Finite Group Databasebeta

← #30571130

Subgroups — #30571130

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#21459686C211yesmaximal, characteristic, complemented
2C2×#16840432C211yesmaximal, characteristic, complemented
3#21458860C214yesmaximal, complemented
4#21458844C214yesmaximal, complemented
5#21458552C214yesmaximal, complemented
6#21458531C214yesmaximal, complemented
7#21456840C214yesmaximal, complemented
8#21456737C211yesmaximal, characteristic, complemented
9#21455958C214yesmaximal, complemented
10#21455823C211yesmaximal, characteristic, complemented
11#21455820C211yesmaximal, characteristic, complemented
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3208C211yesmaximal, characteristic, complemented
14C22×#11758224—21no—
15#16840497C2212yescomplemented
16#16840491C2212yescomplemented
17#16840486C2212yescomplemented
18#16840485—22no—
19#16840473C2212yescomplemented
20#16840450—22no—
21#16840432—22no—
22#16840432—22no—
23#16840432C2214yes—
24#16840432C2214yes—
25#16840361—22no—
26#16840338—21no—
27#16840312C2211yescharacteristic
28C2×#14519773C2212yes—
29#16840298C2214yes—
30C2×#14519749—22no—
31C2×#14519748—21no—
32#16840242—21no—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244C2211yescharacteristic
35C2×D4×Dic61C2212yes—
36#16840148C2218yescomplemented
37#16840138C2212yescomplemented
38#16840125C2218yescomplemented
39#16840108C2212yescomplemented
40#16840090C2212yes—
41#16840076C2212yes—
42#16840060C2218yescomplemented
43#16840049C2212yescomplemented
44#16840043C2212yescomplemented
45#16839979C2218yescomplemented
46#16839964C2212yescomplemented
47#16839960C2218yescomplemented
48#16839958C2212yescomplemented
49#16839956C2218yescomplemented
50#16839954C2214yescomplemented
51#16839912C2212yes—
52#16839898C2212yes—
53#16839896C2214yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839871C2218yescomplemented
57#16839863C2218yescomplemented
58C4×#11758201C2214yescomplemented
59C4×D488C2214yescomplemented
60#16839816C2214yes—
61#16839815C2211yescharacteristic
62C61×#533C2211yescharacteristic
63C2×#127×C61C2211yescharacteristic
64C61×#492C2212yescomplemented
65C61×#486C2212yescomplemented
66C61×#460C2214yescomplemented
67C61×#450C2214yes—
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C2×#11758224—24no—
71C2×#11758224—24no—
72C2×#11758224—48no—
73C22×Dic122D411yescharacteristic
74C22×D244C2311yescharacteristic
75C22×C22⋊D61—22no—
76C22×C22⋊D61—22no—
77C23×Dic61—22no—
78C22×C4×D61—21no—
79C2×#11758201C2314yes—
80C2×D488C2314yes—
81C2×#11758199C2318yes—
82#14519777—22no—
83#14519777C2311yescharacteristic
84#14519774C2311yescharacteristic
85#14519773—24no—
86#14519773C2318yes—
87#14519772—24no—
88#14519772—24no—
89#14519768—24no—
90#14519768C2314yes—
91#14519766C2314yes—
92#14519765—22no—
93#14519765—21no—
94#14519762—24no—
95#14519761—24no—
96#14519757—22no—
97#14519757—22no—
98#14519753—24no—
99#14519751—21no—
100#14519751C2312yes—
101#14519749—28no—
102#14519749—28no—
103#14519748—24no—
104#14519748—24no—
105Q8×Dic61C2314yes—
106C4×Dic122—22no—
107C4×Dic122D412yes—
108C4×Dic122D412yes—
109C4×Dic122C2312yes—
110C4×Dic122C2312yes—
111C4×D244—21no—
112C4×D244—21no—
113C4×D244—21no—
114C4×D244—22no—
115C4×D244C2312yes—
116C4×D244C2312yes—
117C4×D244C2312yes—
118D4×Dic61—24no—
119D4×Dic61C2318yes—
120#14519739C2314yes—
121#14519737—24no—
122#14519735—22no—
123#14519735—24no—
124C2×C4⋊Dic61D411yescharacteristic
125C2×C4⋊Dic61C2311yescharacteristic
126C2×#11758196D412yes—
127C4⋊C4×D61—22no—
128C2×C22⋊Dic61—22no—
129C2×#11758198—21no—
130C2×C4×Dic61C2312yes—
131#14519711C2314yes—
132#14519708C2314yes—
133C4⋊#92418C2312yes—
134#14519706C2318yes—
135#14519701C2314yes—
136#14519699C2314yes—
137#14519699C2314yes—
138#14519698C2318yes—
139#14519697C2314yes—
140C22⋊#92418C2312yes—
141#14519687C2318yes—
142C4×#92418C2312yes—
143C22×D4×C61—21no—
144C22×D4×C61—21no—
145C22×D4×C61C2312yes—
146C2×SD16×C61C2318yes—
147C4⋊Q8×C61C2311yescharacteristic
148C4⋊Q8×C61C2311yescharacteristic
149#127×C61—21no—
150#127×C61—21no—
151#127×C61C2312yes—
152#127×C61C2312yes—
153#125×C61—21no—
154C22⋊Q8×C61—21no—
155C22⋊Q8×C61C2312yes—
156C4×Q8×C61—21no—
157C4×Q8×C61C2312yes—
158#117×C61—21no—
159C2×C42×C61C2311yescharacteristic
160C4⋊C8×C61C2312yes—
161#102×C61C2318yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C24×C61—22no—
165C23×D61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61C2412yes—
168C2×D4×C61—22no—
169C2×D4×C61—22no—
170C2×D4×C61—22no—
171C2×D4×C61—22no—
172C2×D4×C61—22no—
173C2×D4×C61—44no—
174C2×D4×C61—44no—
175C2×D4×C61—82no—
176C2×D4×C61C2414yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2411yescharacteristic
180C2×C22⋊D61—22no—
181C2×C22⋊D61—28no—
182C2×C22⋊D61—28no—
183C2×C22⋊D61—48no—
184C2×C22⋊D61—84no—
185C22×Dic61—21no—
186C22×Dic61—28no—
187C22×Dic61—48no—
188C22×Dic61C2×D412yes—
189C22×Dic61C2×D412yes—
190#11758224—416no—
191#11758224—416no—
192#11758222—44no—
193C2×D244—21no—
194C2×D244—21no—
195C2×D244—21no—
196C2×D244—42no—
197C2×D244C2412yes—
198C2×D244C2412yes—
199C2×C4×D61—22no—
200C2×C4×D61—22no—
201C2×C4×D61—22no—
202C2×C4×D61—24no—
203C2×C4×D61—44no—
204C2×Dic122—24no—
205C2×Dic122—21no—
206C2×Dic122C2×D412yes—
207C2×Dic122C2×D412yes—
208SD16×C61—48no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—22no—
215C4⋊C4×C61C2412yes—
216C22⋊C4×C61—22no—
217C42×C61—22no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221C42×C61C2411yescharacteristic
222C22⋊Dic61—28no—
223#11758201—44no—
224#11758199—48no—
225#11758198—22no—
226#11758198—22no—
227#11758198—22no—
228#11758198—22no—
229#11758198—24no—
230#11758198—24no—
231#11758198—24no—
232C4⋊Dic61—21no—
233C4⋊Dic61—21no—
234C4⋊Dic61—24no—
235C4⋊Dic61—24no—
236C4⋊Dic61C2×D412yes—
237C4⋊Dic61C2×D412yes—
238C4⋊Dic61C2412yes—
239C4⋊Dic61C2412yes—
240#11758196—24no—
241#11758196C2×D414yes—
242#11758196C2×D414yes—
243C4×Dic61—24no—
244C4×Dic61C2×D414yes—
245C4×Dic61C2414yes—
246C2×#92418C2414yes—
247D488—44no—
248C23×C61—24no—
249C23×C61—44no—
250C23×C61—44no—
251C23×C61—44no—
252C23×C61C22×D411yescharacteristic
253C22×D61—22no—
254C22×D61—22no—
255C22×D61—42no—
256C22×D61—42no—
257C22×D61—42no—
258Q8×C61—24no—
259Q8×C61—81no—
260D4×C61—28no—
261D4×C61—44no—
262D4×C61—44no—
263D4×C61—44no—
264D4×C61—82no—
265D4×C61—82no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—22no—
269C2×C4×C61—41no—
270C2×C4×C61AGL(1,ℤ/8)14yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2511yescharacteristic
277C22⋊D61—416no—
278C22⋊D61—44no—
279C22⋊D61—88no—
280C2×Dic61—24no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—42no—
284C2×Dic61—416no—
285C2×Dic61—42no—
286C2×Dic61C22×D414yes—
287C2×Dic61C22×D414yes—
288D244—22no—
289D244—24no—
290D244—22no—
291D244—42no—
292D244—42no—
293D244—81no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61—84no—
298Dic122—24no—
299Dic122—24no—
300Dic122—42no—
301C8×C61—24no—
302#92418—24no—
303#61890—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—44no—
307C22×C61—44no—
308C22×C61—44no—
309C22×C61—88no—
310C22×C61C2×AGL(1,ℤ/8)12yes—
311C22×C61C23×D411yescharacteristic
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—84no—
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#3208D6111yescharacteristic, complemented
339#3205—611no—
340#3114—614no—
341#3101—611no—
342#3100—611no—
343#2983—611no—
344C2×C61—22no—
345C2×C61—84no—
346C2×C61#348711yescharacteristic
347C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
348C2×C61#358511yescharacteristic
349D61—82no—
350D61—82no—
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#533C2×D6111yescharacteristic
365#532—611no—
366C2×#127—612no—
367C2×#127C2×D6111yescharacteristic
368#529—614no—
369C2×#121—1221no—
370C2×#121—1222no—
371#520—1221no—
372C2×C4×Q8—611no—
373C2×C4×D4—612no—
374C2×C4×D4—611no—
375#496—618no—
376#493—612no—
377#492C2×D6112yescomplemented
378#490—618no—
379#486—612no—
380#486C2×D6112yescomplemented
381#483—612no—
382#480—612no—
383#478—618no—
384#476—612no—
385#474—612no—
386#463—618no—
387#461—612no—
388#460—618no—
389#460C2×D6114yescomplemented
390#459—614no—
391#459—612no—
392#459—618no—
393#450—612no—
394#450C2×D6114yes—
395#447—614no—
396#447—612no—
397#446—614no—
398#446—614no—
399#442—618no—
400#440—618no—
401C4×SD16—614no—
402C4×D8—614no—
403#432—614no—
404#431—611no—
405#431C2×D6111yescharacteristic
406C61#6189011yescharacteristic, complemented, Sylow
407C2×#41—1224no—
408C2×#41—1224no—
409C2×#41—1221no—
410C2×#41—2448no—
411C22×Q8—611no—
412C22×D4—21no—
413C22×D4—21no—
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4—1222no—
417C22×D4C22×D6112yes—
418C23×C4—1222no—
419C23×C4—1221no—
420C2×SD16—614no—
421C2×SD16C22×D6118yes—
422C2×D8—618no—
423C2×D8—614no—
424C4⋊Q8—611no—
425C4⋊Q8—1222no—
426C4⋊Q8C22×D6111yescharacteristic
427C4⋊Q8C22×D6111yescharacteristic
428#127—21no—
429#127—21no—
430#127—618no—
431#127—611no—
432#127—1224no—
433#127C22×D6112yes—
434#127C22×D6112yes—
435#126—1224no—
436#126—1224no—
437#125—21no—
438#125—614no—
439#125—1224no—
440#124—614no—
441#124—1222no—
442#124—1221no—
443#123—1224no—
444#123—1224no—
445C22⋊Q8—21no—
446C22⋊Q8—1222no—
447C22⋊Q8—1222no—
448C22⋊Q8—1224no—
449C22⋊Q8C22×D6112yes—
450#121—612no—
451#121—1224no—
452#121—1224no—
453#121—1228no—
454#121—1228no—
455#121—1221no—
456C4×Q8—21no—
457C4×Q8—612no—
458C4×Q8—612no—
459C4×Q8—612no—
460C4×Q8—612no—
461C4×Q8—614no—
462C4×Q8—1222no—
463C4×Q8C22×D6112yes—
464C4×D4—618no—
465C4×D4—614no—
466C4×D4—612no—
467C4×D4—612no—
468C4×D4—612no—
469C4×D4—1222no—
470C4×D4—1224no—
471C4×D4—1224no—
472C4×D4—1221no—
473C4×D4—1221no—
474C4×D4—1221no—
475#117—21no—
476#117—1224no—
477#117—1222no—
478C2×C4⋊C4—611no—
479C2×C4⋊C4—612no—
480C2×C4⋊C4—611no—
481C2×C4⋊C4—1222no—
482C2×C22⋊C4—1222no—
483C2×C22⋊C4—1221no—
484C2×C42—612no—
485C2×C42C22×D6111yescharacteristic
486#107—614no—
487#106—614no—
488C4⋊C8—618no—
489C4⋊C8—612no—
490C4⋊C8C22×D6112yes—
491#103—614no—
492#102—618no—
493#102—614no—
494#102—614no—
495#102—614no—
496#102C22×D6118yes—
497C22⋊C8—612no—
498C22⋊C8C22×D6112yes—
499#97—618no—
500C4×C8—612no—
501C4×C8C22×D6112yes—
502C24—22no—
503C24—1221no—
504#41—24416no—
505#41—24416no—
506#41—2444no—
507C2×Q8—21no—
508C2×Q8—612no—
509C2×Q8—612no—
510C2×Q8—1221no—
511C2×Q8—1224no—
512C2×Q8C23×D6112yes—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—44no—
519C2×D4—44no—
520C2×D4—82no—
521C2×D4—612no—
522C2×D4—612no—
523C2×D4—1228no—
524C2×D4—1228no—
525C2×D4—1222no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4—1221no—
529C2×D4—4884no—
530C2×D4—2442no—
531C2×D4—2448no—
532C2×D4C23×D6114yes—
533C22×C4—612no—
534C22×C4—612no—
535C22×C4—1224no—
536C22×C4—1228no—
537C22×C4—1221no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—2448no—
542C22×C4—2444no—
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546SD16—48no—
547SD16—2444no—
548D8—2444no—
549D8—2448no—
550C2×C8—614no—
551C2×C8C23×D6114yes—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—612no—
558C4⋊C4—614no—
559C4⋊C4—614no—
560C4⋊C4—612no—
561C4⋊C4—612no—
562C4⋊C4—612no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1224no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4C23×D6112yes—
569C22⋊C4—22no—
570C22⋊C4—1224no—
571C22⋊C4—1224no—
572C22⋊C4—1224no—
573C22⋊C4—1228no—
574C22⋊C4—1222no—
575C22⋊C4—1222no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C42—22no—
579C42—614no—
580C42—614no—
581C42—1224no—
582C42D4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C42C23×D6111yescharacteristic
586C23—24no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—1222no—
591C23—1222no—
592C23—2442no—
593C23—2442no—
594C23—2442no—
595C23C2×D4×D6111yescharacteristic
596Q8—24no—
597Q8—81no—
598Q8—1224no—
599Q8—1224no—
600Q8—2442no—
601D4—28no—
602D4—44no—
603D4—44no—
604D4—44no—
605D4—82no—
606D4—82no—
607D4—1224no—
608D4—1222no—
609D4—1222no—
610D4—4881no—
611D4—4888no—
612D4—2442no—
613D4—24416no—
614D4—2444no—
615D4—2442no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—41no—
620C2×C4—614no—
621C2×C4—614no—
622C2×C4—1224no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—4884no—
626C2×C4—2442no—
627C2×C4—24416no—
628C2×C4—2442no—
629C2×C4—2444no—
630C2×C4—2444no—
631C2×C4—2444no—
632C2×C4#1451980414yes—
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—44no—
644C22—44no—
645C22—44no—
646C22—88no—
647C22—4884no—
648C22—2442no—
649C22—2442no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22—2442no—
654C22C2×#1451980412yes—
655C22C22×D4×D6111yescharacteristic
656C4—21no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—42no—
661C4—1224no—
662C4—1224no—
663C4—2442no—
664C4—2442no—
665C4C2×#1451980412yes—
666C4C2×#1451980412yes—
667C4#1684059712yes—
668C2—22no—
669C2—84no—
670C2—4882no—
671C2—4882no—
672C2#2145947511yescharacteristic
673C2C22×#1451980411yescharacteristic
674C2#2146059511yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3208C2
socleC22×C61C23×D4
hypercenter#3208D61
hypocenterC61#61890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21459686C2semidirect
2. C2×#16840432C2semidirect
3. #21458860C2semidirect
4. #21458844C2semidirect
5. #21458552C2semidirect
6. #21458531C2semidirect
7. #21456840C2semidirect
8. #21456737C2semidirect
9. #21455958C2semidirect
10. #21455823C2semidirect
11. #21455820C2semidirect
12. #21454902C2semidirect
13. C61×#3208C2semidirect
15. #16840497C22semidirect
16. #16840491C22semidirect
17. #16840486C22semidirect
19. #16840473C22semidirect
36. #16840148C22semidirect
37. #16840138C22semidirect
38. #16840125C22semidirect
39. #16840108C22semidirect
42. #16840060C22semidirect
43. #16840049C22semidirect
44. #16840043C22semidirect
45. #16839979C22semidirect
46. #16839964C22semidirect
47. #16839960C22semidirect
48. #16839958C22semidirect
49. #16839956C22semidirect
50. #16839954C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839871C22semidirect
57. #16839863C22semidirect
58. C4×#11758201C22semidirect
59. C4×D488C22semidirect
64. C61×#492C22semidirect
65. C61×#486C22semidirect
66. C61×#460C22semidirect
338. #3208D61semidirect
377. #492C2×D61semidirect
380. #486C2×D61semidirect
389. #460C2×D61semidirect
406. C61#61890semidirect

Maximal subgroups

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

subgroupindex
#214596862
C2×#168404322
#214588602
#214588442
#214585522
#214585312
#214568402
#214567372
#214559582
#214558232
#214558202
#214549022
C61×#32082
#6189061