Finite Group Databasebeta

← #30571128

Subgroups — #30571128

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#21459684C211yesmaximal, characteristic, complemented
2C2×#16840434C211yesmaximal, characteristic, complemented
3#21458863C214yesmaximal, complemented
4#21458841C214yesmaximal, complemented
5#21458550C214yesmaximal, complemented
6#21458528C214yesmaximal, complemented
7#21456839C214yesmaximal, complemented
8#21456762C211yesmaximal, characteristic, complemented
9#21456739C211yesmaximal, characteristic, complemented
10#21455959C214yesmaximal, complemented
11#21455828C211yesmaximal, characteristic, complemented
12#21455820C211yesmaximal, characteristic, complemented
13C61×#2983C211yesmaximal, characteristic, complemented
14C22×#11758226—21no—
15#16840497C2212yescomplemented
16#16840490C2212yescomplemented
17#16840485C2212yescomplemented
18#16840484—22no—
19#16840472C2212yescomplemented
20#16840453—22no—
21#16840434—22no—
22#16840434—22no—
23#16840434C2214yes—
24#16840434C2214yes—
25#16840359—22no—
26#16840336—21no—
27#16840319C2211yescharacteristic
28#16840312C2211yescharacteristic
29C2×#14519774C2211yescharacteristic
30C2×#14519774C2212yes—
31#16840296C2214yes—
32C2×#14519752—21no—
33C2×#14519752—22no—
34#16840243—21no—
35C2×C4×D244C2212yes—
36C2×C4×D244C2211yescharacteristic
37#16840151C2218yescomplemented
38#16840139C2212yescomplemented
39#16840135C2212yescomplemented
40#16840127C2218yescomplemented
41#16840109C2212yescomplemented
42#16840109C2212yescomplemented
43#16840093C2212yes—
44#16840076C2212yes—
45#16840063C2218yescomplemented
46#16840053C2212yescomplemented
47#16840043C2212yescomplemented
48#16839978C2218yescomplemented
49#16839968C2212yescomplemented
50#16839963C2214yescomplemented
51#16839963C2218yescomplemented
52#16839958C2214yescomplemented
53#16839958C2218yescomplemented
54#16839958C2212yescomplemented
55#16839915C2214yes—
56#16839915C2212yes—
57#16839898C2214yes—
58#16839898C2212yes—
59#16839875C2218yescomplemented
60#16839864C2218yescomplemented
61C4×#11758190C2214yescomplemented
62C4×D488C2214yescomplemented
63C2×C4×Q8×C61C2211yescharacteristic
64C61×#446C2214yescomplemented
65C61×#446C2214yescomplemented
66C61×#432C2214yes—
67C61×#431C2211yescharacteristic
68C61×#431C2211yescharacteristic
69C2×#11758226—24no—
70C2×#11758226—24no—
71C2×#11758226—48no—
72C2×#11758222—21no—
73C22×D244—22no—
74C22×D244—21no—
75C22×D244—22no—
76C22×D244—21no—
77C22×D244C2312yes—
78C22×D244C2311yescharacteristic
79C22×C4×D61—22no—
80C22×C4×D61—21no—
81C2×#11758190C2314yes—
82C2×#11758190C2318yes—
83C2×D488C2314yes—
84C2×D488C2318yes—
85#14519777C2311yescharacteristic
86#14519777C2311yescharacteristic
87#14519777C2311yescharacteristic
88#14519775—22no—
89#14519774—24no—
90#14519774—21no—
91#14519774—21no—
92#14519774C2311yescharacteristic
93#14519774C2312yes—
94#14519774C2318yes—
95#14519774C2312yes—
96#14519772—24no—
97#14519771—24no—
98#14519769—21no—
99#14519769C2314yes—
100#14519767—24no—
101#14519766—22no—
102#14519765—21no—
103#14519765C2314yes—
104#14519762—24no—
105#14519760—24no—
106#14519756—21no—
107#14519756C2312yes—
108#14519755—24no—
109#14519753—22no—
110#14519753—22no—
111#14519752—24no—
112#14519752—24no—
113#14519752—28no—
114#14519752—28no—
115#14519751—21no—
116#14519751C2312yes—
117C4×Dic122—21no—
118C4×Dic122C2314yes—
119C4×Dic122C2312yes—
120#14519742—22no—
121C4×D244—21no—
122C4×D244—24no—
123C4×D244—21no—
124C4×D244—21no—
125C4×D244C2312yes—
126C4×D244C2314yes—
127C4×D244C2312yes—
128C4×D244C2312yes—
129C4×D244C2318yes—
130#14519739—22no—
131C4×C22⋊D61—24no—
132#14519736—21no—
133#14519735—22no—
134#14519734—24no—
135C2×C4⋊Dic61C2311yescharacteristic
136C4⋊C4×D61—22no—
137C2×#11758198—22no—
138C2×#11758198—21no—
139#14519711C2314yes—
140#14519709C2314yes—
141#14519702C2314yes—
142#14519699C2314yes—
143#14519699C2318yes—
144#14519699C2318yes—
145#14519699C2314yes—
146#14519699C2314yes—
147C22×Q8×C61D411yescharacteristic
148C4×Q8×C61D412yes—
149C4×Q8×C61D412yes—
150C4×Q8×C61C2312yes—
151C4×Q8×C61C2312yes—
152C2×C4⋊C4×C61D411yescharacteristic
153C2×C4⋊C4×C61D412yes—
154C2×C42×C61C2312yes—
155C2×C42×C61C2311yescharacteristic
156C4⋊C8×C61C2312yes—
157C4⋊C8×C61C2312yes—
158C4⋊C8×C61C2318yes—
159C22⋊C8×C61C2312yes—
160C22⋊C8×C61C2312yes—
161#97×C61C2318yes—
162C4×C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C23×D61—22no—
165C23×D61—21no—
166C2×Q8×C61—24no—
167C2×Q8×C61C2×D412yes—
168C2×Q8×C61C2×D412yes—
169C22×C4×C61C2×D412yes—
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2411yescharacteristic
174C2×C22⋊D61—22no—
175C22×Dic61—21no—
176#11758226—416no—
177#11758226—416no—
178#11758222—44no—
179C2×D244—28no—
180C2×D244—22no—
181C2×D244—22no—
182C2×D244—22no—
183C2×D244—21no—
184C2×D244—21no—
185C2×D244—28no—
186C2×D244—22no—
187C2×D244—22no—
188C2×D244—21no—
189C2×D244—42no—
190C2×D244—44no—
191C2×D244—48no—
192C2×D244—44no—
193C2×D244—82no—
194C2×D244—84no—
195C2×D244C2412yes—
196C2×D244C2414yes—
197C2×D244C2412yes—
198C2×C4×D61—28no—
199C2×C4×D61—22no—
200C2×C4×D61—22no—
201C2×C4×D61—22no—
202C2×C4×D61—24no—
203C2×C4×D61—48no—
204C2×C4×D61—44no—
205C2×Dic122—21no—
206C2×Dic122—21no—
207C2×Dic122C2412yes—
208C2×C8×C61C2414yes—
209C2×C8×C61C2414yes—
210C4⋊C4×C61C2×D412yes—
211C4⋊C4×C61C2×D412yes—
212C4⋊C4×C61C2×D414yes—
213C4⋊C4×C61C2×D414yes—
214C42×C61C2×D414yes—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2411yescharacteristic
218C42×C61C2411yescharacteristic
219C42×C61C2414yes—
220C22⋊Dic61—22no—
221#11758198—28no—
222#11758198—22no—
223#11758198—22no—
224#11758198—22no—
225#11758198—24no—
226#11758198—24no—
227#11758198—22no—
228#11758198—24no—
229C4⋊Dic61—21no—
230C4⋊Dic61—21no—
231C4⋊Dic61—21no—
232C4⋊Dic61—21no—
233C4⋊Dic61—21no—
234C4⋊Dic61C2412yes—
235C4⋊Dic61C2412yes—
236C4⋊Dic61C2412yes—
237#11758196—24no—
238#11758196—22no—
239#11758196—24no—
240#11758196—24no—
241#11758196—22no—
242C4×Dic61—24no—
243C4×Dic61—22no—
244D488—44no—
245D488—48no—
246#11758190—48no—
247#11758190—44no—
248C23×C61C22×D411yescharacteristic
249C22×D61—22no—
250C22×D61—22no—
251C22×D61—24no—
252C22×D61—42no—
253C22×D61—44no—
254C22×D61—44no—
255C22×D61—42no—
256C22×D61—44no—
257C22×D61—42no—
258Q8×C61—24no—
259Q8×C61—24no—
260C2×C4×C61—24no—
261C2×C4×C61—22no—
262C2×C4×C61—41no—
263C2×C4×C61—42no—
264C2×C4×C61AGL(1,ℤ/8)14yes—
265C2×C4×C61C22×D414yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D414yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2511yescharacteristic
273C22⋊D61—44no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—42no—
279D244—22no—
280D244—28no—
281D244—24no—
282D244—22no—
283D244—42no—
284D244—44no—
285D244—416no—
286D244—44no—
287D244—44no—
288D244—42no—
289D244—81no—
290D244—82no—
291D244—88no—
292D244—82no—
293C4×D61—416no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61—84no—
298Dic122—24no—
299Dic122—42no—
300Dic122—81no—
301C8×C61—24no—
302C8×C61—24no—
303#61890—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61C2×AGL(1,ℤ/8)12yes—
307C22×C61C23×D411yescharacteristic
308C2×D61—42no—
309C2×D61—44no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—44no—
313C2×D61—42no—
314C2×D61—44no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—88no—
318C2×D61—84no—
319C4×C61—21no—
320C4×C61—24no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—24no—
324C4×C61C2×AGL(1,ℤ/8)12yes—
325C4×C61C2×AGL(1,ℤ/8)12yes—
326C4×C61#58512yes—
327Dic61—42no—
328Dic61—42no—
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#3100—611no—
343#2983D6111yescharacteristic, complemented
344C2×C61—22no—
345C2×C61#348711yescharacteristic
346C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
347C2×C61#358511yescharacteristic
348D61—82no—
349D61—82no—
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#532—611no—
366C2×#127—611no—
367C2×#127—612no—
368#529—614no—
369C2×#121—1221no—
370C2×#121—1222no—
371#520—1221no—
372C2×C4×Q8C2×D6111yescharacteristic
373C2×C4×D4—611no—
374C2×C4×D4—612no—
375#496—618no—
376#493—612no—
377#492—612no—
378#490—618no—
379#486—612no—
380#486—612no—
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#460—614no—
390#459—612no—
391#459—614no—
392#459—618no—
393#450—612no—
394#450—614no—
395#447—612no—
396#447—614no—
397#446C2×D6114yescomplemented
398#446C2×D6114yescomplemented
399#442—618no—
400#440—618no—
401C4×SD16—614no—
402C4×D8—614no—
403#432C2×D6114yes—
404#431C2×D6111yescharacteristic
405#431C2×D6111yescharacteristic
406C61#6189011yescharacteristic, complemented, Sylow
407C2×#41—1224no—
408C2×#41—1224no—
409C2×#41—1221no—
410C2×#41—2448no—
411C22×Q8D24411yescharacteristic
412C22×D4—611no—
413C22×D4—612no—
414C22×D4—1222no—
415C22×D4—1221no—
416C22×D4—1222no—
417C22×D4—1221no—
418C23×C4—1222no—
419C23×C4—1221no—
420C2×SD16—614no—
421C2×SD16—618no—
422C2×D8—614no—
423C2×D8—618no—
424C4⋊Q8—611no—
425C4⋊Q8—611no—
426C4⋊Q8—611no—
427C4⋊Q8—1222no—
428#127—612no—
429#127—618no—
430#127—612no—
431#127—611no—
432#127—1224no—
433#127—1221no—
434#127—1221no—
435#126—1224no—
436#126—1224no—
437#125—614no—
438#125—1224no—
439#125—1221no—
440#124—614no—
441#124—1222no—
442#124—1221no—
443#123—1224no—
444#123—1224no—
445C22⋊Q8—612no—
446C22⋊Q8—1222no—
447C22⋊Q8—1222no—
448C22⋊Q8—1221no—
449C22⋊Q8—1224no—
450#121—612no—
451#121—1221no—
452#121—1224no—
453#121—1224no—
454#121—1228no—
455#121—1228no—
456C4×Q8—614no—
457C4×Q8—612no—
458C4×Q8—1222no—
459C4×Q8—1221no—
460C4×Q8D24412yes—
461C4×Q8D24412yes—
462C4×Q8C22×D6112yes—
463C4×Q8C22×D6112yes—
464C4×D4—612no—
465C4×D4—612no—
466C4×D4—618no—
467C4×D4—614no—
468C4×D4—612no—
469C4×D4—1224no—
470C4×D4—1222no—
471C4×D4—1224no—
472C4×D4—1221no—
473C4×D4—1221no—
474C4×D4—1221no—
475#117—1221no—
476#117—1222no—
477#117—1224no—
478C2×C4⋊C4—611no—
479C2×C4⋊C4—1222no—
480C2×C4⋊C4D24411yescharacteristic
481C2×C4⋊C4D24412yes—
482C2×C22⋊C4—1222no—
483C2×C22⋊C4—1221no—
484C2×C42C22×D6112yes—
485C2×C42C22×D6111yescharacteristic
486#107—614no—
487#106—614no—
488C4⋊C8C22×D6112yes—
489C4⋊C8C22×D6112yes—
490C4⋊C8C22×D6118yes—
491#103—614no—
492#102—614no—
493#102—614no—
494#102—614no—
495#102—618no—
496#102—618no—
497C22⋊C8C22×D6112yes—
498C22⋊C8C22×D6112yes—
499#97C22×D6118yes—
500C4×C8C22×D6112yes—
501C4×C8C22×D6112yes—
502C24—1222no—
503C24—1221no—
504#41—24416no—
505#41—24416no—
506#41—2444no—
507C2×Q8—24no—
508C2×Q8—612no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8C2×D24412yes—
512C2×Q8C2×D24412yes—
513C2×D4—612no—
514C2×D4—614no—
515C2×D4—612no—
516C2×D4—1222no—
517C2×D4—1221no—
518C2×D4—1228no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1221no—
522C2×D4—1222no—
523C2×D4—1221no—
524C2×D4—1228no—
525C2×D4—1222no—
526C2×D4—1222no—
527C2×D4—4884no—
528C2×D4—4882no—
529C2×D4—2442no—
530C2×D4—2444no—
531C2×D4—2448no—
532C2×D4—2444no—
533C22×C4—1221no—
534C22×C4—1224no—
535C22×C4—1222no—
536C22×C4—1228no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—2448no—
540C22×C4—2444no—
541C22×C4C2×D24412yes—
542C22×C4C2×D24412yes—
543C22×C4C2×D24411yescharacteristic
544C22×C4C2×D24411yescharacteristic
545C22×C4C23×D6111yescharacteristic
546SD16—2444no—
547SD16—2448no—
548D8—2448no—
549D8—2444no—
550C2×C8C23×D6114yes—
551C2×C8C23×D6114yes—
552C4⋊C4—612no—
553C4⋊C4—612no—
554C4⋊C4—612no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1224no—
558C4⋊C4—1222no—
559C4⋊C4—1224no—
560C4⋊C4—1222no—
561C4⋊C4—1224no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4C2×D24412yes—
566C4⋊C4C2×D24412yes—
567C4⋊C4C2×D24414yes—
568C4⋊C4C2×D24414yes—
569C22⋊C4—1222no—
570C22⋊C4—1228no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—1224no—
575C22⋊C4—1224no—
576C22⋊C4—1222no—
577C22⋊C4—1224no—
578C42—1224no—
579C42—1222no—
580C42C2×D24414yes—
581C42C2×D24411yescharacteristic
582C42C2×D24411yescharacteristic
583C42C23×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C42C23×D6114yes—
586C23—1222no—
587C23—1222no—
588C23—1224no—
589C23—2444no—
590C23—2442no—
591C23—2442no—
592C23—2444no—
593C23—2444no—
594C23—2442no—
595C23C22×D24411yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—1224no—
599Q8—4881no—
600Q8—2442no—
601D4—1224no—
602D4—1222no—
603D4—1228no—
604D4—1222no—
605D4—4881no—
606D4—4888no—
607D4—4882no—
608D4—4882no—
609D4—2442no—
610D4—2444no—
611D4—24416no—
612D4—2444no—
613D4—2444no—
614D4—2444no—
615D4—2442no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—4884no—
625C2×C4—24416no—
626C2×C4—2442no—
627C2×C4—2444no—
628C2×C4—2444no—
629C2×C4—2444no—
630C2×C4#1451980514yes—
631C2×C4C22×D24414yes—
632C2×C4C22×D24411yescharacteristic
633C2×C4C22×D24411yescharacteristic
634C2×C4C22×D24411yescharacteristic
635C2×C4C22×D24414yes—
636C2×C4C22×D24411yescharacteristic
637C2×C4C22×D24411yescharacteristic
638C2×C4C24×D6111yescharacteristic
639C8—24no—
640C8—24no—
641C22—21no—
642C22—21no—
643C22—4888no—
644C22—4884no—
645C22—2442no—
646C22—2444no—
647C22—2442no—
648C22—2444no—
649C22—2442no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22—2444no—
654C22C2×#1451980512yes—
655C22C23×D24411yescharacteristic
656C4—21no—
657C4—24no—
658C4—24no—
659C4—21no—
660C4—21no—
661C4—2442no—
662C4—2442no—
663C4—2442no—
664C4—2442no—
665C4C2×#1451980512yes—
666C4C2×#1451980512yes—
667C4#1684059912yes—
668C2—22no—
669C2—4882no—
670C2—4884no—
671C2—4882no—
672C2#2145947811yescharacteristic
673C2C22×#1451980511yescharacteristic
674C2#2146060011yescharacteristic
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2983C2
socleC22×C61C23×D4
hypercenter#2983D61
hypocenterC61#61890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2983

Complements

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

NH ≅ G/Ntype
1. #21459684C2semidirect
2. C2×#16840434C2semidirect
3. #21458863C2semidirect
4. #21458841C2semidirect
5. #21458550C2semidirect
6. #21458528C2semidirect
7. #21456839C2semidirect
8. #21456762C2semidirect
9. #21456739C2semidirect
10. #21455959C2semidirect
11. #21455828C2semidirect
12. #21455820C2semidirect
13. C61×#2983C2semidirect
15. #16840497C22semidirect
16. #16840490C22semidirect
17. #16840485C22semidirect
19. #16840472C22semidirect
37. #16840151C22semidirect
38. #16840139C22semidirect
39. #16840135C22semidirect
40. #16840127C22semidirect
41. #16840109C22semidirect
42. #16840109C22semidirect
45. #16840063C22semidirect
46. #16840053C22semidirect
47. #16840043C22semidirect
48. #16839978C22semidirect
49. #16839968C22semidirect
50. #16839963C22semidirect
51. #16839963C22semidirect
52. #16839958C22semidirect
53. #16839958C22semidirect
54. #16839958C22semidirect
59. #16839875C22semidirect
60. #16839864C22semidirect
61. C4×#11758190C22semidirect
62. C4×D488C22semidirect
64. C61×#446C22semidirect
65. C61×#446C22semidirect
343. #2983D61semidirect
397. #446C2×D61semidirect
398. #446C2×D61semidirect
406. C61#61890semidirect

Maximal subgroups

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

subgroupindex
#214596842
C2×#168404342
#214588632
#214588412
#214585502
#214585282
#214568392
#214567622
#214567392
#214559592
#214558282
#214558202
C61×#29832
#6189061