Finite Group Databasebeta

← #30616097

Subgroups — #30616097

Order 15616.

780 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462506C211yesmaximal, characteristic, complemented
2#21462391C211yesmaximal, characteristic, complemented
3#21461901C211yesmaximal, characteristic, complemented
4#21461900C211yesmaximal, characteristic, complemented
5#21461887C211yesmaximal, characteristic, complemented
6#21461884C211yesmaximal, characteristic, complemented
7#21461811C211yesmaximal, characteristic
8#21461811C211yesmaximal, characteristic
9#21461789C211yesmaximal, characteristic, complemented
10#21461787C211yesmaximal, characteristic, complemented
11#21461639C211yesmaximal, characteristic, complemented
12C4×#14519842C211yesmaximal, characteristic, complemented
13C4×#14519841C211yesmaximal, characteristic, complemented
14#21461535C211yesmaximal, characteristic, complemented
15#21454614C211yesmaximal, characteristic
16C2×#14519865C2211yescharacteristic
17C2×#14519865C2211yescharacteristic
18#16840786—21no—
19#16840786—21no—
20#16840783—21no—
21#16840782—21no—
22C2×#14519848C2211yescharacteristic
23C2×#14519847C2211yescharacteristic
24#16840769C2211yescharacteristic
25#16840769C2211yescharacteristic
26C2×#14519842C2211yescharacteristic
27C2×#14519842C2211yescharacteristic
28C2×#14519841C2211yescharacteristic
29C2×#14519841C2211yescharacteristic
30#16840736C2211yescharacteristic
31#16840736C2211yescharacteristic
32#16840727—21no—
33#16840727—21no—
34#16840727C2211yescharacteristic
35#16840727C2211yescharacteristic
36C4×#11758213—21no—
37C4×#11758213C2211yescharacteristic
38C4×#11758213C2211yescharacteristic
39#16840721—21no—
40#16840721C2211yescharacteristic
41C42×#6923C2211yescharacteristic
42#16840638C2211yescharacteristic
43#16840638C2211yescharacteristic
44#16840632C2211yescharacteristic
45#16840632C2211yescharacteristic
46#16840632C2211yescharacteristic
47#16840632C2211yescharacteristic
48(C2×#6923)⋊C8C2211yescharacteristic
49(C2×#6923)⋊C8C2211yescharacteristic
50(C2×#6923)⋊C8C2211yescharacteristic
51(C2×#6923)⋊C8C2211yescharacteristic
52C2×#14519798C2211yescharacteristic
53C4×#11758226C2211yescharacteristic
54C4×#11758224C2211yescharacteristic
55#16839872C411yescharacteristic, complemented
56#16839871C411yescharacteristic, complemented
57#16839867C411yescharacteristic, complemented
58#16839866C411yescharacteristic, complemented
59#16839864C411yescharacteristic, complemented
60#16839863C411yescharacteristic, complemented
61#16839861C411yescharacteristic, complemented
62#16839799C2211yescharacteristic
63#16839757C2211yescharacteristic
64#16839755C2211yescharacteristic
65#97×D61C2211yescharacteristic
66C61×#440C411yescharacteristic, complemented
67#14519865—22no—
68#14519865—22no—
69#14519865—22no—
70#14519865—22no—
71C2×#11758213—21no—
72C2×#11758213—21no—
73C2×#11758213C2311yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758213C2311yescharacteristic
76C2×#11758213C2311yescharacteristic
77C2×D61⋊C8—21no—
78C2×D61⋊C8—21no—
79C2×D61⋊C8C2311yescharacteristic
80C2×D61⋊C8C2311yescharacteristic
81C2×C4×#6923—21no—
82C2×C4×#6923—21no—
83C2×C4×#6923C2311yescharacteristic
84C2×C4×#6923C2311yescharacteristic
85#14519848—22no—
86#14519847—22no—
87#14519844—21no—
88#14519844—21no—
89C4⋊#92420—21no—
90C4⋊#92420—21no—
91#14519842—22no—
92#14519842—22no—
93#14519841—22no—
94#14519841—22no—
95C22⋊#92420—21no—
96C22⋊#92420—21no—
97C22⋊#92420—21no—
98C22⋊#92420—21no—
99#14519832—21no—
100#14519832—21no—
101#14519832—21no—
102#14519832—21no—
103#14519832D411yescharacteristic
104#14519832D411yescharacteristic
105#14519832D411yescharacteristic
106#14519832D411yescharacteristic
107C4×#92420—21no—
108C4×#92420—21no—
109C4×#92420—21no—
110C4×#92420—21no—
111C2×#11758226C2311yescharacteristic
112C2×#11758224C2311yescharacteristic
113#14519798—22no—
114#14519798—22no—
115C2×#11758202C2×C411yescharacteristic
116C2×#11758190C2×C411yescharacteristic
117C2×#11758199C2×C411yescharacteristic
118C2×C8×D61C2311yescharacteristic
119C2×C8×D61C2311yescharacteristic
120Q8×Dic61C2×C411yescharacteristic
121C4×Dic122C2×C411yescharacteristic
122#14519742C2×C411yescharacteristic
123C4×D244—21no—
124C4×D244C2×C411yescharacteristic
125D4×Dic61C2×C411yescharacteristic
126#14519739—21no—
127#14519739C2×C411yescharacteristic
128C4×C22⋊D61—21no—
129#14519737—21no—
130#14519735—21no—
131#14519735C2311yescharacteristic
132#14519735C2311yescharacteristic
133#14519733—21no—
134C2×C4×Dic61—21no—
135C42×D61—21no—
136C42×D61C2311yescharacteristic
137#14519711C2×C411yescharacteristic
138#14519708C2×C411yescharacteristic
139#14519708C2×C411yescharacteristic
140#14519702C2×C411yescharacteristic
141#14519701C2×C411yescharacteristic
142#14519700C2×C411yescharacteristic
143#14519699C2×C411yescharacteristic
144#14519698C2×C411yescharacteristic
145#14519697C2×C411yescharacteristic
146#14519688C2×C411yescharacteristic
147#14519687C2×C411yescharacteristic
148#14519687C2×C411yescharacteristic
149C2×SD16×C61C2×C411yescharacteristic
150C4×Q8×C61C2×C411yescharacteristic
151C4×D4×C61C2×C411yescharacteristic
152#107×C61C2×C411yescharacteristic
153#103×C61C2×C411yescharacteristic
154#102×C61C2×C411yescharacteristic
155#97×C61C2×C411yescharacteristic
156C22×#6923—21no—
157C22×#6923—21no—
158C2×Q8×C61C22×C411yescharacteristic
159C2×D4×C61C22×C411yescharacteristic
160C22×C4×C61—21no—
161C2×C22⋊D61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164#11758226—21no—
165#11758226—21no—
166#11758226#4112yes—
167#11758226#4112yes—
168#11758224—21no—
169#11758224—21no—
170#11758224—21no—
171#11758224—21no—
172C2×D244—21no—
173C2×D244C22×C411yescharacteristic
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61#4111yescharacteristic
178C2×C4×D61#4111yescharacteristic
179C2×C4×D61C2411yescharacteristic
180C2×Dic122C22×C411yescharacteristic
181C2×#92420—21no—
182C2×#92420—21no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420C2×D411yescharacteristic
186C2×#92420C2×D411yescharacteristic
187C2×#92420C2×D411yescharacteristic
188C2×#92420C2×D411yescharacteristic
189C4×#6923—21no—
190C4×#6923—21no—
191C4×#6923—21no—
192C4×#6923—21no—
193C4×#6923—42no—
194C4×#6923—42no—
195#11758213—24no—
196#11758213—24no—
197#11758213—21no—
198#11758213—21no—
199#11758213—21no—
200#11758213—21no—
201#11758213—21no—
202#11758213—21no—
203#11758213#4112yes—
204#11758213#4112yes—
205D61⋊C8—24no—
206D61⋊C8—24no—
207D61⋊C8—21no—
208D61⋊C8—21no—
209D61⋊C8—21no—
210D61⋊C8—21no—
211SD16×C61—22no—
212C2×C8×C61C22×C411yescharacteristic
213C2×C8×C61C22×C411yescharacteristic
214C4⋊C4×C61—21no—
215C4⋊C4×C61C22×C411yescharacteristic
216C4⋊C4×C61C22×C411yescharacteristic
217C22⋊C4×C61—21no—
218C42×C61—21no—
219C42×C61C22×C411yescharacteristic
220C22⋊Dic61—21no—
221#11758202—22no—
222#11758199—22no—
223#11758198—21no—
224#11758198—21no—
225#11758198—21no—
226#11758198—21no—
227C4⋊Dic61—21no—
228C4⋊Dic61C22×C411yescharacteristic
229C4⋊Dic61C22×C411yescharacteristic
230#11758196—21no—
231#11758196—21no—
232C4×Dic61—21no—
233C4×Dic61—21no—
234C4×Dic61—21no—
235C4×Dic61—21no—
236C4×Dic61—21no—
237C4×Dic61C22×C411yescharacteristic
238C4×Dic61C2×D411yescharacteristic
239C4×Dic61C2×D411yescharacteristic
240C2×#92418C22×C411yescharacteristic
241C2×#92418C22×C411yescharacteristic
242#11758190—22no—
243C8×D61—21no—
244C8×D61—21no—
245C8×D61—21no—
246C8×D61—21no—
247C23×C61—21no—
248C22×D61—21no—
249C22×D61C2×#4111yescharacteristic
250C2×#6923—22no—
251C2×#6923—22no—
252C2×#6923—41no—
253C2×#6923—41no—
254C2×#6923—41no—
255C2×#6923—41no—
256Q8×C61—21no—
257Q8×C61#11712yes—
258D4×C61—21no—
259D4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—81no—
264C2×C4×C61C4×D411yescharacteristic
265C2×C4×C61C4×D411yescharacteristic
266C2×C4×C61C23×C411yescharacteristic
267C22⋊D61—41no—
268C22⋊D61—41no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—81no—
275C2×Dic61C4×D411yescharacteristic
276C2×Dic61C4×D411yescharacteristic
277C2×Dic61C22×D411yescharacteristic
278D244—22no—
279D244—22no—
280D244—21no—
281D244#11712yes—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—81no—
287C4×D61—81no—
288C4×D61C2×#4111yescharacteristic
289C4×D61C2×#4111yescharacteristic
290C4×D61C2×#4111yescharacteristic
291C4×D61C2×#4111yescharacteristic
292Dic122—21no—
293Dic122—21no—
294#92420—21no—
295#92420—21no—
296#92420—21no—
297#92420—21no—
298#92420—22no—
299#92420—22no—
300#92420—41no—
301#92420—41no—
302C8×C61—21no—
303C8×C61—21no—
304#92418—21no—
305#92418—21no—
306#21139—611nomaximal, Sylow
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61C2×C4×D411yescharacteristic
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61#52911yescharacteristic
317C2×D61#52911yescharacteristic
318#6923—42no—
319#6923—42no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—22no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61C2×#11711yescharacteristic
326C4×C61C4×#4111yescharacteristic
327Dic61—21no—
328Dic61—21no—
329Dic61—41no—
330Dic61—41no—
331Dic61—41no—
332Dic61#52511yescharacteristic
333Dic61#52511yescharacteristic
334#2943—611no—
335#2923—611no—
336#2875—611no—
337#1929—611no—
338#1929—611no—
339#1922—611no—
340#1922—611no—
341#1876—611no—
342#1876—611no—
343#1866—611no—
344#1866—611no—
345#1778—611no—
346C4×#104—611no—
347C4×#104—611no—
348#1745—611no—
349C2×C61—81no—
350C2×C61#229011yescharacteristic
351C2×C61#295611yescharacteristic
352C2×C61#295611yescharacteristic
353D61—21no—
354D61—21no—
355D61—42no—
356C2×#135—611no—
357C2×#131—611no—
358C2×#131—611no—
359C4×#41—611no—
360C4×#41—611no—
361#442—611no—
362#442—611no—
363#441—611no—
364#441—611no—
365#440—611no—
366#440—611no—
367#440—611no—
368#440#692311yescharacteristic, complemented
369#433—1221no—
370#433—1221no—
371#432—1221no—
372#432—1221no—
373C2×#108—611no—
374C2×#108—611no—
375#427—611no—
376#424—611no—
377#424—611no—
378C2×#104—611no—
379C2×#104—611no—
380C2×#104—611no—
381C2×#104—611no—
382#416—611no—
383#416—611no—
384#408—611no—
385#408—611no—
386#405—611no—
387#405—611no—
388#405—1221no—
389#405—1221no—
390C4×M16—611no—
391C4×M16—611no—
392C4×M16—1221no—
393C2×#97—611no—
394C2×#97—611no—
395C2×#97—1221no—
396C43—611no—
397#341—611no—
398#341—611no—
399#339—611no—
400#339—611no—
401#339—611no—
402#339—611no—
403#336—611no—
404#336—611no—
405#336—611no—
406#336—611no—
407C61#2113911yescharacteristic, complemented, Sylow
408C2×#41—611no—
409C2×#41—611no—
410#135—1222no—
411#135—1222no—
412C2×Q16—611no—
413C2×SD16—611no—
414C2×SD16C2×#692311yescharacteristic
415C2×D8—611no—
416#131—1222no—
417#131—1222no—
418#131—1222no—
419#131—1222no—
420C2×M16—611no—
421C2×M16—611no—
422C2×M16—611no—
423C2×M16—611no—
424C2×M16—1221no—
425C2×M16—1221no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8—611no—
429C22×C8—611no—
430C22×C8—1221no—
431C22×C8—1221no—
432C4×Q8—611no—
433C4×Q8—611no—
434C4×Q8—611no—
435C4×Q8C2×#692311yescharacteristic
436C4×D4—611no—
437C4×D4—611no—
438C4×D4—611no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4—1221no—
442C4×D4—1221no—
443C4×D4C2×#692311yescharacteristic
444#117—611no—
445#117—611no—
446#117—1221no—
447#117—1221no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455#108—1222no—
456#108—1222no—
457#107—611no—
458#107C2×#692311yescharacteristic
459#106—611no—
460#106—611no—
461C4⋊C8—1221no—
462C4⋊C8—1221no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465#104—1222no—
466#104—1222no—
467#104—1222no—
468#104—1222no—
469#103—611no—
470#103—611no—
471#103—611no—
472#103C2×#692311yescharacteristic
473#102—611no—
474#102—611no—
475#102—611no—
476#102C2×#692311yescharacteristic
477C22⋊C8—1221no—
478C22⋊C8—1221no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481#97—611no—
482#97—611no—
483#97—611no—
484#97—611no—
485#97—611no—
486#97—611no—
487#97—611no—
488#97—1221no—
489#97—1221no—
490#97—1221no—
491#97—1221no—
492#97C2×#692311yescharacteristic
493C4×C8—1221no—
494C4×C8—1221no—
495C4×C8—1221no—
496C4×C8—1221no—
497#41—612no—
498#41—612no—
499#41—1221no—
500#41—1221no—
501#41—1221no—
502#41—1221no—
503#41—1221no—
504#41—1221no—
505C2×Q8—611no—
506C2×Q8C22×#692311yescharacteristic
507C2×D4—611no—
508C2×D4—1221no—
509C2×D4—1221no—
510C2×D4C22×#692311yescharacteristic
511C22×C4—21no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522Q16—1222no—
523SD16—22no—
524SD16—1222no—
525D8—1222no—
526M16—612no—
527M16—612no—
528M16—1221no—
529M16—1221no—
530M16—1224no—
531M16—1224no—
532M16—1221no—
533M16—1221no—
534M16—1221no—
535M16—1221no—
536C2×C8—611no—
537C2×C8—611no—
538C2×C8—611no—
539C2×C8—611no—
540C2×C8—611no—
541C2×C8—611no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1224no—
553C2×C8—1224no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8C22×#692311yescharacteristic
557C2×C8C22×#692311yescharacteristic
558C4⋊C4—21no—
559C4⋊C4—611no—
560C4⋊C4—611no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4C22×#692311yescharacteristic
565C4⋊C4C22×#692311yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C22⋊C4—1221no—
571C22⋊C4—1221no—
572C42—21no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—1221no—
577C42—1221no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42—1221no—
584C42—1221no—
585C42—2442no—
586C42—2442no—
587C42C22×#692311yescharacteristic
588C23—21no—
589C23—611no—
590C23—1221no—
591Q8—21no—
592Q8—1221no—
593Q8—1221no—
594Q8#1451985612yes—
595D4—21no—
596D4—21no—
597D4—612no—
598D4—1222no—
599D4—1222no—
600D4—1221no—
601D4—2441no—
602D4—2441no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—81no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—4881no—
622C2×C4—4881no—
623C2×C4—4881no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4D4×#692311yescharacteristic
633C2×C4D4×#692311yescharacteristic
634C2×C4C23×#692311yescharacteristic
635C8—21no—
636C8—21no—
637C8—1221no—
638C8—1221no—
639C8—1221no—
640C8—1221no—
641C8—1222no—
642C8—1222no—
643C8—1221no—
644C8—1221no—
645C8—2441no—
646C8—2441no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—611no—
651C22—611no—
652C22—1221no—
653C22—1221no—
654C22—1222no—
655C22—2441no—
656C22—2441no—
657C22C2×D4×#692311yescharacteristic
658C4—22no—
659C4—21no—
660C4—21no—
661C4—41no—
662C4—41no—
663C4—611no—
664C4—611no—
665C4—1221no—
666C4—1221no—
667C4—2442no—
668C4—2442no—
669C4—2441no—
670C4—2441no—
671C4—2441no—
672C4C2×#1451985611yescharacteristic
673C4#41×#692311yescharacteristic
674C2—81no—
675C2—1221no—
676C2—1221no—
677C2—2442no—
678C2#2146227311yescharacteristic
679C2#2146259311yescharacteristic
680C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#440C4
socleC22×C61C2×C4×D4
prime radicalC2×#14519798C22
cosocleC2×C4×D61C24
hypercenter#440#6923
hypocenterC61#21139
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #21462506C2semidirect
2. #21462391C2semidirect
3. #21461901C2semidirect
4. #21461900C2semidirect
5. #21461887C2semidirect
6. #21461884C2semidirect
9. #21461789C2semidirect
10. #21461787C2semidirect
11. #21461639C2semidirect
12. C4×#14519842C2semidirect
13. C4×#14519841C2semidirect
14. #21461535C2semidirect
55. #16839872C4semidirect
56. #16839871C4semidirect
57. #16839867C4semidirect
58. #16839866C4semidirect
59. #16839864C4semidirect
60. #16839863C4semidirect
61. #16839861C4semidirect
66. C61×#440C4semidirect
368. #440#6923semidirect
407. C61#21139semidirect

Maximal subgroups

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

subgroupindex
#214625062
#214623912
#214619012
#214619002
#214618872
#214618842
#214618112
#214617892
#214617872
#214616392
C4×#145198422
C4×#145198412
#214615352
#214546142
#2113961