Finite Group Databasebeta

← #35226327

Subgroups — #35226327

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#23782476C211yesmaximal, characteristic, complemented
2#23782473C211yesmaximal, characteristic, complemented
3#23782472C211yesmaximal, characteristic, complemented
4#23782036C211yesmaximal, characteristic, complemented
5#23781930C211yesmaximal, characteristic, complemented
6#23781928C211yesmaximal, characteristic, complemented
7#23781926C211yesmaximal, characteristic, complemented
8#23781925C211yesmaximal, characteristic, complemented
9#23781848C211yesmaximal, characteristic, complemented
10#23781846C211yesmaximal, characteristic, complemented
11#23781833C211yesmaximal, characteristic, complemented
12#23781831C211yesmaximal, characteristic, complemented
13#23781782C211yesmaximal, characteristic, complemented
14#23781780C211yesmaximal, characteristic, complemented
15#577×C163C211yesmaximal, characteristic, complemented
16#19104782C2211yescharacteristic
17#19104781C2211yescharacteristic, complemented
18#19104767—21no—
19#19104766—21no—
20#19104766C2211yescharacteristic, complemented
21#19104765—21no—
22#19104765C2211yescharacteristic, complemented
23#19104765C2211yescharacteristic, complemented
24#19104761—21no—
25#19104761C2211yescharacteristic
26#19104760—21no—
27#19104760C2211yescharacteristic
28#19104760C2211yescharacteristic
29#19104759—21no—
30#19104756—21no—
31#19104755—21no—
32#19104755—21no—
33#19104755C2211yescharacteristic
34C2×#14579688C2211yescharacteristic, complemented
35C2×#14579687—21no—
36C2×#14579687C2211yescharacteristic, complemented
37C2×#14579686—21no—
38C2×#14579686C2211yescharacteristic, complemented
39SD16×D163—21no—
40#19104742C2211yescharacteristic, complemented
41#19104742C2211yescharacteristic, complemented
42#19104741C2211yescharacteristic, complemented
43#19104733C2211yescharacteristic, complemented
44#19104731C2211yescharacteristic
45#19104671C2211yescharacteristic, complemented
46#19104670C2211yescharacteristic, complemented
47#19104662C2211yescharacteristic, complemented
48#19104662C2211yescharacteristic, complemented
49#19104662C2211yescharacteristic, complemented
50#19104661C2211yescharacteristic, complemented
51#19104654C2211yescharacteristic, complemented
52#19104653C2211yescharacteristic, complemented
53#19104652C2211yescharacteristic, complemented
54#19104651C2211yescharacteristic, complemented
55C8×Dic163C2211yescharacteristic, complemented
562−1+4×C163C2211yescharacteristic
57H(5,𝔽2)×C163C2211yescharacteristic, complemented
58#137×C163—21no—
59#137×C163C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C163—21no—
61AGL(1,ℤ/8)×C163C2211yescharacteristic
62#135×C163—21no—
63#135×C163C2211yescharacteristic
64C2×SD16×C163—21no—
65C2×SD16×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#14579689—21no—
84#14579689D411yescharacteristic, complemented
85#14579688—21no—
86#14579688—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#14579686—21no—
96#14579686—21no—
97#14579686—21no—
98#14579686D411yescharacteristic, complemented
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—
113Dic163⋊C4—41no—
114C4×Dic163C2311yescharacteristic
115#41×C163—21no—
116#41×C163—21no—
117#41×C163—21no—
118#41×C163—21no—
119#41×C163—41no—
120#41×C163C2311yescharacteristic
121#41×C163C2311yescharacteristic
122#41×C163C2311yescharacteristic
123C2×Q8×C163—21no—
124C2×Q8×C163—21no—
125C2×Q8×C163C2311yescharacteristic
126C2×D4×C163—21no—
127C2×D4×C163—21no—
128C2×D4×C163—41no—
129C2×D4×C163C2311yescharacteristic
130Q16×C163—21no—
131Q16×C163D411yescharacteristic, complemented
132SD16×C163—21no—
133SD16×C163—21no—
134SD16×C163—21no—
135SD16×C163—21no—
136SD16×C163D411yescharacteristic, complemented
137SD16×C163D411yescharacteristic, complemented
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—41no—
171Q8×C163—21no—
172Q8×C163—21no—
173Q8×C163—41no—
174Q8×C163C2×D411yescharacteristic
175Q8×C163C2×D411yescharacteristic
176Q8×C163C2×D411yescharacteristic
177D4×C163—21no—
178D4×C163—21no—
179D4×C163—21no—
180D4×C163—21no—
181D4×C163—21no—
182D4×C163—41no—
183D4×C163—41no—
184D4×C163—41no—
185D4×C163C2×D411yescharacteristic
186D4×C163C2×D411yescharacteristic
187D4×C163C2×D411yescharacteristic
188C2×C4×C163—21no—
189C2×C4×C163—21no—
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163—41no—
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—81no—
204C22×C163—161no—
205C22×C163H(5,𝔽2)11yescharacteristic
206C2×D163—21no—
207C2×D163—41no—
208C2×D163—81no—
209C2×D163—161no—
210C4×C163—21no—
211C4×C163—21no—
212C4×C163—21no—
213C4×C163—41no—
214C4×C163C22×D411yescharacteristic
215C4×C163C22×D411yescharacteristic
216Dic163—22no—
217Dic163—21no—
218Dic163—41no—
219C2×C163—21no—
220C2×C163—41no—
221C2×C163—41no—
222C2×C163—41no—
223C2×C163—81no—
224C2×C163D4211yescharacteristic
225D163—41no—
226D163—81no—
227C163#329111yescharacteristic, complemented, Sylow
228#3291—1631nomaximal, Sylow
229#577—1631no—
230#577—1631no—
231#577—1631no—
232#577D16311yescharacteristic, complemented
233#492—1631no—
234#471—1631no—
235#471—1631no—
236#471—1631no—
237#471—1631no—
238#456—1631no—
239#456—1631no—
240#453—1631no—
241#453—1631no—
242#443—1631no—
243#443—1631no—
2442−1+4—1631no—
2452−1+4C2×D16311yescharacteristic
246H(5,𝔽2)—1631no—
247H(5,𝔽2)C2×D16311yescharacteristic, complemented
248#137—21no—
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)—21no—
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—21no—
265#135—3261no—
266#135—3261no—
267#135—3261no—
268#135—1631no—
269#135C2×D16311yescharacteristic
270C2×SD16—21no—
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#108—1631no—
286#104—1631no—
287#104—1631no—
288#104—1631no—
289#104—1631no—
290#101—1631no—
291#101—1631no—
292#100—1631no—
293#100—1631no—
294C4×C8—1631no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—3261no—
301#41—3261no—
302#41—3261no—
303#41—3261no—
304#41—6521no—
305#41—1631no—
306#41C22×D16311yescharacteristic
307#41C22×D16311yescharacteristic
308#41C22×D16311yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—3261no—
312C2×Q8—3261no—
313C2×Q8—1631no—
314C2×Q8C22×D16311yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—3261no—
319C2×D4—3261no—
320C2×D4—6521no—
321C2×D4—6521no—
322C2×D4—1631no—
323C2×D4C22×D16311yescharacteristic
324Q16—21no—
325Q16—3261no—
326Q16—3261no—
327Q16—3261no—
328Q16—1631no—
329Q16C22⋊D16311yescharacteristic, complemented
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—
348SD16C22⋊D16311yescharacteristic, complemented
349SD16C22⋊D16311yescharacteristic, complemented
350D8—21no—
351D8—3261no—
352D8—3261no—
353D8—3261no—
354D8—1631no—
355D8C22⋊D16311yescharacteristic, complemented
356M16—21no—
357M16—3261no—
358M16—3261no—
359M16—3261no—
360M16—1631no—
361M16—1631no—
362M16—1631no—
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—41no—
374C23—3261no—
375C23—6521no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—3261no—
380Q8—3261no—
381Q8—6521no—
382Q8—1631no—
383Q8D4×D16311yescharacteristic
384Q8C2×C22⋊D16311yescharacteristic
385Q8C2×C22⋊D16311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—13042no—
395D4—3261no—
396D4—3261no—
397D4—3261no—
398D4—6521no—
399D4—6521no—
400D4—6521no—
401D4—1631no—
402D4D4×D16311yescharacteristic
403D4C2×C22⋊D16311yescharacteristic
404D4C2×C22⋊D16311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—3261no—
413C2×C4—3261no—
414C2×C4—3261no—
415C2×C4—6521no—
416C2×C4—6521no—
417C2×C4—6521no—
418C2×C4C23×D16311yescharacteristic
419C8—21no—
420C8—3261no—
421C8—3261no—
422C8—3261no—
423C8—1631no—
424C8—1631no—
425C8C2×C22⋊D16311yescharacteristic
426C8C2×C22⋊D16311yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—13041no—
434C22—26081no—
435C22—3261no—
436C22—6521no—
437C22#1910478011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—3262no—
443C4—3261no—
444C4—6521no—
445C4C2×D4×D16311yescharacteristic
446C4C22×C22⋊D16311yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—13041no—
453C2—6521no—
454C2D4×C22⋊D16311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#577×C163C2
socleC2×C163D42
hypercenter#577D163
hypocenterC163#3291
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #23782476C2semidirect
2. #23782473C2semidirect
3. #23782472C2semidirect
4. #23782036C2semidirect
5. #23781930C2semidirect
6. #23781928C2semidirect
7. #23781926C2semidirect
8. #23781925C2semidirect
9. #23781848C2semidirect
10. #23781846C2semidirect
11. #23781833C2semidirect
12. #23781831C2semidirect
13. #23781782C2semidirect
14. #23781780C2semidirect
15. #577×C163C2semidirect
17. #19104781C22semidirect
20. #19104766C22semidirect
22. #19104765C22semidirect
23. #19104765C22semidirect
34. C2×#14579688C22semidirect
36. C2×#14579687C22semidirect
38. C2×#14579686C22semidirect
40. #19104742C22semidirect
41. #19104742C22semidirect
42. #19104741C22semidirect
43. #19104733C22semidirect
45. #19104671C22semidirect
46. #19104670C22semidirect
47. #19104662C22semidirect
48. #19104662C22semidirect
49. #19104662C22semidirect
50. #19104661C22semidirect
51. #19104654C22semidirect
52. #19104653C22semidirect
53. #19104652C22semidirect
54. #19104651C22semidirect
55. C8×Dic163C22semidirect
57. H(5,𝔽2)×C163C22semidirect
59. #137×C163C22semidirect
65. C2×SD16×C163C22semidirect
66. #131×C163C22semidirect
84. #14579689D4semidirect
92. #14579687D4semidirect
98. #14579686D4semidirect
102. #14579684D4semidirect
131. Q16×C163D4semidirect
136. SD16×C163D4semidirect
137. SD16×C163D4semidirect
139. D8×C163D4semidirect
227. C163#3291semidirect
232. #577D163semidirect
247. H(5,𝔽2)C2×D163semidirect
255. #137C2×D163semidirect
277. C2×SD16C2×D163semidirect
281. #131C2×D163semidirect
329. Q16C22⋊D163semidirect
348. SD16C22⋊D163semidirect
349. SD16C22⋊D163semidirect
355. D8C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
#237824762
#237824732
#237824722
#237820362
#237819302
#237819282
#237819262
#237819252
#237818482
#237818462
#237818332
#237818312
#237817822
#237817802
#577×C1632
#3291163