Finite Group Databasebeta

← #35226328

Subgroups — #35226328

Order 20864.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782477C211yesmaximal, characteristic, complemented
2#23782475C211yesmaximal, characteristic, complemented
3#23782474C211yesmaximal, characteristic, complemented
4#23782471C211yesmaximal, characteristic, complemented
5#23782036C211yesmaximal, characteristic, complemented
6#23781931C211yesmaximal, characteristic, complemented
7#23781929C211yesmaximal, characteristic, complemented
8#23781927C211yesmaximal, characteristic, complemented
9#23781850C211yesmaximal, characteristic, complemented
10#23781846C211yesmaximal, characteristic, complemented
11#23781835C211yesmaximal, characteristic, complemented
12#23781831C211yesmaximal, characteristic, complemented
13#23781781C211yesmaximal, characteristic, complemented
14#23781780C211yesmaximal, characteristic, complemented
15#471×C163C211yesmaximal, characteristic, complemented
16#19104783C2211yescharacteristic
17#19104782C2211yescharacteristic
18#19104781C2211yescharacteristic, complemented
19#19104780C2211yescharacteristic, complemented
20#19104768—21no—
21#19104768C2211yescharacteristic, complemented
22#19104768C2211yescharacteristic, complemented
23#19104767—21no—
24#19104766C2211yescharacteristic, complemented
25#19104764—21no—
26#19104764—21no—
27#19104763—21no—
28#19104763C2211yescharacteristic
29#19104763C2211yescharacteristic
30#19104762—21no—
31#19104762—21no—
32#19104761C2211yescharacteristic
33#19104759—21no—
34#19104758—21no—
35#19104757—21no—
36#19104757C2211yescharacteristic
37#19104756—21no—
38#19104755C2211yescharacteristic
39#19104754—21no—
40C2×#14579688—21no—
41C2×#14579688C2211yescharacteristic, complemented
42C2×#14579687C2211yescharacteristic, complemented
43C2×#14579686C2211yescharacteristic, complemented
44SD16×D163—21no—
45SD16×D163—21no—
46SD16×D163—21no—
47#19104743C2211yescharacteristic, complemented
48#19104743C2211yescharacteristic, complemented
49#19104741C2211yescharacteristic, complemented
50#19104733C2211yescharacteristic, complemented
51#19104731C2211yescharacteristic
52#19104670C2211yescharacteristic, complemented
53#19104662C2211yescharacteristic, complemented
54#19104661C2211yescharacteristic, complemented
55#19104661C2211yescharacteristic, complemented
56#19104661C2211yescharacteristic, complemented
57#19104653C2211yescharacteristic, complemented
58#19104651C2211yescharacteristic, complemented
59C8×Dic163C2211yescharacteristic, complemented
60#137×C163C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C163C2211yescharacteristic
62C2×SD16×C163C2211yescharacteristic, complemented
63#131×C163C2211yescharacteristic, complemented
64#108×C163C2211yescharacteristic, complemented
65#101×C163C2211yescharacteristic, complemented
66#100×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579700—21no—
69#14579700—21no—
70#14579699—21no—
71#14579699—21no—
72#14579699—41no—
73#14579699C2311yescharacteristic
74#14579699C2311yescharacteristic
75#14579699C2311yescharacteristic
76#14579698—21no—
77#14579698—21no—
78#14579698—21no—
79#14579698—41no—
80C2×Dic326—21no—
81C2×Dic326C2311yescharacteristic
82Q8×D163—21no—
83Q8×D163—21no—
84Q8×D163—21no—
85C2×D652—21no—
86C2×D652C2311yescharacteristic
87C2×C22⋊D163—41no—
88C2×C22⋊D163—41no—
89D4×D163—21no—
90D4×D163—21no—
91D4×D163—21no—
92D4×D163—41no—
93Dic652—21no—
94Dic652—21no—
95Dic652D411yescharacteristic, complemented
96#14579689—21no—
97#14579689D411yescharacteristic, complemented
98#14579688—21no—
99#14579688—21no—
100#14579688—21no—
101#14579688—21no—
102#14579688—21no—
103#14579688—21no—
104#14579688D411yescharacteristic, complemented
105#14579688D411yescharacteristic, complemented
106#14579687—21no—
107#14579687—21no—
108#14579687—21no—
109#14579687D411yescharacteristic, complemented
110#14579686—21no—
111#14579686—21no—
112#14579686—21no—
113#14579686D411yescharacteristic, complemented
114D1304—21no—
115D1304—21no—
116D1304D411yescharacteristic, complemented
117#14579684—21no—
118#14579684D411yescharacteristic, complemented
119#14579683C2311yescharacteristic
120#14579682—21no—
121#14579682—21no—
122#14579682—21no—
123C2×C163⋊C8C2311yescharacteristic
124C8×D163—21no—
125C8×D163—21no—
126C8×D163—21no—
127Dic163⋊C4—41no—
128C4×Dic163C2311yescharacteristic
129#41×C163C2311yescharacteristic
130C2×Q8×C163C2311yescharacteristic
131C2×D4×C163C2311yescharacteristic
132Q16×C163—21no—
133SD16×C163—21no—
134SD16×C163—21no—
135SD16×C163—21no—
136SD16×C163—21no—
137D8×C163—21no—
138M16×C163—21no—
139M16×C163C2311yescharacteristic
140M16×C163C2311yescharacteristic
141M16×C163C2311yescharacteristic
142C2×C8×C163—21no—
143C2×C8×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—41no—
146C22×D163—41no—
147Dic326—21no—
148Dic326—21no—
149Dic326—41no—
150Dic326—41no—
151Dic326C2×D411yescharacteristic
152Dic326C2×D411yescharacteristic
153Dic326C2×D411yescharacteristic
154D652—21no—
155D652—21no—
156D652—41no—
157D652—41no—
158D652C2×D411yescharacteristic
159D652C2×D411yescharacteristic
160D652C2×D411yescharacteristic
161C22⋊D163—21no—
162C22⋊D163—21no—
163C22⋊D163—21no—
164C22⋊D163—41no—
165C22⋊D163—41no—
166C22⋊D163—41no—
167C22⋊D163—41no—
168C22⋊D163—82no—
169C2×Dic163—21no—
170C2×Dic163—21no—
171C2×Dic163—41no—
172C2×Dic163—41no—
173C4×D163—21no—
174C4×D163—21no—
175C4×D163—21no—
176C4×D163—41no—
177C4×D163—41no—
178C4×D163—41no—
179C4×D163—41no—
180C163⋊C8—21no—
181C163⋊C8C2×D411yescharacteristic
182C163⋊C8C2×D411yescharacteristic
183C23×C163—21no—
184Q8×C163—21no—
185Q8×C163—21no—
186Q8×C163C2×D411yescharacteristic
187D4×C163—21no—
188D4×C163—21no—
189D4×C163—21no—
190D4×C163C2×D411yescharacteristic
191C2×C4×C163—21no—
192C2×C4×C163—21no—
193C2×C4×C163C2411yescharacteristic
194C8×C163—21no—
195C8×C163—21no—
196C8×C163—21no—
197C8×C163C2×D411yescharacteristic
198C8×C163C2×D411yescharacteristic
199C22×C163—21no—
200C22×C163—41no—
201C22×C163—81no—
202C22×C163H(5,𝔽2)11yescharacteristic
203C2×D163—21no—
204C2×D163—21no—
205C2×D163—21no—
206C2×D163—41no—
207C2×D163—81no—
208C2×D163—161no—
209C2×D163—161no—
210C4×C163—21no—
211C4×C163—41no—
212C4×C163C22×D411yescharacteristic
213C4×C163C22×D411yescharacteristic
214Dic163—22no—
215Dic163—21no—
216Dic163—21no—
217Dic163—21no—
218Dic163—41no—
219C2×C163—21no—
220C2×C163—41no—
221C2×C163—81no—
222C2×C163D4211yescharacteristic
223D163—41no—
224D163—41no—
225D163—41no—
226D163—81no—
227C163#329111yescharacteristic, complemented, Sylow
228#3291—1631nomaximal, Sylow
229#577—1631no—
230#577—1631no—
231#577—1631no—
232#577—1631no—
233#492—1631no—
234#471—1631no—
235#471—1631no—
236#471—1631no—
237#471D16311yescharacteristic, complemented
238#456—1631no—
239#456—1631no—
240#453—1631no—
241#453—1631no—
242#443—1631no—
243#443—1631no—
2442−1+4—1631no—
2452−1+4—1631no—
246H(5,𝔽2)—1631no—
247H(5,𝔽2)—1631no—
248#137—3261no—
249#137—3261no—
250#137—3261no—
251#137—3261no—
252#137—1631no—
253#137—1631no—
254#137—1631no—
255#137C2×D16311yescharacteristic, complemented
256AGL(1,ℤ/8)—3261no—
257AGL(1,ℤ/8)—3261no—
258AGL(1,ℤ/8)—3261no—
259AGL(1,ℤ/8)—3261no—
260AGL(1,ℤ/8)—1631no—
261AGL(1,ℤ/8)—1631no—
262AGL(1,ℤ/8)—1631no—
263AGL(1,ℤ/8)C2×D16311yescharacteristic
264#135—3261no—
265#135—3261no—
266#135—3261no—
267#135—3261no—
268#135—1631no—
269#135—1631no—
270C2×SD16—3261no—
271C2×SD16—3261no—
272C2×SD16—3261no—
273C2×SD16—3261no—
274C2×SD16—1631no—
275C2×SD16—1631no—
276C2×SD16—1631no—
277C2×SD16C2×D16311yescharacteristic, complemented
278#131—1631no—
279#131—1631no—
280#131—1631no—
281#131C2×D16311yescharacteristic, complemented
282C4⋊Q8—1631no—
283#127—1631no—
284#108—1631no—
285#108C2×D16311yescharacteristic, complemented
286#104—1631no—
287#104—1631no—
288#104—1631no—
289#104—1631no—
290#101—1631no—
291#101C2×D16311yescharacteristic, complemented
292#100—1631no—
293#100C2×D16311yescharacteristic, complemented
294C4×C8—1631no—
295#41—3261no—
296#41—3261no—
297#41—3261no—
298#41—3261no—
299#41—3261no—
300#41—3261no—
301#41—3261no—
302#41—3261no—
303#41—6521no—
304#41—6521no—
305#41—1631no—
306#41—1631no—
307#41—1631no—
308#41C22×D16311yescharacteristic
309C2×Q8—3261no—
310C2×Q8—3261no—
311C2×Q8—3261no—
312C2×Q8—3261no—
313C2×Q8—1631no—
314C2×Q8C22×D16311yescharacteristic
315C2×D4—3261no—
316C2×D4—3261no—
317C2×D4—3261no—
318C2×D4—3261no—
319C2×D4—6521no—
320C2×D4—6521no—
321C2×D4—6521no—
322C2×D4—1631no—
323C2×D4C22×D16311yescharacteristic
324Q16—21no—
325Q16—3261no—
326Q16—3261no—
327Q16—3261no—
328Q16—1631no—
329Q16—1631no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338SD16—3261no—
339SD16—3261no—
340SD16—3261no—
341SD16—3261no—
342SD16—3261no—
343SD16—3261no—
344SD16—3261no—
345SD16—3261no—
346SD16—1631no—
347SD16—1631no—
348SD16—1631no—
349SD16—1631no—
350D8—21no—
351D8—3261no—
352D8—3261no—
353D8—3261no—
354D8—1631no—
355D8—1631no—
356M16—21no—
357M16—3261no—
358M16—3261no—
359M16—3261no—
360M16—1631no—
361M16C22×D16311yescharacteristic
362M16C22×D16311yescharacteristic
363M16C22×D16311yescharacteristic
364C2×C8—21no—
365C2×C8—3261no—
366C2×C8—3261no—
367C2×C8—3261no—
368C2×C8—1631no—
369C2×C8C22×D16311yescharacteristic
370C4⋊C4—6521no—
371C42—1631no—
372C23—21no—
373C23—3261no—
374C23—6521no—
375C23—6521no—
376Q8—21no—
377Q8—21no—
378Q8—3261no—
379Q8—3261no—
380Q8—6521no—
381Q8—6521no—
382Q8—1631no—
383Q8—1631no—
384Q8—1631no—
385Q8D4×D16311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—13042no—
390D4—3261no—
391D4—3261no—
392D4—3261no—
393D4—3261no—
394D4—3261no—
395D4—6521no—
396D4—6521no—
397D4—6521no—
398D4—6521no—
399D4—6521no—
400D4—6521no—
401D4—1631no—
402D4—1631no—
403D4—1631no—
404D4D4×D16311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—3261no—
411C2×C4—3261no—
412C2×C4—6521no—
413C2×C4—6521no—
414C2×C4—6521no—
415C2×C4—6521no—
416C2×C4—6521no—
417C2×C4—6521no—
418C2×C4C23×D16311yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—3261no—
423C8—1631no—
424C8—1631no—
425C8D4×D16311yescharacteristic
426C8D4×D16311yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—13041no—
431C22—26081no—
432C22—26081no—
433C22—3261no—
434C22—3261no—
435C22—3261no—
436C22—6521no—
437C22#1910478111yescharacteristic
438C4—21no—
439C4—41no—
440C4—3262no—
441C4—3261no—
442C4—3261no—
443C4—3261no—
444C4—6521no—
445C4C2×D4×D16311yescharacteristic
446C4C2×D4×D16311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—13041no—
451C2—6521no—
452C2—6521no—
453C2—6521no—
454C2#2378229911yescharacteristic
subgroupG/H
centerC2#23782299
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#471×C163C2
socleC2×C163D42
hypercenter#471D163
hypocenterC163#3291
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #23782477C2semidirect
2. #23782475C2semidirect
3. #23782474C2semidirect
4. #23782471C2semidirect
5. #23782036C2semidirect
6. #23781931C2semidirect
7. #23781929C2semidirect
8. #23781927C2semidirect
9. #23781850C2semidirect
10. #23781846C2semidirect
11. #23781835C2semidirect
12. #23781831C2semidirect
13. #23781781C2semidirect
14. #23781780C2semidirect
15. #471×C163C2semidirect
18. #19104781C22semidirect
19. #19104780C22semidirect
21. #19104768C22semidirect
22. #19104768C22semidirect
24. #19104766C22semidirect
41. C2×#14579688C22semidirect
42. C2×#14579687C22semidirect
43. C2×#14579686C22semidirect
47. #19104743C22semidirect
48. #19104743C22semidirect
49. #19104741C22semidirect
50. #19104733C22semidirect
52. #19104670C22semidirect
53. #19104662C22semidirect
54. #19104661C22semidirect
55. #19104661C22semidirect
56. #19104661C22semidirect
57. #19104653C22semidirect
58. #19104651C22semidirect
59. C8×Dic163C22semidirect
60. #137×C163C22semidirect
62. C2×SD16×C163C22semidirect
63. #131×C163C22semidirect
64. #108×C163C22semidirect
65. #101×C163C22semidirect
66. #100×C163C22semidirect
95. Dic652D4semidirect
97. #14579689D4semidirect
104. #14579688D4semidirect
105. #14579688D4semidirect
109. #14579687D4semidirect
113. #14579686D4semidirect
116. D1304D4semidirect
118. #14579684D4semidirect
227. C163#3291semidirect
237. #471D163semidirect
255. #137C2×D163semidirect
277. C2×SD16C2×D163semidirect
281. #131C2×D163semidirect
285. #108C2×D163semidirect
291. #101C2×D163semidirect
293. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824772
#237824752
#237824742
#237824712
#237820362
#237819312
#237819292
#237819272
#237818502
#237818462
#237818352
#237818312
#237817812
#237817802
#471×C1632
#3291163