Finite Group Databasebeta

← #15845777

Subgroups — #15845777

Order 3328.

766 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#13125289C211yesmaximal, characteristic
2C2×#11718688C211yesmaximal, characteristic, complemented
3Q8×#6099C211yesmaximal, characteristic, complemented
4#13114318C212yesmaximal, complemented
5#13114310C212yesmaximal, complemented
6#13114194C212yesmaximal, complemented
7#13114186C212yesmaximal, complemented
8#13113589C211yesmaximal, characteristic, complemented
9#13113513C211yesmaximal, characteristic, complemented
10#13113503C211yesmaximal, characteristic, complemented
11C13×#3103C211yesmaximal, characteristic, complemented
12C13×#532C2211yescharacteristic
13C2×C4×Q8×C13C2211yescharacteristic
14C2×#88203—21no—
15C2×#88203—21no—
16C22×#6106—21no—
17C2×#88128C2211yescharacteristic
18C13×#481—21no—
19C13×#475C2211yescharacteristic, complemented
20C13×#475C2211yescharacteristic, complemented
21C13×#460C2211yescharacteristic, complemented
22C13×#460C2211yescharacteristic, complemented
23C13×#448—21no—
24C4×C13×SD16—21no—
25C4×C13×SD16—21no—
26C13×#431C2211yescharacteristic
27#11718746C2212yes—
28#11718734C2212yes—
29#11718719—22no—
30#11718688—21no—
31#11718688—21no—
32#11718688C2212yes—
33#11718688C2212yes—
34#11718685—22no—
35#11718685C2212yescomplemented
36Q8×#1055C2212yescomplemented
37C2×#88049—21no—
38#11718585—21no—
39#11718585C2211yescharacteristic
40C2×C4⋊#1055C2211yescharacteristic
41C4×#6099—21no—
42C4×#6099C2211yescharacteristic
43#11718139C2212yescomplemented
44#11718136C2212yescomplemented
45#11718077C2212yes—
46#11718077C2212yes—
47#11718072C2212yescomplemented
48#11718071C2212yescomplemented
49C2×C13×#41—21no—
50C22×Q8×C13D411yescharacteristic
51C22×D52—21no—
52C22×D52—21no—
53C22×D52—21no—
54C2×C13×SD16—21no—
55C2×C13×SD16—21no—
56C13×C4⋊Q8C2311yescharacteristic
57C13×#127C2311yescharacteristic
58C13×#124—21no—
59C13×#121—21no—
60C13×#121—21no—
61C4×Q8×C13—21no—
62C4×Q8×C13—21no—
63C4×Q8×C13D411yescharacteristic
64C4×Q8×C13D411yescharacteristic
65C4×Q8×C13C2311yescharacteristic
66C4×Q8×C13C2311yescharacteristic
67C4×D4×C13—21no—
68C4×D4×C13—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4D411yescharacteristic
71C2×C42×C13—21no—
72C2×C42×C13C2311yescharacteristic
73#88203—44no—
74#88203—44no—
75C2×#6106—22no—
76C2×#6106—22no—
77C2×#6106—22no—
78C2×#6106—22no—
79C2×#6106—44no—
80C2×#6104—22no—
81C2×#6104—22no—
82C2×#6099—21no—
83C2×#6099D411yescharacteristic
84C2×#6099D411yescharacteristic
85C22×#1055—21no—
86#88128—21no—
87#88128—21no—
88#88128—21no—
89#88128C2312yes—
90C13×#106—21no—
91C13×#106—21no—
92C13×C4⋊C8C2311yescharacteristic
93C13×C4⋊C8C2311yescharacteristic
94C13×#103—21no—
95C13×#103—21no—
96C13×#102—21no—
97C13×#102—21no—
98C13×C22⋊C8—21no—
99C4×C8×C13—21no—
100#88049—22no—
101#88049—22no—
102C4⋊#1055—21no—
103C4⋊#1055—21no—
104C4⋊#1055—22no—
105C4⋊#1055—22no—
106C4⋊#1055C2312yes—
107C4⋊#1055C2312yes—
108#88043—22no—
109#88043D412yes—
110C4×#1055—22no—
111C4×#1055C2312yes—
112#19170—131nomaximal, Sylow
113C23×D13—41no—
114C13×#41—42no—
115C13×#41—42no—
116C2×Q8×C13—21no—
117C2×Q8×C13—22no—
118C2×Q8×C13—21no—
119C2×Q8×C13—41no—
120C2×Q8×C13C2×D411yescharacteristic
121C2×Q8×C13C2×D411yescharacteristic
122C2×D4×C13—21no—
123C2×D4×C13—21no—
124C2×D4×C13—21no—
125C2×D4×C13—41no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—41no—
129C22×C4×C13C2×D411yescharacteristic
130C22×C4×C13C2×D411yescharacteristic
131C22×C4×C13#4111yescharacteristic
132C2×D52—22no—
133C2×D52—22no—
134C2×D52—22no—
135C2×D52—42no—
136C2×D52—42no—
137C2×D52—42no—
138C2×D52—42no—
139C2×D52—41no—
140C2×D52—41no—
141C2×D52—41no—
142C2×D52—82no—
143C13×SD16—42no—
144C13×SD16—42no—
145C2×C8×C13—21no—
146C2×C8×C13—21no—
147C13×C4⋊C4—21no—
148C13×C4⋊C4—21no—
149C13×C4⋊C4—21no—
150C13×C4⋊C4—21no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4C2×D411yescharacteristic
155C13×C4⋊C4C2×D411yescharacteristic
156C13×C22⋊C4—41no—
157C13×C22⋊C4—41no—
158C42×C13—21no—
159C42×C13—21no—
160C42×C13—21no—
161C42×C13—21no—
162C42×C13C2×D411yescharacteristic
163C42×C13C2×D411yescharacteristic
164C42×C13#4111yescharacteristic
165C42×C13C2411yescharacteristic
166#6106—44no—
167#6106—44no—
168#6106—44no—
169#6106—44no—
170#6104—44no—
171#6104—44no—
172#6099—22no—
173#6099—42no—
174#6099SD1614yescomplemented
175C2×#1055—22no—
176C2×#1055—22no—
177C2×#1055—42no—
178C2×#1055C2×D412yes—
179C2×#1055C2×D412yes—
180#3151—131no—
181#3103D1311yescharacteristic, complemented
182C2×#460—131no—
183Q8×M16—131no—
184#1688—132no—
185#1686—132no—
186#1668—132no—
187#1666—132no—
188#1565—131no—
189#1547—131no—
190#1545—131no—
191C23×C13—41no—
192C23×C13#12111yescharacteristic
193C22×D13—42no—
194C22×D13—82no—
195C22×D13—82no—
196C22×D13—82no—
197Q8×C13—21no—
198Q8×C13—21no—
199Q8×C13—21no—
200Q8×C13—42no—
201Q8×C13—41no—
202Q8×C13—41no—
203Q8×C13AGL(1,ℤ/8)12yescomplemented
204D4×C13—41no—
205D4×C13—41no—
206D4×C13—41no—
207D4×C13—41no—
208D4×C13—81no—
209D4×C13—82no—
210C2×C4×C13—21no—
211C2×C4×C13—21no—
212C2×C4×C13—21no—
213C2×C4×C13—21no—
214C2×C4×C13—21no—
215C2×C4×C13—21no—
216C2×C4×C13—41no—
217C2×C4×C13—42no—
218C2×C4×C13—41no—
219C2×C4×C13—81no—
220C2×C4×C13—82no—
221C2×C4×C13#12111yescharacteristic
222C2×C4×C13#12111yescharacteristic
223C2×C4×C13#12111yescharacteristic
224C2×C4×C13C2×SD1612yes—
225C2×C4×C13AGL(1,ℤ/8)12yes—
226C2×C4×C13C22×D411yescharacteristic
227C2×C4×C13C22×D411yescharacteristic
228C2×C4×C13C2×#4111yescharacteristic
229D52—42no—
230D52—42no—
231D52—42no—
232D52—42no—
233D52—82no—
234D52—82no—
235D52—82no—
236D52—82no—
237C8×C13—41no—
238C8×C13—41no—
239#1055—22no—
240#1055—42no—
241#1055—42no—
242#1055C2×SD1614yescomplemented
243C2×AGL(1,ℤ/8)—261no—
244C2×AGL(1,ℤ/8)—261no—
245C22×SD16—261no—
246#532C2×D1311yescharacteristic
247C2×#127—131no—
248C2×C4×Q8C2×D1311yescharacteristic
249#496—262no—
250#493—132no—
251#492—132no—
252#481—21no—
253#475C2×D1311yescharacteristic, complemented
254#475C2×D1311yescharacteristic, complemented
255#460—132no—
256#460—132no—
257#460—261no—
258#460—261no—
259#460C2×D1311yescharacteristic, complemented
260#460C2×D1311yescharacteristic, complemented
261#448—21no—
262#446—132no—
263#446—262no—
264C8×Q8—132no—
265C4×SD16—21no—
266C4×SD16—21no—
267#431C2×D1311yescharacteristic
268#423—131no—
269#423—261no—
270C2×C4⋊C8—131no—
271C2×#102—261no—
272C4×M16—131no—
273C4×M16—261no—
274#335—132no—
275#334—132no—
276#332—132no—
277#331—132no—
278#331—132no—
279#331—132no—
280C22×C13—21no—
281C22×C13—21no—
282C22×C13—81no—
283C22×C13—81no—
284C22×C13—81no—
285C22×C13#46012yes—
286C22×C13C2×#12111yescharacteristic
287C2×D13—82no—
288C2×D13—82no—
289C2×D13—82no—
290C2×D13—164no—
291C4×C13—21no—
292C4×C13—22no—
293C4×C13—21no—
294C4×C13—21no—
295C4×C13—41no—
296C4×C13—41no—
297C4×C13—41no—
298C4×C13—81no—
299C4×C13#46012yes—
300C4×C13C22×SD1611yescharacteristic
301C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C13#57511yescharacteristic
304C2×#41—21no—
305C22×Q8C22⋊D1311yescharacteristic
306C22×D4—261no—
307C22×D4—261no—
308C22×D4—261no—
309AGL(1,ℤ/8)—524no—
310AGL(1,ℤ/8)—524no—
311C2×SD16—21no—
312C2×SD16—21no—
313C2×SD16—262no—
314C2×SD16—262no—
315C2×SD16—262no—
316C2×SD16—262no—
317C2×SD16—524no—
318C2×D8—262no—
319C2×D8—262no—
320C2×M16—131no—
321C2×M16—131no—
322C2×M16—261no—
323C22×C8—261no—
324C4⋊Q8C22×D1311yescharacteristic
325#127—132no—
326#127—261no—
327#127—261no—
328#127—261no—
329#127C22×D1311yescharacteristic
330#124—21no—
331#121—21no—
332#121—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8C22⋊D1311yescharacteristic
336C4×Q8C22⋊D1311yescharacteristic
337C4×Q8C22×D1311yescharacteristic
338C4×Q8C22×D1311yescharacteristic
339C4×D4—21no—
340C4×D4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C22⋊D1311yescharacteristic
343C2×C42—21no—
344C2×C42C22×D1311yescharacteristic
345#106—21no—
346#106—21no—
347C4⋊C8—132no—
348C4⋊C8—132no—
349C4⋊C8—262no—
350C4⋊C8—262no—
351C4⋊C8—261no—
352C4⋊C8—261no—
353C4⋊C8C22×D1311yescharacteristic
354C4⋊C8C22×D1311yescharacteristic
355#103—21no—
356#103—21no—
357#102—21no—
358#102—21no—
359#102—262no—
360#102—262no—
361C22⋊C8—21no—
362#97—132no—
363#97—262no—
364C4×C8—21no—
365C4×C8—132no—
366C4×C8—262no—
367C2×C13—22no—
368C2×C13—161no—
369C2×C13C2×#46011yescharacteristic
370C2×C13#305511yescharacteristic
371C2×C13#306811yescharacteristic
372D13—162no—
373C24—521no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—41no—
380C2×Q8C2×C22⋊D1311yescharacteristic
381C2×Q8C2×C22⋊D1311yescharacteristic
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—262no—
387C2×D4—262no—
388C2×D4—262no—
389C2×D4—522no—
390C2×D4—522no—
391C2×D4—522no—
392C2×D4—522no—
393C2×D4—521no—
394C2×D4—521no—
395C2×D4—521no—
396C2×D4—1042no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4D4×D1311yescharacteristic
401C22×C4#612911yescharacteristic
402C22×C4C2×C22⋊D1311yescharacteristic
403SD16—42no—
404SD16—42no—
405SD16—524no—
406SD16—524no—
407SD16—524no—
408SD16—524no—
409D8—524no—
410D8—524no—
411M16—134no—
412M16—262no—
413M16—522no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—132no—
417C2×C8—132no—
418C2×C8—262no—
419C2×C8—262no—
420C2×C8—522no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4C2×C22⋊D1311yescharacteristic
429C4⋊C4C2×C22⋊D1311yescharacteristic
430C22⋊C4—41no—
431C22⋊C4—41no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42D4×D1311yescharacteristic
437C42#612911yescharacteristic
438C42C2×C22⋊D1311yescharacteristic
439C42C23×D1311yescharacteristic
440C13#1917011yescharacteristic, complemented, Sylow
441C23—41no—
442C23—522no—
443C23—1042no—
444C23—1042no—
445C23—1042no—
446C23#8819311yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8#8820312yescomplemented
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—82no—
460D4—522no—
461D4—522no—
462D4—522no—
463D4—522no—
464D4—1042no—
465D4—1042no—
466D4—1042no—
467D4—1042no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4SD16×D1312yes—
480C2×C4#8816812yes—
481C2×C4#8819311yescharacteristic
482C2×C4#8819311yescharacteristic
483C2×C4#8819311yescharacteristic
484C2×C4C2×D4×D1311yescharacteristic
485C2×C4C2×#612911yescharacteristic
486C2×C4C22×C22⋊D1311yescharacteristic
487C8—41no—
488C8—41no—
489C8—134no—
490C8—262no—
491C8—522no—
492C8—522no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—1042no—
499C22—1042no—
500C22—1042no—
501C22—2084no—
502C22#1171882412yes—
503C22C2×#8819311yescharacteristic
504C4—21no—
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4#1171882412yes—
513C4C2×SD16×D1311yescharacteristic
514C4C2×#8816811yescharacteristic
515C4C2×#8820311yescharacteristic
516C4#1171952711yescharacteristic
517C2—22no—
518C2—161no—
519C2—2082no—
520C2C2×#1171882411yescharacteristic
521C2#1312445711yescharacteristic
522C2#1312456611yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3103C2
socleC22×C13C2×#121
prime radical#13125289C2
hypercenter#3103D13
hypocenterC13#19170
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. C2×#11718688C2semidirect
3. Q8×#6099C2semidirect
4. #13114318C2semidirect
5. #13114310C2semidirect
6. #13114194C2semidirect
7. #13114186C2semidirect
8. #13113589C2semidirect
9. #13113513C2semidirect
10. #13113503C2semidirect
11. C13×#3103C2semidirect
19. C13×#475C22semidirect
20. C13×#475C22semidirect
21. C13×#460C22semidirect
22. C13×#460C22semidirect
35. #11718685C22semidirect
36. Q8×#1055C22semidirect
43. #11718139C22semidirect
44. #11718136C22semidirect
47. #11718072C22semidirect
48. #11718071C22semidirect
174. #6099SD16semidirect
181. #3103D13semidirect
203. Q8×C13AGL(1,ℤ/8)semidirect
242. #1055C2×SD16semidirect
253. #475C2×D13semidirect
254. #475C2×D13semidirect
259. #460C2×D13semidirect
260. #460C2×D13semidirect
440. C13#19170semidirect
453. Q8#88203semidirect

Maximal subgroups

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

subgroupindex
#131252892
C2×#117186882
Q8×#60992
#131143182
#131143102
#131141942
#131141862
#131135892
#131135132
#131135032
C13×#31032
#1917013