Finite Group Databasebeta

← #30571129

Subgroups — #30571129

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×#16840433C211yesmaximal, characteristic, complemented
3#21458859C214yesmaximal, complemented
4#21458838C214yesmaximal, complemented
5#21458546C214yesmaximal, complemented
6#21458525C214yesmaximal, complemented
7#21456840C214yesmaximal, complemented
8#21456757C211yesmaximal, characteristic, complemented
9#21456739C211yesmaximal, characteristic, complemented
10#21455960C214yesmaximal, complemented
11#21455819C211yesmaximal, characteristic, complemented
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3101C211yesmaximal, characteristic, complemented
14C22×#11758222—21no—
15#16840497C2212yescomplemented
16#16840491C2212yescomplemented
17#16840486C2212yescomplemented
18#16840485—22no—
19#16840473C2212yescomplemented
20#16840450—22no—
21#16840433—22no—
22#16840433—22no—
23#16840433C2214yes—
24#16840433C2214yes—
25#16840361—22no—
26#16840338—21no—
27#16840312C2211yescharacteristic
28C2×#14519774C2211yescharacteristic
29C2×#14519773C2212yes—
30#16840298C2214yes—
31C2×#14519751—21no—
32C2×#14519750—22no—
33#16840242—21no—
34C2×C4×Dic122C2211yescharacteristic
35C2×#14519739C2212yes—
36#16840148C2218yescomplemented
37#16840139C2212yescomplemented
38#16840136C2212yescomplemented
39#16840125C2218yescomplemented
40#16840110C2212yescomplemented
41#16840109C2212yescomplemented
42#16840075C2212yes—
43#16840061C2218yescomplemented
44#16840042C2212yescomplemented
45#16839980C2218yescomplemented
46#16839962C2214yescomplemented
47#16839960C2218yescomplemented
48#16839958C2214yescomplemented
49#16839957C2212yescomplemented
50#16839956C2218yescomplemented
51#16839912C2214yes—
52#16839898C2214yes—
53#16839896C2212yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839873C2218yescomplemented
57#16839865C2218yescomplemented
58C4×#11758199C2214yescomplemented
59#16839816C2214yes—
60#16839815C2211yescharacteristic
61C61×#533C2211yescharacteristic
62C2×C4×D4×C61C2211yescharacteristic
63C61×#483C2212yes—
64C61×#476C2212yescomplemented
65C61×#461C2212yescomplemented
66C61×#450C2212yes—
67C4×SD16×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C2×#11758222—24no—
71C2×#11758222—24no—
72C2×#11758222—48no—
73C22×Dic122D411yescharacteristic
74C22×D244—21no—
75C22×D244—21no—
76C22×D244C2312yes—
77C22×C22⋊D61—22no—
78C22×C22⋊D61—22no—
79C22×C4×D61—22no—
80C2×#11758201C2318yes—
81C2×D488C2318yes—
82C2×#11758199C2314yes—
83#14519777—22no—
84#14519777C2311yescharacteristic
85#14519774—21no—
86#14519774—21no—
87#14519774C2311yescharacteristic
88#14519774C2312yes—
89#14519774C2312yes—
90#14519773—24no—
91#14519773C2318yes—
92#14519772—24no—
93#14519772—24no—
94#14519768—24no—
95#14519768C2314yes—
96#14519766C2314yes—
97#14519765—22no—
98#14519765—21no—
99#14519762—24no—
100#14519761—24no—
101#14519757—22no—
102#14519757—22no—
103#14519753—24no—
104#14519751—21no—
105#14519751—24no—
106#14519751—24no—
107#14519751C2312yes—
108#14519750—28no—
109#14519750—28no—
110Q8×Dic61C2314yes—
111C4×Dic122—22no—
112C4×Dic122D412yes—
113C4×Dic122D412yes—
114C4×Dic122C2312yes—
115C4×Dic122C2312yes—
116C4×D244—21no—
117C4×D244—22no—
118C4×D244C2312yes—
119#14519739—24no—
120#14519739C2314yes—
121#14519739C2318yes—
122#14519737—24no—
123#14519735—22no—
124#14519735—24no—
125C2×C4⋊Dic61D411yescharacteristic
126C2×#11758196D412yes—
127C4⋊C4×D61—22no—
128C2×#11758198—22no—
129C2×C4×Dic61C2312yes—
130#14519710C2314yes—
131C4⋊#92418C2312yes—
132#14519706C2318yes—
133#14519699C2318yes—
134#14519698C2314yes—
135#14519697C2318yes—
136#14519697C2314yes—
137C22⋊#92418C2312yes—
138#14519687C2318yes—
139C4×#92418C2312yes—
140C22×D4×C61C2311yescharacteristic
141C23×C4×C61—21no—
142C2×SD16×C61C2314yes—
143C4⋊Q8×C61C2311yescharacteristic
144C4⋊Q8×C61C2311yescharacteristic
145#125×C61—21no—
146C22⋊Q8×C61—21no—
147C22⋊Q8×C61C2312yes—
148C4×Q8×C61—21no—
149C4×Q8×C61C2312yes—
150C4×D4×C61—21no—
151C4×D4×C61—21no—
152C4×D4×C61C2312yes—
153C4×D4×C61C2312yes—
154#117×C61—21no—
155C2×C4⋊C4×C61C2311yescharacteristic
156C2×C22⋊C4×C61—21no—
157C2×C42×C61C2311yescharacteristic
158#106×C61C2314yes—
159C4⋊C8×C61C2312yes—
160#103×C61C2314yes—
161#102×C61C2314yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C24×C61—21no—
165C23×D61—22no—
166C2×Q8×C61—21no—
167C2×Q8×C61C2412yes—
168C2×D4×C61—21no—
169C2×D4×C61—42no—
170C2×D4×C61C2412yes—
171C22×C4×C61—21no—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—44no—
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2411yescharacteristic
178C2×C22⋊D61—22no—
179C2×C22⋊D61—28no—
180C2×C22⋊D61—28no—
181C2×C22⋊D61—48no—
182C2×C22⋊D61—84no—
183C22×Dic61C2×D412yes—
184C22×Dic61C2×D412yes—
185#11758222—416no—
186#11758222—416no—
187#11758222—44no—
188C2×D244—22no—
189C2×D244—22no—
190C2×D244—22no—
191C2×D244—21no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—21no—
195C2×D244—44no—
196C2×D244—44no—
197C2×D244—82no—
198C2×D244C2414yes—
199C2×D244C2412yes—
200C2×C4×D61—22no—
201C2×C4×D61—28no—
202C2×C4×D61—24no—
203C2×C4×D61—48no—
204C2×Dic122—24no—
205C2×Dic122—21no—
206C2×Dic122C2×D412yes—
207C2×Dic122C2×D412yes—
208SD16×C61—44no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61—21no—
214C4⋊C4×C61—22no—
215C4⋊C4×C61—22no—
216C4⋊C4×C61C2412yes—
217C4⋊C4×C61C2412yes—
218C22⋊C4×C61—22no—
219C22⋊C4×C61—22no—
220C22⋊C4×C61—22no—
221C42×C61—22no—
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2411yescharacteristic
225C42×C61C2411yescharacteristic
226#11758201—48no—
227#11758199—44no—
228#11758198—22no—
229#11758198—22no—
230#11758198—28no—
231#11758198—24no—
232#11758198—24no—
233#11758198—24no—
234C4⋊Dic61—21no—
235C4⋊Dic61—24no—
236C4⋊Dic61—24no—
237C4⋊Dic61C2×D412yes—
238C4⋊Dic61C2×D412yes—
239C4⋊Dic61C2412yes—
240#11758196—24no—
241#11758196C2×D414yes—
242#11758196C2×D414yes—
243C4×Dic61—24no—
244C4×Dic61C2×D414yes—
245C4×Dic61C2414yes—
246C2×#92418C2414yes—
247D488—48no—
248C23×C61—22no—
249C23×C61—42no—
250C23×C61—42no—
251C23×C61—42no—
252C23×C61C22×D411yescharacteristic
253C22×D61—22no—
254C22×D61—24no—
255C22×D61—44no—
256C22×D61—44no—
257C22×D61—44no—
258Q8×C61—24no—
259Q8×C61—81no—
260D4×C61—22no—
261D4×C61—22no—
262D4×C61—42no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—41no—
268C2×C4×C61—42no—
269C2×C4×C61—44no—
270C2×C4×C61—44no—
271C2×C4×C61AGL(1,ℤ/8)14yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C2511yescharacteristic
278C22⋊D61—416no—
279C22⋊D61—44no—
280C22⋊D61—88no—
281C2×Dic61—24no—
282C2×Dic61—22no—
283C2×Dic61—42no—
284C2×Dic61C22×D414yes—
285C2×Dic61C22×D414yes—
286D244—24no—
287D244—28no—
288D244—44no—
289D244—44no—
290D244—44no—
291D244—42no—
292D244—81no—
293D244—82no—
294D244—82no—
295C4×D61—416no—
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—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—84no—
310C22×C61C2×AGL(1,ℤ/8)12yes—
311C22×C61C23×D411yescharacteristic
312C2×D61—42no—
313C2×D61—44no—
314C2×D61—44no—
315C2×D61—42no—
316C2×D61—44no—
317C2×D61—42no—
318C2×D61—88no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—42no—
323C4×C61—42no—
324C4×C61—42no—
325C4×C61C2×AGL(1,ℤ/8)12yes—
326C4×C61C2×AGL(1,ℤ/8)12yes—
327C4×C61#58512yes—
328Dic61—24no—
329Dic61—24no—
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#3101D6111yescharacteristic, complemented
342#3100—611no—
343#2983—611no—
344C2×C61—22no—
345C2×C61—82no—
346C2×C61#348711yescharacteristic
347C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
348C2×C61#358511yescharacteristic
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#533C2×D6111yescharacteristic
365#532—611no—
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#483C2×D6112yes—
382#480—612no—
383#478—618no—
384#476C2×D6112yescomplemented
385#474—612no—
386#463—618no—
387#461C2×D6112yescomplemented
388#460—618no—
389#460—614no—
390#459—612no—
391#459—614no—
392#459—618no—
393#450—614no—
394#450C2×D6112yes—
395#447—612no—
396#447—614no—
397#446—614no—
398#446—614no—
399#442—618no—
400#440—618no—
401C4×SD16C2×D6114yescomplemented
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—612no—
413C22×D4—1222no—
414C22×D4—1222no—
415C22×D4—1221no—
416C22×D4—1221no—
417C22×D4C22×D6111yescharacteristic
418C23×C4—21no—
419C23×C4—1222no—
420C2×SD16—618no—
421C2×SD16C22×D6114yes—
422C2×D8—614no—
423C2×D8—618no—
424C4⋊Q8—611no—
425C4⋊Q8—1222no—
426C4⋊Q8C22×D6111yescharacteristic
427C4⋊Q8C22×D6111yescharacteristic
428#127—618no—
429#127—612no—
430#127—612no—
431#127—611no—
432#127—1224no—
433#127—1221no—
434#127—1221no—
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—1228no—
452#121—1228no—
453#121—1224no—
454#121—1224no—
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—21no—
465C4×D4—21no—
466C4×D4—618no—
467C4×D4—614no—
468C4×D4—612no—
469C4×D4—1222no—
470C4×D4—1224no—
471C4×D4—1224no—
472C4×D4—1221no—
473C4×D4C22×D6112yes—
474C4×D4C22×D6112yes—
475#117—21no—
476#117—1224no—
477#117—1222no—
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#107—614no—
487#106C22×D6114yes—
488C4⋊C8—618no—
489C4⋊C8—612no—
490C4⋊C8C22×D6112yes—
491#103C22×D6114yes—
492#102—614no—
493#102—618no—
494#102—614no—
495#102—618no—
496#102C22×D6114yes—
497C22⋊C8—612no—
498C22⋊C8C22×D6112yes—
499#97—618no—
500C4×C8—612no—
501C4×C8C22×D6112yes—
502C24—21no—
503C24—1222no—
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—21no—
514C2×D4—42no—
515C2×D4—614no—
516C2×D4—612no—
517C2×D4—1228no—
518C2×D4—1228no—
519C2×D4—1222no—
520C2×D4—1221no—
521C2×D4—1222no—
522C2×D4—1222no—
523C2×D4—1221no—
524C2×D4—1222no—
525C2×D4—1222no—
526C2×D4—1222no—
527C2×D4—4882no—
528C2×D4—4884no—
529C2×D4—2444no—
530C2×D4—2448no—
531C2×D4—2444no—
532C2×D4C23×D6112yes—
533C22×C4—21no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—44no—
537C22×C4—612no—
538C22×C4—612no—
539C22×C4—1224no—
540C22×C4—1228no—
541C22×C4—1222no—
542C22×C4—2448no—
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546SD16—44no—
547SD16—2448no—
548D8—2448no—
549D8—2444no—
550C2×C8—614no—
551C2×C8C23×D6114yes—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—612no—
559C4⋊C4—614no—
560C4⋊C4—614no—
561C4⋊C4—612no—
562C4⋊C4—612no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1224no—
566C4⋊C4—1221no—
567C4⋊C4C23×D6112yes—
568C4⋊C4C23×D6112yes—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—1228no—
573C22⋊C4—1224no—
574C22⋊C4—1224no—
575C22⋊C4—1224no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C42—22no—
579C42—614no—
580C42—614no—
581C42—1224no—
582C42D4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C42C23×D6111yescharacteristic
586C23—22no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—1222no—
591C23—1224no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23C2×D4×D6111yescharacteristic
596Q8—24no—
597Q8—81no—
598Q8—1224no—
599Q8—1224no—
600Q8—2442no—
601D4—22no—
602D4—22no—
603D4—42no—
604D4—1224no—
605D4—1228no—
606D4—4881no—
607D4—4882no—
608D4—4882no—
609D4—4888no—
610D4—2444no—
611D4—24416no—
612D4—2444no—
613D4—2444no—
614D4—2444no—
615D4—2442no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—614no—
625C2×C4—614no—
626C2×C4—1224no—
627C2×C4—1222no—
628C2×C4—4884no—
629C2×C4—24416no—
630C2×C4—2442no—
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—42no—
644C22—42no—
645C22—42no—
646C22—84no—
647C22—4888no—
648C22—2444no—
649C22—2442no—
650C22—2444no—
651C22—2442no—
652C22—2442no—
653C22—2444no—
654C22C2×#1451980412yes—
655C22C22×D4×D6111yescharacteristic
656C4—21no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—1224no—
663C4—1224no—
664C4—2442no—
665C4C2×#1451980412yes—
666C4C2×#1451980412yes—
667C4#1684059812yes—
668C2—22no—
669C2—82no—
670C2—4884no—
671C2—4882no—
672C2#2145947311yescharacteristic
673C2C22×#1451980411yescharacteristic
674C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3101C2
socleC22×C61C23×D4
hypercenter#3101D61
hypocenterC61#61890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #21459686C2semidirect
2. C2×#16840433C2semidirect
3. #21458859C2semidirect
4. #21458838C2semidirect
5. #21458546C2semidirect
6. #21458525C2semidirect
7. #21456840C2semidirect
8. #21456757C2semidirect
9. #21456739C2semidirect
10. #21455960C2semidirect
11. #21455819C2semidirect
12. #21454902C2semidirect
13. C61×#3101C2semidirect
15. #16840497C22semidirect
16. #16840491C22semidirect
17. #16840486C22semidirect
19. #16840473C22semidirect
36. #16840148C22semidirect
37. #16840139C22semidirect
38. #16840136C22semidirect
39. #16840125C22semidirect
40. #16840110C22semidirect
41. #16840109C22semidirect
43. #16840061C22semidirect
44. #16840042C22semidirect
45. #16839980C22semidirect
46. #16839962C22semidirect
47. #16839960C22semidirect
48. #16839958C22semidirect
49. #16839957C22semidirect
50. #16839956C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839873C22semidirect
57. #16839865C22semidirect
58. C4×#11758199C22semidirect
64. C61×#476C22semidirect
65. C61×#461C22semidirect
67. C4×SD16×C61C22semidirect
341. #3101D61semidirect
384. #476C2×D61semidirect
387. #461C2×D61semidirect
401. C4×SD16C2×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×#168404332
#214588592
#214588382
#214585462
#214585252
#214568402
#214567572
#214567392
#214559602
#214558192
#214549022
C61×#31012
#6189061