Finite Group Databasebeta

← #30616093

Subgroups — #30616093

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462503C211yesmaximal, characteristic, complemented
2C4×#14519865C211yesmaximal, 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
12#21461583C211yesmaximal, characteristic, complemented
13#21461582C211yesmaximal, characteristic, complemented
14#21461539C211yesmaximal, characteristic, complemented
15#21454614C211yesmaximal, characteristic
16C2×#14519865C2211yescharacteristic
17C2×#14519865C2211yescharacteristic
18#16840786—21no—
19#16840786—21no—
20#16840779—21no—
21#16840778—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
39C4×D61⋊C8—21no—
40C4×D61⋊C8C2211yescharacteristic
41#16840698C2211yescharacteristic
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×#6923C2311yescharacteristic
82C2×C4×#6923C2311yescharacteristic
83#14519848—22no—
84#14519847—22no—
85#14519844—21no—
86#14519844—21no—
87C4⋊#92420—21no—
88C4⋊#92420—21no—
89#14519842—22no—
90#14519842—22no—
91#14519841—22no—
92#14519841—22no—
93C22⋊#92420—21no—
94C22⋊#92420—21no—
95C22⋊#92420—21no—
96C22⋊#92420—21no—
97#14519832—21no—
98#14519832—21no—
99#14519832—21no—
100#14519832—21no—
101C4×#92420—21no—
102C4×#92420—21no—
103C4×#92420—21no—
104C4×#92420—21no—
105C4×#92420D411yescharacteristic
106C4×#92420D411yescharacteristic
107C4×#92420D411yescharacteristic
108C4×#92420D411yescharacteristic
109(C2×#6923)⋊C4—21no—
110(C2×#6923)⋊C4—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—
193#11758213—24no—
194#11758213—24no—
195#11758213—21no—
196#11758213—21no—
197#11758213—21no—
198#11758213—21no—
199#11758213—21no—
200#11758213—21no—
201#11758213#4112yes—
202#11758213#4112yes—
203D61⋊C8—24no—
204D61⋊C8—24no—
205D61⋊C8—21no—
206D61⋊C8—21no—
207D61⋊C8—21no—
208D61⋊C8—21no—
209SD16×C61—22no—
210C2×C8×C61C22×C411yescharacteristic
211C2×C8×C61C22×C411yescharacteristic
212C4⋊C4×C61—21no—
213C4⋊C4×C61C22×C411yescharacteristic
214C4⋊C4×C61C22×C411yescharacteristic
215C22⋊C4×C61—21no—
216C42×C61—21no—
217C42×C61C22×C411yescharacteristic
218C22⋊Dic61—21no—
219#11758202—22no—
220#11758199—22no—
221#11758198—21no—
222#11758198—21no—
223#11758198—21no—
224#11758198—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61C22×C411yescharacteristic
227C4⋊Dic61C22×C411yescharacteristic
228#11758196—21no—
229#11758196—21no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61—21no—
233C4×Dic61—21no—
234C4×Dic61—21no—
235C4×Dic61C22×C411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238C2×#92418C22×C411yescharacteristic
239C2×#92418C22×C411yescharacteristic
240#11758190—22no—
241C8×D61—21no—
242C8×D61—21no—
243C8×D61—21no—
244C8×D61—21no—
245C23×C61—21no—
246C22×D61—21no—
247C22×D61C2×#4111yescharacteristic
248C2×#6923—41no—
249C2×#6923—41no—
250C2×#6923—41no—
251C2×#6923—41no—
252C2×#6923—41no—
253C2×#6923—41no—
254Q8×C61—21no—
255Q8×C61#11712yes—
256D4×C61—21no—
257D4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—42no—
262C2×C4×C61C4×D411yescharacteristic
263C2×C4×C61C4×D411yescharacteristic
264C2×C4×C61C23×C411yescharacteristic
265C22⋊D61—41no—
266C22⋊D61—41no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—42no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61C4×D411yescharacteristic
274C2×Dic61C4×D411yescharacteristic
275C2×Dic61C22×D411yescharacteristic
276D244—22no—
277D244—22no—
278D244—21no—
279D244#11712yes—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61C2×#4111yescharacteristic
287C4×D61C2×#4111yescharacteristic
288C4×D61C2×#4111yescharacteristic
289C4×D61C2×#4111yescharacteristic
290Dic122—21no—
291Dic122—21no—
292#92420—21no—
293#92420—21no—
294#92420—21no—
295#92420—21no—
296#92420—22no—
297#92420—22no—
298#92420—41no—
299#92420—41no—
300C8×C61—21no—
301C8×C61—21no—
302#92418—21no—
303#92418—21no—
304#21138—611nomaximal, Sylow
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61C2×C4×D411yescharacteristic
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61#52911yescharacteristic
315C2×D61#52911yescharacteristic
316#6923—81no—
317#6923—81no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—22no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61C2×#11711yescharacteristic
324C4×C61C4×#4111yescharacteristic
325Dic61—21no—
326Dic61—21no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61#52511yescharacteristic
331Dic61#52511yescharacteristic
332#2943—611no—
333#2922—611no—
334C4×#131—611no—
335#1929—611no—
336#1929—611no—
337#1922—611no—
338#1922—611no—
339#1876—611no—
340#1876—611no—
341#1866—611no—
342#1866—611no—
343#1778—611no—
344#1759—611no—
345#1759—611no—
346#1746—611no—
347C2×C61—81no—
348C2×C61#229011yescharacteristic
349C2×C61#295511yescharacteristic
350C2×C61#295511yescharacteristic
351D61—21no—
352D61—21no—
353D61—42no—
354C2×#135—611no—
355C2×#131—611no—
356C2×#131—611no—
357C4×#41—611no—
358C4×#41—611no—
359#442—611no—
360#442—611no—
361#441—611no—
362#441—611no—
363#440—611no—
364#440—611no—
365#440—611no—
366#440#692311yescharacteristic, complemented
367#433—1221no—
368#433—1221no—
369#431—1221no—
370#431—1221no—
371C2×#108—611no—
372C2×#108—611no—
373#427—611no—
374#424—611no—
375#424—611no—
376C2×#104—611no—
377C2×#104—611no—
378C2×#104—611no—
379C2×#104—611no—
380#416—611no—
381#416—611no—
382#408—611no—
383#408—611no—
384#405—611no—
385#405—611no—
386#405—1221no—
387#405—1221no—
388C4×M16—611no—
389C4×M16—611no—
390C4×M16—1221no—
391C2×#97—611no—
392C2×C4×C8—611no—
393C2×C4×C8—1221no—
394#376—611no—
395#341—611no—
396#341—611no—
397#339—611no—
398#339—611no—
399#339—611no—
400#339—611no—
401#336—611no—
402#336—611no—
403#336—611no—
404#336—611no—
405C61#2113811yescharacteristic, complemented, Sylow
406C2×#41—611no—
407C2×#41—611no—
408#135—1222no—
409#135—1222no—
410C2×Q16—611no—
411C2×SD16—611no—
412C2×SD16C2×#692311yescharacteristic
413C2×D8—611no—
414#131—1222no—
415#131—1222no—
416#131—1222no—
417#131—1222no—
418C2×M16—611no—
419C2×M16—611no—
420C2×M16—611no—
421C2×M16—611no—
422C2×M16—1221no—
423C2×M16—1221no—
424C22×C8—611no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8—1221no—
429C22×C8—1221no—
430C4×Q8—611no—
431C4×Q8—611no—
432C4×Q8—611no—
433C4×Q8C2×#692311yescharacteristic
434C4×D4—611no—
435C4×D4—611no—
436C4×D4—611no—
437C4×D4—1221no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4C2×#692311yescharacteristic
442#117—611no—
443#117—611no—
444#117—1221no—
445#117—1221no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42—1221no—
451#108—1222no—
452#108—1222no—
453#107—611no—
454#107C2×#692311yescharacteristic
455#106—611no—
456#106—611no—
457C4⋊C8—1221no—
458C4⋊C8—1221no—
459C4⋊C8—1221no—
460C4⋊C8—1221no—
461#104—1222no—
462#104—1222no—
463#104—1222no—
464#104—1222no—
465#103—611no—
466#103—611no—
467#103—611no—
468#103C2×#692311yescharacteristic
469#102—611no—
470#102—611no—
471#102—611no—
472#102C2×#692311yescharacteristic
473C22⋊C8—1221no—
474C22⋊C8—1221no—
475C22⋊C8—1221no—
476C22⋊C8—1221no—
477#97—611no—
478#97—611no—
479#97—611no—
480#97—1221no—
481#97—1221no—
482#97—1221no—
483#97—1221no—
484#97C2×#692311yescharacteristic
485C4×C8—611no—
486C4×C8—611no—
487C4×C8—611no—
488C4×C8—611no—
489C4×C8—1221no—
490C4×C8—1221no—
491C4×C8—1221no—
492C4×C8—1221no—
493#95—1221no—
494#95—1221no—
495#41—612no—
496#41—612no—
497#41—1221no—
498#41—1221no—
499#41—1221no—
500#41—1221no—
501#41—1221no—
502#41—1221no—
503C2×Q8—611no—
504C2×Q8C22×#692311yescharacteristic
505C2×D4—611no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4C22×#692311yescharacteristic
509C22×C4—21no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520Q16—1222no—
521SD16—22no—
522SD16—1222no—
523D8—1222no—
524M16—612no—
525M16—612no—
526M16—1221no—
527M16—1221no—
528M16—1224no—
529M16—1224no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533M16—1221no—
534C2×C8—611no—
535C2×C8—611no—
536C2×C8—611no—
537C2×C8—611no—
538C2×C8—611no—
539C2×C8—611no—
540C2×C8—1221no—
541C2×C8—1221no—
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—1224no—
551C2×C8—1224no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8C22×#692311yescharacteristic
555C2×C8C22×#692311yescharacteristic
556C4⋊C4—21no—
557C4⋊C4—611no—
558C4⋊C4—611no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4C22×#692311yescharacteristic
563C4⋊C4C22×#692311yescharacteristic
564C22⋊C4—21no—
565C22⋊C4—1221no—
566C22⋊C4—1221no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C42—21no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—1221no—
575C42—1221no—
576C42—1221no—
577C42—1221no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42C22×#692311yescharacteristic
584C23—21no—
585C23—611no—
586C23—1221no—
587Q8—21no—
588Q8—1221no—
589Q8—1221no—
590Q8#1451985612yes—
591D4—21no—
592D4—21no—
593D4—612no—
594D4—1222no—
595D4—1222no—
596D4—1221no—
597D4—2441no—
598D4—2441no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—42no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—611no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2442no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4D4×#692311yescharacteristic
629C2×C4D4×#692311yescharacteristic
630C2×C4C23×#692311yescharacteristic
631C8—21no—
632C8—21no—
633C8—1221no—
634C8—1221no—
635C8—1221no—
636C8—1221no—
637C8—1222no—
638C8—1222no—
639C8—1221no—
640C8—1221no—
641C8—2441no—
642C8—2441no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—611no—
647C22—611no—
648C22—1221no—
649C22—1221no—
650C22—1222no—
651C22—2441no—
652C22—2441no—
653C22C2×D4×#692311yescharacteristic
654C4—22no—
655C4—21no—
656C4—21no—
657C4—41no—
658C4—41no—
659C4—611no—
660C4—611no—
661C4—1221no—
662C4—1221no—
663C4—4881no—
664C4—4881no—
665C4—2441no—
666C4—2441no—
667C4—2441no—
668C4C2×#1451985611yescharacteristic
669C4#41×#692311yescharacteristic
670C2—81no—
671C2—1221no—
672C2—1221no—
673C2—2442no—
674C2#2146227311yescharacteristic
675C2#2146259111yescharacteristic
676C2#2146259211yescharacteristic
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#21138
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. #21462503C2semidirect
2. C4×#14519865C2semidirect
3. #21461901C2semidirect
4. #21461900C2semidirect
5. #21461887C2semidirect
6. #21461884C2semidirect
9. #21461789C2semidirect
10. #21461787C2semidirect
11. #21461639C2semidirect
12. #21461583C2semidirect
13. #21461582C2semidirect
14. #21461539C2semidirect
55. #16839872C4semidirect
56. #16839871C4semidirect
57. #16839867C4semidirect
58. #16839866C4semidirect
59. #16839864C4semidirect
60. #16839863C4semidirect
61. #16839861C4semidirect
66. C61×#440C4semidirect
366. #440#6923semidirect
405. C61#21138semidirect

Maximal subgroups

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

subgroupindex
#214625032
C4×#145198652
#214619012
#214619002
#214618872
#214618842
#214618112
#214617892
#214617872
#214616392
#214615832
#214615822
#214615392
#214546142
#2113861