Finite Group Databasebeta

← #29663920

Subgroups — #29663920

Order 15616.

716 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#21456245C211yesmaximal, characteristic, complemented
2#21455970C211yesmaximal, characteristic, complemented
3#41×#92418C211yesmaximal, characteristic, complemented
4#21445187C211yesmaximal, characteristic, complemented
5#21445182C211yesmaximal, characteristic, complemented
6#21445177C211yesmaximal, characteristic, complemented
7#21445172C211yesmaximal, characteristic, complemented
8#21445169C211yesmaximal, characteristic, complemented
9#21445165C211yesmaximal, characteristic, complemented
10#21445162C211yesmaximal, characteristic, complemented
11#21445158C211yesmaximal, characteristic, complemented
12#21444580C211yesmaximal, characteristic, complemented
13#21444525C211yesmaximal, characteristic, complemented
14#21444524C211yesmaximal, characteristic, complemented
15C61×#3115C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17C2×#14519797—21no—
18#16840312C2211yescharacteristic
19#16840144—21no—
20#16840144—21no—
21#16840141C2211yescharacteristic
22#16840138C2211yescharacteristic
23#16840136C2211yescharacteristic
24#16840134C2211yescharacteristic
25#16840086—21no—
26#16840047C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16839962C2211yescharacteristic
29#16839957C2211yescharacteristic
30#16839907—21no—
31Q8×#92418C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839872—21no—
34#16839860—21no—
35D4×#92418—21no—
36D4×#92418—21no—
37D4×#92418C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839819C2211yescharacteristic
40#16839815—21no—
41#16839815C2211yescharacteristic
42C2×C4×#92418—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51#16839507C2211yescharacteristic
52#16839506C2211yescharacteristic
53#16839506C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#482—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#442—21no—
65C61×#440—21no—
66C61×#432C2211yescharacteristic
67C2×#11758222—21no—
68#14519797—44no—
69#14519797—44no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C22×#92418—21no—
81C22×#92418—21no—
82C22×#92418D411yescharacteristic
83C22×#92418D411yescharacteristic
84#14519777C2311yescharacteristic
85#14519774C2311yescharacteristic
86#14519765—21no—
87#14519751—21no—
88#14519751—21no—
89C4×D244—21no—
90C4×D244—21no—
91#14519710—21no—
92#14519708—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519698—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519688—21no—
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418D411yescharacteristic
112C4×#92418D411yescharacteristic
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×#41×C61D411yescharacteristic
117C2×SD16×C61—21no—
118C2×D8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61C2311yescharacteristic
121#124×C61—21no—
122#121×C61—21no—
123#121×C61—21no—
124C4×Q8×C61D411yescharacteristic
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61D411yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61D411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136#107×C61—21no—
137#106×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144C22⋊C8×C61—21no—
145#97×C61—21no—
146#41×C61—22no—
147#41×C61—22no—
148#41×C61—42no—
149#41×C61—42no—
150C2×Q8×C61—21no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61#4111yescharacteristic
165C2×C22⋊D61—41no—
166#11758222—42no—
167#11758222—42no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173SD16×C61—42no—
174D8×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61#4111yescharacteristic
194C42×C61C2411yescharacteristic
195#11758202—42no—
196#11758202—42no—
197#11758201—42no—
198#11758201—42no—
199#11758201—42no—
200#11758200—42no—
201#11758200—42no—
202#11758199—42no—
203#11758199—42no—
204#11758199—42no—
205#11758198—41no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—42no—
216C2×#92418—42no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C23×C61—21no—
222C23×C61—41no—
223C23×C61#12111yescharacteristic
224C22×D61—41no—
225Q8×C61—21no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61#13512yescomplemented
229D4×C61—24no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236D4×C61—82no—
237D4×C61#13512yescomplemented
238C2×C4×C61—24no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—82no—
252C2×C4×C61#12111yescharacteristic
253C2×C4×C61#12111yescharacteristic
254C2×C4×C61#12111yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—82no—
259C2×Dic61—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—81no—
265C4×D61—82no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—41no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418AGL(1,ℤ/8)12yes—
277#92418AGL(1,ℤ/8)12yes—
278#19269—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—22no—
282C22×C61—21no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61C2×#12111yescharacteristic
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—22no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61C2×#13511yescharacteristic
301C4×C61C2×#13511yescharacteristic
302C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
304Dic61—81no—
305#3147—611no—
306#3115—611no—
307#3115D6111yescharacteristic, complemented
308C8×#41—611no—
309#1670—611no—
310#1669—611no—
311#1668—611no—
312#1667—611no—
313#1666—611no—
314#1666—611no—
315#1665—611no—
316#1665—611no—
317#1568—611no—
318#1553—611no—
319#1553—611no—
320C2×C61—41no—
321C2×C61—42no—
322C2×C61—161no—
323C2×C61#303811yescharacteristic
324C2×C61#305011yescharacteristic
325C2×C61#306711yescharacteristic
326D61—161no—
327C2×#135—1221no—
328C2×#135—1221no—
329#532—611no—
330#532C2×D6111yescharacteristic
331C4×#41C2×D6111yescharacteristic
332#495—1221no—
333#495—1221no—
334#494—611no—
335#493—611no—
336#492—611no—
337#492—611no—
338#482—21no—
339#482—1221no—
340#475—611no—
341#475C2×D6111yescharacteristic
342#474—611no—
343#474C2×D6111yescharacteristic
344#460—611no—
345#460C2×D6111yescharacteristic
346#459—611no—
347#459C2×D6111yescharacteristic
348#449—21no—
349#449—1221no—
350C8×Q8—611no—
351C8×Q8—611no—
352#442—21no—
353#442—1221no—
354#440—21no—
355#440—1221no—
356C8×D4—611no—
357C8×D4—611no—
358C8×D4—1221no—
359C8×D4—1221no—
360#432—611no—
361#432C2×D6111yescharacteristic
362#431—611no—
363#431—1221no—
364C2×C4×C8—611no—
365C2×C4×C8—1221no—
366#335—611no—
367#335—611no—
368#334—611no—
369#334—611no—
370#332—611no—
371#332—611no—
372#332—611no—
373#332—611no—
374#331—611no—
375#331—611no—
376#331—611no—
377#331—611no—
378C61#1926911yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C2×#41—1221no—
381C2×#41C22⋊D6111yescharacteristic
382#135—2444no—
383#135—2444no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×SD16—21no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—1221no—
392C2×D8—21no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×D8—1221no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—1221no—
399C22×C8—1221no—
400C4⋊Q8—611no—
401C4⋊Q8C22×D6111yescharacteristic
402#127—611no—
403#127C22×D6111yescharacteristic
404#124—21no—
405#124—1221no—
406#121—21no—
407#121—21no—
408#121—1221no—
409#121—1221no—
410C4×Q8C22⋊D6111yescharacteristic
411C4×Q8C22×D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4C22⋊D6111yescharacteristic
419C4×D4C22×D6111yescharacteristic
420#117—21no—
421#117C22⋊D6111yescharacteristic
422C2×C42—21no—
423C2×C42C22×D6111yescharacteristic
424#107—21no—
425#107—1221no—
426#106—21no—
427#106—1221no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—21no—
437#103—1221no—
438#102—21no—
439#102—21no—
440#102—21no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444C22⋊C8—21no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449#97—21no—
450#97—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—1221no—
456C4×C8—1221no—
457C4×C8—1221no—
458#41—22no—
459#41—22no—
460#41—42no—
461#41—42no—
462#41—2442no—
463#41—2442no—
464C2×Q8—21no—
465C2×Q8—1221no—
466C2×Q8C2×C22⋊D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—41no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—2441no—
476C2×D4C2×C22⋊D6111yescharacteristic
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4—2441no—
482C22×C4D4×D6111yescharacteristic
483C22×C4#1175822411yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485Q16—2442no—
486Q16—2442no—
487SD16—42no—
488SD16—2442no—
489SD16—2442no—
490SD16—2442no—
491SD16—2442no—
492SD16—2442no—
493D8—42no—
494D8—2442no—
495D8—2442no—
496D8—2442no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—2442no—
510C2×C8—2442no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42D4×D6111yescharacteristic
530C42#1175822411yescharacteristic
531C42C2×C22⋊D6111yescharacteristic
532C42C23×D6111yescharacteristic
533C23—21no—
534C23—41no—
535C23—2441no—
536C23#1451974911yescharacteristic
537Q8—21no—
538Q8—41no—
539Q8—41no—
540Q8—2441no—
541Q8—2441no—
542Q8#1451979712yescomplemented
543D4—24no—
544D4—21no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—81no—
550D4—82no—
551D4—4881no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4#1451979712yescomplemented
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—4882no—
573C2×C4—2441no—
574C2×C4#1451974911yescharacteristic
575C2×C4#1451974911yescharacteristic
576C2×C4#1451974911yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C8—41no—
581C8—41no—
582C8—612no—
583C8—612no—
584C8—1221no—
585C8—1221no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C8—2441no—
590C22—21no—
591C22—22no—
592C22—21no—
593C22—21no—
594C22—41no—
595C22—41no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22C2×#1451974911yescharacteristic
603C4—21no—
604C4—22no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—4881no—
612C4C2×#1451979711yescharacteristic
613C4C2×#1451979711yescharacteristic
614C4AGL(1,ℤ/8)×D6111yescharacteristic
615C4AGL(1,ℤ/8)×D6111yescharacteristic
616C2—41no—
617C2—42no—
618C2—161no—
619C2—9761no—
620C2#2145529711yescharacteristic
621C2#2145539711yescharacteristic
622C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3115C2
socleC22×C61C2×#121
hypercenter#3115D61
hypocenterC61#19269
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #21456245C2semidirect
2. #21455970C2semidirect
3. #41×#92418C2semidirect
4. #21445187C2semidirect
5. #21445182C2semidirect
6. #21445177C2semidirect
7. #21445172C2semidirect
8. #21445169C2semidirect
9. #21445165C2semidirect
10. #21445162C2semidirect
11. #21445158C2semidirect
12. #21444580C2semidirect
13. #21444525C2semidirect
14. #21444524C2semidirect
15. C61×#3115C2semidirect
228. Q8×C61#135semidirect
237. D4×C61#135semidirect
307. #3115D61semidirect
378. C61#19269semidirect
542. Q8#14519797semidirect
557. D4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214562452
#214559702
#41×#924182
#214451872
#214451822
#214451772
#214451722
#214451692
#214451652
#214451622
#214451582
#214445802
#214445252
#214445242
C61×#31152
#1926961