Finite Group Databasebeta

← #30616418

Subgroups — #30616418

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462504C211yesmaximal, characteristic, complemented
2#21462503C211yesmaximal, characteristic, complemented
3#21461975C211yesmaximal, characteristic
4#21461971C211yesmaximal, characteristic
5#21461928C211yesmaximal, characteristic, complemented
6#21461927C211yesmaximal, characteristic, complemented
7#21461915C211yesmaximal, characteristic, complemented
8#21461914C211yesmaximal, characteristic, complemented
9#21461913C211yesmaximal, characteristic, complemented
10#21461912C211yesmaximal, characteristic, complemented
11#21461639C211yesmaximal, characteristic, complemented
12#21461583C211yesmaximal, characteristic, complemented
13#21461582C211yesmaximal, characteristic, complemented
14#21461477C211yesmaximal, characteristic, complemented
15#21454614C211yesmaximal, characteristic
16#16840787—21no—
17#16840787—21no—
18#16840786—21no—
19#16840786—21no—
20#16840781—21no—
21#16840780—21no—
22#16840779—21no—
23#16840778—21no—
24C2×#14519848C2211yescharacteristic
25C2×#14519847C2211yescharacteristic
26#16840769C2211yescharacteristic
27#16840769C2211yescharacteristic
28#16840769C2211yescharacteristic
29#16840769C2211yescharacteristic
30C2×#14519842C2211yescharacteristic
31C2×#14519842C2211yescharacteristic
32C2×#14519841C2211yescharacteristic
33C2×#14519841C2211yescharacteristic
34#16840737C2211yescharacteristic
35#16840737C2211yescharacteristic
36#16840736C2211yescharacteristic
37#16840736C2211yescharacteristic
38C4×#11758213C2211yescharacteristic
39C4×#11758213C2211yescharacteristic
40C2×#14519831C2211yescharacteristic
41C2×#14519831C2211yescharacteristic
42C4×D61⋊C8C2211yescharacteristic
43C2×C8×#6923C2211yescharacteristic
44C2×C8×#6923C2211yescharacteristic
45#16840698C2211yescharacteristic
46#16840638C2211yescharacteristic
47#16840638C2211yescharacteristic
48#16840632C2211yescharacteristic
49#16840632C2211yescharacteristic
50#16840632C2211yescharacteristic
51#16840632C2211yescharacteristic
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
67C2×#11758213C2311yescharacteristic
68C2×#11758213C2311yescharacteristic
69C2×#11758213C2311yescharacteristic
70C2×#11758213C2311yescharacteristic
71C2×D61⋊C8C2311yescharacteristic
72C2×D61⋊C8C2311yescharacteristic
73C2×C4×#6923C2311yescharacteristic
74C2×C4×#6923C2311yescharacteristic
75#14519848—22no—
76#14519847—22no—
77#14519844—21no—
78#14519844—21no—
79#14519844—21no—
80#14519844—21no—
81C4⋊#92420—21no—
82C4⋊#92420—21no—
83C4⋊#92420—21no—
84C4⋊#92420—21no—
85#14519842—22no—
86#14519842—22no—
87#14519841—22no—
88#14519841—22no—
89#14519834—21no—
90#14519834—21no—
91#14519834—21no—
92#14519834—21no—
93C22⋊#92420—21no—
94C22⋊#92420—21no—
95C22⋊#92420—21no—
96C22⋊#92420—21no—
97#14519832—21no—
98#14519832—21no—
99#14519831—22no—
100#14519831—22no—
101C4×#92420—21no—
102C4×#92420—21no—
103C4×#92420D411yescharacteristic
104C4×#92420D411yescharacteristic
105C4×#92420D411yescharacteristic
106C4×#92420D411yescharacteristic
107C8×#6923—22no—
108C8×#6923—22no—
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—21no—
167#11758226—21no—
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#4112yes—
192C4×#6923#4112yes—
193#11758213—21no—
194#11758213—21no—
195#11758213—21no—
196#11758213—21no—
197#11758213—21no—
198#11758213—21no—
199#11758213—21no—
200#11758213—21no—
201D61⋊C8—21no—
202D61⋊C8—21no—
203D61⋊C8—21no—
204D61⋊C8—21no—
205SD16×C61—22no—
206C2×C8×C61C22×C411yescharacteristic
207C2×C8×C61C22×C411yescharacteristic
208C4⋊C4×C61—21no—
209C4⋊C4×C61C22×C411yescharacteristic
210C4⋊C4×C61C22×C411yescharacteristic
211C22⋊C4×C61—21no—
212C42×C61—21no—
213C42×C61C22×C411yescharacteristic
214C22⋊Dic61—21no—
215#11758202—22no—
216#11758199—22no—
217#11758198—21no—
218#11758198—21no—
219#11758198—21no—
220#11758198—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61C22×C411yescharacteristic
223C4⋊Dic61C22×C411yescharacteristic
224#11758196—21no—
225#11758196—21no—
226C4×Dic61—21no—
227C4×Dic61—21no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61C22×C411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C2×#92418C22×C411yescharacteristic
235C2×#92418C22×C411yescharacteristic
236#11758190—22no—
237C8×D61—21no—
238C8×D61—21no—
239C8×D61#4112yes—
240C8×D61#4112yes—
241C23×C61—21no—
242C22×D61—21no—
243C22×D61C2×#4111yescharacteristic
244C2×#6923—22no—
245C2×#6923—22no—
246C2×#6923—41no—
247C2×#6923—41no—
248C2×#6923—41no—
249C2×#6923—41no—
250Q8×C61—21no—
251Q8×C61—21no—
252D4×C61—21no—
253D4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—42no—
258C2×C4×C61C4×D411yescharacteristic
259C2×C4×C61C4×D411yescharacteristic
260C2×C4×C61C23×C411yescharacteristic
261C22⋊D61—41no—
262C22⋊D61—41no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61C4×D411yescharacteristic
270C2×Dic61C4×D411yescharacteristic
271C2×Dic61C22×D411yescharacteristic
272D244—21no—
273D244—21no—
274D244—41no—
275D244—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61C2×#4111yescharacteristic
283C4×D61C2×#4111yescharacteristic
284C4×D61C2×#4111yescharacteristic
285C4×D61C2×#4111yescharacteristic
286Dic122—21no—
287Dic122—21no—
288#92420—21no—
289#92420—21no—
290#92420—21no—
291#92420—21no—
292#92420—41no—
293#92420—41no—
294#92420—41no—
295#92420—41no—
296C8×C61—21no—
297C8×C61#11712yes—
298#92418—21no—
299#92418#11712yes—
300#21296—611nomaximal, Sylow
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61C2×C4×D411yescharacteristic
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61#52911yescharacteristic
311C2×D61#52911yescharacteristic
312#6923—42no—
313#6923—42no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61C2×#11711yescharacteristic
320C4×C61C4×#4111yescharacteristic
321Dic61—21no—
322Dic61—21no—
323Dic61—41no—
324Dic61—41no—
325Dic61—41no—
326Dic61#53211yescharacteristic
327Dic61#53211yescharacteristic
328#2943—611no—
329#2922—611no—
330#2922—611no—
331#1975—611no—
332#1974—611no—
333#1946—611no—
334#1945—611no—
335#1938—611no—
336#1938—611no—
337#1937—611no—
338#1937—611no—
339#1778—611no—
340#1759—611no—
341#1759—611no—
342#1726—611no—
343C2×C61—81no—
344C2×C61#229311yescharacteristic
345C2×C61#295511yescharacteristic
346C2×C61#295511yescharacteristic
347D61—21no—
348D61—21no—
349D61—81no—
350C2×#135—611no—
351C4×#41—611no—
352C4×#41—611no—
353#442—611no—
354#442—611no—
355#441—611no—
356#441—611no—
357#440—611no—
358#440—611no—
359#440—611no—
360#440#692311yescharacteristic, complemented
361#433—1221no—
362#433—1221no—
363#433—1221no—
364#433—1221no—
365#431—1221no—
366#431—1221no—
367#431—1221no—
368#431—1221no—
369C2×#108—611no—
370C2×#108—611no—
371#427—611no—
372#424—611no—
373#424—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#408—611no—
385#408—611no—
386C4×M16—611no—
387C4×M16—611no—
388C2×#97—611no—
389C2×#97—611no—
390C2×#97—611no—
391C2×C4×C8—611no—
392C2×C4×C8—611no—
393C2×C4×C8—611no—
394#376—611no—
395#341—611no—
396#341—611no—
397#339—611no—
398#339—611no—
399#339—611no—
400#339—611no—
401C61#2129611yescharacteristic, complemented, Sylow
402C2×#41—611no—
403C2×#41—611no—
404#135—1222no—
405#135—1222no—
406C2×Q16—611no—
407C2×SD16—611no—
408C2×SD16C2×#692311yescharacteristic
409C2×D8—611no—
410C2×M16—611no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16—611no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8—611no—
417C22×C8—611no—
418C4×Q8—611no—
419C4×Q8—611no—
420C4×Q8—611no—
421C4×Q8C2×#692311yescharacteristic
422C4×D4—611no—
423C4×D4—611no—
424C4×D4—611no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4C2×#692311yescharacteristic
430#117—611no—
431#117—611no—
432#117—1221no—
433#117—1221no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439#108—1222no—
440#108—1222no—
441#107—611no—
442#107C2×#692311yescharacteristic
443#106—611no—
444#106—611no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453#104—1222no—
454#104—1222no—
455#104—1222no—
456#104—1222no—
457#103—611no—
458#103—611no—
459#103—611no—
460#103C2×#692311yescharacteristic
461#102—611no—
462#102—611no—
463#102—611no—
464#102C2×#692311yescharacteristic
465C22⋊C8—1221no—
466C22⋊C8—1221no—
467C22⋊C8—1221no—
468C22⋊C8—1221no—
469C22⋊C8—1221no—
470C22⋊C8—1221no—
471C22⋊C8—1221no—
472C22⋊C8—1221no—
473#97—611no—
474#97—611no—
475#97—611no—
476#97—1222no—
477#97—1221no—
478#97—1221no—
479#97—1222no—
480#97C2×#692311yescharacteristic
481C4×C8—611no—
482C4×C8—611no—
483C4×C8—611no—
484C4×C8—611no—
485C4×C8—1222no—
486C4×C8—1221no—
487C4×C8—1221no—
488C4×C8—1222no—
489#95—1221no—
490#95—1221no—
491#41—1221no—
492#41—1221no—
493#41—1221no—
494#41—1221no—
495#41—1221no—
496#41—1221no—
497#41—1221no—
498#41—1221no—
499C2×Q8—611no—
500C2×Q8C22×#692311yescharacteristic
501C2×D4—611no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4C22×#692311yescharacteristic
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516Q16—1222no—
517SD16—22no—
518SD16—1222no—
519D8—1222no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16—1221no—
524M16—1221no—
525M16—1221no—
526M16—1221no—
527M16—1221no—
528C2×C8—611no—
529C2×C8—612no—
530C2×C8—612no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—611no—
535C2×C8—611no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8C22×#692311yescharacteristic
547C2×C8C22×#692311yescharacteristic
548C4⋊C4—21no—
549C4⋊C4—611no—
550C4⋊C4—611no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4C22×#692311yescharacteristic
555C4⋊C4C22×#692311yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—1221no—
558C22⋊C4—1221no—
559C22⋊C4—1221no—
560C22⋊C4—1221no—
561C22⋊C4—1221no—
562C42—21no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—612no—
567C42—612no—
568C42—1221no—
569C42—1221no—
570C42—1221no—
571C42—1221no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42C22×#692311yescharacteristic
576C23—21no—
577C23—611no—
578C23—1221no—
579Q8—21no—
580Q8—21no—
581Q8—1221no—
582Q8—1221no—
583D4—21no—
584D4—21no—
585D4—1221no—
586D4—1221no—
587D4—2441no—
588D4—2441no—
589D4—2441no—
590D4—2441no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—42no—
595C2×C4—611no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2442no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2442no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2442no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4D4×#692311yescharacteristic
621C2×C4D4×#692311yescharacteristic
622C2×C4C23×#692311yescharacteristic
623C8—21no—
624C8—612no—
625C8—1221no—
626C8—1221no—
627C8—1221no—
628C8—1221no—
629C8—1221no—
630C8—2441no—
631C8—2441no—
632C8—2441no—
633C8—2441no—
634C8#1451985612yes—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—611no—
639C22—611no—
640C22—1221no—
641C22—1221no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22C2×D4×#692311yescharacteristic
646C4—21no—
647C4—21no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—611no—
652C4—611no—
653C4—1221no—
654C4—1221no—
655C4—2442no—
656C4—2442no—
657C4—2441no—
658C4—2441no—
659C4—2441no—
660C4C2×#1451985611yescharacteristic
661C4#41×#692311yescharacteristic
662C2—81no—
663C2—1221no—
664C2—1221no—
665C2—4881no—
666C2#2146227611yescharacteristic
667C2#2146259111yescharacteristic
668C2#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#21296
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. #21462504C2semidirect
2. #21462503C2semidirect
5. #21461928C2semidirect
6. #21461927C2semidirect
7. #21461915C2semidirect
8. #21461914C2semidirect
9. #21461913C2semidirect
10. #21461912C2semidirect
11. #21461639C2semidirect
12. #21461583C2semidirect
13. #21461582C2semidirect
14. #21461477C2semidirect
55. #16839872C4semidirect
56. #16839871C4semidirect
57. #16839867C4semidirect
58. #16839866C4semidirect
59. #16839864C4semidirect
60. #16839863C4semidirect
61. #16839861C4semidirect
66. C61×#440C4semidirect
360. #440#6923semidirect
401. C61#21296semidirect

Maximal subgroups

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

subgroupindex
#214625042
#214625032
#214619752
#214619712
#214619282
#214619272
#214619152
#214619142
#214619132
#214619122
#214616392
#214615832
#214615822
#214614772
#214546142
#2129661