Finite Group Databasebeta

← #30571123

Subgroups — #30571123

Order 15616.

2042 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459681C211yesmaximal, characteristic, complemented
2C2×#16840433C211yesmaximal, characteristic, complemented
3#21458865C214yesmaximal, complemented
4#21458837C214yesmaximal, complemented
5#21458547C214yesmaximal, complemented
6#21458524C214yesmaximal, complemented
7#21456835C214yesmaximal, complemented
8#21456762C211yesmaximal, characteristic, complemented
9#21456734C211yesmaximal, characteristic, complemented
10#21455954C214yesmaximal, complemented
11#21455827C211yesmaximal, characteristic, complemented
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3100C211yesmaximal, characteristic, complemented
14C22×#11758222—21no—
15#16840498C2212yescomplemented
16#16840487C2212yescomplemented
17#16840482C2212yescomplemented
18#16840481—22no—
19#16840469C2212yescomplemented
20#16840452—22no—
21#16840433—22no—
22#16840433—22no—
23#16840433C2214yes—
24#16840433C2214yes—
25#16840357—22no—
26#16840333—21no—
27#16840319C2211yescharacteristic
28C2×#14519774C2211yescharacteristic
29C2×#14519773C2212yes—
30#16840293C2214yes—
31C2×#14519751—21no—
32C2×#14519750—22no—
33#16840245—21no—
34C2×C4×Dic122C2211yescharacteristic
35C2×#14519739C2212yes—
36#16840150C2218yescomplemented
37#16840138C2212yescomplemented
38#16840135C2212yescomplemented
39#16840125C2218yescomplemented
40#16840110C2212yescomplemented
41#16840109C2212yescomplemented
42#16840092C2212yes—
43#16840065C2218yescomplemented
44#16840052C2212yescomplemented
45#16839976C2218yescomplemented
46#16839967C2212yescomplemented
47#16839963C2214yescomplemented
48#16839961C2218yescomplemented
49#16839957C2214yescomplemented
50#16839955C2218yescomplemented
51#16839915C2214yes—
52#16839913C2212yes—
53#16839895C2214yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839871C2218yescomplemented
57#16839861C2218yescomplemented
58C4×#11758200C2214yescomplemented
59#16839816C2214yes—
60#16839815C2211yescharacteristic
61C61×#532C2211yescharacteristic
62C2×C4×D4×C61C2211yescharacteristic
63C61×#480C2212yes—
64C61×#474C2212yescomplemented
65C61×#459C2212yescomplemented
66C61×#447C2212yes—
67C4×D8×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758222—24no—
70C2×#11758222—24no—
71C2×#11758222—48no—
72C22×Dic122D411yescharacteristic
73C22×D244—21no—
74C22×D244—21no—
75C22×D244C2312yes—
76C22×C22⋊D61—22no—
77C22×C22⋊D61—22no—
78C22×C4×D61—22no—
79C2×#11758190C2318yes—
80C2×#11758200C2314yes—
81C2×#11758199C2318yes—
82#14519777C2311yescharacteristic
83#14519777C2311yescharacteristic
84#14519775—22no—
85#14519774—21no—
86#14519774—21no—
87#14519774C2312yes—
88#14519774C2312yes—
89#14519773—24no—
90#14519773C2318yes—
91#14519770—24no—
92#14519770—24no—
93#14519769—21no—
94#14519768C2314yes—
95#14519767—24no—
96#14519764—22no—
97#14519764C2314yes—
98#14519758—24no—
99#14519758—24no—
100#14519756—21no—
101#14519756—22no—
102#14519756—22no—
103#14519756C2312yes—
104#14519754—24no—
105#14519751—24no—
106#14519751—24no—
107#14519750—28no—
108#14519750—28no—
109Q8×Dic61—22no—
110C4×Dic122—21no—
111C4×Dic122D412yes—
112C4×Dic122D412yes—
113C4×Dic122C2312yes—
114C4×Dic122C2312yes—
115C4×Dic122C2312yes—
116#14519742C2314yes—
117D4×Dic61—24no—
118D4×Dic61—22no—
119D4×Dic61C2314yes—
120#14519739—24no—
121#14519739C2318yes—
122#14519736—21no—
123#14519736—22no—
124#14519733—24no—
125C2×C4⋊Dic61—22no—
126C2×C4⋊Dic61D411yescharacteristic
127C2×#11758196D412yes—
128C2×#11758198—22no—
129C2×C4×Dic61C2312yes—
130#14519708C2314yes—
131C4⋊#92418C2312yes—
132#14519706C2318yes—
133#14519700C2314yes—
134#14519699C2318yes—
135#14519698C2314yes—
136#14519697C2318yes—
137C22⋊#92418C2312yes—
138#14519687C2318yes—
139C4×#92418C2312yes—
140C2×#41×C61—21no—
141C22×D4×C61C2311yescharacteristic
142C23×C4×C61—21no—
143C2×D8×C61C2314yes—
144C4⋊Q8×C61C2311yescharacteristic
145#127×C61C2311yescharacteristic
146#124×C61—21no—
147#121×C61—21no—
148#121×C61C2312yes—
149C4×D4×C61—21no—
150C4×D4×C61—21no—
151C4×D4×C61—21no—
152C4×D4×C61C2312yes—
153C4×D4×C61C2312yes—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61C2311yescharacteristic
156C2×C22⋊C4×C61—21no—
157C2×C42×C61C2311yescharacteristic
158#107×C61C2314yes—
159C4⋊C8×C61C2312yes—
160#102×C61C2314yes—
161#102×C61C2314yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C24×C61—21no—
165C23×D61—22no—
166#41×C61—44no—
167C2×Q8×C61—21no—
168C2×D4×C61—22no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61—42no—
173C2×D4×C61C2412yes—
174C2×D4×C61C2412yes—
175C22×C4×C61—21no—
176C22×C4×C61—22no—
177C22×C4×C61—22no—
178C22×C4×C61—22no—
179C22×C4×C61—44no—
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2411yescharacteristic
183C2×C22⋊D61—28no—
184C2×C22⋊D61—28no—
185C2×C22⋊D61—48no—
186C2×C22⋊D61—84no—
187C22×Dic61—24no—
188C22×Dic61C2×D412yes—
189C22×Dic61C2×D412yes—
190#11758222—416no—
191#11758222—416no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—22no—
195C2×D244—22no—
196C2×D244—22no—
197C2×D244—44no—
198C2×D244—44no—
199C2×D244—82no—
200C2×D244C2414yes—
201C2×C4×D61—28no—
202C2×C4×D61—48no—
203C2×Dic122—24no—
204C2×Dic122—21no—
205C2×Dic122C2×D412yes—
206C2×Dic122C2×D412yes—
207C2×Dic122C2412yes—
208D8×C61—44no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61C2412yes—
213C4⋊C4×C61C2412yes—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—22no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221C42×C61C2411yescharacteristic
222C22⋊Dic61—22no—
223C22⋊Dic61—24no—
224C22⋊Dic61—24no—
225C22⋊Dic61—24no—
226#11758200—44no—
227#11758199—48no—
228#11758198—28no—
229C4⋊Dic61—21no—
230C4⋊Dic61—21no—
231C4⋊Dic61—21no—
232C4⋊Dic61—24no—
233C4⋊Dic61C2×D412yes—
234C4⋊Dic61C2×D412yes—
235C4⋊Dic61C2412yes—
236#11758196—22no—
237#11758196—22no—
238#11758196—24no—
239#11758196—24no—
240#11758196C2×D414yes—
241#11758196C2×D414yes—
242C4×Dic61—22no—
243C4×Dic61—24no—
244C4×Dic61C2×D414yes—
245C4×Dic61C2414yes—
246C2×#92418C2414yes—
247#11758190—48no—
248C23×C61—22no—
249C23×C61—22no—
250C23×C61—42no—
251C23×C61—42no—
252C23×C61—42no—
253C23×C61C22×D411yescharacteristic
254C22×D61—24no—
255C22×D61—44no—
256C22×D61—44no—
257C22×D61—44no—
258Q8×C61—42no—
259D4×C61—24no—
260D4×C61—22no—
261D4×C61—22no—
262D4×C61—42no—
263D4×C61—44no—
264D4×C61—42no—
265D4×C61—81no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—22no—
269C2×C4×C61—41no—
270C2×C4×C61—42no—
271C2×C4×C61—44no—
272C2×C4×C61—44no—
273C2×C4×C61—44no—
274C2×C4×C61AGL(1,ℤ/8)14yes—
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C2511yescharacteristic
281C22⋊D61—416no—
282C22⋊D61—88no—
283C2×Dic61—24no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—42no—
287C2×Dic61—84no—
288C2×Dic61C22×D414yes—
289C2×Dic61C22×D414yes—
290D244—28no—
291D244—44no—
292D244—44no—
293D244—44no—
294D244—82no—
295D244—82no—
296C4×D61—416no—
297Dic122—24no—
298Dic122—24no—
299Dic122—24no—
300Dic122—81no—
301C8×C61—24no—
302#92418—24no—
303#61890—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—84no—
313C22×C61C2×AGL(1,ℤ/8)12yes—
314C22×C61C23×D411yescharacteristic
315C2×D61—44no—
316C2×D61—44no—
317C2×D61—44no—
318C2×D61—88no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—42no—
323C4×C61—42no—
324C4×C61C2×AGL(1,ℤ/8)12yes—
325C4×C61C2×AGL(1,ℤ/8)12yes—
326C4×C61#58512yes—
327Dic61—24no—
328Dic61—24no—
329Dic61—42no—
330Dic61—42no—
331#3507—611no—
332C2×#550—611no—
333#3399—614no—
334#3397—614no—
335#3368—614no—
336#3366—614no—
337#3217—614no—
338#3208—611no—
339#3205—611no—
340#3114—614no—
341#3101—611no—
342#3100D6111yescharacteristic, complemented
343#2983—611no—
344C2×C61—22no—
345C2×C61—82no—
346C2×C61—82no—
347C2×C61#348711yescharacteristic
348C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
349C2×C61#358511yescharacteristic
350D61—84no—
351C22×#41—1221no—
352#559—612no—
353#557—612no—
354#556—612no—
355#556—1222no—
356#554—612no—
357#552—1222no—
358#550—614no—
359#550—614no—
360#550—1222no—
361#550—1222no—
362#539—1222no—
363#536—1221no—
364#533—611no—
365#532C2×D6111yescharacteristic
366C2×#127—612no—
367C2×#127—611no—
368#529—614no—
369C2×#121—1222no—
370C2×#121—1221no—
371#520—1221no—
372C2×C4×Q8—611no—
373C2×C4×D4—612no—
374C2×C4×D4C2×D6111yescharacteristic
375#496—618no—
376#493—612no—
377#492—612no—
378#490—618no—
379#486—612no—
380#486—612no—
381#483—612no—
382#480C2×D6112yes—
383#478—618no—
384#476—612no—
385#474C2×D6112yescomplemented
386#463—618no—
387#461—612no—
388#460—614no—
389#460—618no—
390#459—618no—
391#459—614no—
392#459C2×D6112yescomplemented
393#450—612no—
394#450—614no—
395#447—614no—
396#447C2×D6112yes—
397#446—614no—
398#446—614no—
399#442—618no—
400#440—618no—
401C4×SD16—614no—
402C4×D8C2×D6114yescomplemented
403#432—614no—
404#431—611no—
405#431C2×D6111yescharacteristic
406C61#6189011yescharacteristic, complemented, Sylow
407C2×#41—21no—
408C2×#41—1224no—
409C2×#41—1224no—
410C2×#41—2448no—
411C22×Q8—611no—
412C22×D4—612no—
413C22×D4—1222no—
414C22×D4—1222no—
415C22×D4—1221no—
416C22×D4—1221no—
417C22×D4C22×D6111yescharacteristic
418C23×C4—21no—
419C23×C4—1222no—
420C2×SD16—614no—
421C2×SD16—618no—
422C2×D8—618no—
423C2×D8C22×D6114yes—
424C4⋊Q8—611no—
425C4⋊Q8—611no—
426C4⋊Q8—1222no—
427C4⋊Q8C22×D6111yescharacteristic
428#127—618no—
429#127—612no—
430#127—612no—
431#127—1224no—
432#127—1221no—
433#127—1221no—
434#127C22×D6111yescharacteristic
435#126—1224no—
436#126—1224no—
437#125—614no—
438#125—1224no—
439#125—1221no—
440#124—21no—
441#124—614no—
442#124—1222no—
443#123—1224no—
444#123—1224no—
445C22⋊Q8—612no—
446C22⋊Q8—1224no—
447C22⋊Q8—1222no—
448C22⋊Q8—1222no—
449C22⋊Q8—1221no—
450#121—21no—
451#121—1228no—
452#121—1228no—
453#121—1224no—
454#121—1224no—
455#121C22×D6112yes—
456C4×Q8—612no—
457C4×Q8—612no—
458C4×Q8—612no—
459C4×Q8—612no—
460C4×Q8—614no—
461C4×Q8—612no—
462C4×Q8—1222no—
463C4×Q8—1221no—
464C4×D4—21no—
465C4×D4—21no—
466C4×D4—21no—
467C4×D4—614no—
468C4×D4—618no—
469C4×D4—1224no—
470C4×D4—1222no—
471C4×D4—1224no—
472C4×D4C22×D6112yes—
473C4×D4C22×D6112yes—
474C4×D4C22×D6112yes—
475#117—1224no—
476#117—1222no—
477#117—1221no—
478C2×C4⋊C4—611no—
479C2×C4⋊C4—612no—
480C2×C4⋊C4—1222no—
481C2×C4⋊C4C22×D6111yescharacteristic
482C2×C22⋊C4—21no—
483C2×C22⋊C4—1222no—
484C2×C42—612no—
485C2×C42C22×D6111yescharacteristic
486#107C22×D6114yes—
487#106—614no—
488C4⋊C8—618no—
489C4⋊C8—612no—
490C4⋊C8C22×D6112yes—
491#103—614no—
492#102—614no—
493#102—618no—
494#102—618no—
495#102C22×D6114yes—
496#102C22×D6114yes—
497C22⋊C8—612no—
498C22⋊C8C22×D6112yes—
499#97—618no—
500C4×C8—612no—
501C4×C8C22×D6112yes—
502C24—21no—
503C24—1222no—
504#41—44no—
505#41—24416no—
506#41—24416no—
507C2×Q8—21no—
508C2×Q8—612no—
509C2×Q8—612no—
510C2×Q8—612no—
511C2×Q8—1221no—
512C2×Q8—1224no—
513C2×D4—22no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—21no—
517C2×D4—42no—
518C2×D4—614no—
519C2×D4—1228no—
520C2×D4—1228no—
521C2×D4—1222no—
522C2×D4—1222no—
523C2×D4—1222no—
524C2×D4—1222no—
525C2×D4—1222no—
526C2×D4—4882no—
527C2×D4—4884no—
528C2×D4—2444no—
529C2×D4—2448no—
530C2×D4—2444no—
531C2×D4C23×D6112yes—
532C2×D4C23×D6112yes—
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—22no—
537C22×C4—44no—
538C22×C4—612no—
539C22×C4—612no—
540C22×C4—1228no—
541C22×C4—1224no—
542C22×C4—2448no—
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546SD16—2448no—
547SD16—2444no—
548D8—44no—
549D8—2448no—
550C2×C8—614no—
551C2×C8C23×D6114yes—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—612no—
555C4⋊C4—614no—
556C4⋊C4—614no—
557C4⋊C4—612no—
558C4⋊C4—612no—
559C4⋊C4—1224no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1221no—
563C4⋊C4—1222no—
564C4⋊C4—1222no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4C23×D6112yes—
568C4⋊C4C23×D6112yes—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—1228no—
574C22⋊C4—1224no—
575C22⋊C4—1224no—
576C22⋊C4—1224no—
577C22⋊C4—1222no—
578C42—614no—
579C42—614no—
580C42—1224no—
581C42—1222no—
582C42D4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C42C23×D6111yescharacteristic
586C23—22no—
587C23—22no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—1224no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23C2×D4×D6111yescharacteristic
596Q8—42no—
597Q8—1224no—
598Q8—1224no—
599Q8—1224no—
600Q8—4881no—
601D4—22no—
602D4—24no—
603D4—22no—
604D4—42no—
605D4—44no—
606D4—42no—
607D4—81no—
608D4—1228no—
609D4—4882no—
610D4—4882no—
611D4—4888no—
612D4—2444no—
613D4—24416no—
614D4—2444no—
615D4—2444no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—614no—
625C2×C4—614no—
626C2×C4—1224no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—4884no—
630C2×C4—24416no—
631C2×C4—2442no—
632C2×C4#1451980114yes—
633C2×C4C2×D4×D6111yescharacteristic
634C2×C4C2×D4×D6111yescharacteristic
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C24×D6111yescharacteristic
639C8—24no—
640C8—1224no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—84no—
650C22—4888no—
651C22—2444no—
652C22—2444no—
653C22—2444no—
654C22C2×#1451980112yes—
655C22C22×D4×D6111yescharacteristic
656C4—21no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—42no—
661C4—1224no—
662C4—1224no—
663C4—2442no—
664C4—2442no—
665C4C2×#1451980112yes—
666C4C2×#1451980112yes—
667C4#1684059812yes—
668C2—22no—
669C2—82no—
670C2—82no—
671C2—4884no—
672C2#2145947311yescharacteristic
673C2C22×#1451980111yescharacteristic
674C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3100C2
socleC22×C61C23×D4
hypercenter#3100D61
hypocenterC61#61890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21459681C2semidirect
2. C2×#16840433C2semidirect
3. #21458865C2semidirect
4. #21458837C2semidirect
5. #21458547C2semidirect
6. #21458524C2semidirect
7. #21456835C2semidirect
8. #21456762C2semidirect
9. #21456734C2semidirect
10. #21455954C2semidirect
11. #21455827C2semidirect
12. #21454902C2semidirect
13. C61×#3100C2semidirect
15. #16840498C22semidirect
16. #16840487C22semidirect
17. #16840482C22semidirect
19. #16840469C22semidirect
36. #16840150C22semidirect
37. #16840138C22semidirect
38. #16840135C22semidirect
39. #16840125C22semidirect
40. #16840110C22semidirect
41. #16840109C22semidirect
43. #16840065C22semidirect
44. #16840052C22semidirect
45. #16839976C22semidirect
46. #16839967C22semidirect
47. #16839963C22semidirect
48. #16839961C22semidirect
49. #16839957C22semidirect
50. #16839955C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839871C22semidirect
57. #16839861C22semidirect
58. C4×#11758200C22semidirect
64. C61×#474C22semidirect
65. C61×#459C22semidirect
67. C4×D8×C61C22semidirect
342. #3100D61semidirect
385. #474C2×D61semidirect
392. #459C2×D61semidirect
402. C4×D8C2×D61semidirect
406. C61#61890semidirect

Maximal subgroups

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

subgroupindex
#214596812
C2×#168404332
#214588652
#214588372
#214585472
#214585242
#214568352
#214567622
#214567342
#214559542
#214558272
#214549022
C61×#31002
#6189061