Finite Group Databasebeta

← #35226020

Subgroups — #35226020

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
2C22×#14579684C211yesmaximal, characteristic, complemented
3#23782306C211yesmaximal, characteristic, complemented
4D4×C22⋊D163C211yesmaximal, characteristic, complemented
5#23782051C211yesmaximal, characteristic, complemented
6#23781917C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781882C211yesmaximal, characteristic, complemented
9#23781859C211yesmaximal, characteristic, complemented
10#23781809C211yesmaximal, characteristic, complemented
11#23781801C211yesmaximal, characteristic, complemented
12#23781800C211yesmaximal, characteristic, complemented
13#23781774C211yesmaximal, characteristic, complemented
14#23781733C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic, complemented
17C22×D652C2211yescharacteristic, complemented
18C22×C22⋊D163—21no—
19C2×D4×D163C2211yescharacteristic, complemented
20#19104762—24no—
21C2×#14579686C2211yescharacteristic, complemented
22C2×D1304C2211yescharacteristic, complemented
23C2×D1304C2211yescharacteristic, complemented
24C2×#14579684—24no—
25C2×#14579684C2211yescharacteristic
26C2×#14579684C2211yescharacteristic
27C2×#14579684C2211yescharacteristic, complemented
28C2×#14579684C2211yescharacteristic, complemented
29C2×#14579682C2211yescharacteristic, complemented
30C22×C163⋊C8C2211yescharacteristic, complemented
31#19104731C2211yescharacteristic, complemented
32#19104724C2211yescharacteristic, complemented
33#19104720—21no—
34#19104710—21no—
35#19104710C2211yescharacteristic, complemented
36#19104709C2211yescharacteristic
37#19104707—21no—
38#19104706C2211yescharacteristic, complemented
39#19104705—21no—
40#19104704—21no—
41#19104703—21no—
42D4×Dic163C2211yescharacteristic, complemented
43#19104697C2211yescharacteristic, complemented
44C4×C22⋊D163C2211yescharacteristic
45C22⋊C4×D163—21no—
46#19104666C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104659C2211yescharacteristic, complemented
49#19104657C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic, complemented
51#19104655C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104645C2211yescharacteristic, complemented
54C22×D4×C163C2211yescharacteristic
55C2×SD16×C163C2211yescharacteristic, complemented
56C2×SD16×C163C2211yescharacteristic, complemented
57C22⋊Q8×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59#102×C163C2211yescharacteristic, complemented
60C22⋊C8×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63#14579700—22no—
64#14579700—41no—
65C2×D652—22no—
66C2×D652—21no—
67C2×D652—21no—
68C2×D652—21no—
69C2×D652—41no—
70C2×D652C2311yescharacteristic
71C2×D652C2311yescharacteristic
72C2×D652C2311yescharacteristic
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163—21no—
77C2×C22⋊D163—21no—
78C2×C22⋊D163—44no—
79C2×C22⋊D163D411yescharacteristic
80D4×D163—22no—
81D4×D163—41no—
82C22×Dic163—21no—
83C2×C4×D163—21no—
84C2×C4×D163—21no—
85C2×C4×D163C2311yescharacteristic
86#14579686—22no—
87D1304—22no—
88D1304—41no—
89#14579684—22no—
90#14579684—22no—
91#14579684—22no—
92#14579684D414yescomplemented
93#14579682—22no—
94C2×C163⋊C8—22no—
95C2×C163⋊C8C2311yescharacteristic
96C2×C163⋊C8C2311yescharacteristic
97C4⋊Dic163C2311yescharacteristic
98Dic163⋊C4D411yescharacteristic
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163D411yescharacteristic
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4D411yescharacteristic
108C4×Dic163C2311yescharacteristic
109C24×C163—21no—
110C2×Q8×C163C2311yescharacteristic
111C2×D4×C163—22no—
112C2×D4×C163—21no—
113C2×D4×C163—41no—
114C2×D4×C163C2311yescharacteristic
115C2×D4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117SD16×C163—22no—
118SD16×C163—41no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163—21no—
122C4⋊C4×C163C2311yescharacteristic
123C22⋊C4×C163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—81no—
132C22×D163—81no—
133C22×D163C2×D411yescharacteristic
134D652—21no—
135D652—21no—
136D652—21no—
137D652—21no—
138D652—21no—
139D652—42no—
140D652—42no—
141D652—81no—
142D652C2×D412yescomplemented
143C22⋊D163—22no—
144C22⋊D163—42no—
145C22⋊D163—42no—
146C22⋊D163—42no—
147C22⋊D163—42no—
148C22⋊D163—81no—
149C2×Dic163—21no—
150C2×Dic163—42no—
151C2×Dic163C2×D411yescharacteristic
152C2×Dic163C2×D411yescharacteristic
153C2×Dic163C2×D411yescharacteristic
154C4×D163—21no—
155C4×D163—21no—
156C4×D163—42no—
157C4×D163—81no—
158C163⋊C8—21no—
159C163⋊C8C2×D412yescomplemented
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—42no—
163C23×C163—42no—
164C23×C163—81no—
165C23×C163C2×D411yescharacteristic
166Q8×C163—21no—
167Q8×C163—21no—
168D4×C163—21no—
169D4×C163—21no—
170D4×C163—21no—
171D4×C163—42no—
172D4×C163C2×D412yes—
173C2×C4×C163—21no—
174C2×C4×C163—21no—
175C2×C4×C163—21no—
176C2×C4×C163—21no—
177C2×C4×C163C2×D411yescharacteristic
178C2×C4×C163C2411yescharacteristic
179C8×C163—21no—
180C8×C163—21no—
181C22×C163—21no—
182C22×C163—21no—
183C22×C163—22no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—84no—
190C22×C163AGL(1,ℤ/8)12yescomplemented
191C22×C163C22×D411yescharacteristic
192C2×D163—21no—
193C2×D163—22no—
194C2×D163—21no—
195C2×D163—21no—
196C2×D163—41no—
197C2×D163—41no—
198C2×D163—41no—
199C2×D163—41no—
200C2×D163—41no—
201C2×D163—41no—
202C2×D163—41no—
203C2×D163—41no—
204C2×D163—84no—
205C2×D163—84no—
206C4×C163—21no—
207C4×C163—41no—
208C4×C163—41no—
209C4×C163C22×D411yescharacteristic
210C4×C163H(5,𝔽2)11yescharacteristic
211Dic163—21no—
212Dic163—21no—
213Dic163—21no—
214Dic163—41no—
215C2×C163—22no—
216C2×C163—42no—
217C2×C163—81no—
218C2×C163D4211yescharacteristic
219C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C163#57611yescharacteristic
221D163—41no—
222D163—42no—
223D163—81no—
224D163—81no—
225C163#326411yescharacteristic, complemented, Sylow
226#3264—1631nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1631no—
228C22×D8—1631no—
229#546—1631no—
230D42—1631no—
231#496—1631no—
232#469—1631no—
233#465—1631no—
234#463—1631no—
235#459—1631no—
236#450D16311yescharacteristic, complemented
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×D4C2×D16311yescharacteristic
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×D8—1631no—
257C2×D8—1631no—
258C2×M16—1631no—
259C22×C8—1631no—
260#127—1631no—
261#124—1631no—
262#123—3261no—
263C22⋊Q8C2×D16311yescharacteristic, complemented
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#102C2×D16311yescharacteristic, complemented
283#102C2×D16311yescharacteristic, complemented
284C22⋊C8—1631no—
285C22⋊C8C2×D16311yescharacteristic, complemented
286#97—1631no—
287C24—21no—
288C24—3261no—
289C24—3261no—
290#41—3262no—
291#41—6521no—
292C2×Q8C22×D16311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—3261no—
297C2×D4—3262no—
298C2×D4—3261no—
299C2×D4—3262no—
300C2×D4—3261no—
301C2×D4—3261no—
302C2×D4—3261no—
303C2×D4—3261no—
304C2×D4—3261no—
305C2×D4—3261no—
306C2×D4—6521no—
307C2×D4—6521no—
308C2×D4—6524no—
309C2×D4—1631no—
310C2×D4—1631no—
311C2×D4—1631no—
312C2×D4—1631no—
313C2×D4C22×D16311yescharacteristic
314C2×D4C22×D16311yescharacteristic
315C22×C4—3261no—
316C22×C4—3261no—
317C22×C4—3261no—
318C22×C4—1631no—
319C22×C4C22×D16311yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—3262no—
323D8—3262no—
324D8—3262no—
325D8—3262no—
326D8—3262no—
327D8—6521no—
328D8—1634no—
329M16—3262no—
330C2×C8—3262no—
331C2×C8—1631no—
332C2×C8—1631no—
333C2×C8C22×D16311yescharacteristic
334C2×C8C22×D16311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1631no—
337C4⋊C4—1631no—
338C4⋊C4C22×D16311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3261no—
341C22⋊C4—3261no—
342C22⋊C4—3261no—
343C22⋊C4—3261no—
344C22⋊C4—3261no—
345C22⋊C4—3261no—
346C22⋊C4—3261no—
347C22⋊C4—1631no—
348C22⋊C4—1631no—
349C42—1631no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—13041no—
356C23—13041no—
357C23—3261no—
358C23—3261no—
359C23—3261no—
360C23—6522no—
361C23—6522no—
362C23—6522no—
363C23—6522no—
364C23—1631no—
365C23D4×D16311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—13041no—
373D4—13041no—
374D4—3262no—
375D4—3261no—
376D4—3261no—
377D4—3261no—
378D4—3261no—
379D4—3261no—
380D4—6522no—
381D4—6522no—
382D4—6522no—
383D4—6522no—
384D4—6522no—
385D4—6522no—
386D4—1632no—
387D4D4×D16312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—13041no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—6522no—
397C2×C4—6522no—
398C2×C4—1631no—
399C2×C4—1631no—
400C2×C4—1631no—
401C2×C4D4×D16311yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C8—21no—
404C8—21no—
405C8—3261no—
406C8—1632no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—13044no—
417C22—13044no—
418C22—3261no—
419C22—3261no—
420C22—3261no—
421C22—3262no—
422C22—6521no—
423C22—6521no—
424C22—6521no—
425C22—6521no—
426C22—6521no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22#1910476212yescomplemented
431C22C2×D4×D16311yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3261no—
436C4—3261no—
437C4—3261no—
438C4—6521no—
439C4C2×D4×D16311yescharacteristic
440C4#1910478011yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—13041no—
445C2—13041no—
446C2—6521no—
447C2—6522no—
448C2#2378229711yescharacteristic
449C2C2×#1910476211yescharacteristic
450C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
hypercenter#450D163
hypocenterC163#3264
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19104762C2semidirect
2. C22×#14579684C2semidirect
3. #23782306C2semidirect
4. D4×C22⋊D163C2semidirect
5. #23782051C2semidirect
6. #23781917C2semidirect
7. #23781896C2semidirect
8. #23781882C2semidirect
9. #23781859C2semidirect
10. #23781809C2semidirect
11. #23781801C2semidirect
12. #23781800C2semidirect
13. #23781774C2semidirect
14. #23781733C2semidirect
15. #450×C163C2semidirect
16. C2×#14579700C22semidirect
17. C22×D652C22semidirect
19. C2×D4×D163C22semidirect
21. C2×#14579686C22semidirect
22. C2×D1304C22semidirect
23. C2×D1304C22semidirect
27. C2×#14579684C22semidirect
28. C2×#14579684C22semidirect
29. C2×#14579682C22semidirect
30. C22×C163⋊C8C22semidirect
31. #19104731C22semidirect
32. #19104724C22semidirect
35. #19104710C22semidirect
38. #19104706C22semidirect
42. D4×Dic163C22semidirect
43. #19104697C22semidirect
46. #19104666C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104659C22semidirect
49. #19104657C22semidirect
50. #19104657C22semidirect
51. #19104655C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104645C22semidirect
55. C2×SD16×C163C22semidirect
56. C2×SD16×C163C22semidirect
57. C22⋊Q8×C163C22semidirect
58. #102×C163C22semidirect
59. #102×C163C22semidirect
60. C22⋊C8×C163C22semidirect
92. #14579684D4semidirect
142. D652C2×D4semidirect
159. C163⋊C8C2×D4semidirect
190. C22×C163AGL(1,ℤ/8)semidirect
225. C163#3264semidirect
236. #450D163semidirect
249. C2×SD16C2×D163semidirect
250. C2×SD16C2×D163semidirect
263. C22⋊Q8C2×D163semidirect
282. #102C2×D163semidirect
283. #102C2×D163semidirect
285. C22⋊C8C2×D163semidirect
430. C22#19104762semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C22×#145796842
#237823062
D4×C22⋊D1632
#237820512
#237819172
#237818962
#237818822
#237818592
#237818092
#237818012
#237818002
#237817742
#237817332
#450×C1632
#3264163