Finite Group Databasebeta

← #30626535

Subgroups — #30626535

Order 15616.

2130 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
1#21462753C211yesmaximal, characteristic, complemented
2#21462751C211yesmaximal, characteristic, complemented
3#21462741C213yesmaximal, complemented
4#21462737C211yesmaximal, characteristic, complemented
5#21462682C213yesmaximal, complemented
6#21462680C213yesmaximal, complemented
7#21462677C213yesmaximal, complemented
8#21462665C211yesmaximal, characteristic, complemented
9#21462663C211yesmaximal, characteristic, complemented
10#21462630C213yesmaximal, complemented
11#21462625C211yesmaximal, characteristic, complemented
12#21462560C213yesmaximal, complemented
13#21462559C213yesmaximal, complemented
14D61⋊#436C213yesmaximal, complemented
15#21459025C211yesmaximal, characteristic
16C2×D61⋊M16C2216yescomplemented
17#16840814C2211yescharacteristic, complemented
18#16840814C2213yescomplemented
19Q8×#92420C2211yescharacteristic, complemented
20Q8×#92420C2211yescharacteristic, complemented
21Q8×#92420C2211yescharacteristic, complemented
22Q8×#92420C2211yescharacteristic, complemented
23#16840797C2213yescomplemented
24#16840797C2213yescomplemented
25#16840797C2213yescomplemented
26#16840797C2213yescomplemented
27#16840796C2213yescomplemented
28#16840796C2213yescomplemented
29#16840796C2213yescomplemented
30#16840796C2213yescomplemented
31#16840795C2213yescomplemented
32#16840795C2216yescomplemented
33#16840795C2213yescomplemented
34#16840794C2213yescomplemented
35#16840794C2216yescomplemented
36#16840794C2213yescomplemented
37#16840792C2213yescomplemented
38#16840792C2216yescomplemented
39#16840792C2213yescomplemented
40#16840790C2213yescomplemented
41#16840790C2213yescomplemented
42#16840790C2213yescomplemented
43#16840790C2213yescomplemented
44#16840784C2213yescomplemented
45#16840783C2213yescomplemented
46#16840781C2213yescomplemented
47#16840781C2213yescomplemented
48#16840780C2213yescomplemented
49#16840779C2213yescomplemented
50#16840766C2213yescomplemented
51#16840765C2213yescomplemented
52D61⋊(C4⋊C8)C2213yescomplemented
53#16840734C2216yescomplemented
54#16840733C2216yescomplemented
55D61⋊(C22⋊C8)C2216yescomplemented
56C4×#11758213C2213yescomplemented
57C4×#11758217C2213yescomplemented
58C4×D61⋊C8C2213yescomplemented
59C2×#14519823C411yescharacteristic
60#16840244C411yescharacteristic
61#16840244C411yescharacteristic
62#16840243C413yes—
63#16840236C413yes—
64#16840234C413yes—
65#16840234C413yes—
66#41×Dic61C2211yescharacteristic
67C4×#11758226C2211yescharacteristic
68C4×#11758226C2211yescharacteristic
69#16840230C2213yes—
70C2×#14519739C2213yes—
71C4×D4×D61C2213yes—
72C4×D4×D61C2213yes—
73C61×#520C411yescharacteristic
74C2×#11758213C2313yes—
75C2×#11758213C2313yes—
76C2×#11758213C2316yes—
77D61⋊M16—412no—
78C2×#11758217C2313yes—
79C2×#11758217C2313yes—
80C2×#11758217C2316yes—
81C2×D61⋊C8C2313yes—
82C2×D61⋊C8C2313yes—
83C2×D61⋊C8C2316yes—
84#14519844C2313yes—
85#14519844C2313yes—
86#14519844C2311yescharacteristic
87#14519844C2311yescharacteristic
88C4⋊#92420C2313yes—
89C4⋊#92420C2313yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2313yes—
92C4⋊#92420C2311yescharacteristic
93C4⋊#92420C2313yes—
94#14519834C2313yes—
95#14519834C2316yes—
96#14519834C2313yes—
97#14519834C2316yes—
98#14519834C2313yes—
99#14519834C2313yes—
100C22⋊#92420C2313yes—
101C22⋊#92420C2316yes—
102C22⋊#92420C2313yes—
103#14519832C2311yescharacteristic
104#14519832C2313yes—
105#14519832C2311yescharacteristic
106#14519832C2313yes—
107C4×#92420C2311yescharacteristic
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2313yes—
111C4×#92420C2313yes—
112C4×#92420C2313yes—
113#14519823—28no—
114C2×#11758226C2×C411yescharacteristic
115C2×#11758226C2×C411yescharacteristic
116C2×#11758222C2×C413yes—
117C22×D244C2×C413yes—
118C2×D4×D61C2×C413yes—
119C2×D4×D61C2×C413yes—
120C22×C4×D61C2316yes—
121Q8×Dic61C2×C411yescharacteristic
122Q8×Dic61C2311yescharacteristic
123#14519742C2×C413yes—
124#14519742C2311yescharacteristic
125C4×D244C2×C413yes—
126C4×D244C2×C413yes—
127C4×D244C2×C413yes—
128C4×D244C2×C413yes—
129D4×Dic61C2×C413yes—
130D4×Dic61C2313yes—
131#14519739C2×C413yes—
132#14519739C2×C413yes—
133#14519739C2×C416yes—
134#14519739C2×C413yes—
135#14519739C2313yes—
136#14519739C2313yes—
137#14519739C2313yes—
138C4×C22⋊D61C2×C416yes—
139C4×C22⋊D61C2×C416yes—
140#14519737C2×C413yes—
141#14519737C2×C413yes—
142#14519737C2313yes—
143#14519737C2313yes—
144#14519736C2×C413yes—
145#14519735C2×C413yes—
146#14519735C2×C413yes—
147#14519734C2×C413yes—
148#14519734C2×C413yes—
149C4⋊C4×D61C2×C413yes—
150C4⋊C4×D61C2×C413yes—
151C2×#11758198C2×C416yes—
152C22⋊C4×D61C2×C416yes—
153C22⋊C4×D61C2×C416yes—
154C2×C4×Dic61C2313yes—
155C42×D61C2313yes—
156C42×D61C2313yes—
157C2×#41×C61C2×C411yescharacteristic
158C4×Q8×C61C2×C411yescharacteristic
159C4×Q8×C61C2×C411yescharacteristic
160C4×D4×C61C2×C413yes—
161C4×D4×C61C2×C413yes—
162#117×C61C2×C413yes—
163C2×C4⋊C4×C61C2×C413yes—
164C23×D61C22×C416yes—
165#41×C61—24no—
166C2×Q8×C61C22×C411yescharacteristic
167C2×D4×C61C22×C413yes—
168C22×C4×C61C22×C413yes—
169C22×C4×C61C22×C416yes—
170C2×C22⋊D61C22×C413yes—
171C2×C22⋊D61C22×C413yes—
172C22×Dic61C22×C413yes—
173C22×Dic61C2413yes—
174#11758226—24no—
175#11758226M1618yescomplemented
176D4×D61—212no—
177D4×D61—212no—
178#11758222—212no—
179C2×D244—212no—
180C2×D244C22×C413yes—
181C2×D244C22×C413yes—
182C2×D244C22×C413yes—
183C2×C4×D61—46no—
184C2×C4×D61—46no—
185C2×C4×D61C22×C413yes—
186C2×C4×D61C22×C416yes—
187C2×C4×D61C22×C416yes—
188C2×C4×D61C22×C413yes—
189C2×C4×D61C22×C413yes—
190C2×C4×D61C22×C413yes—
191C2×C4×D61C2413yes—
192C2×C4×D61C2413yes—
193C2×Dic122C22×C411yescharacteristic
194#11758217—26no—
195#11758217—43no—
196#11758217—46no—
197C2×#92420C2411yescharacteristic
198C2×#92420C2411yescharacteristic
199C2×#92420C2411yescharacteristic
200C2×#92420C2413yes—
201C2×#92420C2413yes—
202C2×#92420C2413yes—
203C2×#92420C2413yes—
204C2×#92420C2411yescharacteristic
205#11758213—26no—
206#11758213—46no—
207#11758213—43no—
208D61⋊C8—26no—
209D61⋊C8—43no—
210D61⋊C8—46no—
211C4⋊C4×C61C22×C413yes—
212C4⋊C4×C61C22×C413yes—
213C4⋊C4×C61C22×C413yes—
214C4⋊C4×C61C22×C411yescharacteristic
215C22⋊C4×C61C22×C413yes—
216C22⋊C4×C61C22×C413yes—
217C42×C61C22×C413yes—
218C42×C61C22×C413yes—
219C22⋊Dic61C22×C416yes—
220#11758198C22×C413yes—
221#11758198C22×C413yes—
222#11758198C22×C416yes—
223#11758198C22×C416yes—
224#11758198C22×C413yes—
225#11758198C22×C413yes—
226C4⋊Dic61C22×C413yes—
227C4⋊Dic61C22×C413yes—
228#11758196C22×C413yes—
229#11758196C22×C413yes—
230#11758196C22×C411yescharacteristic
231#11758196C22×C411yescharacteristic
232C4×Dic61C22×C413yes—
233C4×Dic61C22×C413yes—
234C4×Dic61C22×C413yes—
235C4×Dic61C22×C413yes—
236C4×Dic61C2413yes—
237C4×Dic61C2411yescharacteristic
238C4×Dic61C2411yescharacteristic
239C4×Dic61C2411yescharacteristic
240C23×C61C23×C413yes—
241C22×D61—212no—
242C22×D61—46no—
243C22×D61—46no—
244C22×D61C23×C413yes—
245C22×D61C23×C413yes—
246Q8×C61C2×M1614yescomplemented
247D4×C61—26no—
248C2×C4×C61—26no—
249C2×C4×C61—46no—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C23×C413yes—
252C2×C4×C61C23×C413yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C23×C413yes—
256C22⋊D61—26no—
257C22⋊D61—26no—
258C2×Dic61—23no—
259C2×Dic61—23no—
260C2×Dic61—43no—
261C2×Dic61C23×C411yescharacteristic
262C2×Dic61C23×C411yescharacteristic
263C2×Dic61C23×C413yes—
264C2×Dic61C23×C413yes—
265C2×Dic61C23×C411yescharacteristic
266C2×Dic61C2511yescharacteristic
267D244—26no—
268D244—26no—
269D244C2×M16112yescomplemented
270C4×D61—26no—
271C4×D61—23no—
272C4×D61—23no—
273C4×D61—23no—
274C4×D61—23no—
275C4×D61—46no—
276C4×D61—46no—
277C4×D61—43no—
278C4×D61—43no—
279C4×D61C2×M16112yescomplemented
280Dic122—22no—
281#92420—23no—
282#92420—23no—
283#92420—23no—
284#92420—21no—
285#92420—23no—
286#92420—21no—
287#92420—21no—
288#92420H(5,𝔽2)12yescomplemented
289#60325—611nomaximal, Sylow
290C22×C61—23no—
291C22×C61—23no—
292C22×C61—23no—
293C22×C61C24×C411yescharacteristic
294C2×D61—23no—
295C2×D61—23no—
296C2×D61—23no—
297C2×D61—23no—
298C2×D61—23no—
299C2×D61—812no—
300C2×D61C22×M1616yes—
301C4×C61—23no—
302C4×C61—23no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61C22×M1616yes—
307Dic61—23no—
308Dic61—21no—
309Dic61—23no—
310Dic61—21no—
311Dic61C22×M1612yes—
312Dic61C2×H(5,𝔽2)11yescharacteristic
313Dic61#58511yescharacteristic
314C4×H(5,𝔽2)—611no—
315#2991—611no—
316#2991—611no—
317#2989—613no—
318#2988—611no—
319#2975—613no—
320#2974—613no—
321#2974—613no—
322#2971—611no—
323#2971—611no—
324#2965—613no—
325C8×#41—611no—
326D4×M16—613no—
327D4×M16—613no—
328C2×#436—613no—
329C2×C61—43no—
330C2×C61C4×H(5,𝔽2)11yescharacteristic
331C2×C61C23×M1611yescharacteristic
332C2×C61#357311yescharacteristic
333D61—26no—
334D61—43no—
335C2×H(5,𝔽2)—611no—
336C22×M16—616no—
337#520—611no—
338#520—611no—
339#520—613no—
340#520#692311yescharacteristic
341#518—613no—
342#518—613no—
343#518—613no—
344C4×#41—611no—
345C4×#41—611no—
346C4×#41—611no—
347C4×#41—613no—
348C2×C4×D4—613no—
349C2×C4×D4—613no—
350C2×C4×D4—613no—
351#446—611no—
352#446—613no—
353C8×Q8—611no—
354C8×Q8—611no—
355C8×Q8—611no—
356C8×Q8—611no—
357#436—613no—
358#436—616no—
359#436—613no—
360#436—613no—
361#436—613no—
362#436—613no—
363#436—613no—
364#436—613no—
365#436—613no—
366#436—613no—
367#436—613no—
368#435—613no—
369#435—616no—
370#435—613no—
371#435—613no—
372#435—616no—
373#435—613no—
374C8×D4—613no—
375C8×D4—613no—
376C8×D4—613no—
377C8×D4—613no—
378#432—613no—
379#432—613no—
380#431—613no—
381#431—613no—
382#431—613no—
383#431—613no—
384#423—613no—
385#423—613no—
386C2×C4⋊C8—613no—
387#407—616no—
388#407—616no—
389C2×C22⋊C8—616no—
390C4×M16—613no—
391C4×M16—613no—
392C2×C4×C8—613no—
393C61#6032511yescharacteristic, complemented, Sylow
394H(5,𝔽2)—1228no—
395C2×#41—611no—
396C2×#41—611no—
397C2×#41—613no—
398C2×#41C2×#692311yescharacteristic
399C22×D4—613no—
400C22×D4—613no—
401C22×D4—613no—
402C23×C4—616no—
403C2×M16—613no—
404C2×M16—613no—
405C2×M16—616no—
406C2×M16—613no—
407C2×M16—616no—
408C2×M16—613no—
409C2×M16—24412no—
410C22×C8—613no—
411C22×C8—616no—
412C22×C8—613no—
413C4×Q8—611no—
414C4×Q8—611no—
415C4×Q8—611no—
416C4×Q8—613no—
417C4×Q8C2×#692311yescharacteristic
418C4×Q8C2×#692311yescharacteristic
419C4×D4—613no—
420C4×D4—613no—
421C4×D4—616no—
422C4×D4—616no—
423C4×D4—613no—
424C4×D4—613no—
425C4×D4—613no—
426C4×D4—613no—
427C4×D4—613no—
428C4×D4—616no—
429C4×D4—613no—
430C4×D4—613no—
431C4×D4—613no—
432C4×D4—613no—
433C4×D4—613no—
434C4×D4—613no—
435C4×D4—613no—
436C4×D4—613no—
437C4×D4—613no—
438C4×D4C2×#692313yes—
439C4×D4C2×#692313yes—
440#117—613no—
441#117—613no—
442#117—613no—
443#117—613no—
444#117—613no—
445#117C2×#692313yes—
446C2×C4⋊C4—613no—
447C2×C4⋊C4—613no—
448C2×C4⋊C4C2×#692313yes—
449C2×C22⋊C4—616no—
450C2×C22⋊C4—616no—
451C2×C22⋊C4—616no—
452C2×C42—613no—
453C2×C42—613no—
454C2×C42—613no—
455C4⋊C8—613no—
456C4⋊C8—613no—
457C4⋊C8—613no—
458C4⋊C8—613no—
459C4⋊C8—611no—
460C4⋊C8—613no—
461C4⋊C8—613no—
462C4⋊C8—613no—
463C4⋊C8—611no—
464C4⋊C8—611no—
465C22⋊C8—613no—
466C22⋊C8—616no—
467C22⋊C8—613no—
468C22⋊C8—613no—
469C22⋊C8—616no—
470C22⋊C8—613no—
471C22⋊C8—613no—
472C22⋊C8—616no—
473C22⋊C8—613no—
474#97—611no—
475#97—611no—
476#97—613no—
477#97—613no—
478C4×C8—611no—
479C4×C8—613no—
480C4×C8—613no—
481C4×C8—613no—
482C4×C8—613no—
483C4×C8—613no—
484C24—616no—
485#41—24no—
486#41—618no—
487#41—12212no—
488#41—1224no—
489C2×Q8—611no—
490C2×Q8C22×#692311yescharacteristic
491C2×D4—613no—
492C2×D4—613no—
493C2×D4—613no—
494C2×D4—613no—
495C2×D4—613no—
496C2×D4—12212no—
497C2×D4—12212no—
498C2×D4—12212no—
499C2×D4C22×#692313yes—
500C22×C4—613no—
501C22×C4—613no—
502C22×C4—613no—
503C22×C4—613no—
504C22×C4—613no—
505C22×C4—613no—
506C22×C4—613no—
507C22×C4—613no—
508C22×C4—616no—
509C22×C4—616no—
510C22×C4—2446no—
511C22×C4—2446no—
512C22×C4C22×#692313yes—
513C22×C4C22×#692316yes—
514M16—1226no—
515M16—1226no—
516M16—2446no—
517M16—2446no—
518M16—2443no—
519M16—2443no—
520C2×C8—611no—
521C2×C8—611no—
522C2×C8—611no—
523C2×C8—611no—
524C2×C8—613no—
525C2×C8—613no—
526C2×C8—613no—
527C2×C8—613no—
528C2×C8—1226no—
529C2×C8—2446no—
530C2×C8—2443no—
531C4⋊C4—613no—
532C4⋊C4—613no—
533C4⋊C4—613no—
534C4⋊C4—611no—
535C4⋊C4—611no—
536C4⋊C4—613no—
537C4⋊C4C22×#692313yes—
538C4⋊C4C22×#692313yes—
539C4⋊C4C22×#692313yes—
540C4⋊C4C22×#692311yescharacteristic
541C22⋊C4—616no—
542C22⋊C4—613no—
543C22⋊C4—616no—
544C22⋊C4—613no—
545C22⋊C4—616no—
546C22⋊C4—613no—
547C22⋊C4—613no—
548C22⋊C4C22×#692313yes—
549C22⋊C4C22×#692313yes—
550C42—611no—
551C42—613no—
552C42—611no—
553C42—611no—
554C42—613no—
555C42—613no—
556C42—613no—
557C42—613no—
558C42C22×#692313yes—
559C42C22×#692313yes—
560C23—613no—
561C23—613no—
562C23—12212no—
563C23—2446no—
564C23—2446no—
565C23C23×#692313yes—
566Q8—1222no—
567Q8D61⋊M1614yescomplemented
568D4—26no—
569D4—6112no—
570D4—1226no—
571D4—1226no—
572D4—1226no—
573D4—1226no—
574C2×C4—26no—
575C2×C4—46no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—6112no—
579C2×C4—611no—
580C2×C4—613no—
581C2×C4—611no—
582C2×C4—613no—
583C2×C4—1223no—
584C2×C4—1223no—
585C2×C4—1223no—
586C2×C4—1223no—
587C2×C4—1223no—
588C2×C4—1223no—
589C2×C4—1226no—
590C2×C4—2443no—
591C2×C4—2443no—
592C2×C4—2446no—
593C2×C4—2446no—
594C2×C4—2443no—
595C2×C4C23×#692311yescharacteristic
596C2×C4C23×#692311yescharacteristic
597C2×C4C23×#692311yescharacteristic
598C2×C4C23×#692313yes—
599C2×C4C23×#692313yes—
600C2×C4C23×#692313yes—
601C8—612no—
602C8—1223no—
603C8—1221no—
604C8—1223no—
605C8—1223no—
606C8—1223no—
607C8—1221no—
608C8—1221no—
609C22—23no—
610C22—23no—
611C22—23no—
612C22—616no—
613C22—1223no—
614C22—1223no—
615C22—1223no—
616C22—1223no—
617C22—1223no—
618C22—48812no—
619C22C24×#692311yescharacteristic
620C4—23no—
621C4—23no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—611no—
626C4—611no—
627C4—612no—
628C4—1223no—
629C4—1221no—
630C4—1223no—
631C4—1221no—
632C4C2×D61⋊M1616yes—
633C2—43no—
634C2—1226no—
635C2—2443no—
636C2H(5,𝔽2)×#692311yescharacteristic
637C2C22×D61⋊M1611yescharacteristic
638C2#2146281511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#520C4
prime radicalC2×#14519823C4
cosocleC2×Dic61C25
hypercenter#520#6923
hypocenterC61#60325
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #21462753C2semidirect
2. #21462751C2semidirect
3. #21462741C2semidirect
4. #21462737C2semidirect
5. #21462682C2semidirect
6. #21462680C2semidirect
7. #21462677C2semidirect
8. #21462665C2semidirect
9. #21462663C2semidirect
10. #21462630C2semidirect
11. #21462625C2semidirect
12. #21462560C2semidirect
13. #21462559C2semidirect
14. D61⋊#436C2semidirect
16. C2×D61⋊M16C22semidirect
17. #16840814C22semidirect
18. #16840814C22semidirect
19. Q8×#92420C22semidirect
20. Q8×#92420C22semidirect
21. Q8×#92420C22semidirect
22. Q8×#92420C22semidirect
23. #16840797C22semidirect
24. #16840797C22semidirect
25. #16840797C22semidirect
26. #16840797C22semidirect
27. #16840796C22semidirect
28. #16840796C22semidirect
29. #16840796C22semidirect
30. #16840796C22semidirect
31. #16840795C22semidirect
32. #16840795C22semidirect
33. #16840795C22semidirect
34. #16840794C22semidirect
35. #16840794C22semidirect
36. #16840794C22semidirect
37. #16840792C22semidirect
38. #16840792C22semidirect
39. #16840792C22semidirect
40. #16840790C22semidirect
41. #16840790C22semidirect
42. #16840790C22semidirect
43. #16840790C22semidirect
44. #16840784C22semidirect
45. #16840783C22semidirect
46. #16840781C22semidirect
47. #16840781C22semidirect
48. #16840780C22semidirect
49. #16840779C22semidirect
50. #16840766C22semidirect
51. #16840765C22semidirect
52. D61⋊(C4⋊C8)C22semidirect
53. #16840734C22semidirect
54. #16840733C22semidirect
55. D61⋊(C22⋊C8)C22semidirect
56. C4×#11758213C22semidirect
57. C4×#11758217C22semidirect
58. C4×D61⋊C8C22semidirect
175. #11758226M16semidirect
246. Q8×C61C2×M16semidirect
269. D244C2×M16semidirect
279. C4×D61C2×M16semidirect
288. #92420H(5,𝔽2)semidirect
393. C61#60325semidirect
567. Q8D61⋊M16semidirect

Maximal subgroups

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

subgroupindex
#214627532
#214627512
#214627412
#214627372
#214626822
#214626802
#214626772
#214626652
#214626632
#214626302
#214626252
#214625602
#214625592
D61⋊#4362
#214590252
#6032561