Finite Group Databasebeta

← #30616421

Subgroups — #30616421

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462507C211yesmaximal, characteristic, complemented
2#21462506C211yesmaximal, 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
12C4×#14519842C211yesmaximal, characteristic, complemented
13C4×#14519841C211yesmaximal, characteristic, complemented
14#97×#6923C211yesmaximal, characteristic, complemented
15#21454614C211yesmaximal, characteristic
16#16840787—21no—
17#16840787—21no—
18#16840786—21no—
19#16840786—21no—
20#16840785—21no—
21#16840784—21no—
22#16840783—21no—
23#16840782—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
40#16840721C2211yescharacteristic
41C2×#14519831C2211yescharacteristic
42C2×#14519831C2211yescharacteristic
43C2×C8×#6923C2211yescharacteristic
44C2×C8×#6923C2211yescharacteristic
45C42×#6923C2211yescharacteristic
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×#6923—21no—
74C2×C4×#6923—21no—
75C2×C4×#6923C2311yescharacteristic
76C2×C4×#6923C2311yescharacteristic
77#14519848—22no—
78#14519847—22no—
79#14519844—21no—
80#14519844—21no—
81#14519844—21no—
82#14519844—21no—
83C4⋊#92420—21no—
84C4⋊#92420—21no—
85C4⋊#92420—21no—
86C4⋊#92420—21no—
87#14519842—22no—
88#14519842—22no—
89#14519841—22no—
90#14519841—22no—
91#14519834—21no—
92#14519834—21no—
93#14519834—21no—
94#14519834—21no—
95C22⋊#92420—21no—
96C22⋊#92420—21no—
97C22⋊#92420—21no—
98C22⋊#92420—21no—
99#14519832—21no—
100#14519832—21no—
101#14519832D411yescharacteristic
102#14519832D411yescharacteristic
103#14519832D411yescharacteristic
104#14519832D411yescharacteristic
105#14519831—22no—
106#14519831—22no—
107C4×#92420—21no—
108C4×#92420—21no—
109C8×#6923—22no—
110C8×#6923—22no—
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—42no—
192C4×#6923—42no—
193C4×#6923#4112yes—
194C4×#6923#4112yes—
195#11758213—21no—
196#11758213—21no—
197#11758213—21no—
198#11758213—21no—
199#11758213—21no—
200#11758213—21no—
201#11758213—21no—
202#11758213—21no—
203D61⋊C8—21no—
204D61⋊C8—21no—
205D61⋊C8—21no—
206D61⋊C8—21no—
207SD16×C61—22no—
208C2×C8×C61C22×C411yescharacteristic
209C2×C8×C61C22×C411yescharacteristic
210C4⋊C4×C61—21no—
211C4⋊C4×C61C22×C411yescharacteristic
212C4⋊C4×C61C22×C411yescharacteristic
213C22⋊C4×C61—21no—
214C42×C61—21no—
215C42×C61C22×C411yescharacteristic
216C22⋊Dic61—21no—
217#11758202—22no—
218#11758199—22no—
219#11758198—21no—
220#11758198—21no—
221#11758198—21no—
222#11758198—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61C22×C411yescharacteristic
225C4⋊Dic61C22×C411yescharacteristic
226#11758196—21no—
227#11758196—21no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61—21no—
233C4×Dic61C22×C411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C2×#92418C22×C411yescharacteristic
237C2×#92418C22×C411yescharacteristic
238#11758190—22no—
239C8×D61—21no—
240C8×D61—21no—
241C8×D61#4112yes—
242C8×D61#4112yes—
243C23×C61—21no—
244C22×D61—21no—
245C22×D61C2×#4111yescharacteristic
246C2×#6923—22no—
247C2×#6923—22no—
248C2×#6923—22no—
249C2×#6923—22no—
250C2×#6923—41no—
251C2×#6923—41no—
252Q8×C61—21no—
253Q8×C61—21no—
254D4×C61—21no—
255D4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—81no—
260C2×C4×C61C4×D411yescharacteristic
261C2×C4×C61C4×D411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C22⋊D61—41no—
264C22⋊D61—41no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—81no—
271C2×Dic61C4×D411yescharacteristic
272C2×Dic61C4×D411yescharacteristic
273C2×Dic61C22×D411yescharacteristic
274D244—21no—
275D244—21no—
276D244—41no—
277D244—41no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—81no—
283C4×D61—81no—
284C4×D61C2×#4111yescharacteristic
285C4×D61C2×#4111yescharacteristic
286C4×D61C2×#4111yescharacteristic
287C4×D61C2×#4111yescharacteristic
288Dic122—21no—
289Dic122—21no—
290#92420—21no—
291#92420—21no—
292#92420—21no—
293#92420—21no—
294#92420—41no—
295#92420—41no—
296#92420—41no—
297#92420—41no—
298C8×C61—21no—
299C8×C61#11712yes—
300#92418—21no—
301#92418#11712yes—
302#21297—611nomaximal, Sylow
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61C2×C4×D411yescharacteristic
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61#52911yescharacteristic
313C2×D61#52911yescharacteristic
314#6923—24no—
315#6923—42no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61C2×#11711yescharacteristic
322C4×C61C4×#4111yescharacteristic
323Dic61—21no—
324Dic61—21no—
325Dic61—41no—
326Dic61—41no—
327Dic61—41no—
328Dic61#53211yescharacteristic
329Dic61#53211yescharacteristic
330#2943—611no—
331#2923—611no—
332#2923—611no—
333#1975—611no—
334#1974—611no—
335#1946—611no—
336#1945—611no—
337#1938—611no—
338#1938—611no—
339#1937—611no—
340#1937—611no—
341#1778—611no—
342C4×#104—611no—
343C4×#104—611no—
344C4×#97—611no—
345C2×C61—81no—
346C2×C61#229311yescharacteristic
347C2×C61#295611yescharacteristic
348C2×C61#295611yescharacteristic
349D61—21no—
350D61—21no—
351D61—81no—
352C2×#135—611no—
353C4×#41—611no—
354C4×#41—611no—
355#442—611no—
356#442—611no—
357#441—611no—
358#441—611no—
359#440—611no—
360#440—611no—
361#440—611no—
362#440#692311yescharacteristic, complemented
363#433—1221no—
364#433—1221no—
365#433—1221no—
366#433—1221no—
367#432—1221no—
368#432—1221no—
369#432—1221no—
370#432—1221no—
371C2×#108—611no—
372C2×#108—611no—
373#427—611no—
374#424—611no—
375#424—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#408—611no—
387#408—611no—
388C4×M16—611no—
389C4×M16—611no—
390C2×#97—611no—
391C2×#97—611no—
392C2×#97—611no—
393C2×#97—611no—
394C2×C4×C8—611no—
395C2×C4×C8—611no—
396C43—611no—
397#341—611no—
398#341—611no—
399#339—611no—
400#339—611no—
401#339—611no—
402#339—611no—
403C61#2129711yescharacteristic, complemented, Sylow
404C2×#41—611no—
405C2×#41—611no—
406#135—1222no—
407#135—1222no—
408C2×Q16—611no—
409C2×SD16—611no—
410C2×SD16C2×#692311yescharacteristic
411C2×D8—611no—
412C2×M16—611no—
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—611no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8—611no—
420C4×Q8—611no—
421C4×Q8—611no—
422C4×Q8—611no—
423C4×Q8C2×#692311yescharacteristic
424C4×D4—611no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—1221no—
430C4×D4—1221no—
431C4×D4C2×#692311yescharacteristic
432#117—611no—
433#117—611no—
434#117—1221no—
435#117—1221no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443#108—1222no—
444#108—1222no—
445#107—611no—
446#107C2×#692311yescharacteristic
447#106—611no—
448#106—611no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456C4⋊C8—1221no—
457#104—1222no—
458#104—1222no—
459#104—1222no—
460#104—1222no—
461#103—611no—
462#103—611no—
463#103—611no—
464#103C2×#692311yescharacteristic
465#102—611no—
466#102—611no—
467#102—611no—
468#102C2×#692311yescharacteristic
469C22⋊C8—1221no—
470C22⋊C8—1221no—
471C22⋊C8—1221no—
472C22⋊C8—1221no—
473C22⋊C8—1221no—
474C22⋊C8—1221no—
475C22⋊C8—1221no—
476C22⋊C8—1221no—
477#97—611no—
478#97—611no—
479#97—611no—
480#97—611no—
481#97—611no—
482#97—611no—
483#97—611no—
484#97—1222no—
485#97—1221no—
486#97—1221no—
487#97—1222no—
488#97C2×#692311yescharacteristic
489C4×C8—1222no—
490C4×C8—1221no—
491C4×C8—1221no—
492C4×C8—1222no—
493#41—1221no—
494#41—1221no—
495#41—1221no—
496#41—1221no—
497#41—1221no—
498#41—1221no—
499#41—1221no—
500#41—1221no—
501C2×Q8—611no—
502C2×Q8C22×#692311yescharacteristic
503C2×D4—611no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4C22×#692311yescharacteristic
507C22×C4—21no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518Q16—1222no—
519SD16—22no—
520SD16—1222no—
521D8—1222no—
522M16—1221no—
523M16—1221no—
524M16—1221no—
525M16—1221no—
526M16—1221no—
527M16—1221no—
528M16—1221no—
529M16—1221no—
530C2×C8—611no—
531C2×C8—612no—
532C2×C8—612no—
533C2×C8—611no—
534C2×C8—611no—
535C2×C8—611no—
536C2×C8—611no—
537C2×C8—611no—
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×C8—1221no—
547C2×C8—1221no—
548C2×C8C22×#692311yescharacteristic
549C2×C8C22×#692311yescharacteristic
550C4⋊C4—21no—
551C4⋊C4—611no—
552C4⋊C4—611no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4C22×#692311yescharacteristic
557C4⋊C4C22×#692311yescharacteristic
558C22⋊C4—21no—
559C22⋊C4—1221no—
560C22⋊C4—1221no—
561C22⋊C4—1221no—
562C22⋊C4—1221no—
563C22⋊C4—1221no—
564C42—21no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—612no—
569C42—612no—
570C42—1221no—
571C42—1221no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42—1221no—
576C42—1221no—
577C42—2442no—
578C42—2442no—
579C42C22×#692311yescharacteristic
580C23—21no—
581C23—611no—
582C23—1221no—
583Q8—21no—
584Q8—21no—
585Q8—1221no—
586Q8—1221no—
587D4—21no—
588D4—21no—
589D4—1221no—
590D4—1221no—
591D4—2441no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—81no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—4881no—
614C2×C4—4881no—
615C2×C4—4881no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4D4×#692311yescharacteristic
625C2×C4D4×#692311yescharacteristic
626C2×C4C23×#692311yescharacteristic
627C8—21no—
628C8—612no—
629C8—1221no—
630C8—1221no—
631C8—1221no—
632C8—1221no—
633C8—1221no—
634C8—2441no—
635C8—2441no—
636C8—2441no—
637C8—2441no—
638C8#1451985612yes—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—611no—
643C22—611no—
644C22—1221no—
645C22—1221no—
646C22—2441no—
647C22—2441no—
648C22—2441no—
649C22C2×D4×#692311yescharacteristic
650C4—21no—
651C4—21no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—611no—
656C4—611no—
657C4—1224no—
658C4—1221no—
659C4—1221no—
660C4—2442no—
661C4—2441no—
662C4—2441no—
663C4—2441no—
664C4C2×#1451985611yescharacteristic
665C4#41×#692311yescharacteristic
666C2—81no—
667C2—1221no—
668C2—1221no—
669C2—4881no—
670C2#2146227611yescharacteristic
671C2#2146259311yescharacteristic
672C2#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#21297
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. #21462507C2semidirect
2. #21462506C2semidirect
5. #21461928C2semidirect
6. #21461927C2semidirect
7. #21461915C2semidirect
8. #21461914C2semidirect
9. #21461913C2semidirect
10. #21461912C2semidirect
11. #21461639C2semidirect
12. C4×#14519842C2semidirect
13. C4×#14519841C2semidirect
14. #97×#6923C2semidirect
55. #16839872C4semidirect
56. #16839871C4semidirect
57. #16839867C4semidirect
58. #16839866C4semidirect
59. #16839864C4semidirect
60. #16839863C4semidirect
61. #16839861C4semidirect
66. C61×#440C4semidirect
362. #440#6923semidirect
403. C61#21297semidirect

Maximal subgroups

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

subgroupindex
#214625072
#214625062
#214619752
#214619712
#214619282
#214619272
#214619152
#214619142
#214619132
#214619122
#214616392
C4×#145198422
C4×#145198412
#97×#69232
#214546142
#2129761