Finite Group Databasebeta

← #35229320

Subgroups — #35229320

Order 20864.

1412 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782501C211yesmaximal, characteristic, complemented
2C2×#19104782C211yesmaximal, characteristic, complemented
3#23782481C214yesmaximal, complemented
4#23782479C214yesmaximal, complemented
5#23782475C214yesmaximal, complemented
6#23782472C214yesmaximal, complemented
7#23782463C211yesmaximal, characteristic, complemented
8#23782461C214yesmaximal, complemented
9C2×#19104768C211yesmaximal, characteristic, complemented
10C2×#19104765C211yesmaximal, characteristic, complemented
11#137×D163C214yesmaximal, complemented
12#23782431C211yesmaximal, characteristic, complemented
13#575×C163C211yesmaximal, characteristic, complemented
14#19104783C2212yescomplemented
15#19104782—22no—
16#19104782—22no—
17#19104782—21no—
18#19104782C2214yescomplemented
19#19104782C2214yescomplemented
20#19104782C2212yescomplemented
21#19104781—21no—
22#19104781C2212yescomplemented
23#19104780—22no—
24#19104780C2212yescomplemented
25C2×#14579699—22no—
26C2×#14579699—21no—
27C2×#14579699C2211yescharacteristic
28C2×#14579699C2211yescharacteristic
29C2×#14579698—22no—
30C2×#14579698C2212yescomplemented
31#41×D163—22no—
32#41×D163C2214yes—
33C22×Dic326—21no—
34C22×Dic326C2211yescharacteristic, complemented
35C2×Q8×D163C2212yescomplemented
36#19104768C2214yescomplemented
37#19104768C2214yescomplemented
38#19104767C2218yescomplemented
39#19104766C2212yescomplemented
40#19104765C2214yescomplemented
41#19104765C2214yescomplemented
42#19104765C2212yescomplemented
43#19104764C2218yescomplemented
44#19104764C2218yescomplemented
45#19104761C2212yescomplemented
46#19104760C2212yescomplemented
47#19104759C2218yescomplemented
48#19104756C2218yescomplemented
49#19104755C2214yescomplemented
50#19104754C2218yescomplemented
51C2×Dic652C2212yescomplemented
52C2×#14579689C2212yescomplemented
53Q16×D163C2218yescomplemented
54C2×#14579688C2212yescomplemented
55C2×#14579687C2212yescomplemented
56SD16×D163C2218yescomplemented
57#19104743C2214yescomplemented
58#19104741C2214yescomplemented
59C2×#14579683C2211yescharacteristic, complemented
60M16×D163C2214yescomplemented
61C2×#41×C163C2211yescharacteristic
62C2×#41×C163C2211yescharacteristic, complemented
63#137×C163C2212yescomplemented
64#137×C163C2212yescomplemented
65AGL(1,ℤ/8)×C163C2212yescomplemented
66AGL(1,ℤ/8)×C163C2212yescomplemented
67#135×C163C2214yescomplemented
68C2×M16×C163C2211yescharacteristic, complemented
69#14579700—22no—
70#14579700C2314yes—
71#14579699—22no—
72#14579699—22no—
73#14579699—24no—
74#14579699—24no—
75#14579699—24no—
76#14579699—24no—
77#14579699—21no—
78#14579699—21no—
79#14579699D412yes—
80#14579699D412yes—
81#14579699C2312yes—
82#14579699C2312yes—
83#14579699C2312yes—
84#14579699C2312yes—
85#14579698—22no—
86#14579698—24no—
87#14579698—28no—
88#14579698—28no—
89#14579698—24no—
90#14579698—24no—
91#14579698C2314yes—
92#14579698C2318yes—
93C2×Dic326—24no—
94C2×Dic326—24no—
95C2×Dic326—22no—
96C2×Dic326—22no—
97C2×Dic326—21no—
98C2×Dic326—21no—
99C2×Dic326—21no—
100C2×Dic326D411yescharacteristic
101C2×Dic326C2312yes—
102C2×Dic326C2312yes—
103C2×Dic326C2312yes—
104C2×Dic326C2311yescharacteristic
105Q8×D163—22no—
106Q8×D163—24no—
107Q8×D163C2318yes—
108Q8×D163C2314yes—
109C2×D652—22no—
110C2×D652D411yescharacteristic
111C2×D652C2311yescharacteristic
112C2×C22⋊D163—22no—
113C2×C22⋊D163—24no—
114C2×C22⋊D163—21no—
115C2×C22⋊D163D412yes—
116D4×D163—24no—
117D4×D163—22no—
118D4×D163—24no—
119D4×D163C2314yes—
120C22×Dic163—22no—
121C22×Dic163—21no—
122C2×C4×D163—22no—
123C2×C4×D163—24no—
124C2×C4×D163—21no—
125C2×C4×D163C2312yes—
126Dic652C2318yes—
127#14579689C2314yes—
128#14579689C2318yes—
129#14579688C2318yes—
130#14579687C2318yes—
131#14579687C2314yes—
132#14579686C2314yes—
133#14579684C2314yes—
134#14579683C2312yes—
135#14579683C2312yes—
136#14579682C2318yes—
137C2×C163⋊C8C2312yes—
138C8×D163C2318yes—
139#41×C163—21no—
140#41×C163—21no—
141#41×C163—21no—
142#41×C163—21no—
143#41×C163C2312yes—
144#41×C163C2312yes—
145#41×C163C2312yes—
146#41×C163C2312yes—
147C2×Q8×C163C2311yescharacteristic
148C2×Q8×C163C2311yescharacteristic
149C2×D4×C163—21no—
150C2×D4×C163—21no—
151C2×D4×C163C2311yescharacteristic
152C2×D4×C163C2311yescharacteristic
153C22×C4×C163—21no—
154C22×C4×C163—21no—
155C22×C4×C163C2311yescharacteristic
156Q16×C163C2314yes—
157SD16×C163C2314yes—
158SD16×C163C2314yes—
159D8×C163C2314yes—
160M16×C163C2312yes—
161M16×C163C2312yes—
162C2×C8×C163C2312yes—
163C22×D163—21no—
164C22×D163—24no—
165C22×D163C2×D412yes—
166Dic326—22no—
167Dic326—22no—
168Dic326—22no—
169Dic326—21no—
170Dic326—28no—
171Dic326—24no—
172Dic326C2×D412yes—
173Dic326C2×D412yes—
174Dic326C2414yes—
175Dic326C2412yes—
176D652—24no—
177D652—21no—
178D652C2×D412yes—
179D652C2×D412yes—
180D652C2412yes—
181C22⋊D163—22no—
182C22⋊D163—22no—
183C22⋊D163—24no—
184C22⋊D163—28no—
185C22⋊D163—24no—
186C22⋊D163C2×D414yes—
187C22⋊D163C2×D414yes—
188C2×Dic163—22no—
189C2×Dic163—21no—
190C2×Dic163—22no—
191C2×Dic163—22no—
192C2×Dic163—24no—
193C2×Dic163—28no—
194C2×Dic163C2×D412yes—
195C4×D163—24no—
196C4×D163—28no—
197C4×D163—22no—
198C4×D163—22no—
199C4×D163—24no—
200C4×D163C2×D414yes—
201C4×D163C2414yes—
202C163⋊C8C2414yes—
203C23×C163—21no—
204C23×C163—21no—
205C23×C163C2×D411yescharacteristic
206Q8×C163—21no—
207Q8×C163—21no—
208Q8×C163C2412yes—
209Q8×C163C2412yes—
210D4×C163—22no—
211D4×C163—22no—
212D4×C163—22no—
213D4×C163—22no—
214D4×C163—21no—
215D4×C163—21no—
216D4×C163C2412yes—
217D4×C163C2412yes—
218C2×C4×C163—21no—
219C2×C4×C163—21no—
220C2×C4×C163—22no—
221C2×C4×C163—22no—
222C2×C4×C163—22no—
223C2×C4×C163—22no—
224C2×C4×C163C2×D411yescharacteristic
225C2×C4×C163C2×D411yescharacteristic
226C2×C4×C163C2×D411yescharacteristic
227C2×C4×C163C2411yescharacteristic
228C2×C4×C163C2411yescharacteristic
229C2×C4×C163C2411yescharacteristic
230C8×C163C2414yes—
231C22×C163—22no—
232C22×C163—22no—
233C22×C163—41no—
234C22×C163—42no—
235C22×C163—81no—
236C22×C163C22×D411yescharacteristic
237C22×C163C22×D411yescharacteristic
238C22×C163C22×D411yescharacteristic
239C2×D163—22no—
240C2×D163—42no—
241C2×D163—84no—
242C2×D163—81no—
243C2×D163C22×D414yes—
244C4×C163—22no—
245C4×C163—22no—
246C4×C163C22×D411yescharacteristic
247C4×C163C22×D411yescharacteristic
248C4×C163C22×D411yescharacteristic
249C4×C163C2511yescharacteristic
250Dic163—22no—
251Dic163—24no—
252Dic163C22×D414yes—
253C2×C163—21no—
254C2×C163—21no—
255C2×C163—21no—
256C2×C163—42no—
257C2×C163—42no—
258C2×C163C23×D411yescharacteristic
259D163—24no—
260D163—42no—
261C163#358611yescharacteristic, complemented, Sylow
262#3586—1631nomaximal, Sylow
263#585—1631no—
264C2×2−1+4—1631no—
265#578—1634no—
266#578—1634no—
267#577—1634no—
268#577—1634no—
269#575—1631no—
270#575—1634no—
271#575D16311yescharacteristic, complemented
272C2×#137—1631no—
273C2×#137—1634no—
274C2×#137—1631no—
275#568—1631no—
2762−1+4—3262no—
2772−1+4—3262no—
2782−1+4—3261no—
2792−1+4—1634no—
2802−1+4—1634no—
2812−1+4—1632no—
2822−1+4—1632no—
283H(5,𝔽2)—3262no—
284H(5,𝔽2)—3261no—
285H(5,𝔽2)—1632no—
286H(5,𝔽2)—1632no—
287C2×#41—3262no—
288C2×#41—3261no—
289C2×#41—3262no—
290C2×#41—3262no—
291C2×#41—1634no—
292C2×#41—1632no—
293C2×#41—1631no—
294C2×#41—1631no—
295C2×#41C2×D16311yescharacteristic
296C2×#41C2×D16311yescharacteristic, complemented
297C22×Q8—3261no—
298C22×Q8—1632no—
299C22×Q8—1631no—
300#137—1634no—
301#137—1634no—
302#137—1632no—
303#137—1632no—
304#137—1638no—
305#137—1638no—
306#137—1638no—
307#137—1634no—
308#137—1634no—
309#137C2×D16312yescomplemented
310#137C2×D16312yescomplemented
311AGL(1,ℤ/8)—1632no—
312AGL(1,ℤ/8)—1632no—
313AGL(1,ℤ/8)—1638no—
314AGL(1,ℤ/8)C2×D16312yescomplemented
315AGL(1,ℤ/8)C2×D16312yescomplemented
316#135—1634no—
317#135—1638no—
318#135—1638no—
319#135C2×D16314yescomplemented
320C2×Q16—1632no—
321C2×Q16—1638no—
322C2×Q16—1632no—
323C2×SD16—1632no—
324C2×SD16—1638no—
325C2×SD16—1632no—
326#131—1634no—
327#131—1634no—
328C2×M16—1631no—
329C2×M16—1634no—
330C2×M16C2×D16311yescharacteristic, complemented
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335#41—3262no—
336#41—3264no—
337#41—3262no—
338#41—3264no—
339#41—3264no—
340#41—3264no—
341#41—3262no—
342#41—3264no—
343#41—3268no—
344#41—3268no—
345#41—3262no—
346#41—3264no—
347#41—3264no—
348#41—3261no—
349#41—3261no—
350#41—1634no—
351#41—1638no—
352#41—1634no—
353#41—1632no—
354#41—1632no—
355#41—1632no—
356#41—1632no—
357#41—1632no—
358#41—1632no—
359#41C22×D16312yes—
360#41C22×D16312yes—
361#41C22×D16312yes—
362#41C22×D16312yes—
363C2×Q8—3264no—
364C2×Q8—3264no—
365C2×Q8—3262no—
366C2×Q8—3262no—
367C2×Q8—3262no—
368C2×Q8—3264no—
369C2×Q8—3261no—
370C2×Q8—3261no—
371C2×Q8—3261no—
372C2×Q8—1638no—
373C2×Q8—1634no—
374C2×Q8—1632no—
375C2×Q8—1632no—
376C2×Q8—1631no—
377C2×Q8—1632no—
378C2×Q8—1631no—
379C2×Q8C22×D16311yescharacteristic
380C2×Q8C22×D16311yescharacteristic
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—3262no—
384C2×D4—3264no—
385C2×D4—3264no—
386C2×D4—3262no—
387C2×D4—3262no—
388C2×D4—3261no—
389C2×D4—3264no—
390C2×D4—1634no—
391C2×D4—1632no—
392C2×D4—1631no—
393C2×D4—1631no—
394C2×D4C22×D16311yescharacteristic
395C2×D4C22×D16311yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—3262no—
399C22×C4—3264no—
400C22×C4—3261no—
401C22×C4—3262no—
402C22×C4—3261no—
403C22×C4—1632no—
404C22×C4C22×D16311yescharacteristic
405Q16—1638no—
406Q16—1634no—
407Q16—1638no—
408Q16C22×D16314yes—
409SD16—1634no—
410SD16—1638no—
411SD16—1634no—
412SD16—1638no—
413SD16C22×D16314yes—
414SD16C22×D16314yes—
415D8—1634no—
416D8C22×D16314yes—
417M16—1632no—
418M16—1632no—
419M16—1638no—
420M16C22×D16312yes—
421M16C22×D16312yes—
422C2×C8—1632no—
423C2×C8—1638no—
424C2×C8C22×D16312yes—
425C23—21no—
426C23—21no—
427C23—3264no—
428C23—3261no—
429C23—1632no—
430C23D4×D16311yescharacteristic
431Q8—21no—
432Q8—21no—
433Q8—3268no—
434Q8—3264no—
435Q8—3262no—
436Q8—3262no—
437Q8—3262no—
438Q8—3261no—
439Q8—1634no—
440Q8—1632no—
441Q8—1632no—
442Q8—1632no—
443Q8C23×D16312yes—
444Q8C23×D16312yes—
445D4—22no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—21no—
450D4—21no—
451D4—3264no—
452D4—3264no—
453D4—3268no—
454D4—3262no—
455D4—3262no—
456D4—3264no—
457D4—3261no—
458D4—1634no—
459D4—1634no—
460D4—1632no—
461D4—1632no—
462D4—1632no—
463D4C23×D16312yes—
464D4C23×D16312yes—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—3264no—
472C2×C4—3268no—
473C2×C4—3264no—
474C2×C4—3262no—
475C2×C4—3261no—
476C2×C4—3262no—
477C2×C4—3262no—
478C2×C4—3264no—
479C2×C4—3268no—
480C2×C4—3262no—
481C2×C4—3262no—
482C2×C4—1632no—
483C2×C4—1634no—
484C2×C4—1634no—
485C2×C4D4×D16311yescharacteristic
486C2×C4D4×D16311yescharacteristic
487C2×C4D4×D16311yescharacteristic
488C2×C4C23×D16311yescharacteristic
489C2×C4C23×D16311yescharacteristic
490C2×C4C23×D16311yescharacteristic
491C8—1634no—
492C8C23×D16314yes—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—42no—
497C22—81no—
498C22—13041no—
499C22—13044no—
500C22—3262no—
501C22—6522no—
502C22—1634no—
503C22C2×D4×D16311yescharacteristic
504C22C2×D4×D16311yescharacteristic
505C22C2×D4×D16311yescharacteristic
506C4—22no—
507C4—22no—
508C4—3262no—
509C4—3264no—
510C4—1634no—
511C4C2×D4×D16311yescharacteristic
512C4C2×D4×D16311yescharacteristic
513C4C2×D4×D16311yescharacteristic
514C4C24×D16311yescharacteristic
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—42no—
520C2—3264no—
521C2—6522no—
522C2C22×D4×D16311yescharacteristic
subgroupG/H
centerC2C22×D4×D163
commutator, cosocleC4×C163C25
FrattiniC4C24×D163
Fitting#575×C163C2
socleC2×C163C23×D4
hypercenter#575D163
hypocenterC163#3586
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #23782501C2semidirect
2. C2×#19104782C2semidirect
3. #23782481C2semidirect
4. #23782479C2semidirect
5. #23782475C2semidirect
6. #23782472C2semidirect
7. #23782463C2semidirect
8. #23782461C2semidirect
9. C2×#19104768C2semidirect
10. C2×#19104765C2semidirect
11. #137×D163C2semidirect
12. #23782431C2semidirect
13. #575×C163C2semidirect
14. #19104783C22semidirect
18. #19104782C22semidirect
19. #19104782C22semidirect
20. #19104782C22semidirect
22. #19104781C22semidirect
24. #19104780C22semidirect
30. C2×#14579698C22semidirect
34. C22×Dic326C22semidirect
35. C2×Q8×D163C22semidirect
36. #19104768C22semidirect
37. #19104768C22semidirect
38. #19104767C22semidirect
39. #19104766C22semidirect
40. #19104765C22semidirect
41. #19104765C22semidirect
42. #19104765C22semidirect
43. #19104764C22semidirect
44. #19104764C22semidirect
45. #19104761C22semidirect
46. #19104760C22semidirect
47. #19104759C22semidirect
48. #19104756C22semidirect
49. #19104755C22semidirect
50. #19104754C22semidirect
51. C2×Dic652C22semidirect
52. C2×#14579689C22semidirect
53. Q16×D163C22semidirect
54. C2×#14579688C22semidirect
55. C2×#14579687C22semidirect
56. SD16×D163C22semidirect
57. #19104743C22semidirect
58. #19104741C22semidirect
59. C2×#14579683C22semidirect
60. M16×D163C22semidirect
62. C2×#41×C163C22semidirect
63. #137×C163C22semidirect
64. #137×C163C22semidirect
65. AGL(1,ℤ/8)×C163C22semidirect
66. AGL(1,ℤ/8)×C163C22semidirect
67. #135×C163C22semidirect
68. C2×M16×C163C22semidirect
261. C163#3586semidirect
271. #575D163semidirect
296. C2×#41C2×D163semidirect
309. #137C2×D163semidirect
310. #137C2×D163semidirect
314. AGL(1,ℤ/8)C2×D163semidirect
315. AGL(1,ℤ/8)C2×D163semidirect
319. #135C2×D163semidirect
330. C2×M16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237825012
C2×#191047822
#237824812
#237824792
#237824752
#237824722
#237824632
#237824612
C2×#191047682
C2×#191047652
#137×D1632
#237824312
#575×C1632
#3586163