Finite Group Databasebeta

← #35226300

Subgroups — #35226300

Order 20864.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782476C211yesmaximal, characteristic, complemented
2#23782472C211yesmaximal, characteristic, complemented
3#23782468C211yesmaximal, characteristic, complemented
4#23782053C211yesmaximal, characteristic, complemented
5#23781933C211yesmaximal, characteristic, complemented
6#23781932C211yesmaximal, characteristic, complemented
7#23781928C211yesmaximal, characteristic, complemented
8#23781926C211yesmaximal, characteristic, complemented
9#23781843C211yesmaximal, characteristic, complemented
10#23781836C211yesmaximal, characteristic, complemented
11#23781833C211yesmaximal, characteristic, complemented
12#23781831C211yesmaximal, characteristic, complemented
13#23781787C211yesmaximal, characteristic, complemented
14#23781785C211yesmaximal, characteristic, complemented
15#576×C163C211yesmaximal, characteristic, complemented
16#19104782C2211yescharacteristic, complemented
17#19104781C2211yescharacteristic, complemented
18#19104767—21no—
19#19104765—21no—
20#19104765C2211yescharacteristic, complemented
21#19104765C2211yescharacteristic, complemented
22#19104761—21no—
23#19104761—21no—
24#19104761C2211yescharacteristic, complemented
25#19104761C2211yescharacteristic, complemented
26#19104760—21no—
27#19104760C2211yescharacteristic
28#19104760C2211yescharacteristic, complemented
29#19104759—21no—
30#19104756—21no—
31#19104755—21no—
32#19104755—21no—
33#19104755C2211yescharacteristic
34C2×#14579688C2211yescharacteristic, complemented
35C2×#14579687—21no—
36C2×#14579687C2211yescharacteristic, complemented
37SD16×D163—21no—
38C2×#14579684—21no—
39C2×#14579684C2211yescharacteristic, complemented
40#19104742C2211yescharacteristic, complemented
41#19104742C2211yescharacteristic, complemented
42#19104741C2211yescharacteristic, complemented
43#19104731C2211yescharacteristic, complemented
44#19104722C2211yescharacteristic, complemented
45#19104671C2211yescharacteristic, complemented
46#19104670C2211yescharacteristic, complemented
47#19104662C2211yescharacteristic, complemented
48#19104662C2211yescharacteristic, complemented
49#19104662C2211yescharacteristic, complemented
50#19104661C2211yescharacteristic, complemented
51#19104653C2211yescharacteristic, complemented
52#19104652C2211yescharacteristic, complemented
53#19104652C2211yescharacteristic, complemented
54#19104651C2211yescharacteristic, complemented
55#19104645C2211yescharacteristic, complemented
56H(5,𝔽2)×C163C2211yescharacteristic
57H(5,𝔽2)×C163C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C163—21no—
59AGL(1,ℤ/8)×C163—21no—
60AGL(1,ℤ/8)×C163C2211yescharacteristic
61AGL(1,ℤ/8)×C163C2211yescharacteristic, complemented
62#135×C163—21no—
63#135×C163C2211yescharacteristic
64C2×D8×C163—21no—
65C2×D8×C163C2211yescharacteristic, complemented
66#131×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579699—21no—
69#14579699—21no—
70#14579699C2311yescharacteristic
71#14579698—21no—
72#14579698—41no—
73C2×Dic326—21no—
74C2×Dic326C2311yescharacteristic
75Q8×D163—21no—
76C2×D652—21no—
77C2×D652C2311yescharacteristic
78C2×C22⋊D163—41no—
79D4×D163—21no—
80D4×D163—41no—
81#14579689—21no—
82#14579689—21no—
83#14579689D411yescharacteristic, complemented
84#14579688—21no—
85#14579688—21no—
86#14579687—21no—
87#14579687—21no—
88#14579687—21no—
89#14579687—21no—
90#14579687—21no—
91#14579687—21no—
92#14579687D411yescharacteristic, complemented
93#14579686—21no—
94#14579686—21no—
95#14579686D411yescharacteristic, complemented
96#14579684—21no—
97#14579684—21no—
98#14579684—21no—
99#14579684—21no—
100#14579684—21no—
101#14579684—21no—
102#14579684D411yescharacteristic, complemented
103#14579683—21no—
104#14579683—21no—
105#14579683C2311yescharacteristic
106#14579683C2311yescharacteristic
107#14579683C2311yescharacteristic
108#14579682—21no—
109C2×C163⋊C8—21no—
110C2×C163⋊C8—21no—
111C2×C163⋊C8C2311yescharacteristic
112C8×D163—21no—
113C22⋊Dic163—41no—
114C4×Dic163C2311yescharacteristic
115#41×C163—21no—
116#41×C163—21no—
117#41×C163—21no—
118#41×C163—21no—
119#41×C163C2311yescharacteristic
120#41×C163C2311yescharacteristic
121#41×C163C2311yescharacteristic
122C2×D4×C163—21no—
123C2×D4×C163—21no—
124C2×D4×C163—21no—
125C2×D4×C163—21no—
126C2×D4×C163—41no—
127C2×D4×C163—41no—
128C2×D4×C163C2311yescharacteristic
129C2×D4×C163C2311yescharacteristic
130Q16×C163D411yescharacteristic, complemented
131SD16×C163—21no—
132SD16×C163—21no—
133SD16×C163D411yescharacteristic, complemented
134SD16×C163D411yescharacteristic, complemented
135D8×C163—21no—
136D8×C163—21no—
137D8×C163—21no—
138D8×C163—21no—
139D8×C163D411yescharacteristic, complemented
140M16×C163—21no—
141M16×C163C2311yescharacteristic
142C2×C8×C163—21no—
143C2×C8×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—41no—
146Dic326—21no—
147Dic326—21no—
148Dic326—41no—
149Dic326C2×D411yescharacteristic
150D652—21no—
151D652—21no—
152D652—41no—
153D652C2×D411yescharacteristic
154C22⋊D163—21no—
155C22⋊D163—41no—
156C22⋊D163—41no—
157C22⋊D163—82no—
158C2×Dic163—21no—
159C2×Dic163—21no—
160C2×Dic163—41no—
161C4×D163—21no—
162C4×D163—41no—
163C4×D163—41no—
164C163⋊C8—21no—
165C163⋊C8—21no—
166C163⋊C8—21no—
167C163⋊C8C2×D411yescharacteristic
168C163⋊C8C2×D411yescharacteristic
169C23×C163—21no—
170C23×C163—21no—
171C23×C163—41no—
172C23×C163—41no—
173Q8×C163C2×D411yescharacteristic
174Q8×C163C2×D411yescharacteristic
175Q8×C163C2×D411yescharacteristic
176D4×C163—21no—
177D4×C163—21no—
178D4×C163—21no—
179D4×C163—21no—
180D4×C163—21no—
181D4×C163—21no—
182D4×C163—21no—
183D4×C163—41no—
184D4×C163—41no—
185D4×C163—41no—
186D4×C163—41no—
187D4×C163C2×D411yescharacteristic
188D4×C163C2×D411yescharacteristic
189D4×C163C2×D411yescharacteristic
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163—21no—
193C2×C4×C163—41no—
194C2×C4×C163—41no—
195C2×C4×C163C2411yescharacteristic
196C8×C163—21no—
197C8×C163C2×D411yescharacteristic
198C8×C163C2×D411yescharacteristic
199C22×C163—21no—
200C22×C163—21no—
201C22×C163—21no—
202C22×C163—41no—
203C22×C163—41no—
204C22×C163—81no—
205C22×C163—81no—
206C22×C163—161no—
207C22×C163—161no—
208C22×C163H(5,𝔽2)11yescharacteristic
209C2×D163—21no—
210C2×D163—41no—
211C2×D163—81no—
212C2×D163—161no—
213C4×C163—21no—
214C4×C163—21no—
215C4×C163—21no—
216C4×C163C22×D411yescharacteristic
217C4×C163C22×D411yescharacteristic
218Dic163—21no—
219Dic163—41no—
220Dic163—41no—
221C2×C163—21no—
222C2×C163—41no—
223C2×C163—41no—
224C2×C163—41no—
225C2×C163—81no—
226C2×C163—81no—
227C2×C163D4211yescharacteristic
228D163—41no—
229D163—81no—
230C163#328811yescharacteristic, complemented, Sylow
231#3288—1631nomaximal, Sylow
232#577—1631no—
233#577—1631no—
234#576—1631no—
235#576D16311yescharacteristic, complemented
236#496—1631no—
237#472—1631no—
238#472—1631no—
239#471—1631no—
240#471—1631no—
241#455—1631no—
242#454—1631no—
243#453—1631no—
244#453—1631no—
245#444—1631no—
246#444—1631no—
2472−1+4—1631no—
248H(5,𝔽2)—1631no—
249H(5,𝔽2)C2×D16311yescharacteristic
250H(5,𝔽2)C2×D16311yescharacteristic, complemented
251#137—3261no—
252#137—3261no—
253#137—1631no—
254#137—1631no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—3261no—
258AGL(1,ℤ/8)—3261no—
259AGL(1,ℤ/8)—3261no—
260AGL(1,ℤ/8)—3261no—
261AGL(1,ℤ/8)—1631no—
262AGL(1,ℤ/8)—1631no—
263AGL(1,ℤ/8)—1631no—
264AGL(1,ℤ/8)—1631no—
265AGL(1,ℤ/8)C2×D16311yescharacteristic
266AGL(1,ℤ/8)C2×D16311yescharacteristic, complemented
267#135—21no—
268#135—3261no—
269#135—3261no—
270#135—3261no—
271#135—1631no—
272#135C2×D16311yescharacteristic
273C2×SD16—3261no—
274C2×SD16—3261no—
275C2×SD16—1631no—
276C2×SD16—1631no—
277C2×D8—21no—
278C2×D8—3261no—
279C2×D8—1631no—
280C2×D8C2×D16311yescharacteristic, complemented
281#131—1631no—
282#131—1631no—
283#131—1631no—
284#131C2×D16311yescharacteristic, complemented
285#127—1631no—
286#124—1631no—
287#108—1631no—
288#108—1631no—
289#104—1631no—
290#104—1631no—
291#104—1631no—
292#104—1631no—
293#101—1631no—
294#100—1631no—
295#100—1631no—
296#100—1631no—
297#97—1631no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—3261no—
303#41—3261no—
304#41—3261no—
305#41—3261no—
306#41—6521no—
307#41—1631no—
308#41C22×D16311yescharacteristic
309#41C22×D16311yescharacteristic
310#41C22×D16311yescharacteristic
311C2×Q8—3261no—
312C2×Q8—3261no—
313C2×Q8—1631no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—3261no—
321C2×D4—3261no—
322C2×D4—6521no—
323C2×D4—6521no—
324C2×D4—1631no—
325C2×D4C22×D16311yescharacteristic
326C2×D4C22×D16311yescharacteristic
327Q16—3261no—
328Q16—3261no—
329Q16—1631no—
330Q16C22⋊D16311yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—3261no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338SD16—3261no—
339SD16—3261no—
340SD16—3261no—
341SD16—3261no—
342SD16—3261no—
343SD16—1631no—
344SD16—1631no—
345SD16C22⋊D16311yescharacteristic, complemented
346SD16C22⋊D16311yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—3261no—
352D8—3261no—
353D8—3261no—
354D8—3261no—
355D8—3261no—
356D8—3261no—
357D8—1631no—
358D8C22⋊D16311yescharacteristic, complemented
359M16—21no—
360M16—3261no—
361M16—3261no—
362M16—3261no—
363M16—1631no—
364M16—1631no—
365M16—1631no—
366M16C22×D16311yescharacteristic
367C2×C8—21no—
368C2×C8—3261no—
369C2×C8—3261no—
370C2×C8—3261no—
371C2×C8—1631no—
372C2×C8C22×D16311yescharacteristic
373C22⋊C4—6521no—
374C42—1631no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—3261no—
380C23—6521no—
381Q8—3261no—
382Q8—3261no—
383Q8—6521no—
384Q8—1631no—
385Q8D4×D16311yescharacteristic
386Q8C2×C22⋊D16311yescharacteristic
387Q8C2×C22⋊D16311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—13042no—
400D4—3261no—
401D4—3261no—
402D4—3261no—
403D4—6521no—
404D4—6521no—
405D4—6521no—
406D4—1631no—
407D4D4×D16311yescharacteristic
408D4C2×C22⋊D16311yescharacteristic
409D4C2×C22⋊D16311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—3261no—
416C2×C4—3261no—
417C2×C4—3261no—
418C2×C4—6521no—
419C2×C4—6521no—
420C2×C4—6521no—
421C2×C4C23×D16311yescharacteristic
422C8—21no—
423C8—3261no—
424C8—3261no—
425C8—3261no—
426C8—1631no—
427C8—1631no—
428C8C2×C22⋊D16311yescharacteristic
429C8C2×C22⋊D16311yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—13041no—
440C22—26081no—
441C22—3261no—
442C22—6521no—
443C22#1910478011yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—3261no—
448C4—6521no—
449C4—6521no—
450C4C2×D4×D16311yescharacteristic
451C4C22×C22⋊D16311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—13041no—
459C2—6521no—
460C2D4×C22⋊D16311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#576×C163C2
socleC2×C163D42
hypercenter#576D163
hypocenterC163#3288
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #23782476C2semidirect
2. #23782472C2semidirect
3. #23782468C2semidirect
4. #23782053C2semidirect
5. #23781933C2semidirect
6. #23781932C2semidirect
7. #23781928C2semidirect
8. #23781926C2semidirect
9. #23781843C2semidirect
10. #23781836C2semidirect
11. #23781833C2semidirect
12. #23781831C2semidirect
13. #23781787C2semidirect
14. #23781785C2semidirect
15. #576×C163C2semidirect
16. #19104782C22semidirect
17. #19104781C22semidirect
20. #19104765C22semidirect
21. #19104765C22semidirect
24. #19104761C22semidirect
25. #19104761C22semidirect
28. #19104760C22semidirect
34. C2×#14579688C22semidirect
36. C2×#14579687C22semidirect
39. C2×#14579684C22semidirect
40. #19104742C22semidirect
41. #19104742C22semidirect
42. #19104741C22semidirect
43. #19104731C22semidirect
44. #19104722C22semidirect
45. #19104671C22semidirect
46. #19104670C22semidirect
47. #19104662C22semidirect
48. #19104662C22semidirect
49. #19104662C22semidirect
50. #19104661C22semidirect
51. #19104653C22semidirect
52. #19104652C22semidirect
53. #19104652C22semidirect
54. #19104651C22semidirect
55. #19104645C22semidirect
57. H(5,𝔽2)×C163C22semidirect
61. AGL(1,ℤ/8)×C163C22semidirect
65. C2×D8×C163C22semidirect
66. #131×C163C22semidirect
83. #14579689D4semidirect
92. #14579687D4semidirect
95. #14579686D4semidirect
102. #14579684D4semidirect
130. Q16×C163D4semidirect
133. SD16×C163D4semidirect
134. SD16×C163D4semidirect
139. D8×C163D4semidirect
230. C163#3288semidirect
235. #576D163semidirect
250. H(5,𝔽2)C2×D163semidirect
266. AGL(1,ℤ/8)C2×D163semidirect
280. C2×D8C2×D163semidirect
284. #131C2×D163semidirect
330. Q16C22⋊D163semidirect
345. SD16C22⋊D163semidirect
346. SD16C22⋊D163semidirect
358. D8C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
#237824762
#237824722
#237824682
#237820532
#237819332
#237819322
#237819282
#237819262
#237818432
#237818362
#237818332
#237818312
#237817872
#237817852
#576×C1632
#3288163