Finite Group Databasebeta

← #35223751

Subgroups — #35223751

Order 20864.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D163C211yesmaximal, characteristic, complemented
2C2×#19104756C211yesmaximal, characteristic, complemented
3C2×#19104754C211yesmaximal, characteristic, complemented
4#23782175C211yesmaximal, characteristic, complemented
5#23781829C211yesmaximal, characteristic, complemented
6#23781827C211yesmaximal, characteristic, complemented
7#23781822C211yesmaximal, characteristic, complemented
8#23781817C211yesmaximal, characteristic, complemented
9#23781808C211yesmaximal, characteristic, complemented
10#23781802C211yesmaximal, characteristic, complemented
11#23781797C211yesmaximal, characteristic, complemented
12#23781660C211yesmaximal, characteristic, complemented
13#23781658C211yesmaximal, characteristic, complemented
14C22⋊C8×D163C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699—21no—
18C2×#14579699C2211yescharacteristic
19C2×#14579698—21no—
20C2×#14579698C2211yescharacteristic
21C2×#14579698C2211yescharacteristic, complemented
22#41×D163—24no—
23#41×D163—24no—
24C2×Q8×D163C2211yescharacteristic
25C2×D4×D163—21no—
26C2×D4×D163C2211yescharacteristic
27C22×C4×D163—21no—
28C22×C4×D163C2211yescharacteristic
29#19104756—24no—
30#19104754—24no—
31C2×Dic652C2211yescharacteristic, complemented
32C2×#14579689C2211yescharacteristic, complemented
33C2×#14579688C2211yescharacteristic, complemented
34C2×#14579687C2211yescharacteristic, complemented
35C2×#14579687C2211yescharacteristic, complemented
36C2×#14579684C2211yescharacteristic, complemented
37C2×C8×D163C2211yescharacteristic, complemented
38C2×C8×D163C2211yescharacteristic, complemented
39#19104717—21no—
40#19104715C2211yescharacteristic
41#19104714C2211yescharacteristic, complemented
42#19104706C2211yescharacteristic
43#19104703—21no—
44D4×Dic163—21no—
45(C22⋊D163)⋊C4—21no—
46#19104693C2211yescharacteristic, complemented
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×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×D163D411yescharacteristic
65#14579700—22no—
66#14579700D414yescomplemented
67#14579699—28no—
68#14579699—22no—
69#14579699—22no—
70#14579698—28no—
71#14579698—22no—
72#14579698—22no—
73#14579698—22no—
74#14579698D414yescomplemented
75C2×Dic326—21no—
76C2×Dic326C2311yescharacteristic
77C2×Dic326C2311yescharacteristic
78Q8×D163—22no—
79Q8×D163—22no—
80C2×D652—21no—
81C2×D652C2311yescharacteristic
82C2×C22⋊D163—21no—
83C2×C22⋊D163—21no—
84C2×C22⋊D163—21no—
85C2×C22⋊D163—21no—
86C2×C22⋊D163—21no—
87D4×D163—28no—
88D4×D163—22no—
89D4×D163—22no—
90C22×Dic163—21no—
91C22×Dic163—21no—
92C22×Dic163D411yescharacteristic
93C2×C4×D163—28no—
94C2×C4×D163—21no—
95C2×C4×D163—21no—
96C2×C4×D163—22no—
97C2×C4×D163—22no—
98C2×C4×D163—21no—
99C2×C4×D163D411yescharacteristic
100C2×C4×D163D411yescharacteristic
101C2×C4×D163C2311yescharacteristic
102C2×C4×D163C2311yescharacteristic
103Dic652—22no—
104#14579689—22no—
105#14579688—22no—
106#14579687—22no—
107#14579687—22no—
108#14579684—22no—
109C2×C163⋊C8C2311yescharacteristic
110C2×C163⋊C8C2311yescharacteristic
111C8×D163—22no—
112C8×D163—22no—
113C4⋊Dic163—21no—
114C4⋊Dic163C2311yescharacteristic
115Dic163⋊C4—21no—
116C22⋊Dic163—21no—
117C22⋊Dic163—21no—
118(C2×D163)⋊C4—21no—
119(C2×D163)⋊C4—21no—
120C4×Dic163—21no—
121#41×C163—22no—
122#41×C163—22no—
123C2×Q8×C163C2311yescharacteristic
124C2×D4×C163—21no—
125C2×D4×C163—21no—
126C2×D4×C163C2311yescharacteristic
127C2×D4×C163C2311yescharacteristic
128C22×C4×C163—21no—
129C22×C4×C163C2311yescharacteristic
130SD16×C163—22no—
131D8×C163—22no—
132C2×C8×C163C2311yescharacteristic
133C2×C8×C163C2311yescharacteristic
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—
149Dic326C2×D412yes—
150D652—24no—
151D652—21no—
152D652C2×D412yes—
153C22⋊D163—24no—
154C22⋊D163—24no—
155C22⋊D163—24no—
156C22⋊D163—42no—
157C22⋊D163—42no—
158C2×Dic163—21no—
159C2×Dic163—24no—
160C2×Dic163—22no—
161C2×Dic163—21no—
162C2×Dic163—21no—
163C2×Dic163—42no—
164C2×Dic163C2×D411yescharacteristic
165C2×Dic163C2×D411yescharacteristic
166C4×D163—24no—
167C4×D163—24no—
168C4×D163—22no—
169C4×D163—21no—
170C4×D163—21no—
171C4×D163—22no—
172C4×D163—24no—
173C4×D163C2×D412yes—
174C4×D163C2×D412yes—
175C163⋊C8—21no—
176C163⋊C8—21no—
177C23×C163—21no—
178C23×C163—21no—
179C23×C163C2×D411yescharacteristic
180Q8×C163—21no—
181Q8×C163C2×D412yes—
182D4×C163—24no—
183D4×C163—21no—
184D4×C163—21no—
185D4×C163—21no—
186D4×C163—42no—
187D4×C163C2×D412yes—
188C2×C4×C163—24no—
189C2×C4×C163—21no—
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163—21no—
193C2×C4×C163C2×D411yescharacteristic
194C2×C4×C163C2411yescharacteristic
195C8×C163—21no—
196C8×C163—21no—
197C22×C163—21no—
198C22×C163—22no—
199C22×C163—21no—
200C22×C163—21no—
201C22×C163—41no—
202C22×C163—41no—
203C22×C163—41no—
204C22×C163—41no—
205C22×C163—41no—
206C22×C163C22×D411yescharacteristic
207C2×D163—21no—
208C2×D163—21no—
209C2×D163—21no—
210C2×D163—21no—
211C2×D163—22no—
212C2×D163—44no—
213C2×D163—48no—
214C2×D163—41no—
215C2×D163—41no—
216C2×D163#12012yes—
217C2×D163#13512yescomplemented
218C4×C163—21no—
219C4×C163—22no—
220C4×C163—41no—
221C4×C163C22×D411yescharacteristic
222C4×C163C22×D411yescharacteristic
223Dic163—22no—
224Dic163—21no—
225Dic163—41no—
226Dic163—41no—
227Dic163#12012yes—
228C2×C163—41no—
229C2×C163—42no—
230C2×C163—81no—
231C2×C163C2×#12011yescharacteristic
232C2×C163C2×#13511yescharacteristic
233C2×C163#57511yescharacteristic
234D163—22no—
235D163—22no—
236D163—42no—
237C163#300211yescharacteristic, complemented, Sylow
238#3002—1631nomaximal, Sylow
239C22×#41—1631no—
240C2×#135—1631no—
241C2×#135—1631no—
242#525—1631no—
243#452—1631no—
244#452—1631no—
245#451—1631no—
246#450—1631no—
247#449—1631no—
248#449D16311yescharacteristic, complemented
249#448—1631no—
250#447—1631no—
251#416—1631no—
252#416—1631no—
253C2×C22⋊C8—1631no—
254C2×#41—3261no—
255C2×#41—3264no—
256C2×#41—3264no—
257C2×#41—3261no—
258C2×#41—1631no—
259C2×#41—1631no—
260C2×#41—1631no—
261C2×#41—1631no—
262C2×#41C2×D16311yescharacteristic
263C22×Q8—1631no—
264C22×D4—3261no—
265C22×D4—1631no—
266C23×C4—3261no—
267C23×C4—1631no—
268#135—3264no—
269#135—3264no—
270C2×Q16—1631no—
271C2×Q16—1631no—
272C2×SD16—1631no—
273C2×SD16—1631no—
274C2×SD16—1631no—
275C2×SD16C2×D16311yescharacteristic, complemented
276C2×D8—1631no—
277C2×D8C2×D16311yescharacteristic, complemented
278C22×C8—1631no—
279C22×C8—1631no—
280#123—3261no—
281C22⋊Q8—1631no—
282C22⋊Q8—1631no—
283#121—1631no—
284#121C2×D16311yescharacteristic, complemented
285#120—3261no—
286C4×D4—3261no—
287C4×D4—3261no—
288#117—1631no—
289#103—1631no—
290#103—1631no—
291#103—1631no—
292#103C2×D16311yescharacteristic, complemented
293#102—1631no—
294#102—1631no—
295#102—1631no—
296#102C2×D16311yescharacteristic, complemented
297C22⋊C8—1631no—
298C22⋊C8—1631no—
299C22⋊C8—1631no—
300C22⋊C8C2×D16311yescharacteristic, complemented
301C24—3261no—
302C24—1631no—
303#41—22no—
304#41—22no—
305#41—3268no—
306#41—3262no—
307#41—3262no—
308#41—3268no—
309#41—3262no—
310#41—3262no—
311#41—3262no—
312#41—3262no—
313#41—1634no—
314#41—1634no—
315C2×Q8—3261no—
316C2×Q8—3262no—
317C2×Q8—3262no—
318C2×Q8—1631no—
319C2×Q8—1631no—
320C2×Q8C22×D16311yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—3261no—
324C2×D4—3268no—
325C2×D4—3262no—
326C2×D4—3261no—
327C2×D4—3261no—
328C2×D4—3262no—
329C2×D4—3261no—
330C2×D4—3261no—
331C2×D4—3261no—
332C2×D4—1631no—
333C2×D4C22×D16311yescharacteristic
334C2×D4C22×D16311yescharacteristic
335C22×C4—21no—
336C22×C4—3268no—
337C22×C4—3261no—
338C22×C4—3261no—
339C22×C4—3262no—
340C22×C4—3262no—
341C22×C4—3261no—
342C22×C4—3261no—
343C22×C4—3261no—
344C22×C4—1631no—
345C22×C4—1631no—
346C22×C4—1631no—
347C22×C4—1631no—
348C22×C4—1631no—
349C22×C4C22×D16311yescharacteristic
350Q16—3262no—
351Q16—3262no—
352SD16—22no—
353SD16—3262no—
354SD16—3262no—
355SD16—3262no—
356D8—22no—
357D8—3262no—
358C2×C8—3262no—
359C2×C8—3262no—
360C2×C8—1631no—
361C2×C8—1631no—
362C2×C8C22×D16311yescharacteristic
363C2×C8C22×D16311yescharacteristic
364C4⋊C4—3261no—
365C4⋊C4—3261no—
366C4⋊C4—1631no—
367C4⋊C4C22×D16311yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—3261no—
370C22⋊C4—3261no—
371C22⋊C4—3261no—
372C22⋊C4—3261no—
373C42—3261no—
374C23—21no—
375C23—21no—
376C23—3264no—
377C23—3262no—
378C23—3262no—
379C23—3261no—
380C23—6521no—
381C23—6522no—
382C23—6522no—
383C23—1631no—
384C23—1631no—
385C23D4×D16311yescharacteristic
386Q8—21no—
387Q8—3264no—
388Q8—3261no—
389Q8—3261no—
390Q8—3261no—
391Q8—1632no—
392Q8D4×D16312yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—3264no—
399D4—3264no—
400D4—3264no—
401D4—3264no—
402D4—3261no—
403D4—6522no—
404D4—6522no—
405D4—1632no—
406D4D4×D16312yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—3264no—
413C2×C4—3264no—
414C2×C4—3262no—
415C2×C4—3262no—
416C2×C4—3264no—
417C2×C4—3261no—
418C2×C4—3264no—
419C2×C4—3262no—
420C2×C4—3261no—
421C2×C4—3261no—
422C2×C4—3261no—
423C2×C4—3261no—
424C2×C4—6522no—
425C2×C4—1631no—
426C2×C4—1631no—
427C2×C4—1632no—
428C2×C4—1632no—
429C2×C4D4×D16311yescharacteristic
430C2×C4C23×D16311yescharacteristic
431C8—21no—
432C8—21no—
433C8—3261no—
434C8—3261no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—3261no—
445C22—3261no—
446C22—3261no—
447C22—3261no—
448C22—3262no—
449C22—6524no—
450C22—6528no—
451C22—6521no—
452C22—6521no—
453C22—1632no—
454C22—1632no—
455C22C2×D4×D16311yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—3262no—
460C4—3261no—
461C4—6521no—
462C4—6521no—
463C4—1632no—
464C4C2×D4×D16311yescharacteristic
465C4C2×D4×D16311yescharacteristic
466C2—41no—
467C2—42no—
468C2—81no—
469C2—3262no—
470C2—3262no—
471C2—6522no—
472C2#120×D16311yescharacteristic
473C2#135×D16311yescharacteristic
474C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3002
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×#19104756C2semidirect
3. C2×#19104754C2semidirect
4. #23782175C2semidirect
5. #23781829C2semidirect
6. #23781827C2semidirect
7. #23781822C2semidirect
8. #23781817C2semidirect
9. #23781808C2semidirect
10. #23781802C2semidirect
11. #23781797C2semidirect
12. #23781660C2semidirect
13. #23781658C2semidirect
14. C22⋊C8×D163C2semidirect
15. #449×C163C2semidirect
21. C2×#14579698C22semidirect
31. C2×Dic652C22semidirect
32. C2×#14579689C22semidirect
33. C2×#14579688C22semidirect
34. C2×#14579687C22semidirect
35. C2×#14579687C22semidirect
36. C2×#14579684C22semidirect
37. C2×C8×D163C22semidirect
38. C2×C8×D163C22semidirect
41. #19104714C22semidirect
46. #19104693C22semidirect
47. #19104660C22semidirect
48. #19104659C22semidirect
49. #19104658C22semidirect
50. #19104657C22semidirect
51. #19104656C22semidirect
52. #19104655C22semidirect
53. C22⋊(C163⋊C8)C22semidirect
57. C2×SD16×C163C22semidirect
58. C2×D8×C163C22semidirect
59. #121×C163C22semidirect
60. #103×C163C22semidirect
61. #102×C163C22semidirect
62. C22⋊C8×C163C22semidirect
66. #14579700D4semidirect
74. #14579698D4semidirect
217. C2×D163#135semidirect
237. C163#3002semidirect
248. #449D163semidirect
275. C2×SD16C2×D163semidirect
277. C2×D8C2×D163semidirect
284. #121C2×D163semidirect
292. #103C2×D163semidirect
296. #102C2×D163semidirect
300. 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×#191047562
C2×#191047542
#237821752
#237818292
#237818272
#237818222
#237818172
#237818082
#237818022
#237817972
#237816602
#237816582
C22⋊C8×D1632
#449×C1632
#3002163