Finite Group Databasebeta

← #35226303

Subgroups — #35226303

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#23782477C211yesmaximal, characteristic, complemented
2#23782472C211yesmaximal, characteristic, complemented
3#23782470C211yesmaximal, characteristic, complemented
4#23782467C211yesmaximal, characteristic, complemented
5#23782054C211yesmaximal, characteristic, complemented
6#23781936C211yesmaximal, characteristic, complemented
7#23781935C211yesmaximal, characteristic, complemented
8#23781929C211yesmaximal, characteristic, complemented
9#23781928C211yesmaximal, characteristic, complemented
10#23781844C211yesmaximal, characteristic, complemented
11#23781839C211yesmaximal, characteristic, complemented
12#23781834C211yesmaximal, characteristic, complemented
13#23781834C211yesmaximal, characteristic, complemented
14#23781788C211yesmaximal, characteristic, complemented
15#444×C163C211yesmaximal, characteristic, complemented
16#19104782C2211yescharacteristic, complemented
17#19104781C2211yescharacteristic
18#19104781C2211yescharacteristic, complemented
19#19104781C2211yescharacteristic, complemented
20#19104768—21no—
21#19104768C2211yescharacteristic, complemented
22#19104767—21no—
23#19104765C2211yescharacteristic, complemented
24#19104763—21no—
25#19104763—21no—
26#19104763—21no—
27#19104763C2211yescharacteristic
28#19104763C2211yescharacteristic, complemented
29#19104763C2211yescharacteristic, complemented
30#19104762—21no—
31#19104762—21no—
32#19104760C2211yescharacteristic
33#19104760C2211yescharacteristic, complemented
34#19104760C2211yescharacteristic, complemented
35#19104759—21no—
36#19104758—21no—
37#19104757—21no—
38#19104757—21no—
39#19104757C2211yescharacteristic
40#19104756—21no—
41C2×#14579688—21no—
42C2×#14579688C2211yescharacteristic, complemented
43C2×#14579688C2211yescharacteristic, complemented
44SD16×D163—21no—
45C2×D1304—21no—
46C2×D1304C2211yescharacteristic, complemented
47C2×D1304C2211yescharacteristic, complemented
48D8×D163—21no—
49#19104741C2211yescharacteristic, complemented
50#19104741C2211yescharacteristic, complemented
51#19104732C2211yescharacteristic, complemented
52#19104723C2211yescharacteristic, complemented
53#19104671C2211yescharacteristic, complemented
54#19104663C2211yescharacteristic, complemented
55#19104663C2211yescharacteristic, complemented
56#19104653C2211yescharacteristic, complemented
57#19104651C2211yescharacteristic, complemented
58#19104651C2211yescharacteristic, complemented
59#19104651C2211yescharacteristic, complemented
60#135×C163C2211yescharacteristic
61#131×C163C2211yescharacteristic, complemented
62#131×C163C2211yescharacteristic, complemented
63#108×C163C2211yescharacteristic, complemented
64#104×C163C2211yescharacteristic, complemented
65#104×C163C2211yescharacteristic, complemented
66#97×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579700—21no—
69#14579700—21no—
70#14579699—21no—
71#14579699—21no—
72#14579699—21no—
73#14579699—21no—
74#14579699C2311yescharacteristic
75#14579699C2311yescharacteristic
76#14579698—21no—
77#14579698—41no—
78C2×Dic326—21no—
79C2×Dic326C2311yescharacteristic
80Q8×D163—21no—
81C2×D652—21no—
82C2×D652—21no—
83C2×D652—21no—
84C2×D652—41no—
85C2×D652C2311yescharacteristic
86C2×D652C2311yescharacteristic
87C2×D652C2311yescharacteristic
88D4×D163—21no—
89D4×D163—21no—
90D4×D163—21no—
91D4×D163—41no—
92D4×D163—41no—
93D4×D163—41no—
94Dic652—21no—
95Dic652D411yescharacteristic, complemented
96#14579689—21no—
97#14579688—21no—
98#14579688—21no—
99#14579688—21no—
100#14579688—21no—
101#14579688—21no—
102#14579688—21no—
103#14579688—21no—
104#14579688—21no—
105#14579688D411yescharacteristic, complemented
106#14579688D411yescharacteristic, complemented
107#14579687—21no—
108#14579686—21no—
109#14579686—21no—
110#14579686—21no—
111D1304—21no—
112D1304—21no—
113D1304—21no—
114D1304—21no—
115D1304—21no—
116D1304—21no—
117D1304—21no—
118D1304D411yescharacteristic, complemented
119#14579684—21no—
120#14579684—21no—
121#14579684—21no—
122#14579683C2311yescharacteristic
123#14579683C2311yescharacteristic
124#14579682—21no—
125#14579682—21no—
126C8×D163—21no—
127C8×D163—21no—
128(C2×D163)⋊C4—41no—
129#41×C163C2311yescharacteristic
130#41×C163C2311yescharacteristic
131Q16×C163D411yescharacteristic, complemented
132SD16×C163D411yescharacteristic, complemented
133SD16×C163D411yescharacteristic, complemented
134D8×C163D411yescharacteristic, complemented
135M16×C163—21no—
136M16×C163—21no—
137M16×C163C2311yescharacteristic
138M16×C163C2311yescharacteristic
139C2×C8×C163—21no—
140C2×C8×C163—21no—
141C2×C8×C163C2311yescharacteristic
142C2×C8×C163C2311yescharacteristic
143C42×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—21no—
146C22×D163—21no—
147C22×D163—41no—
148C22×D163—41no—
149C22×D163—41no—
150Dic326—21no—
151Dic326—21no—
152Dic326—41no—
153Dic326C2×D411yescharacteristic
154Dic326C2×D411yescharacteristic
155D652—21no—
156D652—21no—
157D652—21no—
158D652—21no—
159D652—21no—
160D652—21no—
161D652—41no—
162D652—41no—
163D652—41no—
164D652—82no—
165D652C2×D411yescharacteristic
166D652C2×D411yescharacteristic
167C22⋊D163—21no—
168C22⋊D163—21no—
169C22⋊D163—41no—
170C22⋊D163—41no—
171C22⋊D163—41no—
172C22⋊D163—41no—
173C2×Dic163—21no—
174C2×Dic163—41no—
175C4×D163—21no—
176C4×D163—21no—
177C4×D163—41no—
178C4×D163—41no—
179C4×D163—41no—
180C4×D163—41no—
181C163⋊C8—21no—
182C163⋊C8—21no—
183Q8×C163C2×D411yescharacteristic
184Q8×C163C2×D411yescharacteristic
185D4×C163—21no—
186D4×C163—21no—
187D4×C163C2×D411yescharacteristic
188D4×C163C2×D411yescharacteristic
189C2×C4×C163—21no—
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163C2411yescharacteristic
193C8×C163—21no—
194C8×C163—21no—
195C8×C163C2×D411yescharacteristic
196C8×C163C2×D411yescharacteristic
197C8×C163C2×D411yescharacteristic
198C8×C163C2×D411yescharacteristic
199C22×C163—21no—
200C22×C163—21no—
201C22×C163H(5,𝔽2)11yescharacteristic
202C2×D163—21no—
203C2×D163—21no—
204C2×D163—41no—
205C2×D163—41no—
206C2×D163—41no—
207C2×D163—81no—
208C2×D163—81no—
209C2×D163—81no—
210C2×D163—161no—
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×C163D4211yescharacteristic
225D163—41no—
226D163—41no—
227D163—81no—
228D163—81no—
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#453—1631no—
245#444—1631no—
246#444D16311yescharacteristic, complemented
2472−1+4—1631no—
248H(5,𝔽2)—1631no—
249H(5,𝔽2)—1631no—
250H(5,𝔽2)—1631no—
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)—1631no—
266AGL(1,ℤ/8)—1631no—
267#135—3261no—
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—3261no—
278C2×D8—3261no—
279C2×D8—1631no—
280C2×D8—1631no—
281#131—1631no—
282#131—1631no—
283#131C2×D16311yescharacteristic, complemented
284#131C2×D16311yescharacteristic, complemented
285#127—1631no—
286#124—1631no—
287#108—1631no—
288#108C2×D16311yescharacteristic, complemented
289#104—1631no—
290#104—1631no—
291#104C2×D16311yescharacteristic, complemented
292#104C2×D16311yescharacteristic, complemented
293#101—1631no—
294#100—1631no—
295#100—1631no—
296#100—1631no—
297#97C2×D16311yescharacteristic, complemented
298#41—3261no—
299#41—3261no—
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—3261no—
315C2×D4—3261no—
316C2×D4—3261no—
317C2×D4—3261no—
318C2×D4—3261no—
319C2×D4—3261no—
320C2×D4—6521no—
321C2×D4—6521no—
322C2×D4—6521no—
323C2×D4—6521no—
324C2×D4—1631no—
325C2×D4—1631no—
326C2×D4—1631no—
327Q16—3261no—
328Q16—3261no—
329Q16—1631no—
330Q16D65211yescharacteristic, complemented
331SD16—3261no—
332SD16—3261no—
333SD16—3261no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338SD16—3261no—
339SD16—3261no—
340SD16—3261no—
341SD16—3261no—
342SD16—3261no—
343SD16—1631no—
344SD16—1631no—
345SD16D65211yescharacteristic, complemented
346SD16D65211yescharacteristic, complemented
347D8—3261no—
348D8—3261no—
349D8—3261no—
350D8—3261no—
351D8—3261no—
352D8—3261no—
353D8—3261no—
354D8—3261no—
355D8—3261no—
356D8—3261no—
357D8—1631no—
358D8D65211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—3261no—
362M16—3261no—
363M16—1631no—
364M16—1631no—
365M16C22×D16311yescharacteristic
366M16C22×D16311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—3261no—
370C2×C8—3261no—
371C2×C8C22×D16311yescharacteristic
372C2×C8C22×D16311yescharacteristic
373C22⋊C4—6521no—
374C42C22×D16311yescharacteristic
375C23—3261no—
376C23—3261no—
377C23—3261no—
378C23—6521no—
379C23—6521no—
380C23—6521no—
381Q8—3261no—
382Q8—3261no—
383Q8—6521no—
384Q8—1631no—
385Q8—1631no—
386Q8C2×D65211yescharacteristic
387Q8C2×D65211yescharacteristic
388D4—21no—
389D4—21no—
390D4—13042no—
391D4—3261no—
392D4—3261no—
393D4—3261no—
394D4—3261no—
395D4—3261no—
396D4—3261no—
397D4—3261no—
398D4—3261no—
399D4—6521no—
400D4—6521no—
401D4—6521no—
402D4—6521no—
403D4—6521no—
404D4—6521no—
405D4—6521no—
406D4—1631no—
407D4—1631no—
408D4C2×D65211yescharacteristic
409D4C2×D65211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—3261no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—6521no—
417C2×C4—6521no—
418C2×C4—6521no—
419C2×C4—6521no—
420C2×C4—6521no—
421C2×C4C23×D16311yescharacteristic
422C8—21no—
423C8—21no—
424C8—3261no—
425C8—3261no—
426C8D4×D16311yescharacteristic
427C8D4×D16311yescharacteristic
428C8C2×D65211yescharacteristic
429C8C2×D65211yescharacteristic
430C22—21no—
431C22—21no—
432C22—13041no—
433C22—13041no—
434C22—13041no—
435C22—26081no—
436C22—26081no—
437C22—26081no—
438C22—3261no—
439C22—3261no—
440C22—6521no—
441C22—6521no—
442C22—6521no—
443C22#1910478111yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3261no—
448C4—3261no—
449C4—6521no—
450C4C2×D4×D16311yescharacteristic
451C4C22×D65211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—13041no—
456C2—13041no—
457C2—13041no—
458C2—6521no—
459C2—6521no—
460C2D4×D65211yescharacteristic
subgroupG/H
centerC2D4×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#444×C163C2
socleC2×C163D42
hypercenter#444D163
hypocenterC163#3288
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #23782477C2semidirect
2. #23782472C2semidirect
3. #23782470C2semidirect
4. #23782467C2semidirect
5. #23782054C2semidirect
6. #23781936C2semidirect
7. #23781935C2semidirect
8. #23781929C2semidirect
9. #23781928C2semidirect
10. #23781844C2semidirect
11. #23781839C2semidirect
12. #23781834C2semidirect
13. #23781834C2semidirect
14. #23781788C2semidirect
15. #444×C163C2semidirect
16. #19104782C22semidirect
18. #19104781C22semidirect
19. #19104781C22semidirect
21. #19104768C22semidirect
23. #19104765C22semidirect
28. #19104763C22semidirect
29. #19104763C22semidirect
33. #19104760C22semidirect
34. #19104760C22semidirect
42. C2×#14579688C22semidirect
43. C2×#14579688C22semidirect
46. C2×D1304C22semidirect
47. C2×D1304C22semidirect
49. #19104741C22semidirect
50. #19104741C22semidirect
51. #19104732C22semidirect
52. #19104723C22semidirect
53. #19104671C22semidirect
54. #19104663C22semidirect
55. #19104663C22semidirect
56. #19104653C22semidirect
57. #19104651C22semidirect
58. #19104651C22semidirect
59. #19104651C22semidirect
61. #131×C163C22semidirect
62. #131×C163C22semidirect
63. #108×C163C22semidirect
64. #104×C163C22semidirect
65. #104×C163C22semidirect
66. #97×C163C22semidirect
95. Dic652D4semidirect
105. #14579688D4semidirect
106. #14579688D4semidirect
118. D1304D4semidirect
131. Q16×C163D4semidirect
132. SD16×C163D4semidirect
133. SD16×C163D4semidirect
134. D8×C163D4semidirect
230. C163#3288semidirect
246. #444D163semidirect
283. #131C2×D163semidirect
284. #131C2×D163semidirect
288. #108C2×D163semidirect
291. #104C2×D163semidirect
292. #104C2×D163semidirect
297. #97C2×D163semidirect
330. Q16D652semidirect
345. SD16D652semidirect
346. SD16D652semidirect
358. D8D652semidirect

Maximal subgroups

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

subgroupindex
#237824772
#237824722
#237824702
#237824672
#237820542
#237819362
#237819352
#237819292
#237819282
#237818442
#237818392
#237818342
#237817882
#444×C1632
#3288163