Finite Group Databasebeta

← #35226032

Subgroups — #35226032

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×#19104760C211yesmaximal, characteristic, complemented
2C22×#14579686C211yesmaximal, characteristic, complemented
3D4×D652C211yesmaximal, characteristic, complemented
4#23782055C211yesmaximal, characteristic, complemented
5#23781913C211yesmaximal, characteristic, complemented
6#23781899C211yesmaximal, characteristic, complemented
7#23781877C211yesmaximal, characteristic, complemented
8#23781865C211yesmaximal, characteristic, complemented
9#23781815C211yesmaximal, characteristic, complemented
10#23781810C211yesmaximal, characteristic, complemented
11#23781804C211yesmaximal, characteristic, complemented
12#23781798C211yesmaximal, characteristic, complemented
13#23781763C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×D4×D163—21no—
19#19104760—24no—
20C2×#14579686—24no—
21C2×#14579686C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23C2×#14579686C2211yescharacteristic, complemented
24C2×#14579686C2211yescharacteristic, complemented
25C2×#14579686C2211yescharacteristic, complemented
26C2×#14579686C2211yescharacteristic, complemented
27C2×#14579684C2211yescharacteristic
28C2×#14579684C2211yescharacteristic, complemented
29C2×#14579684C2211yescharacteristic, complemented
30C2×#14579683C2211yescharacteristic, complemented
31C22×C163⋊C8C2211yescharacteristic, complemented
32#19104732C2211yescharacteristic
33#19104710—21no—
34#19104709C2211yescharacteristic
35#19104709C2211yescharacteristic
36#19104705—21no—
37#19104705—21no—
38C4×D652C2211yescharacteristic
39#19104668C2211yescharacteristic, complemented
40C4⋊(C163⋊C8)C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104656C2211yescharacteristic, complemented
44#19104655C2211yescharacteristic
45#19104655C2211yescharacteristic, complemented
46#19104655C2211yescharacteristic, complemented
47C22⋊(C163⋊C8)C2211yescharacteristic, complemented
48C22⋊(C163⋊C8)C2211yescharacteristic, complemented
49#19104646C2211yescharacteristic, complemented
50C2×#41×C163C2211yescharacteristic, complemented
51C22×Q8×C163C2211yescharacteristic, complemented
52#124×C163—21no—
53#124×C163C2211yescharacteristic, complemented
54C22⋊Q8×C163—21no—
55C22⋊Q8×C163C2211yescharacteristic, complemented
56#121×C163—21no—
57#121×C163C2211yescharacteristic, complemented
58C4×Q8×C163C2211yescharacteristic, complemented
59C4×D4×C163—21no—
60C4×D4×C163C2211yescharacteristic
61C23×D163—21no—
62C23×D163—21no—
63C2×D652—22no—
64C2×D652—22no—
65C2×D652—21no—
66C2×D652—21no—
67C2×D652—21no—
68C2×D652—21no—
69C2×D652—41no—
70C2×D652—41no—
71C2×D652C2311yescharacteristic
72C2×D652C2311yescharacteristic
73C2×D652C2311yescharacteristic
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76D4×D163—44no—
77C2×C4×D163—21no—
78#14579686—22no—
79#14579686—22no—
80#14579686—22no—
81#14579686—22no—
82#14579686—41no—
83#14579686D414yescomplemented
84#14579684—22no—
85#14579684—22no—
86#14579684—41no—
87#14579683—22no—
88C2×C163⋊C8—22no—
89C2×C163⋊C8C2311yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C2×C163⋊C8C2311yescharacteristic
92C2×C163⋊C8C2311yescharacteristic
93C4⋊Dic163C2311yescharacteristic
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97#41×C163—22no—
98#41×C163—41no—
99C2×Q8×C163—21no—
100C2×Q8×C163—22no—
101C2×Q8×C163—41no—
102C2×Q8×C163C2311yescharacteristic
103C2×Q8×C163C2311yescharacteristic
104C2×D4×C163—21no—
105C2×D4×C163—21no—
106C2×D4×C163D411yescharacteristic
107C2×D4×C163C2311yescharacteristic
108C22×C4×C163—21no—
109C22×C4×C163—21no—
110C22×C4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C4⋊C4×C163—21no—
113C4⋊C4×C163—21no—
114C4⋊C4×C163D411yescharacteristic
115C4⋊C4×C163C2311yescharacteristic
116C22⋊C4×C163—21no—
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163—21no—
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—81no—
132C22×D163—81no—
133D652—21no—
134D652—21no—
135D652—21no—
136D652—21no—
137D652—21no—
138D652—42no—
139D652—42no—
140D652—42no—
141D652—81no—
142D652C2×D412yes—
143C22⋊D163—42no—
144C22⋊D163—42no—
145C2×Dic163—21no—
146C4×D163—42no—
147C163⋊C8—21no—
148C163⋊C8—21no—
149C163⋊C8—21no—
150C163⋊C8C2×D412yescomplemented
151C23×C163—21no—
152C23×C163C2×D411yescharacteristic
153C23×C163C2×D411yescharacteristic
154Q8×C163—21no—
155Q8×C163—21no—
156Q8×C163—21no—
157Q8×C163—42no—
158Q8×C163C2×D412yescomplemented
159D4×C163—22no—
160D4×C163—21no—
161D4×C163—21no—
162D4×C163—42no—
163D4×C163—42no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—42no—
172C2×C4×C163—42no—
173C2×C4×C163C2×D411yescharacteristic
174C2×C4×C163C2×D411yescharacteristic
175C2×C4×C163C2×D411yescharacteristic
176C2×C4×C163C2×D411yescharacteristic
177C2×C4×C163C2411yescharacteristic
178C22×C163—21no—
179C22×C163—21no—
180C22×C163—21no—
181C22×C163—21no—
182C22×C163—21no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163AGL(1,ℤ/8)12yescomplemented
187C22×C163C22×D411yescharacteristic
188C2×D163—22no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—84no—
198C2×D163—84no—
199C4×C163—21no—
200C4×C163—21no—
201C4×C163—22no—
202C4×C163—21no—
203C4×C163—21no—
204C4×C163—41no—
205C4×C163—41no—
206C4×C163C22×D411yescharacteristic
207C4×C163H(5,𝔽2)11yescharacteristic
208Dic163—41no—
209C2×C163—22no—
210C2×C163—41no—
211C2×C163—81no—
212C2×C163D4211yescharacteristic
213C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C163#57711yescharacteristic
215D163—42no—
216D163—81no—
217D163—81no—
218C163#326511yescharacteristic, complemented, Sylow
219#3265—1631nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1631no—
221C22×SD16—1631no—
222#548D16311yescharacteristic, complemented
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×Q8C2×D16311yescharacteristic, complemented
237C22×D4—3261no—
238C22×D4—1631no—
239C22×D4—1631no—
240AGL(1,ℤ/8)—3264no—
241C2×SD16—3264no—
242C2×SD16—1631no—
243C2×SD16—1631no—
244C2×SD16—1631no—
245C2×SD16—1631no—
246C2×SD16—1631no—
247C2×SD16—1631no—
248C2×D8—1631no—
249C2×D8—1631no—
250C2×D8—1631no—
251C2×M16—1631no—
252C22×C8—1631no—
253#127—1631no—
254#124—21no—
255#124C2×D16311yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D16311yescharacteristic, complemented
258#121—21no—
259#121—3261no—
260#121—1631no—
261#121—1631no—
262#121C2×D16311yescharacteristic, complemented
263#120—3261no—
264#120—3261no—
265C4×Q8C2×D16311yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1631no—
268C4×D4C2×D16311yescharacteristic
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—3261no—
281C24—3261no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D16311yescharacteristic
288C2×Q8C22×D16311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—3262no—
292C2×D4—3262no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—3261no—
296C2×D4—3261no—
297C2×D4—3261no—
298C2×D4—3261no—
299C2×D4—6521no—
300C2×D4—6521no—
301C2×D4—6524no—
302C2×D4—1631no—
303C2×D4—1631no—
304C2×D4—1631no—
305C2×D4C22⋊D16311yescharacteristic
306C2×D4C22×D16311yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—3261no—
310C22×C4C22×D16311yescharacteristic
311C22×C4C22×D16311yescharacteristic
312SD16—3262no—
313SD16—3262no—
314SD16—3262no—
315SD16—3262no—
316SD16—6521no—
317SD16—1634no—
318D8—3262no—
319D8—3262no—
320D8—6521no—
321M16—3262no—
322C2×C8—3262no—
323C2×C8—1631no—
324C2×C8—1631no—
325C2×C8—1631no—
326C2×C8—1631no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1631no—
330C4⋊C4C22⋊D16311yescharacteristic
331C4⋊C4C22×D16311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—3261no—
337C22⋊C4—3261no—
338C22⋊C4—3261no—
339C22⋊C4C22⋊D16311yescharacteristic
340C22⋊C4C22⋊D16311yescharacteristic
341C42—21no—
342C42C22×D16311yescharacteristic
343C23—21no—
344C23—13041no—
345C23—13041no—
346C23—3261no—
347C23—3261no—
348C23—3261no—
349C23—6522no—
350C23—6522no—
351C23—6522no—
352C23—6522no—
353C23C2×C22⋊D16311yescharacteristic
354C23C2×C22⋊D16311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D16312yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—13041no—
366D4—3261no—
367D4—3261no—
368D4—3261no—
369D4—3261no—
370D4—3261no—
371D4—6522no—
372D4—6522no—
373D4—6522no—
374D4—6522no—
375D4—6522no—
376D4—1632no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—3261no—
387C2×C4—6522no—
388C2×C4C2×C22⋊D16311yescharacteristic
389C2×C4C2×C22⋊D16311yescharacteristic
390C2×C4C2×C22⋊D16311yescharacteristic
391C2×C4C2×C22⋊D16311yescharacteristic
392C2×C4C23×D16311yescharacteristic
393C8—3261no—
394C8—3261no—
395C8—3261no—
396C8—1632no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—13044no—
406C22—13044no—
407C22—3262no—
408C22—6521no—
409C22—6521no—
410C22—6521no—
411C22—6521no—
412C22—6521no—
413C22—6521no—
414C22—6521no—
415C22—6521no—
416C22#1910476012yescomplemented
417C22C22×C22⋊D16311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—6521no—
426C4C2×D4×D16311yescharacteristic
427C4#1910478011yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—13041no—
432C2—13041no—
433C2—6522no—
434C2D4×C22⋊D16311yescharacteristic
435C2C2×#1910476011yescharacteristic
436C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#548×C163C2
socleC22×C163C22×D4
hypercenter#548D163
hypocenterC163#3265
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19104760C2semidirect
2. C22×#14579686C2semidirect
3. D4×D652C2semidirect
4. #23782055C2semidirect
5. #23781913C2semidirect
6. #23781899C2semidirect
7. #23781877C2semidirect
8. #23781865C2semidirect
9. #23781815C2semidirect
10. #23781810C2semidirect
11. #23781804C2semidirect
12. #23781798C2semidirect
13. #23781763C2semidirect
14. #23781737C2semidirect
15. #548×C163C2semidirect
23. C2×#14579686C22semidirect
24. C2×#14579686C22semidirect
25. C2×#14579686C22semidirect
26. C2×#14579686C22semidirect
28. C2×#14579684C22semidirect
29. C2×#14579684C22semidirect
30. C2×#14579683C22semidirect
31. C22×C163⋊C8C22semidirect
39. #19104668C22semidirect
40. C4⋊(C163⋊C8)C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
43. #19104656C22semidirect
45. #19104655C22semidirect
46. #19104655C22semidirect
47. C22⋊(C163⋊C8)C22semidirect
48. C22⋊(C163⋊C8)C22semidirect
49. #19104646C22semidirect
50. C2×#41×C163C22semidirect
51. C22×Q8×C163C22semidirect
53. #124×C163C22semidirect
55. C22⋊Q8×C163C22semidirect
57. #121×C163C22semidirect
58. C4×Q8×C163C22semidirect
83. #14579686D4semidirect
150. C163⋊C8C2×D4semidirect
158. Q8×C163C2×D4semidirect
186. C22×C163AGL(1,ℤ/8)semidirect
218. C163#3265semidirect
222. #548D163semidirect
235. C2×#41C2×D163semidirect
236. C22×Q8C2×D163semidirect
255. #124C2×D163semidirect
257. C22⋊Q8C2×D163semidirect
262. #121C2×D163semidirect
265. C4×Q8C2×D163semidirect
359. Q8D4×D163semidirect
416. C22#19104760semidirect

Maximal subgroups

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

subgroupindex
C2×#191047602
C22×#145796862
D4×D6522
#237820552
#237819132
#237818992
#237818772
#237818652
#237818152
#237818102
#237818042
#237817982
#237817632
#237817372
#548×C1632
#3265163