Finite Group Databasebeta

← #35223793

Subgroups — #35223793

Order 20864.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D163C211yesmaximal, characteristic
2C2×#19104767C211yesmaximal, characteristic, complemented
3C2×#19104764C211yesmaximal, characteristic, complemented
4C22⋊Q8×D163C211yesmaximal, characteristic, complemented
5#23781829C211yesmaximal, characteristic, complemented
6#23781827C211yesmaximal, characteristic, complemented
7#23781825C211yesmaximal, characteristic, complemented
8#23781822C211yesmaximal, characteristic, complemented
9#23781819C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781814C211yesmaximal, characteristic, complemented
12#23781665C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14#23781627C211yesmaximal, characteristic, complemented
15#452×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699—21no—
18C2×#14579699C2211yescharacteristic
19C2×#14579698—21no—
20C2×#14579698C2211yescharacteristic
21#41×D163—24no—
22#41×D163—24no—
23C2×Q8×D163C2211yescharacteristic
24C2×Q8×D163C2211yescharacteristic, complemented
25C2×D4×D163—21no—
26C2×D4×D163C2211yescharacteristic
27C22×C4×D163—21no—
28C22×C4×D163C2211yescharacteristic
29#19104767—24no—
30#19104764—24no—
31C2×Dic652C2211yescharacteristic, complemented
32C2×#14579689C2211yescharacteristic, complemented
33C2×#14579689C2211yescharacteristic, complemented
34C2×#14579688C2211yescharacteristic, complemented
35C2×#14579687C2211yescharacteristic, complemented
36C2×#14579686C2211yescharacteristic, complemented
37C2×#14579682C2211yescharacteristic, complemented
38C2×#14579682C2211yescharacteristic, complemented
39#19104715C2211yescharacteristic
40#19104714C2211yescharacteristic, complemented
41#19104713—21no—
42#19104712—21no—
43#19104711C2211yescharacteristic
44C4⋊C4×D163—21no—
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
55(C2×D163)⋊C8C2211yescharacteristic
56C2×#41×C163C2211yescharacteristic
57C2×Q16×C163C2211yescharacteristic, complemented
58C2×SD16×C163C2211yescharacteristic, complemented
59C22⋊Q8×C163C2211yescharacteristic, complemented
60#103×C163C2211yescharacteristic, complemented
61#102×C163C2211yescharacteristic, complemented
62C22⋊C8×C163C2211yescharacteristic, complemented
63C23×D163—21no—
64C23×D163D411yescharacteristic
65#14579700—22no—
66#14579700D414yes—
67#14579699—28no—
68#14579699—22no—
69#14579699—22no—
70#14579698—28no—
71#14579698—22no—
72#14579698D414yes—
73C2×Dic326—21no—
74C2×Dic326—21no—
75C2×Dic326C2311yescharacteristic
76C2×Dic326C2311yescharacteristic
77Q8×D163—22no—
78Q8×D163—22no—
79Q8×D163—22no—
80Q8×D163—22no—
81C2×D652—21no—
82C2×D652C2311yescharacteristic
83C2×C22⋊D163—21no—
84C2×C22⋊D163—21no—
85C2×C22⋊D163—21no—
86D4×D163—28no—
87D4×D163—22no—
88D4×D163—22no—
89C22×Dic163—21no—
90C22×Dic163D411yescharacteristic
91C2×C4×D163—28no—
92C2×C4×D163—21no—
93C2×C4×D163—21no—
94C2×C4×D163—22no—
95C2×C4×D163—22no—
96C2×C4×D163—21no—
97C2×C4×D163—21no—
98C2×C4×D163—21no—
99C2×C4×D163D411yescharacteristic
100C2×C4×D163D411yescharacteristic
101C2×C4×D163C2311yescharacteristic
102C2×C4×D163C2311yescharacteristic
103Dic652—22no—
104#14579689—22no—
105#14579689—22no—
106#14579688—22no—
107#14579687—22no—
108#14579686—22no—
109#14579682—22no—
110#14579682—22no—
111C2×C163⋊C8C2311yescharacteristic
112C2×C163⋊C8C2311yescharacteristic
113C4⋊Dic163—21no—
114C4⋊Dic163C2311yescharacteristic
115Dic163⋊C4—21no—
116Dic163⋊C4—21no—
117Dic163⋊C4—21no—
118C22⋊Dic163—21no—
119(C2×D163)⋊C4—21no—
120(C2×D163)⋊C4—21no—
121#41×C163—22no—
122#41×C163—22no—
123C2×Q8×C163C2311yescharacteristic
124C2×Q8×C163C2311yescharacteristic
125C2×D4×C163—21no—
126C2×D4×C163C2311yescharacteristic
127C22×C4×C163—21no—
128C22×C4×C163C2311yescharacteristic
129Q16×C163—22no—
130SD16×C163—22no—
131C2×C8×C163C2311yescharacteristic
132C2×C8×C163C2311yescharacteristic
133C4⋊C4×C163—21no—
134C4⋊C4×C163C2311yescharacteristic
135C22⋊C4×C163—21no—
136C22×D163—24no—
137C22×D163—21no—
138C22×D163—22no—
139C22×D163—22no—
140C22×D163—41no—
141C22×D163—42no—
142C22×D163—42no—
143C22×D163C2×D411yescharacteristic
144C22×D163C2×D411yescharacteristic
145Dic326—24no—
146Dic326—21no—
147Dic326—21no—
148Dic326—21no—
149Dic326—42no—
150Dic326C2×D412yes—
151D652—24no—
152D652—21no—
153D652C2×D412yes—
154C22⋊D163—24no—
155C22⋊D163—24no—
156C22⋊D163—24no—
157C2×Dic163—21no—
158C2×Dic163—24no—
159C2×Dic163—22no—
160C2×Dic163—21no—
161C2×Dic163—21no—
162C2×Dic163C2×D411yescharacteristic
163C2×Dic163C2×D411yescharacteristic
164C4×D163—24no—
165C4×D163—24no—
166C4×D163—22no—
167C4×D163—21no—
168C4×D163—21no—
169C4×D163—22no—
170C4×D163—24no—
171C4×D163—42no—
172C4×D163—42no—
173C4×D163C2×D412yes—
174C4×D163C2×D412yes—
175C163⋊C8—21no—
176C163⋊C8—21no—
177C23×C163—21no—
178C23×C163C2×D411yescharacteristic
179Q8×C163—21no—
180Q8×C163—21no—
181Q8×C163—21no—
182Q8×C163C2×D412yes—
183D4×C163—24no—
184D4×C163—21no—
185D4×C163C2×D412yes—
186C2×C4×C163—24no—
187C2×C4×C163—21no—
188C2×C4×C163—21no—
189C2×C4×C163—21no—
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163C2×D411yescharacteristic
193C2×C4×C163C2411yescharacteristic
194C8×C163—21no—
195C8×C163—21no—
196C22×C163—21no—
197C22×C163—22no—
198C22×C163—21no—
199C22×C163—21no—
200C22×C163—41no—
201C22×C163—41no—
202C22×C163C22×D411yescharacteristic
203C2×D163—21no—
204C2×D163—21no—
205C2×D163—21no—
206C2×D163—21no—
207C2×D163—22no—
208C2×D163—44no—
209C2×D163—48no—
210C2×D163—41no—
211C2×D163—41no—
212C2×D163#12012yes—
213C2×D163#13712yescomplemented
214C4×C163—21no—
215C4×C163—22no—
216C4×C163—41no—
217C4×C163—41no—
218C4×C163C22×D411yescharacteristic
219C4×C163C22×D411yescharacteristic
220Dic163—22no—
221Dic163—21no—
222Dic163—41no—
223Dic163—41no—
224Dic163#12012yes—
225C2×C163—41no—
226C2×C163—42no—
227C2×C163C2×#12011yescharacteristic
228C2×C163C2×#13711yescharacteristic
229C2×C163#57511yescharacteristic
230D163—22no—
231D163—22no—
232D163—42no—
233C163#300711yescharacteristic, complemented, Sylow
234#3007—1631nomaximal, Sylow
235C22×#41—1631no—
236C2×#137—1631no—
237C2×#137—1631no—
238C2×C22⋊Q8—1631no—
239#452—1631no—
240#452—1631no—
241#452—1631no—
242#452D16311yescharacteristic, complemented
243#451—1631no—
244#451—1631no—
245#450—1631no—
246#450—1631no—
247#417—1631no—
248#417—1631no—
249#407—1631no—
250C2×#41—3261no—
251C2×#41—3264no—
252C2×#41—3264no—
253C2×#41—3261no—
254C2×#41—1631no—
255C2×#41—1631no—
256C2×#41—1631no—
257C2×#41C2×D16311yescharacteristic
258C22×Q8—1631no—
259C22×Q8—1631no—
260C22×D4—3261no—
261C22×D4—1631no—
262C23×C4—3261no—
263C23×C4—1631no—
264#137—3264no—
265#137—3264no—
266C2×Q16—1631no—
267C2×Q16—1631no—
268C2×Q16—1631no—
269C2×Q16C2×D16311yescharacteristic, complemented
270C2×SD16—1631no—
271C2×SD16—1631no—
272C2×SD16—1631no—
273C2×SD16C2×D16311yescharacteristic, complemented
274C2×M16—1631no—
275C2×M16—1631no—
276C22⋊Q8—3261no—
277C22⋊Q8—3261no—
278C22⋊Q8—1631no—
279C22⋊Q8—1631no—
280C22⋊Q8—1631no—
281C22⋊Q8C2×D16311yescharacteristic, complemented
282C2×C4⋊C4—3261no—
283C2×C4⋊C4—1631no—
284C2×C22⋊C4—3261no—
285#103—1631no—
286#103—1631no—
287#103—1631no—
288#103C2×D16311yescharacteristic, complemented
289#102—1631no—
290#102—1631no—
291#102—1631no—
292#102C2×D16311yescharacteristic, complemented
293C22⋊C8—1631no—
294C22⋊C8—1631no—
295C22⋊C8—1631no—
296C22⋊C8C2×D16311yescharacteristic, complemented
297C24—3261no—
298C24—1631no—
299#41—22no—
300#41—22no—
301#41—3268no—
302#41—3262no—
303#41—3262no—
304#41—3268no—
305#41—3262no—
306#41—3262no—
307#41—1634no—
308#41—1634no—
309C2×Q8—3261no—
310C2×Q8—3261no—
311C2×Q8—3262no—
312C2×Q8—3262no—
313C2×Q8—3262no—
314C2×Q8—3262no—
315C2×Q8—1631no—
316C2×Q8—1631no—
317C2×Q8C22×D16311yescharacteristic
318C2×Q8C22×D16311yescharacteristic
319C2×D4—21no—
320C2×D4—3261no—
321C2×D4—3268no—
322C2×D4—3262no—
323C2×D4—3261no—
324C2×D4—3261no—
325C2×D4—3262no—
326C2×D4—3261no—
327C2×D4—1631no—
328C2×D4C22×D16311yescharacteristic
329C22×C4—21no—
330C22×C4—3268no—
331C22×C4—3261no—
332C22×C4—3261no—
333C22×C4—3262no—
334C22×C4—3262no—
335C22×C4—3261no—
336C22×C4—3261no—
337C22×C4—3261no—
338C22×C4—3261no—
339C22×C4—1631no—
340C22×C4—1631no—
341C22×C4—1631no—
342C22×C4—1631no—
343C22×C4—1631no—
344C22×C4C22×D16311yescharacteristic
345Q16—22no—
346Q16—3262no—
347Q16—3262no—
348Q16—3262no—
349SD16—22no—
350SD16—3262no—
351SD16—3262no—
352SD16—3262no—
353M16—3262no—
354M16—3262no—
355C2×C8—1631no—
356C2×C8—1631no—
357C2×C8C22×D16311yescharacteristic
358C2×C8C22×D16311yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—3261no—
361C4⋊C4—3261no—
362C4⋊C4—3261no—
363C4⋊C4—3261no—
364C4⋊C4—1631no—
365C4⋊C4C22×D16311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—3261no—
368C22⋊C4—3261no—
369C22⋊C4—3261no—
370C23—21no—
371C23—3264no—
372C23—3262no—
373C23—3262no—
374C23—3261no—
375C23—6521no—
376C23—6522no—
377C23—6522no—
378C23—1631no—
379C23—1631no—
380C23D4×D16311yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—3264no—
385Q8—3261no—
386Q8—3261no—
387Q8—3261no—
388Q8—6522no—
389Q8—1632no—
390Q8D4×D16312yes—
391D4—24no—
392D4—21no—
393D4—3264no—
394D4—3264no—
395D4—3264no—
396D4—3264no—
397D4—3261no—
398D4—1632no—
399D4D4×D16312yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—3264no—
407C2×C4—3264no—
408C2×C4—3262no—
409C2×C4—3262no—
410C2×C4—3264no—
411C2×C4—3261no—
412C2×C4—3264no—
413C2×C4—3262no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—3261no—
417C2×C4—3261no—
418C2×C4—6522no—
419C2×C4—6522no—
420C2×C4—1631no—
421C2×C4—1631no—
422C2×C4—1632no—
423C2×C4—1632no—
424C2×C4D4×D16311yescharacteristic
425C2×C4C23×D16311yescharacteristic
426C8—21no—
427C8—21no—
428C8—3261no—
429C8—3261no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—3261no—
437C22—3261no—
438C22—3261no—
439C22—3261no—
440C22—3262no—
441C22—6524no—
442C22—6528no—
443C22—6521no—
444C22—6521no—
445C22—1632no—
446C22—1632no—
447C22C2×D4×D16311yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—3262no—
453C4—3261no—
454C4—6521no—
455C4—6521no—
456C4—1632no—
457C4C2×D4×D16311yescharacteristic
458C4C2×D4×D16311yescharacteristic
459C2—41no—
460C2—42no—
461C2—3262no—
462C2—3262no—
463C2—6522no—
464C2#120×D16311yescharacteristic
465C2#137×D16311yescharacteristic
466C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#452×C163C2
socleC22×C163C22×D4
prime radicalC2×#41×D163C2
hypercenter#452D163
hypocenterC163#3007
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#19104767C2semidirect
3. C2×#19104764C2semidirect
4. C22⋊Q8×D163C2semidirect
5. #23781829C2semidirect
6. #23781827C2semidirect
7. #23781825C2semidirect
8. #23781822C2semidirect
9. #23781819C2semidirect
10. #23781817C2semidirect
11. #23781814C2semidirect
12. #23781665C2semidirect
13. #23781663C2semidirect
14. #23781627C2semidirect
15. #452×C163C2semidirect
24. C2×Q8×D163C22semidirect
31. C2×Dic652C22semidirect
32. C2×#14579689C22semidirect
33. C2×#14579689C22semidirect
34. C2×#14579688C22semidirect
35. C2×#14579687C22semidirect
36. C2×#14579686C22semidirect
37. C2×#14579682C22semidirect
38. C2×#14579682C22semidirect
40. #19104714C22semidirect
45. C4⋊C4×D163C22semidirect
47. #19104660C22semidirect
48. #19104659C22semidirect
49. #19104658C22semidirect
50. #19104657C22semidirect
51. #19104656C22semidirect
52. #19104655C22semidirect
53. C22⋊(C163⋊C8)C22semidirect
57. C2×Q16×C163C22semidirect
58. C2×SD16×C163C22semidirect
59. C22⋊Q8×C163C22semidirect
60. #103×C163C22semidirect
61. #102×C163C22semidirect
62. C22⋊C8×C163C22semidirect
213. C2×D163#137semidirect
233. C163#3007semidirect
242. #452D163semidirect
269. C2×Q16C2×D163semidirect
273. C2×SD16C2×D163semidirect
281. C22⋊Q8C2×D163semidirect
288. #103C2×D163semidirect
292. #102C2×D163semidirect
296. C22⋊C8C2×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×#191047672
C2×#191047642
C22⋊Q8×D1632
#237818292
#237818272
#237818252
#237818222
#237818192
#237818172
#237818142
#237816652
#237816632
#237816272
#452×C1632
#3007163