Finite Group Databasebeta

← #35226018

Subgroups — #35226018

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×#19104759C211yesmaximal, characteristic, complemented
2C2×D8×D163C211yesmaximal, characteristic, complemented
3#23782310C211yesmaximal, characteristic, complemented
4D4×D652C211yesmaximal, characteristic, complemented
5#23782054C211yesmaximal, characteristic, complemented
6#23781919C211yesmaximal, characteristic, complemented
7#23781895C211yesmaximal, characteristic, complemented
8#23781880C211yesmaximal, characteristic, complemented
9#23781860C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781811C211yesmaximal, characteristic, complemented
12#23781800C211yesmaximal, characteristic, complemented
13#23781800C211yesmaximal, characteristic, complemented
14#23781738C211yesmaximal, characteristic, complemented
15#442×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic, complemented
17C22×D652—21no—
18C2×D4×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic, complemented
20C2×D4×D163C2211yescharacteristic, complemented
21#19104759—24no—
22C2×#14579688C2211yescharacteristic, complemented
23C2×#14579688C2211yescharacteristic, complemented
24C2×#14579687C2211yescharacteristic, complemented
25C2×D1304C2211yescharacteristic, complemented
26C2×D1304C2211yescharacteristic, complemented
27C2×#14579684C2211yescharacteristic
28C2×#14579684C2211yescharacteristic, complemented
29C2×#14579684C2211yescharacteristic, complemented
30D8×D163—24no—
31C2×#14579682C2211yescharacteristic, complemented
32C2×C8×D163C2211yescharacteristic, complemented
33#19104732C2211yescharacteristic, complemented
34#19104723C2211yescharacteristic, complemented
35#19104719—21no—
36#19104715C2211yescharacteristic, complemented
37#19104710C2211yescharacteristic
38#19104710C2211yescharacteristic, complemented
39#19104710C2211yescharacteristic, complemented
40#19104709—21no—
41#19104709—21no—
42#19104705—21no—
43#19104705—21no—
44#19104705—21no—
45C4×D652C2211yescharacteristic
46C2×(C2×D163)⋊C4—21no—
47C4⋊(C163⋊C8)C2211yescharacteristic, complemented
48#19104658C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic, complemented
50#19104655C2211yescharacteristic, complemented
51#19104655C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53(C2×D163)⋊C8C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic, complemented
57#106×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59#102×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63C23×D163—21no—
64#14579698—22no—
65#14579698—41no—
66C2×Dic326C2311yescharacteristic
67C2×D652—21no—
68C2×D652—21no—
69C2×D652—21no—
70C2×D652—21no—
71C2×D652—21no—
72C2×D652—44no—
73C2×D652D411yescharacteristic
74C2×D652C2311yescharacteristic
75C2×D652C2311yescharacteristic
76C2×D652C2311yescharacteristic
77C2×C22⋊D163—21no—
78C2×C22⋊D163—21no—
79C2×C22⋊D163—21no—
80C2×C22⋊D163—21no—
81D4×D163—22no—
82D4×D163—22no—
83D4×D163—22no—
84D4×D163—41no—
85D4×D163—41no—
86D4×D163—41no—
87C22×Dic163—21no—
88C2×C4×D163C2311yescharacteristic
89C2×C4×D163C2311yescharacteristic
90#14579688—22no—
91#14579688—41no—
92#14579687—22no—
93D1304—22no—
94D1304—41no—
95#14579684—22no—
96#14579684—22no—
97#14579684—22no—
98#14579682—22no—
99C2×C163⋊C8C2311yescharacteristic
100C2×C163⋊C8C2311yescharacteristic
101C8×D163—22no—
102C4⋊Dic163—21no—
103C4⋊Dic163D411yescharacteristic
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4—21no—
109(C2×D163)⋊C4—21no—
110(C2×D163)⋊C4D411yescharacteristic
111(C2×D163)⋊C4D411yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C2×D4×C163C2311yescharacteristic
114C22×C4×C163—21no—
115C22×C4×C163—21no—
116D8×C163D414yescomplemented
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C4⋊C4×C163C2311yescharacteristic
120C4⋊C4×C163C2311yescharacteristic
121C22⋊C4×C163—21no—
122C22⋊C4×C163—21no—
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—42no—
132C22×D163—42no—
133C22×D163—81no—
134C22×D163—81no—
135C22×D163—81no—
136C22×D163C2×D411yescharacteristic
137C22×D163C2×D411yescharacteristic
138Dic326—21no—
139Dic326—21no—
140D652—22no—
141D652—21no—
142D652—21no—
143D652—21no—
144D652—21no—
145D652—21no—
146D652—21no—
147D652—42no—
148D652—42no—
149D652—42no—
150D652—81no—
151D652—81no—
152C22⋊D163—42no—
153C22⋊D163—42no—
154C22⋊D163—42no—
155C22⋊D163—42no—
156C2×Dic163—21no—
157C2×Dic163—42no—
158C2×Dic163C2×D411yescharacteristic
159C2×Dic163C2×D411yescharacteristic
160C4×D163—21no—
161C4×D163—21no—
162C4×D163—21no—
163C4×D163—21no—
164C163⋊C8—21no—
165C163⋊C8—21no—
166C23×C163—21no—
167C23×C163—21no—
168D4×C163—21no—
169D4×C163—21no—
170D4×C163C2×D412yes—
171D4×C163C2×D412yescomplemented
172C2×C4×C163—21no—
173C2×C4×C163—21no—
174C2×C4×C163—42no—
175C2×C4×C163—81no—
176C2×C4×C163C2×D411yescharacteristic
177C2×C4×C163C2×D411yescharacteristic
178C2×C4×C163C2411yescharacteristic
179C8×C163—21no—
180C8×C163C2×D412yescomplemented
181C22×C163—22no—
182C22×C163—22no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163C22×D411yescharacteristic
188C2×D163—21no—
189C2×D163—21no—
190C2×D163—21no—
191C2×D163—21no—
192C2×D163—21no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—41no—
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×C163—41no—
210C4×C163C22×D411yescharacteristic
211C4×C163H(5,𝔽2)11yescharacteristic
212Dic163—21no—
213Dic163—21no—
214Dic163—41no—
215C2×C163—42no—
216C2×C163—42no—
217C2×C163D4211yescharacteristic
218C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C163#57611yescharacteristic
220D163—22no—
221D163—41no—
222D163—81no—
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#450—1631no—
237#449—1631no—
238#447—1631no—
239#447—1631no—
240#442D16311yescharacteristic, complemented
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×SD16—1631no—
250C2×SD16—1631no—
251C2×D8—3264no—
252C2×D8—1631no—
253C2×D8—1631no—
254C2×D8—1631no—
255C2×D8—1631no—
256C2×D8—1631no—
257C2×D8C2×D16311yescharacteristic
258C2×M16—1631no—
259C22×C8—1631no—
260#127—1631no—
261#124—1631no—
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×D4C2×D16311yescharacteristic, complemented
274C4×D4C2×D16311yescharacteristic, complemented
275C2×C22⋊C4—3261no—
276#106C2×D16311yescharacteristic, complemented
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⋊C8—1631no—
286#97C2×D16311yescharacteristic, complemented
287C24—3261no—
288C24—3261no—
289C24—3261no—
290#41—3262no—
291#41—6521no—
292C2×Q8—1631no—
293C2×D4—3261no—
294C2×D4—3261no—
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—6524no—
306C2×D4—6521no—
307C2×D4—6521no—
308C2×D4—6521no—
309C2×D4—1631no—
310C2×D4—1631no—
311C2×D4—1631no—
312C2×D4—1631no—
313C2×D4C22×D16311yescharacteristic
314C2×D4C22×D16311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—3261no—
318C22×C4—1631no—
319C22×C4—1631no—
320SD16—3262no—
321SD16—3262no—
322SD16—6521no—
323D8—3262no—
324D8—3262no—
325D8—3262no—
326D8—3262no—
327D8—6521no—
328D8D65214yescomplemented
329M16—3262no—
330C2×C8—3262no—
331C2×C8—1631no—
332C2×C8—1631no—
333C2×C8C22×D16311yescharacteristic
334C2×C8C22×D16311yescharacteristic
335C4⋊C4—3261no—
336C4⋊C4—1631no—
337C4⋊C4C22×D16311yescharacteristic
338C4⋊C4C22×D16311yescharacteristic
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—13041no—
353C23—13041no—
354C23—13041no—
355C23—3261no—
356C23—3261no—
357C23—3261no—
358C23—6522no—
359C23—6522no—
360C23—6522no—
361C23—6522no—
362C23—6522no—
363C23—6522no—
364C23—1631no—
365C23—1631no—
366Q8—3261no—
367Q8—3261no—
368D4—21no—
369D4—21no—
370D4—13041no—
371D4—13041no—
372D4—3262no—
373D4—3261no—
374D4—3261no—
375D4—3261no—
376D4—3261no—
377D4—3261no—
378D4—3261no—
379D4—6522no—
380D4—6522no—
381D4—6522no—
382D4—6522no—
383D4—6522no—
384D4—6522no—
385D4—6522no—
386D4C2×D65212yes—
387D4C2×D65212yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—3261no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—3261no—
397C2×C4—6522no—
398C2×C4—1631no—
399C2×C4—1631no—
400C2×C4D4×D16311yescharacteristic
401C2×C4D4×D16311yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C8—21no—
404C8—3261no—
405C8—3261no—
406C8C2×D65212yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—13044no—
414C22—13044no—
415C22—13044no—
416C22—3261no—
417C22—3261no—
418C22—3261no—
419C22—3261no—
420C22—3261no—
421C22—6521no—
422C22—6521no—
423C22—6521no—
424C22—6521no—
425C22—6521no—
426C22—6521no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22—1632no—
431C22C2×D4×D16311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—3261no—
437C4—3261no—
438C4—6521no—
439C4C22×D65211yescharacteristic
440C4#1910478111yescharacteristic
441C2—42no—
442C2—42no—
443C2—13041no—
444C2—13041no—
445C2—13041no—
446C2—3262no—
447C2—6521no—
448C2D4×D65211yescharacteristic
449C2AGL(1,ℤ/8)×D16311yescharacteristic
450C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#442×C163C2
socleC22×C163C22×D4
hypercenter#442D163
hypocenterC163#3264
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×D8×D163C2semidirect
3. #23782310C2semidirect
4. D4×D652C2semidirect
5. #23782054C2semidirect
6. #23781919C2semidirect
7. #23781895C2semidirect
8. #23781880C2semidirect
9. #23781860C2semidirect
10. #23781817C2semidirect
11. #23781811C2semidirect
12. #23781800C2semidirect
13. #23781800C2semidirect
14. #23781738C2semidirect
15. #442×C163C2semidirect
16. C2×#14579698C22semidirect
19. C2×D4×D163C22semidirect
20. C2×D4×D163C22semidirect
22. C2×#14579688C22semidirect
23. C2×#14579688C22semidirect
24. C2×#14579687C22semidirect
25. C2×D1304C22semidirect
26. C2×D1304C22semidirect
28. C2×#14579684C22semidirect
29. C2×#14579684C22semidirect
31. C2×#14579682C22semidirect
32. C2×C8×D163C22semidirect
33. #19104732C22semidirect
34. #19104723C22semidirect
36. #19104715C22semidirect
38. #19104710C22semidirect
39. #19104710C22semidirect
47. C4⋊(C163⋊C8)C22semidirect
48. #19104658C22semidirect
49. #19104655C22semidirect
50. #19104655C22semidirect
51. #19104655C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. (C2×D163)⋊C8C22semidirect
55. C4×D4×C163C22semidirect
56. C4×D4×C163C22semidirect
57. #106×C163C22semidirect
58. #102×C163C22semidirect
59. #102×C163C22semidirect
60. #97×C163C22semidirect
116. D8×C163D4semidirect
171. D4×C163C2×D4semidirect
180. C8×C163C2×D4semidirect
205. C2×D163AGL(1,ℤ/8)semidirect
225. C163#3264semidirect
240. #442D163semidirect
273. C4×D4C2×D163semidirect
274. C4×D4C2×D163semidirect
276. #106C2×D163semidirect
282. #102C2×D163semidirect
283. #102C2×D163semidirect
286. #97C2×D163semidirect
328. D8D652semidirect
387. D4C2×D652semidirect
406. C8C2×D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
C2×D8×D1632
#237823102
D4×D6522
#237820542
#237819192
#237818952
#237818802
#237818602
#237818172
#237818112
#237818002
#237817382
#442×C1632
#3264163