Finite Group Databasebeta

← #29717420

Subgroups — #29717420

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455342C212yesmaximal, complemented
2#21455221C211yesmaximal, characteristic, complemented
3#21455105C211yesmaximal, characteristic, complemented
4#21454283C211yesmaximal, characteristic, complemented
5#21447122C212yesmaximal, complemented
6#21446869C212yesmaximal, complemented
7#21446346C211yesmaximal, characteristic, complemented
8#21446315C212yesmaximal, complemented
9#21446105C211yesmaximal, characteristic, complemented
10#21445783C211yesmaximal, characteristic, complemented
11C61×#3586C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic, complemented
13C22×#11758201C2211yescharacteristic, complemented
14#16840326C2211yescharacteristic
15#16840326C2212yes—
16#16840240C2212yescomplemented
17#16839986—24no—
18#16839960—24no—
19#16839949—24no—
20#16839907—22no—
21#16839901—22no—
22#16839790C2212yescomplemented
23#16839779C2212yes—
24#16839771C2211yescharacteristic, complemented
25#16839768C2212yescomplemented
26#16839758C2212yes—
27#16839758C2211yescharacteristic, complemented
28#16839736—24no—
29#16839736C2212yes—
30#16839736C2211yescharacteristic
31#16839736C2211yescharacteristic
32#16839736C2212yescomplemented
33#16839729C2211yescharacteristic
34#16839533C2212yescomplemented
35#16839519C2212yescomplemented
36C61×#585C2211yescharacteristic
37C2×2−1+4×C61C2211yescharacteristic
38C61×#578—24no—
39C61×#577—24no—
40C61×#575—22no—
41C61×#575C2212yes—
42C2×#137×C61—22no—
43C2×#137×C61C2212yes—
44C61×#568C2211yescharacteristic
45C22×D244C2311yescharacteristic
46#14519802—42no—
47C2×#11758201—24no—
48C2×#11758201—22no—
49C2×#11758201—21no—
50C2×#11758199—21no—
51C2×#11758194C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519773—22no—
54#14519773—21no—
55#14519766—22no—
56#14519766—21no—
57#14519751—22no—
58#14519751—42no—
59#14519745—22no—
60#14519745—24no—
61Q8×Dic61—24no—
62Q8×Dic61—42no—
63#14519736—24no—
64#14519736C2312yes—
65#14519736C2311yescharacteristic
66#14519736C2312yes—
67#14519706—24no—
68#14519704—42no—
69#14519701—22no—
70#14519701—21no—
71#14519701—22no—
72#14519699—24no—
73#14519698—22no—
74#14519698—21no—
75#14519692—22no—
76#14519692—24no—
77#14519692—24no—
78#14519692D414yes—
79C22⋊#92418—22no—
802−1+4×C61—24no—
812−1+4×C61—22no—
822−1+4×C61—21no—
832−1+4×C61—42no—
84H(5,𝔽2)×C61—22no—
85H(5,𝔽2)×C61—22no—
86H(5,𝔽2)×C61—21no—
87C2×#41×C61—22no—
88C2×#41×C61—21no—
89C2×#41×C61—22no—
90C2×#41×C61—42no—
91C2×#41×C61D412yescomplemented
92C2×#41×C61C2312yes—
93C2×#41×C61C2311yescharacteristic
94C2×#41×C61C2311yescharacteristic
95C22×Q8×C61—21no—
96C22×Q8×C61D412yescomplemented
97C22×Q8×C61C2311yescharacteristic
98#137×C61—28no—
99#137×C61—24no—
100#137×C61—22no—
101#137×C61—24no—
102#137×C61—24no—
103#137×C61—22no—
104AGL(1,ℤ/8)×C61—24no—
105AGL(1,ℤ/8)×C61—22no—
106AGL(1,ℤ/8)×C61—22no—
107#135×C61—28no—
108#135×C61—24no—
109C2×Q16×C61—24no—
110C2×Q16×C61D414yescomplemented
111C2×SD16×C61—24no—
112C2×SD16×C61D414yescomplemented
113#131×C61—24no—
114C2×M16×C61—22no—
115C2×M16×C61C2312yes—
116C23×D61—21no—
117#41×C61—22no—
118#41×C61—22no—
119#41×C61—24no—
120#41×C61—24no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—42no—
128#41×C61—48no—
129#41×C61—44no—
130#41×C61—42no—
131#41×C61—44no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—42no—
135C2×Q8×C61—22no—
136C2×Q8×C61—21no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—24no—
140C2×Q8×C61—42no—
141C2×Q8×C61—44no—
142C2×Q8×C61—42no—
143C2×Q8×C61C2×D412yes—
144C2×Q8×C61C2×D411yescharacteristic
145C2×Q8×C61C2×D414yes—
146C2×Q8×C61C2×D412yes—
147C2×Q8×C61#4111yescharacteristic
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—21no—
151C2×D4×C61—22no—
152C2×D4×C61—42no—
153C2×D4×C61—44no—
154C2×D4×C61—42no—
155C2×D4×C61C2×D412yes—
156C2×D4×C61C2×D412yes—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61#4111yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—21no—
163C22×C4×C61—22no—
164C22×C4×C61—42no—
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2411yescharacteristic
167C2×C22⋊D61—44no—
168C2×C22⋊D61—42no—
169C2×D244—22no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—22no—
173Q16×C61—24no—
174Q16×C61—28no—
175SD16×C61—24no—
176SD16×C61—24no—
177SD16×C61—28no—
178D8×C61—24no—
179M16×C61—24no—
180M16×C61—22no—
181M16×C61—22no—
182C2×C8×C61—24no—
183C2×C8×C61C2×D414yes—
184C22⋊Dic61—22no—
185C22⋊Dic61—21no—
186C22⋊Dic61C2×D414yes—
187#11758201—42no—
188#11758201—44no—
189#11758199—42no—
190#11758198—44no—
191#11758198—42no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—44no—
196C4×Dic61—22no—
197C4×Dic61—21no—
198C4×Dic61—22no—
199C4×Dic61—44no—
200#11758194—22no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—42no—
207C23×C61#12112yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—161no—
212Q8×C61—22no—
213Q8×C61—24no—
214Q8×C61—22no—
215Q8×C61—21no—
216Q8×C61—22no—
217Q8×C61—21no—
218Q8×C61—22no—
219Q8×C61—44no—
220Q8×C61—42no—
221Q8×C61#12014yes—
222D4×C61—24no—
223D4×C61—21no—
224D4×C61—22no—
225D4×C61—21no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—44no—
229D4×C61—42no—
230D4×C61—44no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—42no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61—44no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61#12112yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D412yes—
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)12yes—
252C22⋊D61—164no—
253C22⋊D61—162no—
254C2×Dic61—24no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257D244—41no—
258D244—41no—
259D244—42no—
260C8×C61—24no—
261#92418—41no—
262#92418—41no—
263#24410—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—41no—
268C22×C61—42no—
269C22×C61—82no—
270C22×C61—81no—
271C22×C61—162no—
272C22×C61C2×#12111yescharacteristic
273C22×C61D4212yes—
274C2×D61—84no—
275C2×D61—82no—
276C2×D61—81no—
277C4×C61—24no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—41no—
281C4×C61C2×#12011yescharacteristic
282C4×C61#53411yescharacteristic
283C4×C61#54612yes—
284Dic61—44no—
285Dic61—82no—
286Dic61—81no—
287#3586D6111yescharacteristic, complemented
288#3043—612no—
289#3027—611no—
290#3010—611no—
291#2884—611no—
292#2053—612no—
293#2008—612no—
294#1910—611no—
295#1905—612no—
296#1868—611no—
297#1794—611no—
298C2×C61—22no—
299C2×C61—21no—
300C2×C61—42no—
301C2×C61—81no—
302C2×C61—82no—
303C2×C61#241011yescharacteristic
304D61—161no—
305#585C2×D6111yescharacteristic
306C2×2−1+4C2×D6111yescharacteristic
307#578—24no—
308#577—24no—
309#575—22no—
310#575C2×D6112yes—
311C2×#137—22no—
312C2×#137C2×D6112yes—
313C2×AGL(1,ℤ/8)—611no—
314C22×SD16—611no—
315#568C2×D6111yescharacteristic
316#535—611no—
317#535—612no—
318#519—612no—
319#464—1224no—
320#460—1224no—
321B(4,𝔽2)—1224no—
322#449—1222no—
323#448—1222no—
324#424—612no—
325#421—612no—
326#419—611no—
327#418—612no—
328#416—612no—
329#416—611no—
330#410—611no—
331#410—611no—
332#410—612no—
333#410—612no—
334#410—1224no—
335#408—611no—
336#344—612no—
337H(3,ℤ/4)—612no—
338C61#2441011yescharacteristic, complemented, Sylow
3392−1+4—24no—
3402−1+4—22no—
3412−1+4—21no—
3422−1+4—42no—
343H(5,𝔽2)—22no—
344H(5,𝔽2)—22no—
345H(5,𝔽2)—21no—
346C2×#41—22no—
347C2×#41—21no—
348C2×#41—22no—
349C2×#41—42no—
350C2×#41C22⋊D6112yescomplemented
351C2×#41C22×D6112yes—
352C2×#41C22×D6111yescharacteristic
353C2×#41C22×D6111yescharacteristic
354C22×Q8—21no—
355C22×Q8C22⋊D6112yescomplemented
356C22×Q8C22×D6111yescharacteristic
357C22×D4—611no—
358#137—28no—
359#137—24no—
360#137—22no—
361#137—24no—
362#137—24no—
363#137—22no—
364AGL(1,ℤ/8)—24no—
365AGL(1,ℤ/8)—22no—
366AGL(1,ℤ/8)—22no—
367AGL(1,ℤ/8)—2442no—
368#135—28no—
369#135—24no—
370C2×Q16—24no—
371C2×Q16C22⋊D6114yescomplemented
372C2×SD16—24no—
373C2×SD16—1222no—
374C2×SD16—1224no—
375C2×SD16—1221no—
376C2×SD16C22⋊D6114yescomplemented
377C2×D8—1221no—
378#131—24no—
379C2×M16—22no—
380C2×M16—611no—
381C2×M16C22×D6112yes—
382C22×C8—611no—
383#127—1221no—
384#127—1222no—
385#124—1221no—
386#124—1222no—
387#121—1222no—
388#121—2442no—
389#120—1224no—
390#120—1222no—
391C4×Q8—1224no—
392C4×Q8—2442no—
393#117—612no—
394#117—612no—
395#117—611no—
396#117—1224no—
397C4⋊C8—1224no—
398#104—2442no—
399#103—1221no—
400#103—1222no—
401#103—1222no—
402#102—1221no—
403#102—1222no—
404#102—1224no—
405#99—614no—
406#99—1224no—
407#99—1224no—
408#99—1222no—
409C22⋊C8—1222no—
410C24—1221no—
411#41—22no—
412#41—22no—
413#41—24no—
414#41—24no—
415#41—22no—
416#41—22no—
417#41—22no—
418#41—22no—
419#41—22no—
420#41—22no—
421#41—42no—
422#41—48no—
423#41—44no—
424#41—42no—
425#41—44no—
426#41—42no—
427#41—42no—
428#41—42no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8#1175822411yescharacteristic
438C2×Q8C2×C22⋊D6112yes—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×Q8C2×C22⋊D6112yes—
441C2×Q8C2×C22⋊D6114yes—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—1222no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—2444no—
454C2×D4—2442no—
455C2×D4D4×D6112yes—
456C2×D4#1175822411yescharacteristic
457C2×D4C2×C22⋊D6112yes—
458C2×D4C2×C22⋊D6111yescharacteristic
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4C2×C22⋊D6112yes—
466C22×C4C23×D6111yescharacteristic
467Q16—24no—
468Q16—28no—
469SD16—24no—
470SD16—24no—
471SD16—28no—
472SD16—2442no—
473SD16—2444no—
474D8—24no—
475D8—2442no—
476M16—24no—
477M16—22no—
478M16—22no—
479M16—1222no—
480C2×C8—24no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1222no—
484C2×C8C2×C22⋊D6114yes—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4—2444no—
489C22⋊C4—614no—
490C22⋊C4—1222no—
491C22⋊C4—1221no—
492C22⋊C4—2444no—
493C22⋊C4—2442no—
494C42—1222no—
495C42—1221no—
496C42—1222no—
497C42—2444no—
498C23—21no—
499C23—22no—
500C23—42no—
501C23—9761no—
502C23—2441no—
503C23—2442no—
504C23#1451974912yes—
505C23C2×D4×D6111yescharacteristic
506Q8—22no—
507Q8—24no—
508Q8—22no—
509Q8—21no—
510Q8—22no—
511Q8—21no—
512Q8—22no—
513Q8—44no—
514Q8—42no—
515Q8#1451974614yes—
516D4—24no—
517D4—21no—
518D4—22no—
519D4—21no—
520D4—22no—
521D4—22no—
522D4—44no—
523D4—42no—
524D4—44no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—9762no—
529D4—9764no—
530D4—2441no—
531D4—2441no—
532D4—2442no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—1224no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4#1451974912yes—
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C22×C22⋊D6112yes—
552C2×C4C2×#1175822411yescharacteristic
553C2×C4#1451982212yes—
554C8—24no—
555C8—2441no—
556C8—2441no—
557C22—22no—
558C22—41no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—82no—
563C22—81no—
564C22—162no—
565C22—4884no—
566C22—4882no—
567C22—4881no—
568C22C2×#1451974911yescharacteristic
569C22D4×C22⋊D6112yes—
570C4—24no—
571C4—22no—
572C4—42no—
573C4—41no—
574C4—4882no—
575C4—4881no—
576C4—2444no—
577C4C2×#1451974611yescharacteristic
578C4#1684032111yescharacteristic
579C4#1684040612yes—
580C2—22no—
581C2—21no—
582C2—42no—
583C2—82no—
584C2—81no—
585C2—9761no—
586C2#2144919311yescharacteristic
subgroupG/H
centerC2#21449193
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3586C2
socleC2×C61#2410
hypercenter#3586D61
hypocenterC61#24410
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #3586

Complements

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

NH ≅ G/Ntype
1. #21455342C2semidirect
2. #21455221C2semidirect
3. #21455105C2semidirect
4. #21454283C2semidirect
5. #21447122C2semidirect
6. #21446869C2semidirect
7. #21446346C2semidirect
8. #21446315C2semidirect
9. #21446105C2semidirect
10. #21445783C2semidirect
11. C61×#3586C2semidirect
12. C2×#14519802C22semidirect
13. C22×#11758201C22semidirect
16. #16840240C22semidirect
22. #16839790C22semidirect
24. #16839771C22semidirect
25. #16839768C22semidirect
27. #16839758C22semidirect
32. #16839736C22semidirect
34. #16839533C22semidirect
35. #16839519C22semidirect
91. C2×#41×C61D4semidirect
96. C22×Q8×C61D4semidirect
110. C2×Q16×C61D4semidirect
112. C2×SD16×C61D4semidirect
287. #3586D61semidirect
338. C61#24410semidirect
350. C2×#41C22⋊D61semidirect
355. C22×Q8C22⋊D61semidirect
371. C2×Q16C22⋊D61semidirect
376. C2×SD16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214553422
#214552212
#214551052
#214542832
#214471222
#214468692
#214463462
#214463152
#214461052
#214457832
C61×#35862
#2441061