Finite Group Databasebeta

← #30622469

Subgroups — #30622469

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462349C211yesmaximal, characteristic
2#21462338C211yesmaximal, characteristic, complemented
3#21461969C212yesmaximal, complemented
4#21461947C211yesmaximal, characteristic
5#21461930C211yesmaximal, characteristic, complemented
6#21461896C212yesmaximal, complemented
7#21461812C212yesmaximal, complemented
8#21461616C211yesmaximal, characteristic, complemented
9#21461590C212yesmaximal, complemented
10C4×#14519840C211yesmaximal, characteristic, complemented
11#460×D61C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13C2×D4×#6923C2212yes—
14C2×#14519846C2212yes—
15C2×#14519840C2211yescharacteristic
16C2×#14519840C2211yescharacteristic
17C2×#14519839C2212yes—
18C2×#14519839C2212yes—
19C2×#14519831C2212yes—
20C4×D61⋊C8C2211yescharacteristic
21(C2×#6923)⋊Q8C2212yes—
22#16840714—21no—
23#16840714C2212yes—
24#16840711—22no—
25#16840710—22no—
26#16840708C2211yescharacteristic
27#16840706—21no—
28C4⋊C4×#6923C2212yes—
29C4×C4⋊#6923—21no—
30C4×C4⋊#6923C2211yescharacteristic
31C4×C4⋊#6923C2211yescharacteristic
32#16840636C2212yes—
33#16840635C2211yescharacteristic
34#16840635C2211yescharacteristic
35(C2×#6923)⋊C8C2212yes—
36C2×SD16×D61C2212yes—
37#127×D61C2211yescharacteristic
38C4×Q8×D61C2211yescharacteristic
39#16839963C411yescharacteristic, complemented
40#16839962C411yescharacteristic, complemented
41#16839961C412yescomplemented
42#16839960C412yescomplemented
43#16839959C411yescharacteristic, complemented
44C4⋊C8×D61C2211yescharacteristic
45#102×D61C2212yes—
46C61×#460C411yescharacteristic, complemented
47C23×#6923—22no—
48C2×D61⋊C8C2311yescharacteristic
49C2×D61⋊C8C2311yescharacteristic
50D4×#6923—24no—
51D4×#6923—42no—
52D4×#6923—44no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923D411yescharacteristic
56C2×C4⋊#6923D411yescharacteristic
57C2×C4⋊#6923C2312yes—
58C2×C4⋊#6923C2311yescharacteristic
59C2×C4⋊#6923C2311yescharacteristic
60C2×C22⋊#6923—22no—
61C2×C22⋊#6923—22no—
62C2×C22⋊#6923—22no—
63C2×C4×#6923—21no—
64C2×C4×#6923—22no—
65C2×C4×#6923C2312yes—
66#14519846—42no—
67#14519840—22no—
68#14519840—22no—
69#14519839—24no—
70#14519839—42no—
71#14519831—42no—
72C4×#92420—21no—
73C4×#92420—21no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—21no—
78(C2×#6923)⋊C4D412yes—
79C2×Q8×D61C2311yescharacteristic
80C2×D4×D61—21no—
81C2×D4×D61—21no—
82C2×D4×D61C2312yes—
83C2×#11758190C2×C412yescomplemented
84C2×#11758200C2×C412yescomplemented
85C2×#11758201C2×C412yescomplemented
86SD16×D61—42no—
87SD16×D61—42no—
88C2×C8×D61C2312yes—
89#14519774C2×C411yescharacteristic
90#14519773—21no—
91#14519773—21no—
92#14519773C2×C412yes—
93Q8×Dic61C2×C411yescharacteristic, complemented
94C4×Dic122—21no—
95C4×Dic122C2×C411yescharacteristic, complemented
96#14519742—21no—
97#14519742C2×C411yescharacteristic, complemented
98C4⋊C4×D61—21no—
99C4⋊C4×D61C2311yescharacteristic
100C42×D61—21no—
101C42×D61C2311yescharacteristic
102C4⋊#92418C2×C411yescharacteristic, complemented
103#14519706C2×C412yescomplemented
104#14519699C2×C412yescomplemented
105#14519698C2×C412yescomplemented
106#14519697C2×C412yescomplemented
107C2×SD16×C61C2×C412yescomplemented
108#127×C61C2×C411yescharacteristic
109C4×Q8×C61C2×C411yescharacteristic, complemented
110C4⋊C8×C61C2×C411yescharacteristic, complemented
111#102×C61C2×C412yescomplemented
112C23×D61—22no—
113C22×#6923—21no—
114C22×#6923—22no—
115C22×#6923—21no—
116C22×#6923—44no—
117C22×#6923—44no—
118C22×#6923—82no—
119C22×#6923C2×D412yes—
120C22×#6923C2×D412yes—
121C2×Q8×C61C22×C411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C22×C412yes—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128Q8×D61—21no—
129Q8×D61—21no—
130Q8×D61—21no—
131Q8×D61—21no—
132D4×D61—22no—
133D4×D61—22no—
134D4×D61—22no—
135D4×D61—22no—
136D4×D61—42no—
137D4×D61—42no—
138D4×D61—81no—
139D4×D61—81no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244C22×C412yes—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122C22×C411yescharacteristic
150C22⋊#6923—44no—
151C22⋊#6923—44no—
152C22⋊#6923—82no—
153C2×#92420—21no—
154C2×#92420—21no—
155C4⋊#6923—22no—
156C4⋊#6923—22no—
157C4⋊#6923—22no—
158C4⋊#6923—21no—
159C4⋊#6923—21no—
160C4⋊#6923—22no—
161C4⋊#6923—21no—
162C4⋊#6923—42no—
163C4⋊#6923—81no—
164C4⋊#6923#4112yes—
165C4×#6923—22no—
166C4×#6923—22no—
167C4×#6923—42no—
168C4×#6923—82no—
169D61⋊C8—21no—
170D61⋊C8—21no—
171D61⋊C8—21no—
172D61⋊C8#4112yes—
173SD16×C61—42no—
174C2×C8×C61C22×C412yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C411yescharacteristic
177C42×C61—21no—
178C42×C61C22×C411yescharacteristic
179#11758201—42no—
180#11758200—42no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C411yescharacteristic
183#11758196—21no—
184#11758196—21no—
185#11758196—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61C22×C411yescharacteristic
190C4×Dic61#4111yescharacteristic
191C4×Dic61#4111yescharacteristic
192C2×#92418C22×C412yes—
193#11758190—42no—
194C8×D61—22no—
195C8×D61—22no—
196C23×C61—22no—
197C22×D61—22no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61C22×D411yescharacteristic
205C2×#6923—22no—
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—22no—
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—41no—
213C2×#6923—41no—
214C2×#6923—42no—
215C2×#6923—41no—
216C2×#6923—41no—
217C2×#6923—42no—
218C2×#6923—42no—
219C2×#6923—41no—
220C2×#6923—88no—
221Q8×C61—21no—
222Q8×C61—21no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—42no—
226D4×C61—81no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61C4×D411yescharacteristic
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—42no—
233C22⋊D61—42no—
234C22⋊D61—44no—
235C22⋊D61—82no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C2×#4111yescharacteristic
241D244—22no—
242D244—22no—
243D244—42no—
244D244—81no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61AGL(1,ℤ/8)12yes—
252C4×D61#13712yes—
253C4×D61C2×#4111yescharacteristic
254C4×D61C2×#4111yescharacteristic
255C4×D61H(5,𝔽2)11yescharacteristic
256C4×D61H(5,𝔽2)11yescharacteristic
257Dic122—21no—
258Dic122—21no—
259Dic122—41no—
260Dic122—41no—
261#92420—22no—
262#92420—41no—
263C8×C61—22no—
264#92418—22no—
265#23101—611nomaximal, Sylow
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61C2×C4×D411yescharacteristic
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—82no—
276C2×D61—82no—
277C2×D61#54611yescharacteristic
278C2×D61#55011yescharacteristic
279C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
280C2×D61C2×#13711yescharacteristic
281C2×D61#57611yescharacteristic
282C2×D61#57711yescharacteristic
283#6923—42no—
284#6923—42no—
285#6923—42no—
286#6923—81no—
287#6923—82no—
288C4×C61—21no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61#44012yes—
292C4×C61C4×#4111yescharacteristic
293C4×C61#51811yescharacteristic
294Dic61—21no—
295Dic61—41no—
296Dic61—41no—
297Dic61#44012yes—
298Dic61#55311yescharacteristic
299Dic61#55511yescharacteristic
300C2×#460—611no—
301#2366—611no—
302#2355—611no—
303#1973—612no—
304#1957—611no—
305#1947—611no—
306#1925—612no—
307#1877—612no—
308#1770—611no—
309#1762—612no—
310C4×#103—611no—
311C2×C61—82no—
312C2×C61#236311yescharacteristic
313C2×C61C2×#44011yescharacteristic
314C2×C61#294611yescharacteristic
315D61—82no—
316D61#332411yescharacteristic
317D61#333311yescharacteristic
318D61#339911yescharacteristic
319D61#340011yescharacteristic
320C22×SD16—612no—
321C2×#127—611no—
322C2×C4×Q8—611no—
323C2×C4×D4—612no—
324C2×C4×D4—1221no—
325#460—612no—
326#460—611no—
327#460—611no—
328#460—611no—
329#460—612no—
330#460#692311yescharacteristic, complemented
331C2×#107—612no—
332C2×C4⋊C8—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#102—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×#97—612no—
339C2×C4×C8—611no—
340#391—612no—
341#390—612no—
342#390—1221no—
343#387—1222no—
344#386—1222no—
345#384—611no—
346#382—1221no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—612no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—1221no—
351#340—612no—
352#340—611no—
353#340—611no—
354#336—612no—
355C61#2310111yescharacteristic, complemented, Sylow
356C22×Q8—611no—
357C22×D4—612no—
358C22×D4—1221no—
359C22×D4—1221no—
360C23×C4—1222no—
361C2×SD16—612no—
362C2×SD16—612no—
363C2×SD16—612no—
364C2×SD16—2442no—
365C2×SD16—2442no—
366C2×SD16C2×#692312yescomplemented
367C22×C8—611no—
368C22×C8—612no—
369C22×C8—611no—
370#127—612no—
371#127—611no—
372#127—1221no—
373#127—1221no—
374#127C2×#692311yescharacteristic
375C4×Q8—611no—
376C4×Q8—611no—
377C4×Q8—611no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C2×#692311yescharacteristic, complemented
381C4×D4—1224no—
382C4×D4—2442no—
383C4×D4—2444no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C42—611no—
397C2×C42—612no—
398C2×C42—1221no—
399C2×C42—1222no—
400C2×C42—1221no—
401#107—2442no—
402C4⋊C8—612no—
403C4⋊C8—611no—
404C4⋊C8C2×#692311yescharacteristic, complemented
405#103—1222no—
406#103—1222no—
407#102—612no—
408#102—612no—
409#102—612no—
410#102—1224no—
411#102—2442no—
412#102C2×#692312yescomplemented
413#97—2442no—
414C4×C8—1221no—
415C4×C8—1221no—
416#95—612no—
417#95—1222no—
418#95—1221no—
419#95—1222no—
420#95—1222no—
421C24—1222no—
422C2×Q8—611no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8C22×#692311yescharacteristic
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—612no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1222no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—4881no—
442C2×D4—4881no—
443C2×D4—2442no—
444C2×D4—2442no—
445C2×D4C22×#692312yes—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—4882no—
457C22×C4—2444no—
458C22×C4—2444no—
459SD16—42no—
460SD16—2442no—
461SD16—2442no—
462SD16—2442no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8C22×#692312yes—
473C4⋊C4—21no—
474C4⋊C4—612no—
475C4⋊C4—611no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—2442no—
489C4⋊C4C22×#692311yescharacteristic
490C22⋊C4—4882no—
491C22⋊C4—2444no—
492C22⋊C4—2444no—
493C42—21no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—1221no—
498C42—1221no—
499C42—1222no—
500C42—1222no—
501C42—1221no—
502C42—4882no—
503C42—2442no—
504C42C22×#692311yescharacteristic
505C23—22no—
506C23—611no—
507C23—1222no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—2442no—
513C23—2442no—
514Q8—21no—
515Q8—21no—
516Q8—1221no—
517Q8—1221no—
518Q8—2441no—
519Q8—2441no—
520D4—22no—
521D4—22no—
522D4—42no—
523D4—81no—
524D4—1222no—
525D4—1222no—
526D4—4881no—
527D4—4882no—
528D4—2442no—
529D4—2442no—
530D4—2444no—
531D4—2442no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—4888no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2442no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4D4×#692311yescharacteristic
568C2×C4D4×#692311yescharacteristic
569C2×C4C23×#692311yescharacteristic
570C8—22no—
571C8—1222no—
572C8—1222no—
573C8—2441no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—611no—
578C22—611no—
579C22—611no—
580C22—611no—
581C22—611no—
582C22—611no—
583C22—4882no—
584C22—4882no—
585C22—4882no—
586C22—4882no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22C2×D4×#692311yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—612no—
595C4—611no—
596C4—611no—
597C4—1221no—
598C4—4882no—
599C4—4881no—
600C4—2442no—
601C4—2442no—
602C4—2442no—
603C4—2441no—
604C4—2441no—
605C4#1684080112yes—
606C4#41×#692311yescharacteristic
607C4#1684082911yescharacteristic
608C2—82no—
609C2—611no—
610C2—611no—
611C2—611no—
612C2—611no—
613C2—4882no—
614C2#2146234611yescharacteristic
615C2C2×#1684080111yescharacteristic
616C2#2146257611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#460C4
socleC22×C61C2×C4×D4
prime radical#127×D61C22
cosocleC2×C4×D61C24
hypercenter#460#6923
hypocenterC61#23101
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #21462338C2semidirect
3. #21461969C2semidirect
5. #21461930C2semidirect
6. #21461896C2semidirect
7. #21461812C2semidirect
8. #21461616C2semidirect
9. #21461590C2semidirect
10. C4×#14519840C2semidirect
39. #16839963C4semidirect
40. #16839962C4semidirect
41. #16839961C4semidirect
42. #16839960C4semidirect
43. #16839959C4semidirect
46. C61×#460C4semidirect
83. C2×#11758190C2×C4semidirect
84. C2×#11758200C2×C4semidirect
85. C2×#11758201C2×C4semidirect
93. Q8×Dic61C2×C4semidirect
95. C4×Dic122C2×C4semidirect
97. #14519742C2×C4semidirect
102. C4⋊#92418C2×C4semidirect
103. #14519706C2×C4semidirect
104. #14519699C2×C4semidirect
105. #14519698C2×C4semidirect
106. #14519697C2×C4semidirect
107. C2×SD16×C61C2×C4semidirect
109. C4×Q8×C61C2×C4semidirect
110. C4⋊C8×C61C2×C4semidirect
111. #102×C61C2×C4semidirect
330. #460#6923semidirect
355. C61#23101semidirect
366. C2×SD16C2×#6923semidirect
380. C4×Q8C2×#6923semidirect
404. C4⋊C8C2×#6923semidirect
412. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623492
#214623382
#214619692
#214619472
#214619302
#214618962
#214618122
#214616162
#214615902
C4×#145198402
#460×D612
#2310161