Finite Group Databasebeta

← #35223785

Subgroups — #35223785

Order 20864.

806 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D163C211yesmaximal, characteristic, complemented
2C2×#19104762C211yesmaximal, characteristic, complemented
3C2×#19104759C211yesmaximal, characteristic, complemented
4#121×D163C211yesmaximal, characteristic, complemented
5#23781812C211yesmaximal, characteristic, complemented
6#23781810C211yesmaximal, characteristic, complemented
7#23781808C211yesmaximal, characteristic, complemented
8#23781805C211yesmaximal, characteristic, complemented
9#23781802C211yesmaximal, characteristic, complemented
10#23781800C211yesmaximal, characteristic, complemented
11#23781797C211yesmaximal, characteristic, complemented
12#23781665C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14#23781627C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699—21no—
18C2×#14579699C2211yescharacteristic, complemented
19C2×#14579698—21no—
20C2×#14579698C2211yescharacteristic
21#41×D163—24no—
22#41×D163—24no—
23C2×Q8×D163C2211yescharacteristic, complemented
24C2×D4×D163—21no—
25C2×D4×D163—21no—
26C2×D4×D163C2211yescharacteristic, complemented
27C2×D4×D163C2211yescharacteristic, complemented
28C22×C4×D163—21no—
29C22×C4×D163C2211yescharacteristic
30#19104762—24no—
31#19104759—24no—
32C2×#14579688C2211yescharacteristic, complemented
33C2×#14579687C2211yescharacteristic, complemented
34C2×#14579686C2211yescharacteristic, complemented
35C2×D1304C2211yescharacteristic, complemented
36C2×#14579684C2211yescharacteristic, complemented
37C2×#14579684C2211yescharacteristic, complemented
38C2×#14579682C2211yescharacteristic, complemented
39C2×#14579682C2211yescharacteristic, complemented
40#19104710C2211yescharacteristic
41#19104709C2211yescharacteristic, complemented
42#19104708—21no—
43#19104707—21no—
44#19104706C2211yescharacteristic
45C4⋊C4×D163C2211yescharacteristic, complemented
46C22⋊C4×D163—21no—
47#19104660C2211yescharacteristic, complemented
48#19104659C2211yescharacteristic, complemented
49#19104658C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic, complemented
51#19104656C2211yescharacteristic, complemented
52#19104655C2211yescharacteristic, complemented
53C22⋊(C163⋊C8)C2211yescharacteristic, complemented
54(C2×D163)⋊C8C2211yescharacteristic, complemented
55(C2×D163)⋊C8C2211yescharacteristic, complemented
56C2×#41×C163C2211yescharacteristic, complemented
57C2×SD16×C163C2211yescharacteristic, complemented
58C2×D8×C163C2211yescharacteristic, complemented
59#121×C163C2211yescharacteristic, complemented
60#103×C163C2211yescharacteristic, complemented
61#102×C163C2211yescharacteristic, complemented
62C22⋊C8×C163C2211yescharacteristic, complemented
63C23×D163—21no—
64C23×D163—21no—
65C23×D163D411yescharacteristic
66#14579700—22no—
67#14579700D414yescomplemented
68#14579699—28no—
69#14579699—22no—
70#14579699—22no—
71#14579698—28no—
72#14579698—22no—
73#14579698D414yescomplemented
74C2×Dic326—21no—
75C2×Dic326C2311yescharacteristic
76Q8×D163—22no—
77Q8×D163—22no—
78C2×D652—21no—
79C2×D652—21no—
80C2×D652C2311yescharacteristic
81C2×D652C2311yescharacteristic
82C2×C22⋊D163—21no—
83C2×C22⋊D163—21no—
84C2×C22⋊D163—21no—
85C2×C22⋊D163—21no—
86C2×C22⋊D163—21no—
87C2×C22⋊D163—21no—
88D4×D163—28no—
89D4×D163—22no—
90D4×D163—22no—
91D4×D163—22no—
92D4×D163—22no—
93D4×D163—44no—
94C22×Dic163—21no—
95C22×Dic163D411yescharacteristic
96C2×C4×D163—28no—
97C2×C4×D163—21no—
98C2×C4×D163—21no—
99C2×C4×D163—22no—
100C2×C4×D163—22no—
101C2×C4×D163—21no—
102C2×C4×D163—21no—
103C2×C4×D163D411yescharacteristic
104C2×C4×D163D411yescharacteristic
105C2×C4×D163C2311yescharacteristic
106C2×C4×D163C2311yescharacteristic
107#14579688—22no—
108#14579687—22no—
109#14579686—22no—
110D1304—22no—
111#14579684—22no—
112#14579684—22no—
113#14579682—22no—
114#14579682—22no—
115C2×C163⋊C8C2311yescharacteristic
116C2×C163⋊C8C2311yescharacteristic
117C4⋊Dic163C2311yescharacteristic
118Dic163⋊C4—21no—
119C22⋊Dic163—21no—
120(C2×D163)⋊C4—21no—
121(C2×D163)⋊C4—21no—
122#41×C163—22no—
123#41×C163—22no—
124C2×Q8×C163C2311yescharacteristic
125C2×D4×C163—21no—
126C2×D4×C163—21no—
127C2×D4×C163C2311yescharacteristic
128C2×D4×C163C2311yescharacteristic
129C22×C4×C163—21no—
130C22×C4×C163C2311yescharacteristic
131SD16×C163—22no—
132D8×C163—22no—
133C2×C8×C163C2311yescharacteristic
134C2×C8×C163C2311yescharacteristic
135C4⋊C4×C163C2311yescharacteristic
136C22⋊C4×C163—21no—
137C22×D163—24no—
138C22×D163—21no—
139C22×D163—22no—
140C22×D163—22no—
141C22×D163—21no—
142C22×D163—41no—
143C22×D163—42no—
144C22×D163—42no—
145C22×D163—42no—
146C22×D163—42no—
147C22×D163—42no—
148C22×D163C2×D411yescharacteristic
149C22×D163C2×D411yescharacteristic
150Dic326—24no—
151Dic326—21no—
152Dic326C2×D412yescomplemented
153D652—24no—
154D652—21no—
155D652—21no—
156D652—21no—
157D652—42no—
158D652C2×D412yescomplemented
159C22⋊D163—24no—
160C22⋊D163—24no—
161C22⋊D163—24no—
162C22⋊D163—42no—
163C22⋊D163—42no—
164C22⋊D163—42no—
165C2×Dic163—21no—
166C2×Dic163—24no—
167C2×Dic163—22no—
168C2×Dic163—21no—
169C2×Dic163C2×D411yescharacteristic
170C2×Dic163C2×D411yescharacteristic
171C4×D163—24no—
172C4×D163—24no—
173C4×D163—22no—
174C4×D163—21no—
175C4×D163—21no—
176C4×D163—22no—
177C4×D163—24no—
178C4×D163—42no—
179C4×D163C2×D412yes—
180C4×D163C2×D412yes—
181C163⋊C8—21no—
182C163⋊C8—21no—
183C23×C163—21no—
184C23×C163—21no—
185C23×C163C2×D411yescharacteristic
186Q8×C163—21no—
187Q8×C163C2×D412yescomplemented
188D4×C163—24no—
189D4×C163—21no—
190D4×C163—21no—
191D4×C163—21no—
192D4×C163—42no—
193D4×C163C2×D412yescomplemented
194C2×C4×C163—24no—
195C2×C4×C163—21no—
196C2×C4×C163—21no—
197C2×C4×C163—21no—
198C2×C4×C163—21no—
199C2×C4×C163C2×D411yescharacteristic
200C2×C4×C163C2411yescharacteristic
201C8×C163—21no—
202C8×C163—21no—
203C22×C163—21no—
204C22×C163—22no—
205C22×C163—21no—
206C22×C163—21no—
207C22×C163—41no—
208C22×C163—41no—
209C22×C163—41no—
210C22×C163—41no—
211C22×C163—41no—
212C22×C163C22×D411yescharacteristic
213C2×D163—21no—
214C2×D163—21no—
215C2×D163—21no—
216C2×D163—21no—
217C2×D163—22no—
218C2×D163—44no—
219C2×D163—48no—
220C2×D163—41no—
221C2×D163—41no—
222C2×D163—41no—
223C2×D163—41no—
224C2×D163—41no—
225C2×D163—84no—
226C2×D163#12012yes—
227C2×D163AGL(1,ℤ/8)12yescomplemented
228C4×C163—21no—
229C4×C163—22no—
230C4×C163—41no—
231C4×C163C22×D411yescharacteristic
232C4×C163C22×D411yescharacteristic
233Dic163—22no—
234Dic163—21no—
235Dic163—41no—
236Dic163#12012yes—
237C2×C163—41no—
238C2×C163—42no—
239C2×C163—81no—
240C2×C163C2×#12011yescharacteristic
241C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
242C2×C163#57511yescharacteristic
243D163—22no—
244D163—22no—
245D163—42no—
246D163—81no—
247C163#300611yescharacteristic, complemented, Sylow
248#3006—1631nomaximal, Sylow
249C22×#41—1631no—
250C2×AGL(1,ℤ/8)—1631no—
251C2×AGL(1,ℤ/8)—1631no—
252C2×#121—1631no—
253#449—1631no—
254#449—1631no—
255#449—1631no—
256#449D16311yescharacteristic, complemented
257#448—1631no—
258#448—1631no—
259#447—1631no—
260#447—1631no—
261#417—1631no—
262#417—1631no—
263#407—1631no—
264C2×#41—3261no—
265C2×#41—3264no—
266C2×#41—3264no—
267C2×#41—3261no—
268C2×#41—1631no—
269C2×#41—1631no—
270C2×#41—1631no—
271C2×#41C2×D16311yescharacteristic, complemented
272C22×Q8—1631no—
273C22×D4—3261no—
274C22×D4—3261no—
275C22×D4—1631no—
276C22×D4—1631no—
277C23×C4—3261no—
278C23×C4—1631no—
279AGL(1,ℤ/8)—3264no—
280AGL(1,ℤ/8)—3264no—
281C2×SD16—1631no—
282C2×SD16—1631no—
283C2×SD16—1631no—
284C2×SD16C2×D16311yescharacteristic, complemented
285C2×D8—1631no—
286C2×D8—1631no—
287C2×D8—1631no—
288C2×D8C2×D16311yescharacteristic, complemented
289C2×M16—1631no—
290C2×M16—1631no—
291#121—3261no—
292#121—3261no—
293#121—1631no—
294#121—1631no—
295#121—1631no—
296#121C2×D16311yescharacteristic, complemented
297C2×C4⋊C4—1631no—
298C2×C22⋊C4—3261no—
299#103—1631no—
300#103—1631no—
301#103—1631no—
302#103C2×D16311yescharacteristic, complemented
303#102—1631no—
304#102—1631no—
305#102—1631no—
306#102C2×D16311yescharacteristic, complemented
307C22⋊C8—1631no—
308C22⋊C8—1631no—
309C22⋊C8—1631no—
310C22⋊C8C2×D16311yescharacteristic, complemented
311C24—3261no—
312C24—3261no—
313C24—1631no—
314#41—22no—
315#41—22no—
316#41—3268no—
317#41—3262no—
318#41—3262no—
319#41—3268no—
320#41—3262no—
321#41—3262no—
322#41—1634no—
323#41—1634no—
324C2×Q8—3261no—
325C2×Q8—3262no—
326C2×Q8—3262no—
327C2×Q8—1631no—
328C2×Q8C22×D16311yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—3261no—
332C2×D4—3261no—
333C2×D4—3268no—
334C2×D4—3262no—
335C2×D4—3261no—
336C2×D4—3261no—
337C2×D4—3262no—
338C2×D4—3261no—
339C2×D4—3261no—
340C2×D4—3262no—
341C2×D4—3262no—
342C2×D4—3261no—
343C2×D4—3261no—
344C2×D4—6524no—
345C2×D4—1631no—
346C2×D4—1631no—
347C2×D4C22×D16311yescharacteristic
348C2×D4C22×D16311yescharacteristic
349C22×C4—21no—
350C22×C4—3268no—
351C22×C4—3261no—
352C22×C4—3261no—
353C22×C4—3262no—
354C22×C4—3262no—
355C22×C4—3261no—
356C22×C4—3261no—
357C22×C4—3261no—
358C22×C4—1631no—
359C22×C4—1631no—
360C22×C4—1631no—
361C22×C4—1631no—
362C22×C4—1631no—
363C22×C4C22×D16311yescharacteristic
364SD16—22no—
365SD16—3262no—
366SD16—3262no—
367SD16—3262no—
368D8—22no—
369D8—3262no—
370D8—3262no—
371D8—3262no—
372M16—3262no—
373M16—3262no—
374C2×C8—1631no—
375C2×C8—1631no—
376C2×C8C22×D16311yescharacteristic
377C2×C8C22×D16311yescharacteristic
378C4⋊C4—3261no—
379C4⋊C4—1631no—
380C4⋊C4C22×D16311yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—3261no—
383C22⋊C4—3261no—
384C22⋊C4—3261no—
385C23—21no—
386C23—21no—
387C23—3264no—
388C23—3262no—
389C23—3262no—
390C23—3261no—
391C23—3261no—
392C23—6521no—
393C23—6522no—
394C23—6522no—
395C23—6522no—
396C23—6522no—
397C23—6522no—
398C23—1631no—
399C23—1631no—
400C23D4×D16311yescharacteristic
401Q8—21no—
402Q8—3264no—
403Q8—3261no—
404Q8—1632no—
405Q8D4×D16312yescomplemented
406D4—24no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—42no—
411D4—3264no—
412D4—3264no—
413D4—3264no—
414D4—3264no—
415D4—3261no—
416D4—3261no—
417D4—3261no—
418D4—6522no—
419D4—6522no—
420D4—6522no—
421D4—6522no—
422D4—1632no—
423D4D4×D16312yescomplemented
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—3264no—
430C2×C4—3264no—
431C2×C4—3262no—
432C2×C4—3262no—
433C2×C4—3264no—
434C2×C4—3261no—
435C2×C4—3264no—
436C2×C4—3262no—
437C2×C4—3261no—
438C2×C4—3261no—
439C2×C4—3261no—
440C2×C4—6522no—
441C2×C4—1631no—
442C2×C4—1631no—
443C2×C4—1632no—
444C2×C4—1632no—
445C2×C4D4×D16311yescharacteristic
446C2×C4C23×D16311yescharacteristic
447C8—21no—
448C8—21no—
449C8—3261no—
450C8—3261no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—13044no—
461C22—3261no—
462C22—3261no—
463C22—3261no—
464C22—3261no—
465C22—3262no—
466C22—6524no—
467C22—6528no—
468C22—6521no—
469C22—6521no—
470C22—6521no—
471C22—6521no—
472C22—6521no—
473C22—1632no—
474C22—1632no—
475C22C2×D4×D16311yescharacteristic
476C4—21no—
477C4—22no—
478C4—41no—
479C4—3262no—
480C4—3261no—
481C4—6521no—
482C4—1632no—
483C4C2×D4×D16311yescharacteristic
484C4C2×D4×D16311yescharacteristic
485C2—41no—
486C2—42no—
487C2—81no—
488C2—13041no—
489C2—3262no—
490C2—3262no—
491C2—6522no—
492C2#120×D16311yescharacteristic
493C2AGL(1,ℤ/8)×D16311yescharacteristic
494C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3006
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D163C2semidirect
2. C2×#19104762C2semidirect
3. C2×#19104759C2semidirect
4. #121×D163C2semidirect
5. #23781812C2semidirect
6. #23781810C2semidirect
7. #23781808C2semidirect
8. #23781805C2semidirect
9. #23781802C2semidirect
10. #23781800C2semidirect
11. #23781797C2semidirect
12. #23781665C2semidirect
13. #23781663C2semidirect
14. #23781627C2semidirect
15. #449×C163C2semidirect
18. C2×#14579699C22semidirect
23. C2×Q8×D163C22semidirect
26. C2×D4×D163C22semidirect
27. C2×D4×D163C22semidirect
32. C2×#14579688C22semidirect
33. C2×#14579687C22semidirect
34. C2×#14579686C22semidirect
35. C2×D1304C22semidirect
36. C2×#14579684C22semidirect
37. C2×#14579684C22semidirect
38. C2×#14579682C22semidirect
39. C2×#14579682C22semidirect
41. #19104709C22semidirect
45. C4⋊C4×D163C22semidirect
47. #19104660C22semidirect
48. #19104659C22semidirect
49. #19104658C22semidirect
50. #19104657C22semidirect
51. #19104656C22semidirect
52. #19104655C22semidirect
53. C22⋊(C163⋊C8)C22semidirect
54. (C2×D163)⋊C8C22semidirect
55. (C2×D163)⋊C8C22semidirect
56. C2×#41×C163C22semidirect
57. C2×SD16×C163C22semidirect
58. C2×D8×C163C22semidirect
59. #121×C163C22semidirect
60. #103×C163C22semidirect
61. #102×C163C22semidirect
62. C22⋊C8×C163C22semidirect
67. #14579700D4semidirect
73. #14579698D4semidirect
152. Dic326C2×D4semidirect
158. D652C2×D4semidirect
187. Q8×C163C2×D4semidirect
193. D4×C163C2×D4semidirect
227. C2×D163AGL(1,ℤ/8)semidirect
247. C163#3006semidirect
256. #449D163semidirect
271. C2×#41C2×D163semidirect
284. C2×SD16C2×D163semidirect
288. C2×D8C2×D163semidirect
296. #121C2×D163semidirect
302. #103C2×D163semidirect
306. #102C2×D163semidirect
310. C22⋊C8C2×D163semidirect
405. Q8D4×D163semidirect
423. D4D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1632
C2×#191047622
C2×#191047592
#121×D1632
#237818122
#237818102
#237818082
#237818052
#237818022
#237818002
#237817972
#237816652
#237816632
#237816272
#449×C1632
#3006163