Finite Group Databasebeta

← #30615947

Subgroups — #30615947

Order 15616.

1000 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#430C211yesmaximal, characteristic, complemented
2C2×#16840760C211yesmaximal, characteristic
3C2×#16840760C211yesmaximal, characteristic
4#21462382C211yesmaximal, characteristic, complemented
5#21461907C218yesmaximal, complemented
6#21461531C211yesmaximal, characteristic, complemented
7#21461501C211yesmaximal, characteristic, complemented
8C2×#16839743C211yesmaximal, characteristic
9C2×D61⋊M16C2211yescharacteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13#16840777—22no—
14#16840777C2214yescomplemented
15D61⋊#108—22no—
16#16840760—22no—
17#16840760—22no—
18#16840760—22no—
19#16840760C2214yes—
20C2×#14519841—22no—
21C2×#14519841C2214yes—
22#16840728—22no—
23#16840725—22no—
24M16×#6923C2214yescomplemented
25#16840645C2214yescomplemented
26#16840634C2214yescomplemented
27C22×#11758224C2211yescharacteristic
28C2×M16×D61C2211yescharacteristic
29C2×M16×D61C2211yescharacteristic
30#16839743C414yescomplemented
31#16839743C2214yes—
32C2×#14519695C411yescharacteristic, complemented
33C2×#14519695C411yescharacteristic, complemented
34C2×#14519694C411yescharacteristic, complemented
35C2×#100×C61C411yescharacteristic, complemented
36C2×#11758213D412yes—
37C2×#11758213C2312yes—
38D61⋊M16—21no—
39D61⋊M16—21no—
40D61⋊M16—21no—
41D61⋊M16—21no—
42D61⋊M16—21no—
43D61⋊M16—21no—
44D61⋊M16D412yes—
45D61⋊M16C2312yes—
46C2×#11758217—21no—
47C2×#11758217—21no—
48C2×D61⋊C8—21no—
49C2×D61⋊C8—21no—
50#14519856—21no—
51#14519856—21no—
52#14519856—21no—
53#14519856—21no—
54#14519856—21no—
55#14519856D412yes—
56#14519856D412yes—
57#14519856C2312yes—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—21no—
60C2×C22⋊#6923—21no—
61C2×C22⋊#6923—21no—
62C2×C4×#6923—21no—
63C2×C4×#6923C2312yes—
64#14519848—24no—
65#14519847—24no—
66#14519841—28no—
67#14519841—28no—
68#14519831—24no—
69C8×#6923—24no—
70C2×#11758224—21no—
71C2×#11758224—21no—
72C2×#11758224—21no—
73C2×#11758224—21no—
74C2×#11758224—21no—
75C2×#11758224—21no—
76C2×#11758224—21no—
77C2×#11758224—21no—
78C2×#11758224—21no—
79C2×#11758224—21no—
80C2×#11758224C2×C412yes—
81C2×#11758224C2312yes—
82C22×Dic122C2×C411yescharacteristic
83C22×C22⋊D61—21no—
84C23×Dic61—21no—
85C22×C4×D61C2311yescharacteristic
86C2×#11758194C2×C411yescharacteristic
87C2×#11758194C2×C411yescharacteristic
88C2×#11758189—21no—
89C2×#11758189—21no—
90M16×D61—21no—
91M16×D61—21no—
92M16×D61—21no—
93M16×D61—21no—
94M16×D61C2×C412yes—
95M16×D61C2×C412yes—
96M16×D61C2312yes—
97M16×D61C2312yes—
98C2×C8×D61—21no—
99C2×C8×D61—21no—
100#14519695C2×C414yescomplemented
101#14519695C2×C414yescomplemented
102#14519694C2×C414yescomplemented
103C22×D4×C61C2×C411yescharacteristic
104C2×M16×C61C2×C411yescharacteristic
105C2×M16×C61C2×C411yescharacteristic
106#100×C61C2×C414yescomplemented
107C24×C61—21no—
108C23×D61#4111yescharacteristic
109C22×#6923—22no—
110C22×#6923—41no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61C22×C412yes—
117C22×C4×C61C22×C411yescharacteristic
118C2×C22⋊D61—22no—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61#4111yescharacteristic
132#11758224—22no—
133#11758224—22no—
134#11758224—22no—
135#11758224—22no—
136#11758224—22no—
137#11758224—22no—
138#11758224—22no—
139#11758224—22no—
140#11758224—41no—
141#11758224—41no—
142#11758224—41no—
143#11758224—41no—
144#11758224—41no—
145#11758224—41no—
146#11758224—41no—
147#11758224—41no—
148C2×C4×D61C22×C411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61#4111yescharacteristic
154C2×C4×D61#4111yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×C4×D61#4111yescharacteristic
157C2×C4×D61#4111yescharacteristic
158C2×C4×D61#4111yescharacteristic
159C2×C4×D61C2411yescharacteristic
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122C22×C412yes—
166C22⋊#6923—22no—
167C22⋊#6923—22no—
168C22⋊#6923—22no—
169C22⋊#6923—41no—
170#11758217—22no—
171#11758217—22no—
172#11758217—41no—
173#11758217—41no—
174C2×#92420—22no—
175C2×#92420—22no—
176C4⋊#6923—22no—
177C4⋊#6923—22no—
178C4⋊#6923—22no—
179C4⋊#6923—41no—
180C4×#6923—22no—
181C4×#6923—22no—
182C4×#6923—41no—
183C4×#6923C2×D414yes—
184#11758213—22no—
185#11758213—22no—
186#11758213—22no—
187#11758213C2×D414yes—
188D61⋊C8—22no—
189D61⋊C8—22no—
190D61⋊C8—41no—
191D61⋊C8—41no—
192M16×C61—21no—
193M16×C61—21no—
194M16×C61C22×C412yes—
195M16×C61C22×C412yes—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198#11758194—21no—
199#11758194—21no—
200#11758194C22×C412yes—
201#11758194C22×C412yes—
202C2×#92418—21no—
203C2×#92418—21no—
204#11758189—22no—
205#11758189—22no—
206#11758189—22no—
207#11758189—22no—
208C8×D61—22no—
209C8×D61—22no—
210C8×D61—22no—
211C8×D61—22no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—82no—
216C23×C61#11711yescharacteristic
217C22×D61—81no—
218C22×D61#11711yescharacteristic
219C22×D61#12311yescharacteristic
220C22×D61#12311yescharacteristic
221C22×D61#12411yescharacteristic
222C22×D61#12411yescharacteristic
223C22×D61C2×#4111yescharacteristic
224C2×#6923—24no—
225C2×#6923—42no—
226C2×#6923—82no—
227C2×#6923—82no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61#11711yescharacteristic
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C4×D411yescharacteristic
235C2×C4×C61C4×D411yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C22⋊D61—42no—
239C22⋊D61—42no—
240C22⋊D61—42no—
241C22⋊D61—42no—
242C2×Dic61—22no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61#11711yescharacteristic
250C2×Dic61#12111yescharacteristic
251C2×Dic61#12111yescharacteristic
252C2×Dic61#12611yescharacteristic
253C2×Dic61#12611yescharacteristic
254C2×Dic61C2×#4111yescharacteristic
255C4×D61C4×D411yescharacteristic
256C4×D61C4×D411yescharacteristic
257C4×D61C4×D411yescharacteristic
258C4×D61C4×D411yescharacteristic
259C4×D61#12111yescharacteristic
260C4×D61#12111yescharacteristic
261C4×D61#12311yescharacteristic
262C4×D61#12311yescharacteristic
263C4×D61#12411yescharacteristic
264C4×D61#12411yescharacteristic
265C4×D61#12611yescharacteristic
266C4×D61#12611yescharacteristic
267C4×D61C22×D411yescharacteristic
268C4×D61C22×D411yescharacteristic
269C4×D61C2×#4111yescharacteristic
270C4×D61C2×#4111yescharacteristic
271Dic122—22no—
272Dic122—22no—
273Dic122—41no—
274Dic122—41no—
275#92420—24no—
276#92420—42no—
277C8×C61—22no—
278C8×C61—22no—
279#92418—22no—
280#92418—22no—
281#21080—611nomaximal, Sylow
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—82no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61#38811yescharacteristic
288C22×C61#38811yescharacteristic
289C22×C61C2×#11711yescharacteristic
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61#38811yescharacteristic
297C2×D61#38811yescharacteristic
298C2×D61C2×#12311yescharacteristic
299C2×D61C2×#12411yescharacteristic
300#6923—44no—
301#6923—82no—
302C4×C61#38811yescharacteristic
303C4×C61#38811yescharacteristic
304C4×C61C2×C4×D411yescharacteristic
305C4×C61C2×C4×D411yescharacteristic
306Dic61—42no—
307Dic61#38811yescharacteristic
308Dic61#38811yescharacteristic
309Dic61C2×#12111yescharacteristic
310Dic61C2×#12611yescharacteristic
311C2×#430—611no—
312C2×#421—611no—
313C2×#421—611no—
314C2×#413—611no—
315#2873—611no—
316#1932—618no—
317C2×#344—611no—
318#1733—611no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—82no—
323C2×C61C2×#38811yescharacteristic
324D61—21no—
325D61—21no—
326D61—21no—
327D61—21no—
328C22×#41—611no—
329C22×M16—611no—
330C22×M16—611no—
331C22×M16—611no—
332C22×M16—611no—
333C2×#117—611no—
334C2×#117—611no—
335#430—614no—
336#430—1222no—
337C2×#108—1222no—
338#421—614no—
339#421—1222no—
340#421—1222no—
341#421—1222no—
342C2×#104—614no—
343C2×#104—1222no—
344#413—614no—
345#413—614no—
346C2×#101—611no—
347C2×#101—611no—
348C2×#100—611no—
349C2×#100#692311yescharacteristic, complemented
350#405—1222no—
351#405—1222no—
352C4×M16—614no—
353#344—614no—
354#339—614no—
355C61#2108011yescharacteristic, complemented, Sylow
356C2×#41—612no—
357C2×#41—612no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C22×Q8—611no—
369C22×D4—1221no—
370C22×D4C2×#692311yescharacteristic
371C23×C4—611no—
372C23×C4—1221no—
373C2×M16—612no—
374C2×M16—612no—
375C2×M16—612no—
376C2×M16—611no—
377C2×M16—611no—
378C2×M16—612no—
379C2×M16—612no—
380C2×M16—612no—
381C2×M16—612no—
382C2×M16—612no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C2×M16—1221no—
394C2×M16—1221no—
395C2×M16—1221no—
396C2×M16—1221no—
397C2×M16C2×#692311yescharacteristic
398C2×M16C2×#692311yescharacteristic
399C22×C8—1221no—
400C22×C8—1221no—
401C22×C8—1221no—
402C22×C8—1221no—
403#117—612no—
404#117—612no—
405#117—612no—
406#117—1221no—
407#117—1221no—
408#117—1221no—
409#117—1221no—
410#117—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—612no—
416C2×C42—1221no—
417#108—1224no—
418#108—1224no—
419#104—1228no—
420#104—1228no—
421#101—614no—
422#101—614no—
423#100—614no—
424#100C2×#692314yescomplemented
425#97—1224no—
426C4×C8—1224no—
427C24—21no—
428C24—611no—
429#41—1222no—
430#41—1222no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—1222no—
437#41—2441no—
438#41—2441no—
439#41—2441no—
440#41—2441no—
441#41—2441no—
442#41—2441no—
443#41—2441no—
444#41—2441no—
445C2×Q8—612no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—2441no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4C22×#692312yes—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—2441no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4C22×#692311yescharacteristic
486M16—21no—
487M16—21no—
488M16—614no—
489M16—612no—
490M16—612no—
491M16—1222no—
492M16—1222no—
493M16—1222no—
494M16—1222no—
495M16—1222no—
496M16—1222no—
497M16—1222no—
498M16—1221no—
499M16—1221no—
500M16—1222no—
501M16—1222no—
502M16—2441no—
503M16—2441no—
504M16C22×#692312yes—
505M16C22×#692312yes—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1222no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1222no—
518C2×C8—2441no—
519C2×C8—2441no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—2441no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—2441no—
528C42—614no—
529C42—1222no—
530C42—1222no—
531C42—2441no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—82no—
536C23—611no—
537C23—611no—
538C23—611no—
539C23—611no—
540C23—611no—
541C23—611no—
542C23—4881no—
543C23#1451985611yescharacteristic
544Q8—1222no—
545Q8—1222no—
546Q8—2441no—
547Q8—2441no—
548D4—22no—
549D4—22no—
550D4—41no—
551D4—41no—
552D4—2442no—
553D4—2442no—
554D4—2442no—
555D4—2442no—
556C2×C4—611no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—4882no—
581C2×C4—4882no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4#1451985611yescharacteristic
590C2×C4D4×#692311yescharacteristic
591C2×C4D4×#692311yescharacteristic
592C2×C4D4×#692311yescharacteristic
593C2×C4D4×#692311yescharacteristic
594C2×C4C23×#692311yescharacteristic
595C8—22no—
596C8—22no—
597C8—1224no—
598C8—1222no—
599C8—1222no—
600C8—2442no—
601C22—41no—
602C22—42no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—611no—
607C22—611no—
608C22—611no—
609C22—611no—
610C22—2441no—
611C22—2441no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22#1684071211yescharacteristic
617C22#1684071211yescharacteristic
618C22C2×#1451985611yescharacteristic
619C4—611no—
620C4—611no—
621C4—611no—
622C4—611no—
623C4—4882no—
624C4—2444no—
625C4—2442no—
626C4#1684071211yescharacteristic
627C4#1684071211yescharacteristic
628C4C2×D4×#692311yescharacteristic
629C4C2×D4×#692311yescharacteristic
630C2—21no—
631C2—21no—
632C2—21no—
633C2—82no—
634C2—1221no—
635C2—1221no—
636C2—1221no—
637C2—1221no—
638C2C2×#1684071211yescharacteristic
subgroupG/H
centerC2C2×#16840712
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#100×C61C4
socleC2×C61C2×#388
prime radicalC22×#11758224C22
cosocleC2×C4×D61C24
hypercenterC2×#100#6923
hypocenterC61#21080
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. D61⋊#430C2semidirect
4. #21462382C2semidirect
5. #21461907C2semidirect
6. #21461531C2semidirect
7. #21461501C2semidirect
14. #16840777C22semidirect
24. M16×#6923C22semidirect
25. #16840645C22semidirect
26. #16840634C22semidirect
30. #16839743C4semidirect
32. C2×#14519695C4semidirect
33. C2×#14519695C4semidirect
34. C2×#14519694C4semidirect
35. C2×#100×C61C4semidirect
100. #14519695C2×C4semidirect
101. #14519695C2×C4semidirect
102. #14519694C2×C4semidirect
106. #100×C61C2×C4semidirect
349. C2×#100#6923semidirect
355. C61#21080semidirect
424. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4302
C2×#168407602
#214623822
#214619072
#214615312
#214615012
C2×#168397432
#2108061