Finite Group Databasebeta

← #35229319

Subgroups — #35229319

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×#19104783C211yesmaximal, characteristic, complemented
3#23782481C214yesmaximal, complemented
4#23782478C214yesmaximal, complemented
5#23782474C214yesmaximal, complemented
6#23782471C214yesmaximal, complemented
7#23782465C211yesmaximal, characteristic, complemented
8#23782464C214yesmaximal, complemented
9#23782463C211yesmaximal, characteristic, complemented
10C2×#19104766C211yesmaximal, characteristic, complemented
11#137×D163C214yesmaximal, complemented
12#23782431C211yesmaximal, characteristic, complemented
13C2×#137×C163C211yesmaximal, characteristic, complemented
14#19104783—22no—
15#19104783—22no—
16#19104783C2214yescomplemented
17#19104783C2214yescomplemented
18#19104783C2212yescomplemented
19#19104782—21no—
20#19104782C2212yescomplemented
21#19104781—21no—
22#19104781C2212yescomplemented
23#19104780—22no—
24#19104780C2212yescomplemented
25C2×#14579700—21no—
26C2×#14579700C2212yescomplemented
27C2×#14579699—22no—
28C2×#14579699—22no—
29C2×#14579699C2211yescharacteristic
30C2×#14579699C2211yescharacteristic
31C2×#14579699C2211yescharacteristic, complemented
32#41×D163—22no—
33#41×D163C2214yes—
34C2×Q8×D163—21no—
35C2×Q8×D163C2212yescomplemented
36#19104768C2212yescomplemented
37#19104768C2212yescomplemented
38#19104767C2218yescomplemented
39#19104767C2218yescomplemented
40#19104766C2214yescomplemented
41#19104766C2214yescomplemented
42#19104766C2212yescomplemented
43#19104765C2212yescomplemented
44#19104764C2218yescomplemented
45#19104763C2212yescomplemented
46#19104763C2212yescomplemented
47#19104762C2218yescomplemented
48#19104761C2212yescomplemented
49#19104760C2212yescomplemented
50#19104758C2218yescomplemented
51#19104757C2214yescomplemented
52#19104756C2218yescomplemented
53#19104755C2214yescomplemented
54C2×#14579689C2212yescomplemented
55Q16×D163C2218yescomplemented
56C2×#14579686C2212yescomplemented
57SD16×D163C2218yescomplemented
58#19104743C2214yescomplemented
59#19104741C2214yescomplemented
60C2×#14579683C2211yescharacteristic, complemented
61M16×D163C2214yescomplemented
62C2×#41×C163C2211yescharacteristic
63C22×Q8×C163C2211yescharacteristic, complemented
64#137×C163C2214yescomplemented
65#137×C163C2214yescomplemented
66C2×Q16×C163C2212yescomplemented
67C2×SD16×C163C2212yescomplemented
68C2×M16×C163C2211yescharacteristic, complemented
69#14579700—22no—
70#14579700—24no—
71#14579700—24no—
72#14579700—24no—
73#14579700C2314yes—
74#14579700C2318yes—
75#14579699—22no—
76#14579699—22no—
77#14579699—24no—
78#14579699—28no—
79#14579699—28no—
80#14579699—24no—
81#14579699—21no—
82#14579699—21no—
83#14579699—21no—
84#14579699—21no—
85#14579699D412yes—
86#14579699D412yes—
87#14579699C2312yes—
88#14579699C2312yes—
89#14579699C2312yes—
90#14579699C2312yes—
91#14579699C2312yes—
92#14579699C2312yes—
93#14579698—22no—
94#14579698—24no—
95#14579698—24no—
96#14579698C2314yes—
97C2×Dic326—22no—
98C2×Dic326—22no—
99C2×Dic326D411yescharacteristic
100C2×Dic326C2311yescharacteristic
101C2×Dic326C2311yescharacteristic
102Q8×D163—24no—
103Q8×D163—24no—
104Q8×D163—22no—
105Q8×D163—24no—
106Q8×D163C2318yes—
107Q8×D163C2314yes—
108C2×D652—22no—
109C2×D652—22no—
110C2×D652D411yescharacteristic
111C2×D652C2311yescharacteristic
112C2×D652C2311yescharacteristic
113C2×C22⋊D163—24no—
114C2×C22⋊D163—21no—
115C2×C22⋊D163—21no—
116C2×C22⋊D163D412yes—
117D4×D163—24no—
118D4×D163—22no—
119D4×D163—24no—
120D4×D163C2314yes—
121C2×C4×D163—22no—
122C2×C4×D163—22no—
123C2×C4×D163—24no—
124C2×C4×D163—21no—
125C2×C4×D163—21no—
126C2×C4×D163C2312yes—
127Dic652C2314yes—
128#14579689C2314yes—
129#14579689C2318yes—
130#14579688C2314yes—
131#14579688C2314yes—
132#14579687C2314yes—
133#14579686C2318yes—
134#14579686C2314yes—
135D1304C2314yes—
136#14579684C2314yes—
137#14579683C2312yes—
138#14579683C2312yes—
139#14579682C2318yes—
140C2×C163⋊C8C2312yes—
141C8×D163C2318yes—
142#41×C163—21no—
143#41×C163—21no—
144#41×C163C2312yes—
145#41×C163C2312yes—
146C2×Q8×C163—21no—
147C2×Q8×C163—21no—
148C2×Q8×C163—21no—
149C2×Q8×C163C2312yes—
150C2×Q8×C163C2312yes—
151C2×Q8×C163C2311yescharacteristic
152C2×Q8×C163C2312yes—
153C2×D4×C163—21no—
154C2×D4×C163C2311yescharacteristic
155C22×C4×C163—21no—
156C22×C4×C163—21no—
157C22×C4×C163C2311yescharacteristic
158Q16×C163C2318yes—
159SD16×C163C2318yes—
160M16×C163C2312yes—
161M16×C163C2312yes—
162C2×C8×C163C2312yes—
163C22×D163—21no—
164C22×D163—21no—
165C22×D163—24no—
166C22×D163C2×D412yes—
167Dic326—21no—
168Dic326—21no—
169Dic326—28no—
170Dic326—24no—
171Dic326C2×D412yes—
172Dic326C2×D412yes—
173Dic326C2412yes—
174Dic326C2412yes—
175D652—24no—
176D652—28no—
177D652—21no—
178D652—21no—
179D652C2×D412yes—
180D652C2×D412yes—
181D652C2412yes—
182D652C2412yes—
183C22⋊D163—22no—
184C22⋊D163—22no—
185C22⋊D163—22no—
186C22⋊D163—22no—
187C22⋊D163—24no—
188C22⋊D163—24no—
189C22⋊D163C2×D414yes—
190C22⋊D163C2×D414yes—
191C2×Dic163—21no—
192C2×Dic163—21no—
193C2×Dic163—24no—
194C2×Dic163C2×D412yes—
195C4×D163—24no—
196C4×D163—28no—
197C4×D163—22no—
198C4×D163—22no—
199C4×D163—22no—
200C4×D163—22no—
201C4×D163—24no—
202C4×D163—28no—
203C4×D163C2×D414yes—
204C4×D163C2414yes—
205C163⋊C8C2414yes—
206C23×C163—21no—
207C23×C163C2×D411yescharacteristic
208Q8×C163—22no—
209Q8×C163—22no—
210Q8×C163—22no—
211Q8×C163—21no—
212Q8×C163C2414yes—
213Q8×C163C2412yes—
214D4×C163—22no—
215D4×C163—22no—
216D4×C163—21no—
217D4×C163C2412yes—
218C2×C4×C163—21no—
219C2×C4×C163—22no—
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—41no—
233C22×C163—81no—
234C22×C163C22×D411yescharacteristic
235C22×C163C22×D411yescharacteristic
236C22×C163C22×D411yescharacteristic
237C2×D163—22no—
238C2×D163—22no—
239C2×D163—42no—
240C2×D163—42no—
241C2×D163—84no—
242C2×D163—81no—
243C2×D163C22×D414yes—
244C4×C163—22no—
245C4×C163—24no—
246C4×C163C22×D411yescharacteristic
247C4×C163C22×D411yescharacteristic
248C4×C163C22×D411yescharacteristic
249C4×C163C2511yescharacteristic
250Dic163—22no—
251Dic163—22no—
252Dic163C22×D414yes—
253C2×C163—21no—
254C2×C163—21no—
255C2×C163—21no—
256C2×C163—42no—
257C2×C163C23×D411yescharacteristic
258D163—24no—
259D163—42no—
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#575—1631no—
272C2×#137—1631no—
273C2×#137—1634no—
274C2×#137D16311yescharacteristic, complemented
275#568—1631no—
2762−1+4—3262no—
2772−1+4—3262no—
2782−1+4—3261no—
2792−1+4—1632no—
2802−1+4—1634no—
2812−1+4—1634no—
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—3262no—
289C2×#41—3262no—
290C2×#41—3261no—
291C2×#41—1634no—
292C2×#41—1632no—
293C2×#41—1631no—
294C2×#41—1631no—
295C2×#41—1631no—
296C2×#41C2×D16311yescharacteristic
297C22×Q8—3261no—
298C22×Q8—1632no—
299C22×Q8C2×D16311yescharacteristic, complemented
300#137—1632no—
301#137—1634no—
302#137—1634no—
303#137—1632no—
304#137—1638no—
305#137—1638no—
306#137—1638no—
307#137—1632no—
308#137—1632no—
309#137C2×D16314yescomplemented
310#137C2×D16314yescomplemented
311AGL(1,ℤ/8)—1632no—
312AGL(1,ℤ/8)—1632no—
313AGL(1,ℤ/8)—1638no—
314AGL(1,ℤ/8)—1632no—
315AGL(1,ℤ/8)—1632no—
316#135—1634no—
317#135—1638no—
318#135—1638no—
319#135—1634no—
320C2×Q16—1632no—
321C2×Q16—1638no—
322C2×Q16C2×D16312yescomplemented
323C2×SD16—1632no—
324C2×SD16—1638no—
325C2×SD16C2×D16312yescomplemented
326#131—1634no—
327#131—1634no—
328C2×M16—1631no—
329C2×M16—1634no—
330C2×M16C2×D16311yescharacteristic, complemented
331#41—21no—
332#41—21no—
333#41—3262no—
334#41—3264no—
335#41—3262no—
336#41—3268no—
337#41—3268no—
338#41—3264no—
339#41—3262no—
340#41—3264no—
341#41—3262no—
342#41—3264no—
343#41—3264no—
344#41—3264no—
345#41—3264no—
346#41—3261no—
347#41—3261no—
348#41—3261no—
349#41—3261no—
350#41—1638no—
351#41—1634no—
352#41—1634no—
353#41—1632no—
354#41—1632no—
355#41—1632no—
356#41—1632no—
357#41—1632no—
358#41—1632no—
359#41—1632no—
360#41—1632no—
361#41C22×D16312yes—
362#41C22×D16312yes—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—3262no—
367C2×Q8—3262no—
368C2×Q8—3264no—
369C2×Q8—3264no—
370C2×Q8—3262no—
371C2×Q8—3264no—
372C2×Q8—1638no—
373C2×Q8—1634no—
374C2×Q8—1631no—
375C2×Q8—1631no—
376C2×Q8—1631no—
377C2×Q8C22×D16312yes—
378C2×Q8C22×D16312yes—
379C2×Q8C22×D16311yescharacteristic
380C2×Q8C22×D16312yes—
381C2×D4—21no—
382C2×D4—3262no—
383C2×D4—3262no—
384C2×D4—3264no—
385C2×D4—3264no—
386C2×D4—3262no—
387C2×D4—3261no—
388C2×D4—3261no—
389C2×D4—3264no—
390C2×D4—1634no—
391C2×D4—1632no—
392C2×D4—1631no—
393C2×D4—1631no—
394C2×D4—1631no—
395C2×D4C22×D16311yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—3262no—
399C22×C4—3262no—
400C22×C4—3264no—
401C22×C4—3261no—
402C22×C4—3261no—
403C22×C4—1632no—
404C22×C4C22×D16311yescharacteristic
405Q16—1634no—
406Q16—1638no—
407Q16—1634no—
408Q16C22×D16318yes—
409SD16—1638no—
410SD16—1634no—
411SD16—1634no—
412SD16—1634no—
413SD16—1634no—
414SD16C22×D16318yes—
415D8—1634no—
416D8—1634no—
417M16—1632no—
418M16—1632no—
419M16—1638no—
420M16C22×D16312yes—
421M16C22×D16312yes—
422C2×C8—1632no—
423C2×C8—1638no—
424C2×C8C22×D16312yes—
425C23—21no—
426C23—3264no—
427C23—3261no—
428C23—3261no—
429C23—1632no—
430C23D4×D16311yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—22no—
434Q8—21no—
435Q8—3268no—
436Q8—3264no—
437Q8—3261no—
438Q8—3261no—
439Q8—1632no—
440Q8—1632no—
441Q8—1632no—
442Q8—1632no—
443Q8C23×D16314yes—
444Q8C23×D16312yes—
445D4—22no—
446D4—22no—
447D4—21no—
448D4—3264no—
449D4—3268no—
450D4—3264no—
451D4—3262no—
452D4—3262no—
453D4—3262no—
454D4—3262no—
455D4—3264no—
456D4—3261no—
457D4—3261no—
458D4—1634no—
459D4—1634no—
460D4—1632no—
461D4—1632no—
462D4—1632no—
463D4—1632no—
464D4C23×D16312yes—
465C2×C4—22no—
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—3268no—
475C2×C4—3261no—
476C2×C4—3261no—
477C2×C4—3264no—
478C2×C4—3262no—
479C2×C4—3262no—
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—41no—
495C22—81no—
496C22—13041no—
497C22—13044no—
498C22—3262no—
499C22—3262no—
500C22—6522no—
501C22—6522no—
502C22—1634no—
503C22C2×D4×D16311yescharacteristic
504C22C2×D4×D16311yescharacteristic
505C22C2×D4×D16311yescharacteristic
506C4—22no—
507C4—24no—
508C4—3262no—
509C4—3262no—
510C4—1634no—
511C4C2×D4×D16311yescharacteristic
512C4C2×D4×D16311yescharacteristic
513C4C2×D4×D16311yescharacteristic
514C4C24×D16311yescharacteristic
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—3264no—
520C2—6522no—
521C2—6522no—
522C2C22×D4×D16311yescharacteristic
subgroupG/H
centerC2C22×D4×D163
commutator, cosocleC4×C163C25
FrattiniC4C24×D163
FittingC2×#137×C163C2
socleC2×C163C23×D4
hypercenterC2×#137D163
hypocenterC163#3586
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. #23782501C2semidirect
2. C2×#19104783C2semidirect
3. #23782481C2semidirect
4. #23782478C2semidirect
5. #23782474C2semidirect
6. #23782471C2semidirect
7. #23782465C2semidirect
8. #23782464C2semidirect
9. #23782463C2semidirect
10. C2×#19104766C2semidirect
11. #137×D163C2semidirect
12. #23782431C2semidirect
13. C2×#137×C163C2semidirect
16. #19104783C22semidirect
17. #19104783C22semidirect
18. #19104783C22semidirect
20. #19104782C22semidirect
22. #19104781C22semidirect
24. #19104780C22semidirect
26. C2×#14579700C22semidirect
31. C2×#14579699C22semidirect
35. C2×Q8×D163C22semidirect
36. #19104768C22semidirect
37. #19104768C22semidirect
38. #19104767C22semidirect
39. #19104767C22semidirect
40. #19104766C22semidirect
41. #19104766C22semidirect
42. #19104766C22semidirect
43. #19104765C22semidirect
44. #19104764C22semidirect
45. #19104763C22semidirect
46. #19104763C22semidirect
47. #19104762C22semidirect
48. #19104761C22semidirect
49. #19104760C22semidirect
50. #19104758C22semidirect
51. #19104757C22semidirect
52. #19104756C22semidirect
53. #19104755C22semidirect
54. C2×#14579689C22semidirect
55. Q16×D163C22semidirect
56. C2×#14579686C22semidirect
57. SD16×D163C22semidirect
58. #19104743C22semidirect
59. #19104741C22semidirect
60. C2×#14579683C22semidirect
61. M16×D163C22semidirect
63. C22×Q8×C163C22semidirect
64. #137×C163C22semidirect
65. #137×C163C22semidirect
66. C2×Q16×C163C22semidirect
67. C2×SD16×C163C22semidirect
68. C2×M16×C163C22semidirect
261. C163#3586semidirect
274. C2×#137D163semidirect
299. C22×Q8C2×D163semidirect
309. #137C2×D163semidirect
310. #137C2×D163semidirect
322. C2×Q16C2×D163semidirect
325. C2×SD16C2×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×#191047832
#237824812
#237824782
#237824742
#237824712
#237824652
#237824642
#237824632
C2×#191047662
#137×D1632
#237824312
C2×#137×C1632
#3586163