Finite Group Databasebeta

← #35226297

Subgroups — #35226297

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#23782475C211yesmaximal, characteristic, complemented
3#23782469C211yesmaximal, characteristic, complemented
4#23782468C211yesmaximal, characteristic, complemented
5#23782052C211yesmaximal, characteristic, complemented
6#23781934C211yesmaximal, characteristic, complemented
7#23781933C211yesmaximal, characteristic, complemented
8#23781927C211yesmaximal, characteristic, complemented
9#23781926C211yesmaximal, characteristic, complemented
10#23781844C211yesmaximal, characteristic, complemented
11#23781839C211yesmaximal, characteristic, complemented
12#23781832C211yesmaximal, characteristic, complemented
13#23781784C211yesmaximal, characteristic, complemented
14#23781784C211yesmaximal, characteristic, complemented
15#453×C163C211yesmaximal, characteristic, complemented
16#19104782C2211yescharacteristic, complemented
17#19104781C2211yescharacteristic
18#19104780C2211yescharacteristic, complemented
19#19104768C2211yescharacteristic, complemented
20#19104765—21no—
21#19104765C2211yescharacteristic, complemented
22#19104764—21no—
23#19104763C2211yescharacteristic, complemented
24#19104762—21no—
25#19104761—21no—
26#19104761—21no—
27#19104761C2211yescharacteristic
28#19104761C2211yescharacteristic, complemented
29#19104760—21no—
30#19104760C2211yescharacteristic, complemented
31#19104759—21no—
32#19104759—21no—
33#19104756—21no—
34#19104755—21no—
35#19104755—21no—
36#19104755C2211yescharacteristic
37#19104755C2211yescharacteristic
38#19104754—21no—
39C2×#14579687—21no—
40C2×#14579687C2211yescharacteristic, complemented
41C2×#14579687C2211yescharacteristic, complemented
42SD16×D163—21no—
43C2×#14579684—21no—
44C2×#14579684C2211yescharacteristic, complemented
45C2×#14579684C2211yescharacteristic, complemented
46D8×D163—21no—
47#19104743C2211yescharacteristic, complemented
48#19104743C2211yescharacteristic, complemented
49#19104742C2211yescharacteristic, complemented
50#19104742C2211yescharacteristic, complemented
51#19104723C2211yescharacteristic, complemented
52#19104670C2211yescharacteristic, complemented
53#19104670C2211yescharacteristic, complemented
54#19104663C2211yescharacteristic, complemented
55#19104663C2211yescharacteristic, complemented
56#19104653C2211yescharacteristic, complemented
57#19104652C2211yescharacteristic, complemented
58#19104651C2211yescharacteristic, complemented
59#19104646C2211yescharacteristic, complemented
60H(5,𝔽2)×C163C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C163C2211yescharacteristic
62AGL(1,ℤ/8)×C163C2211yescharacteristic, complemented
63#127×C163C2211yescharacteristic, complemented
64#104×C163C2211yescharacteristic, complemented
65#104×C163C2211yescharacteristic, complemented
66#100×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579699—21no—
69#14579699—21no—
70#14579699C2311yescharacteristic
71#14579699C2311yescharacteristic
72#14579698—21no—
73#14579698—21no—
74#14579698—21no—
75#14579698—41no—
76C2×Dic326—21no—
77C2×Dic326C2311yescharacteristic
78Q8×D163—21no—
79C2×D652—21no—
80C2×D652C2311yescharacteristic
81C2×C22⋊D163—41no—
82D4×D163—21no—
83D4×D163—21no—
84D4×D163—21no—
85D4×D163—41no—
86Dic652—21no—
87#14579689—21no—
88#14579689D411yescharacteristic, complemented
89#14579689D411yescharacteristic, complemented
90#14579688—21no—
91#14579688—21no—
92#14579687—21no—
93#14579687—21no—
94#14579687—21no—
95#14579687—21no—
96#14579687—21no—
97#14579687—21no—
98#14579687—21no—
99#14579687D411yescharacteristic, complemented
100#14579687D411yescharacteristic, complemented
101#14579686—21no—
102#14579686D411yescharacteristic, complemented
103#14579686D411yescharacteristic, complemented
104D1304—21no—
105#14579684—21no—
106#14579684—21no—
107#14579684—21no—
108#14579684—21no—
109#14579684—21no—
110#14579684—21no—
111#14579684—21no—
112#14579684D411yescharacteristic, complemented
113#14579684D411yescharacteristic, complemented
114#14579683—21no—
115#14579683—21no—
116#14579683C2311yescharacteristic
117#14579683C2311yescharacteristic
118#14579682—21no—
119#14579682—21no—
120C2×C163⋊C8—21no—
121C2×C163⋊C8—21no—
122C2×C163⋊C8C2311yescharacteristic
123C2×C163⋊C8C2311yescharacteristic
124C8×D163—21no—
125C8×D163—21no—
126(C2×D163)⋊C4—41no—
127#41×C163—21no—
128#41×C163—21no—
129#41×C163C2311yescharacteristic
130#41×C163C2311yescharacteristic
131C2×D4×C163—21no—
132C2×D4×C163—21no—
133C2×D4×C163—41no—
134C2×D4×C163—41no—
135C2×D4×C163C2311yescharacteristic
136C2×D4×C163C2311yescharacteristic
137SD16×C163—21no—
138SD16×C163—21no—
139D8×C163—21no—
140D8×C163—21no—
141M16×C163C2311yescharacteristic
142M16×C163C2311yescharacteristic
143C42×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—41no—
146C22×D163—41no—
147Dic326—21no—
148Dic326—21no—
149Dic326—41no—
150Dic326C2×D411yescharacteristic
151Dic326C2×D411yescharacteristic
152D652—21no—
153D652—21no—
154D652—41no—
155D652C2×D411yescharacteristic
156D652C2×D411yescharacteristic
157C22⋊D163—21no—
158C22⋊D163—21no—
159C22⋊D163—41no—
160C22⋊D163—41no—
161C22⋊D163—41no—
162C22⋊D163—41no—
163C2×Dic163—21no—
164C2×Dic163—41no—
165C2×Dic163—41no—
166C4×D163—21no—
167C4×D163—21no—
168C4×D163—41no—
169C4×D163—41no—
170C163⋊C8—21no—
171C163⋊C8—21no—
172C163⋊C8C2×D411yescharacteristic
173C163⋊C8C2×D411yescharacteristic
174C163⋊C8C2×D411yescharacteristic
175C163⋊C8C2×D411yescharacteristic
176C23×C163—21no—
177C23×C163—21no—
178C23×C163—41no—
179Q8×C163C2×D411yescharacteristic
180Q8×C163C2×D411yescharacteristic
181D4×C163—21no—
182D4×C163—21no—
183D4×C163—21no—
184D4×C163—21no—
185D4×C163—21no—
186D4×C163—21no—
187D4×C163—41no—
188D4×C163—41no—
189D4×C163—82no—
190D4×C163C2×D411yescharacteristic
191D4×C163C2×D411yescharacteristic
192C2×C4×C163—21no—
193C2×C4×C163—21no—
194C2×C4×C163—21no—
195C2×C4×C163—41no—
196C2×C4×C163C2411yescharacteristic
197C8×C163—21no—
198C8×C163—21no—
199C22×C163—21no—
200C22×C163—21no—
201C22×C163—41no—
202C22×C163—41no—
203C22×C163—81no—
204C22×C163—81no—
205C22×C163—161no—
206C22×C163H(5,𝔽2)11yescharacteristic
207C2×D163—21no—
208C2×D163—21no—
209C2×D163—41no—
210C2×D163—81no—
211C2×D163—161no—
212C2×D163—161no—
213C4×C163—21no—
214C4×C163—21no—
215C4×C163—41no—
216C4×C163C22×D411yescharacteristic
217C4×C163C22×D411yescharacteristic
218Dic163—21no—
219Dic163—21no—
220Dic163—41no—
221C2×C163—21no—
222C2×C163—41no—
223C2×C163—41no—
224C2×C163—81no—
225C2×C163—81no—
226C2×C163D4211yescharacteristic
227D163—41no—
228D163—41no—
229D163—81no—
230C163#328811yescharacteristic, complemented, Sylow
231#3288—1631nomaximal, Sylow
232#577—1631no—
233#577—1631no—
234#576—1631no—
235#576—1631no—
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#453D16311yescharacteristic, complemented
245#444—1631no—
246#444—1631no—
2472−1+4—1631no—
248H(5,𝔽2)—1631no—
249H(5,𝔽2)—1631no—
250H(5,𝔽2)C2×D16311yescharacteristic, complemented
251#137—3261no—
252#137—3261no—
253#137—1631no—
254#137—1631no—
255AGL(1,ℤ/8)—3261no—
256AGL(1,ℤ/8)—3261no—
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—3261no—
268#135—3261no—
269#135—3261no—
270#135—3261no—
271#135—1631no—
272#135—1631no—
273C2×SD16—3261no—
274C2×SD16—3261no—
275C2×SD16—1631no—
276C2×SD16—1631no—
277C2×D8—3261no—
278C2×D8—3261no—
279C2×D8—1631no—
280C2×D8—1631no—
281#131—1631no—
282#131—1631no—
283#131—1631no—
284#131—1631no—
285#127C2×D16311yescharacteristic, complemented
286#124—1631no—
287#108—1631no—
288#108—1631no—
289#104—1631no—
290#104—1631no—
291#104C2×D16311yescharacteristic, complemented
292#104C2×D16311yescharacteristic, complemented
293#101—1631no—
294#100—1631no—
295#100—1631no—
296#100C2×D16311yescharacteristic, complemented
297#97—1631no—
298#41—21no—
299#41—21no—
300#41—3261no—
301#41—3261no—
302#41—3261no—
303#41—3261no—
304#41—3261no—
305#41—3261no—
306#41—6521no—
307#41—1631no—
308#41—1631no—
309#41C22×D16311yescharacteristic
310#41C22×D16311yescharacteristic
311C2×Q8—3261no—
312C2×Q8—3261no—
313C2×Q8—1631no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—3261no—
319C2×D4—3261no—
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—
330Q16—1631no—
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—
345SD16—1631no—
346SD16—1631no—
347D8—21no—
348D8—21no—
349D8—3261no—
350D8—3261no—
351D8—3261no—
352D8—3261no—
353D8—3261no—
354D8—3261no—
355D8—3261no—
356D8—3261no—
357D8—1631no—
358D8—1631no—
359M16—3261no—
360M16—3261no—
361M16—3261no—
362M16—3261no—
363M16—1631no—
364M16—1631no—
365M16C22×D16311yescharacteristic
366M16C22×D16311yescharacteristic
367C2×C8—3261no—
368C2×C8—3261no—
369C2×C8—3261no—
370C2×C8—3261no—
371C2×C8—1631no—
372C2×C8—1631no—
373C22⋊C4—6521no—
374C42C22×D16311yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—3261no—
379C23—6521no—
380C23—6521no—
381Q8—3261no—
382Q8—3261no—
383Q8—6521no—
384Q8—1631no—
385Q8—1631no—
386Q8D4×D16311yescharacteristic
387Q8D4×D16311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—3261no—
398D4—3261no—
399D4—3261no—
400D4—3261no—
401D4—6521no—
402D4—6521no—
403D4—6521no—
404D4—6521no—
405D4—6521no—
406D4—1631no—
407D4—1631no—
408D4D4×D16311yescharacteristic
409D4D4×D16311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—3261no—
417C2×C4—6521no—
418C2×C4—6521no—
419C2×C4—6521no—
420C2×C4—6521no—
421C2×C4C23×D16311yescharacteristic
422C8—21no—
423C8—21no—
424C8—3261no—
425C8—3261no—
426C8—1631no—
427C8—1631no—
428C8—1631no—
429C8—1631no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—13041no—
438C22—26081no—
439C22—26081no—
440C22—3261no—
441C22—3261no—
442C22—6521no—
443C22#1910478011yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3261no—
448C4—3261no—
449C4—6521no—
450C4C2×D4×D16311yescharacteristic
451C4C2×D4×D16311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—13041no—
458C2—6521no—
459C2—6521no—
460C2#2378229711yescharacteristic
subgroupG/H
centerC2#23782297
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#453×C163C2
socleC2×C163D42
hypercenter#453D163
hypocenterC163#3288
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23782476C2semidirect
2. #23782475C2semidirect
3. #23782469C2semidirect
4. #23782468C2semidirect
5. #23782052C2semidirect
6. #23781934C2semidirect
7. #23781933C2semidirect
8. #23781927C2semidirect
9. #23781926C2semidirect
10. #23781844C2semidirect
11. #23781839C2semidirect
12. #23781832C2semidirect
13. #23781784C2semidirect
14. #23781784C2semidirect
15. #453×C163C2semidirect
16. #19104782C22semidirect
18. #19104780C22semidirect
19. #19104768C22semidirect
21. #19104765C22semidirect
23. #19104763C22semidirect
28. #19104761C22semidirect
30. #19104760C22semidirect
40. C2×#14579687C22semidirect
41. C2×#14579687C22semidirect
44. C2×#14579684C22semidirect
45. C2×#14579684C22semidirect
47. #19104743C22semidirect
48. #19104743C22semidirect
49. #19104742C22semidirect
50. #19104742C22semidirect
51. #19104723C22semidirect
52. #19104670C22semidirect
53. #19104670C22semidirect
54. #19104663C22semidirect
55. #19104663C22semidirect
56. #19104653C22semidirect
57. #19104652C22semidirect
58. #19104651C22semidirect
59. #19104646C22semidirect
60. H(5,𝔽2)×C163C22semidirect
62. AGL(1,ℤ/8)×C163C22semidirect
63. #127×C163C22semidirect
64. #104×C163C22semidirect
65. #104×C163C22semidirect
66. #100×C163C22semidirect
88. #14579689D4semidirect
89. #14579689D4semidirect
99. #14579687D4semidirect
100. #14579687D4semidirect
102. #14579686D4semidirect
103. #14579686D4semidirect
112. #14579684D4semidirect
113. #14579684D4semidirect
230. C163#3288semidirect
244. #453D163semidirect
250. H(5,𝔽2)C2×D163semidirect
266. AGL(1,ℤ/8)C2×D163semidirect
285. #127C2×D163semidirect
291. #104C2×D163semidirect
292. #104C2×D163semidirect
296. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824762
#237824752
#237824692
#237824682
#237820522
#237819342
#237819332
#237819272
#237819262
#237818442
#237818392
#237818322
#237817842
#453×C1632
#3288163