
24. C2×#14579684C22semidirect
25. C2×C8×D163C22semidirect
28. #19104722C22semidirect
30. #19104714C22semidirect
35. #19104706C22semidirect
38. Q8×Dic163C22semidirect
42. #19104669C22semidirect
43. Dic163⋊C8C22semidirect
44. #19104660C22semidirect
45. #19104659C22semidirect
46. #19104657C22semidirect
48. #19104656C22semidirect
49. #19104656C22semidirect
50. C22⋊(C163⋊C8)C22semidirect
51. (C2×D163)⋊C8C22semidirect
52. #19104645C22semidirect
53. C2×#41×C163C22semidirect
57. C2×SD16×C163C22semidirect
59. C2×D8×C163C22semidirect
60. C2×M16×C163C22semidirect
118. SD16×C163D4semidirect
161. Q8×C163C2×D4semidirect
177. C8×C163C2×D4semidirect
198. C2×D163AGL(1,ℤ/8)semidirect
218. C163#3265semidirect
220. C2×AGL(1,ℤ/8)D163semidirect
235. C2×#41C2×D163semidirect
247. C2×SD16C2×D163semidirect
250. C2×D8C2×D163semidirect
251. C2×M16C2×D163semidirect
317. SD16C22⋊D163semidirect
359. Q8C2×C22⋊D163semidirect
396. C8C2×C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
#237823222
D4×C22⋊D1632
#237820532
#237819112
#237818982
#237818782
#237818632
#237818162
#237818102
#237818022
#237817972
#237817662
#237817342
C2×AGL(1,ℤ/8)×C1632
#3265163