Finite Group Databasebeta

← #35226015

Subgroups — #35226015

Order 20864.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic, complemented
2C2×D8×D163C211yesmaximal, characteristic, complemented
3#23782308C211yesmaximal, characteristic, complemented
4#23782300C211yesmaximal, characteristic, complemented
5#23781915C211yesmaximal, characteristic, complemented
6#23781891C211yesmaximal, characteristic, complemented
7#23781877C211yesmaximal, characteristic, complemented
8#23781857C211yesmaximal, characteristic, complemented
9#23781814C211yesmaximal, characteristic, complemented
10#23781809C211yesmaximal, characteristic, complemented
11#23781797C211yesmaximal, characteristic, complemented
12#23781797C211yesmaximal, characteristic, complemented
13#23781778C211yesmaximal, characteristic, complemented
14#23781738C211yesmaximal, characteristic, complemented
15#496×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic, complemented
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic, complemented
20C2×D4×D163C2211yescharacteristic, complemented
21#19104762—24no—
22C2×#14579686C2211yescharacteristic, complemented
23C2×D1304C2211yescharacteristic
24C2×#14579684C2211yescharacteristic
25C2×#14579684C2211yescharacteristic, complemented
26C2×#14579684C2211yescharacteristic, complemented
27D8×D163—24no—
28C2×#14579682C2211yescharacteristic, complemented
29C2×C8×D163C2211yescharacteristic, complemented
30#19104717—21no—
31#19104711C2211yescharacteristic, complemented
32#19104708—21no—
33#19104706—21no—
34#19104706C2211yescharacteristic
35#19104706C2211yescharacteristic, complemented
36#19104706C2211yescharacteristic, complemented
37#19104705—21no—
38#19104703—21no—
39#19104703—21no—
40C4×D652C2211yescharacteristic
41C4×D652C2211yescharacteristic, complemented
42C4×D652C2211yescharacteristic, complemented
43C22⋊C4×D163—21no—
44#19104667C2211yescharacteristic, complemented
45C4⋊(C163⋊C8)C2211yescharacteristic, complemented
46#19104659C2211yescharacteristic, complemented
47#19104657C2211yescharacteristic, complemented
48#19104657C2211yescharacteristic, complemented
49#19104656C2211yescharacteristic, complemented
50#19104656C2211yescharacteristic, complemented
51#19104656C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53(C2×D163)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×SD16×C163C2211yescharacteristic, complemented
56C2×D8×C163C2211yescharacteristic, complemented
57C2×D8×C163C2211yescharacteristic, complemented
58#127×C163C2211yescharacteristic, complemented
59#124×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63C23×D163—21no—
64#14579700—22no—
65#14579700—41no—
66C2×D652—21no—
67C2×D652D411yescharacteristic
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76D4×D163—22no—
77D4×D163—22no—
78D4×D163—22no—
79D4×D163—41no—
80D4×D163—41no—
81D4×D163—41no—
82D4×D163—44no—
83C2×C4×D163—21no—
84C2×C4×D163—21no—
85C2×C4×D163—21no—
86C2×C4×D163C2311yescharacteristic
87C2×C4×D163C2311yescharacteristic
88#14579686—22no—
89D1304D414yescomplemented
90#14579684—22no—
91#14579684—22no—
92#14579684—22no—
93#14579682—22no—
94C2×C163⋊C8C2311yescharacteristic
95C2×C163⋊C8C2311yescharacteristic
96C8×D163—22no—
97C4⋊Dic163D411yescharacteristic
98C4⋊Dic163C2311yescharacteristic
99C4⋊Dic163C2311yescharacteristic
100Dic163⋊C4—21no—
101C22⋊Dic163—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4D411yescharacteristic
108(C2×D163)⋊C4D411yescharacteristic
109C2×Q8×C163C2311yescharacteristic
110C2×D4×C163—21no—
111C2×D4×C163—21no—
112C2×D4×C163C2311yescharacteristic
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115SD16×C163—22no—
116SD16×C163—41no—
117D8×C163—22no—
118D8×C163—41no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C22⋊C4×C163—21no—
122C22⋊C4×C163—21no—
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—42no—
132C22×D163—81no—
133C22×D163—81no—
134C22×D163—81no—
135C22×D163C2×D411yescharacteristic
136C22×D163C2×D411yescharacteristic
137D652—22no—
138D652—21no—
139D652—21no—
140D652—42no—
141D652C2×D412yes—
142D652C2×D412yescomplemented
143C22⋊D163—42no—
144C22⋊D163—42no—
145C22⋊D163—42no—
146C22⋊D163—42no—
147C22⋊D163—42no—
148C22⋊D163—81no—
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163C2×D411yescharacteristic
152C2×Dic163C2×D411yescharacteristic
153C4×D163—21no—
154C4×D163—21no—
155C4×D163—21no—
156C4×D163—21no—
157C4×D163—42no—
158C4×D163—42no—
159C4×D163—81no—
160C163⋊C8—21no—
161C163⋊C8—21no—
162C23×C163—21no—
163C23×C163—21no—
164C23×C163—21no—
165Q8×C163—21no—
166Q8×C163—21no—
167D4×C163—21no—
168D4×C163—21no—
169D4×C163—21no—
170D4×C163—21no—
171D4×C163—21no—
172D4×C163—21no—
173D4×C163—42no—
174D4×C163—81no—
175C2×C4×C163—21no—
176C2×C4×C163C2×D411yescharacteristic
177C2×C4×C163C2×D411yescharacteristic
178C2×C4×C163C2411yescharacteristic
179C8×C163—21no—
180C8×C163C2×D412yescomplemented
181C22×C163—41no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163C22×D411yescharacteristic
191C2×D163—21no—
192C2×D163—21no—
193C2×D163—21no—
194C2×D163—22no—
195C2×D163—22no—
196C2×D163—21no—
197C2×D163—21no—
198C2×D163—41no—
199C2×D163—41no—
200C2×D163—41no—
201C2×D163—41no—
202C2×D163—84no—
203C2×D163—84no—
204C2×D163—84no—
205C2×D163AGL(1,ℤ/8)12yescomplemented
206C4×C163—21no—
207C4×C163—21no—
208C4×C163—41no—
209C4×C163C22×D411yescharacteristic
210C4×C163H(5,𝔽2)11yescharacteristic
211Dic163—21no—
212Dic163—21no—
213Dic163—41no—
214Dic163—41no—
215C2×C163—81no—
216C2×C163—81no—
217C2×C163—81no—
218C2×C163D4211yescharacteristic
219C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C163#57611yescharacteristic
221D163—22no—
222D163—41no—
223D163—42no—
224D163—42no—
225C163#326411yescharacteristic, complemented, Sylow
226#3264—1631nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1631no—
228C22×D8—1631no—
229#546—1631no—
230D42—1631no—
231#496D16311yescharacteristic, complemented
232#469—1631no—
233#465—1631no—
234#463—1631no—
235#459—1631no—
236#450—1631no—
237#449—1631no—
238#447—1631no—
239#447—1631no—
240#442—1631no—
241#435—1631no—
242C2×#41—1631no—
243C22×D4—3261no—
244C22×D4—1631no—
245C22×D4—1631no—
246C22×D4—1631no—
247AGL(1,ℤ/8)—3264no—
248C2×SD16—1631no—
249C2×SD16C2×D16311yescharacteristic, complemented
250C2×SD16C2×D16311yescharacteristic, complemented
251C2×D8—3264no—
252C2×D8—1631no—
253C2×D8—1631no—
254C2×D8—1631no—
255C2×D8—1631no—
256C2×D8C2×D16311yescharacteristic, complemented
257C2×D8C2×D16311yescharacteristic, complemented
258C2×M16—1631no—
259C22×C8—1631no—
260#127C2×D16311yescharacteristic, complemented
261#124C2×D16311yescharacteristic, complemented
262#123—3261no—
263C22⋊Q8—1631no—
264#121—3261no—
265#121—3261no—
266#121—1631no—
267#121—1631no—
268#121—1631no—
269#120—3261no—
270#120—3261no—
271#120—3261no—
272C4×D4—1631no—
273C4×D4—1631no—
274C4×D4—1631no—
275C2×C22⋊C4—3261no—
276#106—1631no—
277C4⋊C8—1631no—
278#103—1631no—
279#102—1631no—
280#102—1631no—
281#102—1631no—
282#102—1631no—
283#102—1631no—
284C22⋊C8—1631no—
285C22⋊C8—1631no—
286#97C2×D16311yescharacteristic, complemented
287C24—3261no—
288C24—3261no—
289C24—3261no—
290#41—3262no—
291#41—6521no—
292C2×Q8C22×D16311yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3261no—
296C2×D4—3262no—
297C2×D4—3262no—
298C2×D4—3262no—
299C2×D4—3261no—
300C2×D4—3261no—
301C2×D4—3261no—
302C2×D4—3261no—
303C2×D4—3261no—
304C2×D4—3261no—
305C2×D4—6521no—
306C2×D4—6521no—
307C2×D4—6521no—
308C2×D4—6524no—
309C2×D4—1631no—
310C2×D4—1631no—
311C2×D4—1631no—
312C2×D4C22×D16311yescharacteristic
313C2×D4C22×D16311yescharacteristic
314C2×D4C22×D16311yescharacteristic
315C22×C4—3261no—
316C22×C4—3261no—
317C22×C4—3261no—
318C22×C4—1631no—
319C22×C4—1631no—
320SD16—22no—
321SD16—41no—
322SD16—3262no—
323D8—22no—
324D8—41no—
325D8—3262no—
326D8—3262no—
327D8—3262no—
328D8—1634no—
329M16—3262no—
330C2×C8—3262no—
331C2×C8—1631no—
332C2×C8—1631no—
333C2×C8C22×D16311yescharacteristic
334C2×C8C22×D16311yescharacteristic
335C4⋊C4—3261no—
336C4⋊C4—1631no—
337C4⋊C4—1631no—
338C4⋊C4—1631no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3261no—
342C22⋊C4—3261no—
343C22⋊C4—3261no—
344C22⋊C4—3261no—
345C22⋊C4—3261no—
346C22⋊C4—3261no—
347C22⋊C4—1631no—
348C22⋊C4—1631no—
349C42C22×D16311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—13041no—
354C23—13041no—
355C23—13041no—
356C23—3261no—
357C23—3261no—
358C23—6522no—
359C23—6522no—
360C23—6522no—
361C23—6522no—
362C23—6522no—
363C23—6522no—
364C23—1631no—
365C23—1631no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—13041no—
377D4—3262no—
378D4—3261no—
379D4—3261no—
380D4—6522no—
381D4—6522no—
382D4—6522no—
383D4—6522no—
384D4—6522no—
385D4—6522no—
386D4—1632no—
387D4—1632no—
388C2×C4—21no—
389C2×C4—13041no—
390C2×C4—3261no—
391C2×C4—3261no—
392C2×C4—3261no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—6522no—
397C2×C4—6522no—
398C2×C4—1631no—
399C2×C4—1631no—
400C2×C4D4×D16311yescharacteristic
401C2×C4D4×D16311yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C8—21no—
404C8—3261no—
405C8—3261no—
406C8D4×D16312yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—13044no—
417C22—13044no—
418C22—13044no—
419C22—3261no—
420C22—3261no—
421C22—3261no—
422C22—3261no—
423C22—3261no—
424C22—3262no—
425C22—3262no—
426C22—6521no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22—1632no—
431C22C2×D4×D16311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3261no—
436C4—3261no—
437C4—6521no—
438C4—6521no—
439C4C2×D4×D16311yescharacteristic
440C4#1910478011yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—3262no—
445C2—6521no—
446C2—6522no—
447C2—6522no—
448C2#2378230011yescharacteristic
449C2AGL(1,ℤ/8)×D16311yescharacteristic
450C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#496×C163C2
socleC22×C163C22×D4
hypercenter#496D163
hypocenterC163#3264
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19104762C2semidirect
2. C2×D8×D163C2semidirect
3. #23782308C2semidirect
4. #23782300C2semidirect
5. #23781915C2semidirect
6. #23781891C2semidirect
7. #23781877C2semidirect
8. #23781857C2semidirect
9. #23781814C2semidirect
10. #23781809C2semidirect
11. #23781797C2semidirect
12. #23781797C2semidirect
13. #23781778C2semidirect
14. #23781738C2semidirect
15. #496×C163C2semidirect
16. C2×#14579700C22semidirect
19. C2×D4×D163C22semidirect
20. C2×D4×D163C22semidirect
22. C2×#14579686C22semidirect
25. C2×#14579684C22semidirect
26. C2×#14579684C22semidirect
28. C2×#14579682C22semidirect
29. C2×C8×D163C22semidirect
31. #19104711C22semidirect
35. #19104706C22semidirect
36. #19104706C22semidirect
41. C4×D652C22semidirect
42. C4×D652C22semidirect
44. #19104667C22semidirect
45. C4⋊(C163⋊C8)C22semidirect
46. #19104659C22semidirect
47. #19104657C22semidirect
48. #19104657C22semidirect
49. #19104656C22semidirect
50. #19104656C22semidirect
51. #19104656C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. (C2×D163)⋊C8C22semidirect
54. C2×SD16×C163C22semidirect
55. C2×SD16×C163C22semidirect
56. C2×D8×C163C22semidirect
57. C2×D8×C163C22semidirect
58. #127×C163C22semidirect
59. #124×C163C22semidirect
60. #97×C163C22semidirect
89. D1304D4semidirect
142. D652C2×D4semidirect
180. C8×C163C2×D4semidirect
205. C2×D163AGL(1,ℤ/8)semidirect
225. C163#3264semidirect
231. #496D163semidirect
249. C2×SD16C2×D163semidirect
250. C2×SD16C2×D163semidirect
256. C2×D8C2×D163semidirect
257. C2×D8C2×D163semidirect
260. #127C2×D163semidirect
261. #124C2×D163semidirect
286. #97C2×D163semidirect
406. C8D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C2×D8×D1632
#237823082
#237823002
#237819152
#237818912
#237818772
#237818572
#237818142
#237818092
#237817972
#237817782
#237817382
#496×C1632
#3264163