Finite Group Databasebeta

← #29663792

Subgroups — #29663792

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456380C211yesmaximal, characteristic, complemented
2#21455617C211yesmaximal, characteristic, complemented
3#21455284C211yesmaximal, characteristic
4C8×D4×D61C211yesmaximal, characteristic, complemented
5#21445245C211yesmaximal, characteristic, complemented
6#21445244C211yesmaximal, characteristic, complemented
7#21445231C211yesmaximal, characteristic, complemented
8#21445230C211yesmaximal, characteristic, complemented
9#21445191C211yesmaximal, characteristic, complemented
10#21445161C211yesmaximal, characteristic, complemented
11#21445160C211yesmaximal, characteristic, complemented
12#21444573C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#21444495C211yesmaximal, characteristic, complemented
15C61×#1671C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C22×C8×D61—21no—
18#16840317C2211yescharacteristic
19#16840313C2211yescharacteristic
20C4×D4×D61C2211yescharacteristic
21#16840135C2211yescharacteristic, complemented
22#16840114C2211yescharacteristic, complemented
23#16840110C2211yescharacteristic, complemented
24#16840061—21no—
25#16840060—21no—
26#16840049C2211yescharacteristic
27#16840043C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16839992—21no—
30#16839992—21no—
31#16839980—21no—
32#16839979—21no—
33#16839964C2211yescharacteristic
34#16839958C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839849—21no—
37#16839849—21no—
38C8×D244C2211yescharacteristic
39D4×#92418C2211yescharacteristic
40#16839827C2211yescharacteristic
41C8×C22⋊D61—21no—
42#16839825—21no—
43#16839801—21no—
44#16839784C2211yescharacteristic
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839765—21no—
48#16839762—21no—
49C22⋊C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#498C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×#11758226—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70#14519804—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×D488—21no—
75C2×#11758199—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—44no—
83C2×C8×D61D411yescharacteristic
84C2×C8×D61D411yescharacteristic
85#14519777C2311yescharacteristic
86#14519774C2311yescharacteristic
87#14519768—21no—
88#14519766—21no—
89#14519757—21no—
90#14519753—21no—
91#14519752—21no—
92#14519752—21no—
93Q8×Dic61—21no—
94#14519742—21no—
95C4×D244C2311yescharacteristic
96D4×Dic61D411yescharacteristic
97#14519739—21no—
98#14519739—21no—
99#14519739D411yescharacteristic
100C4×C22⋊D61—21no—
101#14519737—21no—
102#14519735—21no—
103C4⋊C4×D61D411yescharacteristic
104C22⋊C4×D61—21no—
105C42×D61C2311yescharacteristic
106#14519711—21no—
107#14519710—21no—
108#14519708—21no—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519701—21no—
116#14519700—21no—
117#14519700—21no—
118#14519699—21no—
119#14519699—21no—
120#14519698—21no—
121#14519697—21no—
122#14519697—21no—
123#14519697—21no—
124#14519697—21no—
125C22⋊#92418—21no—
126#14519689—21no—
127#14519689—21no—
128C4×#92418C2311yescharacteristic
129C8×Dic61D411yescharacteristic
130C8×Dic61D411yescharacteristic
131C2×SD16×C61—21no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134C4⋊Q8×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#106×C61—21no—
137#106×C61—21no—
138#106×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#103×C61—21no—
143#102×C61—21no—
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153#11758226—42no—
154#11758226—42no—
155D4×D61—22no—
156D4×D61SD1614yescomplemented
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—41no—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—41no—
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61#4111yescharacteristic
171C2×Dic122—21no—
172SD16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—21no—
187#11758201—42no—
188#11758199—42no—
189#11758198—21no—
190#11758198—21no—
191#11758198—41no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—41no—
200C4×Dic61—41no—
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61#4111yescharacteristic
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210D488—42no—
211#11758190—42no—
212#11758190—42no—
213#11758189—42no—
214#11758189—42no—
215C8×D61—22no—
216C8×D61—41no—
217C8×D61—42no—
218C8×D61—42no—
219C8×D61—42no—
220C23×C61—21no—
221C22×D61—21no—
222C22×D61—24no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—41no—
226C22×D61#12111yescharacteristic
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—81no—
230D4×C61—21no—
231D4×C61C2×SD1612yes—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—24no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61#12111yescharacteristic
245C2×Dic61#12111yescharacteristic
246C2×Dic61#12111yescharacteristic
247D244—21no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—81no—
253D244—82no—
254D244C2×SD1612yes—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—41no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—82no—
263C4×D61C2×SD1612yes—
264C4×D61#13512yes—
265Dic122—41no—
266Dic122—41no—
267C8×C61—21no—
268C8×C61—41no—
269C8×C61—41no—
270C8×C61—41no—
271C8×C61AGL(1,ℤ/8)12yescomplemented
272#92418—21no—
273#92418—21no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#19254—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61C2×#12111yescharacteristic
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—22no—
285C2×D61—48no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61#46512yes—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C22×SD1611yescharacteristic
297C4×C61C2×#13511yescharacteristic
298C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C61#57511yescharacteristic
300Dic61—21no—
301Dic61—21no—
302Dic61—41no—
303Dic61—81no—
304Dic61#46512yes—
305#3167—611no—
306#3077—611no—
307#3036—611no—
308C2×C8×D4—611no—
309#1679—611no—
310#1679—611no—
311#1677—611no—
312#1677—611no—
313#1671—611no—
314#1671D6111yescharacteristic, complemented
315#1665—611no—
316#1665—611no—
317#1566—611no—
318#1550—611no—
319#1547—611no—
320C2×C61—42no—
321C2×C61#303611yescharacteristic
322C2×C61C2×#46511yescharacteristic
323C2×C61#306711yescharacteristic
324D61—22no—
325D61—42no—
326D61—161no—
327C2×AGL(1,ℤ/8)—1221no—
328C23×C8—1221no—
329#533—611no—
330#532—611no—
331C2×C4×D4—611no—
332#498C2×D6111yescharacteristic, complemented
333#492—611no—
334#487—611no—
335#486—611no—
336#478—1221no—
337#478—1221no—
338#476—611no—
339#476C2×D6111yescharacteristic
340#474—611no—
341#474—611no—
342#465—1221no—
343#465—1221no—
344#463—1221no—
345#463—1221no—
346#461—611no—
347#461C2×D6111yescharacteristic
348#459—611no—
349#459—611no—
350C4×SD16—1221no—
351C4×SD16—1221no—
352C8×D4—611no—
353C8×D4—611no—
354C8×D4—611no—
355C8×D4—1221no—
356C8×D4—1221no—
357C8×D4C2×D6111yescharacteristic
358#428—1221no—
359#423—611no—
360#423—611no—
361C2×C4⋊C8—611no—
362#418—1221no—
363#417—1221no—
364C2×C22⋊C8—1221no—
365C2×C4×C8—611no—
366#334—611no—
367#334—611no—
368#334—611no—
369#334C2×D6111yescharacteristic, complemented
370#333—611no—
371#333C2×D6111yescharacteristic, complemented
372#332—611no—
373#332—611no—
374#332—611no—
375#332C2×D6111yescharacteristic, complemented
376#331—611no—
377#331—611no—
378C61#1925411yescharacteristic, complemented, Sylow
379C2×#41—1221no—
380C22×D4—611no—
381C23×C4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—21no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×D8—1221no—
388C2×D8—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C22×C8—21no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—1221no—
395C22×C8—1221no—
396C22×C8—1221no—
397C22×C8—1221no—
398C22×C8—2444no—
399C4⋊Q8—611no—
400C4⋊Q8C22×D6111yescharacteristic
401C4⋊Q8C22×D6111yescharacteristic
402#127—611no—
403#125—1221no—
404#124—1221no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407#121—1221no—
408#121—1221no—
409C4×Q8—1221no—
410C4×Q8—1221no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—611no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4C22×D6111yescharacteristic
419#117—1221no—
420C2×C4⋊C4—611no—
421C2×C22⋊C4—1221no—
422C2×C42—611no—
423#107—1221no—
424#107—1221no—
425#106—21no—
426#106—21no—
427#106—21no—
428#106—1221no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435C4⋊C8C22×D6111yescharacteristic
436C4⋊C8C22×D6111yescharacteristic
437C4⋊C8C22×D6111yescharacteristic
438#103—21no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#102—21no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450C22⋊C8—21no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—611no—
457C4×C8C22×D6111yescharacteristic
458C24—1221no—
459#41—2442no—
460#41—2442no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—1221no—
464C2×D4—614no—
465C2×D4—611no—
466C2×D4—1222no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—2441no—
472C2×D4D4×D6111yescharacteristic
473C22×C4—21no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—2441no—
482C22×C4—2442no—
483C22×C4—2442no—
484SD16—42no—
485SD16—2442no—
486SD16—2442no—
487SD16—2442no—
488D8—2442no—
489D8—2442no—
490M16—2442no—
491M16—2442no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—42no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1222no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—2441no—
503C2×C8—2442no—
504C2×C8—2442no—
505C2×C8—2442no—
506C2×C8—2442no—
507C2×C8D4×D6111yescharacteristic
508C2×C8D4×D6111yescharacteristic
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2441no—
518C4⋊C4—2441no—
519C4⋊C4D4×D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—2441no—
531C42C23×D6111yescharacteristic
532C23—21no—
533C23—611no—
534C23—1224no—
535C23—1221no—
536C23—2442no—
537C23—2442no—
538C23—2441no—
539Q8—41no—
540Q8—41no—
541Q8—81no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—612no—
546D4—1224no—
547D4—1221no—
548D4—4881no—
549D4—4882no—
550D4—2441no—
551D4—2441no—
552D4—2441no—
553D4—2441no—
554D4SD16×D6112yes—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—611no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—4882no—
569C2×C4—2441no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C8—21no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—1221no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8#1451980412yescomplemented
589C22—22no—
590C22—41no—
591C22—41no—
592C22—612no—
593C22—1221no—
594C22—1221no—
595C22—1222no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—2448no—
600C22—2441no—
601C22—2441no—
602C22#121×D6111yescharacteristic
603C4—21no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—612no—
608C4—1221no—
609C4—1221no—
610C4—4881no—
611C4—2441no—
612C4C2×SD16×D6111yescharacteristic
613C4#135×D6111yescharacteristic
614C4C2×#1451980411yescharacteristic
615C4#1684056111yescharacteristic
616C2—42no—
617C2—1222no—
618C2—9761no—
619C2—2442no—
620C2#2145528411yescharacteristic
621C2#465×D6111yescharacteristic
622C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1671C2
socleC22×C61C2×#121
prime radical#21455284C2
hypercenter#1671D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1671

Complements

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

NH ≅ G/Ntype
1. #21456380C2semidirect
2. #21455617C2semidirect
4. C8×D4×D61C2semidirect
5. #21445245C2semidirect
6. #21445244C2semidirect
7. #21445231C2semidirect
8. #21445230C2semidirect
9. #21445191C2semidirect
10. #21445161C2semidirect
11. #21445160C2semidirect
12. #21444573C2semidirect
13. #21444506C2semidirect
14. #21444495C2semidirect
15. C61×#1671C2semidirect
21. #16840135C22semidirect
22. #16840114C22semidirect
23. #16840110C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839508C22semidirect
57. #16839507C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#498C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
156. D4×D61SD16semidirect
271. C8×C61AGL(1,ℤ/8)semidirect
314. #1671D61semidirect
332. #498C2×D61semidirect
369. #334C2×D61semidirect
371. #333C2×D61semidirect
375. #332C2×D61semidirect
378. C61#19254semidirect
588. C8#14519804semidirect

Maximal subgroups

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

subgroupindex
#214563802
#214556172
#214552842
C8×D4×D612
#214452452
#214452442
#214452312
#214452302
#214451912
#214451612
#214451602
#214445732
#214445062
#214444952
C61×#16712
#1925461