Finite Group Databasebeta

← #29729727

Subgroups — #29729727

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839965C211yesmaximal, characteristic
2C2×#16839890C211yesmaximal, characteristic, complemented
3#21451175C211yesmaximal, characteristic
4#21451033C211yesmaximal, characteristic, complemented
5#21447209C211yesmaximal, characteristic, complemented
6#21447200C211yesmaximal, characteristic, complemented
7#21447136C211yesmaximal, characteristic
8#21447132C211yesmaximal, characteristic
9#21446674C211yesmaximal, characteristic
10#21446637C211yesmaximal, characteristic, complemented
11#21446584C211yesmaximal, characteristic, complemented
12#21446509C211yesmaximal, characteristic, complemented
13#21446497C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic
15C61×#1973C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16839965—24no—
21#16839890—24no—
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839704—21no—
35#16839703—21no—
36#16839698—21no—
37#16839674—21no—
38#16839669C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839667C2211yescharacteristic
41#16839666—21no—
42#16839665C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839637—21no—
46#16839634—21no—
47#16839627C2211yescharacteristic
48#16839626—21no—
49C4⋊C4×Dic61C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#97×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#390C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519775—22no—
70#14519775—22no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73D4×Dic61—22no—
74D4×Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519708—41no—
92C4⋊#92418—22no—
93#14519706—22no—
94#14519706D414yes—
95#14519702—22no—
96#14519702—22no—
97#14519701—22no—
98#14519701—41no—
99#14519700—41no—
100#14519698—22no—
101#14519687—22no—
102C8×Dic61—22no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114C22×Q8×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61—41no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#107×C61—41no—
127#103×C61—22no—
128#102×C61—22no—
129#97×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—81no—
157C22×Dic61—81no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—41no—
171SD16×C61—44no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D412yes—
177C2×C8×C61#4112yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—81no—
184C42×C61—21no—
185C42×C61—21no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—81no—
189#11758200—44no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—81no—
198#11758196—22no—
199#11758196—22no—
200#11758196—42no—
201#11758196—42no—
202#11758196—81no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C4×Dic61C2×D412yes—
207C4×Dic61#4112yes—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D412yes—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—84no—
273C2×Dic61#12112yes—
274C2×Dic61#12112yes—
275C2×Dic61#13712yes—
276Dic122—44no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281C8×C61—22no—
282C8×C61—22no—
283#92418—22no—
284#92418—41no—
285#25308—611nomaximal, Sylow
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61#55011yescharacteristic
296C22×C61C2×#13711yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57711yescharacteristic
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61#52911yescharacteristic
306C4×C61#54211yescharacteristic
307C4×C61#54611yescharacteristic
308C4×C61#54811yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—22no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#461—611no—
318C2×#446—611no—
319#2580—611no—
320#2568—611no—
321#2068—611no—
322#2066—611no—
323#2055—611no—
324#2055—611no—
325#1973D6111yescharacteristic
326#1968—611no—
327#1960—611no—
328#1949—611no—
329#1935—611no—
330#1933—611no—
331#1762—611no—
332C2×C61—82no—
333C2×C61#242611yescharacteristic
334C2×C61#305611yescharacteristic
335C2×C61#305711yescharacteristic
336C2×C61#337211yescharacteristic
337C2×C61#337611yescharacteristic
338C2×C61#340011yescharacteristic
339C2×C61#340111yescharacteristic
340C22×SD16—611no—
341C22×SD16C2×D6111yescharacteristic
342C2×C4⋊Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×D4—611no—
345#461—1224no—
346#446—1224no—
347C2×#107C2×D6111yescharacteristic
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103C2×D6111yescharacteristic
358C2×#102—611no—
359C2×#102C2×D6111yescharacteristic
360C2×#97—611no—
361C2×#97C2×D6111yescharacteristic
362C2×C4×C8—611no—
363#400—1221no—
364#400—1221no—
365#399—1221no—
366#393—1221no—
367#391—611no—
368#391—611no—
369#391—611no—
370#391—1221no—
371#391C2×D6111yescharacteristic
372#390—611no—
373#390C2×D6111yescharacteristic
374#389—611no—
375#389—611no—
376#385—1221no—
377#385—1221no—
378#382—611no—
379#382—1221no—
380C4×C4⋊C4—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385C61#2530811yescharacteristic, complemented, Sylow
386C22×Q8—611no—
387C22×Q8—611no—
388C22×Q8C22×D6111yescharacteristic
389C22×D4C22×D6111yescharacteristic
390C23×C4—1221no—
391C2×SD16—22no—
392C2×SD16—22no—
393C2×SD16—41no—
394C2×SD16—1222no—
395C2×SD16—1222no—
396C2×SD16—2441no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C4⋊Q8—1222no—
402C4⋊Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1221no—
427C2×C42C22×D6111yescharacteristic
428#107—41no—
429#106—2441no—
430C4⋊C8—614no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433#103—22no—
434#103—1222no—
435#103—1222no—
436#103—1222no—
437#103—2441no—
438#103—2441no—
439#102—22no—
440#102—1222no—
441#97—22no—
442#97—1222no—
443C4×C8—1222no—
444#95—21no—
445#95—21no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—1222no—
465C2×Q8—1222no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—2441no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4#1175822212yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—4881no—
491C22×C4—4881no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496SD16—44no—
497SD16—2444no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—612no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8D4×D6112yes—
509C2×C8#1175822412yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—4881no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—4881no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—612no—
535C42—612no—
536C42—1221no—
537C42—1221no—
538C42—2442no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×D4×D6111yescharacteristic
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—2444no—
552Q8—2441no—
553Q8—2441no—
554Q8—2441no—
555Q8—2441no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—4884no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—22no—
607C8—22no—
608C8—1222no—
609C8—2441no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#121×D6111yescharacteristic
618C22#1684042111yescharacteristic
619C22#1684043211yescharacteristic
620C22#137×D6111yescharacteristic
621C22#1684055811yescharacteristic
622C22#1684057111yescharacteristic
623C22#1684057211yescharacteristic
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—1222no—
630C4—1222no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—2441no—
637C4#1684029211yescharacteristic
638C4#1684037611yescharacteristic
639C4#1684040211yescharacteristic
640C4#1684041811yescharacteristic
641C2—82no—
642C2#2144938211yescharacteristic
643C2#2145544411yescharacteristic
644C2#2145545611yescharacteristic
645C2#2145860111yescharacteristic
646C2#2145865111yescharacteristic
647C2#2145886911yescharacteristic
648C2#2145888011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1973C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1973D61
hypocenterC61#25308
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. C2×#16839890C2semidirect
4. #21451033C2semidirect
5. #21447209C2semidirect
6. #21447200C2semidirect
10. #21446637C2semidirect
11. #21446584C2semidirect
12. #21446509C2semidirect
13. #21446497C2semidirect
385. C61#25308semidirect

Maximal subgroups

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

subgroupindex
C2×#168399652
C2×#168398902
#214511752
#214510332
#214472092
#214472002
#214471362
#214471322
#214466742
#214466372
#214465842
#214465092
#214464972
#214456462
C61×#19732
#2530861