Finite Group Databasebeta

← #29749990

Subgroups — #29749990

Order 15616.

776 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455771C211yesmaximal, characteristic, complemented
2#21455644C211yesmaximal, characteristic, complemented
3#21454654C211yesmaximal, characteristic, complemented
4#21447179C211yesmaximal, characteristic, complemented
5#21447161C211yesmaximal, characteristic, complemented
6#21447109C211yesmaximal, characteristic, complemented
7#21446832C211yesmaximal, characteristic
8#21446623C211yesmaximal, characteristic, complemented
9#21446525C211yesmaximal, characteristic, complemented
10#21446481C211yesmaximal, characteristic, complemented
11#21446478C211yesmaximal, characteristic, complemented
12#21446214C211yesmaximal, characteristic, complemented
13#21446043C211yesmaximal, characteristic, complemented
14#21445699C211yesmaximal, characteristic, complemented
15C61×#3136C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840236C2211yescharacteristic
20#16840083—21no—
21#16840083—21no—
22#16840078—21no—
23#16840073—21no—
24#16840059—21no—
25#16840059—21no—
26#16840049—21no—
27#16840039—21no—
28#16839871—21no—
29#16839865—21no—
30#16839847—21no—
31#16839842—21no—
32C2×#14519712C2211yescharacteristic
33#16839805C2211yescharacteristic
34#16839805C2211yescharacteristic
35#16839802C2211yescharacteristic
36#16839798C2211yescharacteristic
37#16839789C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39#16839767C2211yescharacteristic
40#16839766C2211yescharacteristic
41C2×#14519699C2211yescharacteristic
42C2×#14519698C2211yescharacteristic
43C2×#14519697C2211yescharacteristic
44C2×#14519694C2211yescharacteristic
45C2×#14519693C2211yescharacteristic
46#16839729C2211yescharacteristic
47#16839717C2211yescharacteristic
48C2×C8×Dic61C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839527C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839521C2211yescharacteristic
56C61×#544C2211yescharacteristic
57C61×#535C2211yescharacteristic
58C61×#490—21no—
59C61×#489—21no—
60C61×#488—21no—
61C61×#486—21no—
62C61×#418C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#416C2211yescharacteristic
65C2×#102×C61C2211yescharacteristic
66C61×#405C2211yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—41no—
69#14519802—22no—
70#14519802—22no—
71C2×#11758201—21no—
72C2×#11758199—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519773—22no—
78#14519773—22no—
79Q8×Dic61—21no—
80C4×D244—21no—
81C4×D244—21no—
82D4×Dic61—21no—
83D4×Dic61—41no—
84#14519739—21no—
85#14519739—21no—
86#14519736—21no—
87#14519736C2311yescharacteristic
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—41no—
91C22⋊C4×D61—41no—
92C2×C4×Dic61C2311yescharacteristic
93#14519712—22no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710D411yescharacteristic, complemented
98#14519710D411yescharacteristic, complemented
99#14519708—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708D411yescharacteristic, complemented
103#14519708D411yescharacteristic, complemented
104C4⋊#92418—21no—
105C4⋊#92418—21no—
106#14519704—22no—
107#14519699—21no—
108#14519699—21no—
109#14519698—21no—
110#14519698—21no—
111#14519698—21no—
112#14519698—21no—
113#14519697—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117#14519694—22no—
118#14519693—22no—
119C22⋊#92418—21no—
120C22⋊#92418—21no—
121#14519687—21no—
122C8×Dic61—21no—
123C8×Dic61—22no—
124C2×#41×C61C2311yescharacteristic
125C22×D4×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4⋊Q8×C61—21no—
129C4⋊Q8×C61—21no—
130#127×C61—21no—
131#125×C61—21no—
132#125×C61—21no—
133#124×C61—21no—
134C22⋊Q8×C61—41no—
135#121×C61—21no—
136#121×C61—21no—
137#120×C61—41no—
138#117×C61C2311yescharacteristic
139#117×C61C2311yescharacteristic
140C2×C4⋊C4×C61C2311yescharacteristic
141#103×C61—21no—
142#103×C61D411yescharacteristic, complemented
143#103×C61D411yescharacteristic, complemented
144#102×C61—21no—
145#102×C61—21no—
146#102×C61—21no—
147#102×C61—21no—
148#102×C61—21no—
149#102×C61D411yescharacteristic, complemented
150#102×C61D411yescharacteristic, complemented
151#97×C61—21no—
152C4×C8×C61—21no—
153C24×C61—21no—
154C23×D61—21no—
155#41×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159C2×Q8×C61—41no—
160C2×Q8×C61C2×D411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61—21no—
168C2×D4×C61—41no—
169C2×D4×C61—41no—
170C2×D4×C61C2×D411yescharacteristic
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61C2411yescharacteristic
174C2×C22⋊D61—44no—
175C2×C22⋊D61—82no—
176C2×C22⋊D61—82no—
177C22×Dic61—21no—
178C22×Dic61—21no—
179C22×Dic61—41no—
180C2×D244—21no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244—21no—
185C2×D244#4112yes—
186C2×C4×D61—41no—
187C2×C4×D61—41no—
188M16×C61—21no—
189M16×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—41no—
198C4⋊C4×C61—41no—
199C4⋊C4×C61—41no—
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C4⋊C4×C61C2×D411yescharacteristic
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—41no—
207C22⋊C4×C61—41no—
208C42×C61—21no—
209C42×C61—21no—
210C22⋊Dic61—21no—
211C22⋊Dic61—41no—
212#11758201—24no—
213#11758199—24no—
214#11758198—41no—
215#11758198—41no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—41no—
219C4⋊Dic61C2×D411yescharacteristic
220C4⋊Dic61C2×D411yescharacteristic
221#11758196—82no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—41no—
225C4×Dic61#4112yes—
226#11758194—21no—
227#11758194—21no—
228#11758194—21no—
229#11758194—21no—
230#11758194—21no—
231#11758194#4112yes—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418C2×D411yescharacteristic
238C2×#92418C2×D411yescharacteristic
239C23×C61—21no—
240C23×C61—41no—
241C23×C61—41no—
242C23×C61—41no—
243C23×C61—81no—
244C23×C61—81no—
245C23×C61H(5,𝔽2)11yescharacteristic
246C22×D61—22no—
247C22×D61—41no—
248C22×D61—41no—
249C22×D61—81no—
250C22×D61—81no—
251Q8×C61—21no—
252Q8×C61—21no—
253Q8×C61—161no—
254D4×C61—21no—
255D4×C61—21no—
256D4×C61—41no—
257D4×C61—41no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—41no—
261D4×C61—41no—
262D4×C61—161no—
263D4×C61—161no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—81no—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C612−1+411yescharacteristic
279C22⋊D61—88no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—41no—
286C2×Dic61—81no—
287C2×Dic61—161no—
288D244—22no—
289D244—22no—
290D244—41no—
291D244—41no—
292C4×D61—161no—
293C4×D61—161no—
294C8×C61—21no—
295C8×C61—21no—
296C8×C61—41no—
297#92418—22no—
298#92418—21no—
299#92418—21no—
300#92418—41no—
301#92418—41no—
302#26850—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—82no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61—161no—
313C22×C61#54211yescharacteristic
314C22×C61D4211yescharacteristic
315C22×C61#55111yescharacteristic
316C2×D61—42no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—82no—
320C2×D61—161no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—81no—
327C4×C61#52511yescharacteristic
328C4×C61#53211yescharacteristic
329C4×C61#54711yescharacteristic
330C4×C61#54811yescharacteristic
331Dic61—24no—
332Dic61—41no—
333Dic61—41no—
334Dic61—81no—
335#3136D6111yescharacteristic, complemented
336#3096—611no—
337#3081—611no—
338#2946—611no—
339#2063—611no—
340#2060—611no—
341#2051—611no—
342#2002—611no—
343#1957—611no—
344#1937—611no—
345#1931—611no—
346#1931—611no—
347#1887—611no—
348#1859—611no—
349#1772—611no—
350C2×C61—21no—
351C2×C61—21no—
352C2×C61—81no—
353C2×C61—161no—
354C2×C61#246111yescharacteristic
355C2×C61#329111yescharacteristic
356C2×C61#329211yescharacteristic
357D61—82no—
358C2×AGL(1,ℤ/8)—611no—
359#544C2×D6111yescharacteristic
360#535C2×D6111yescharacteristic
361C2×#127—611no—
362#520—611no—
363#518—611no—
364#490—21no—
365#489—21no—
366#488—21no—
367#486—21no—
368#482—1221no—
369#482—1221no—
370#481—1221no—
371#480—1221no—
372#478—1221no—
373#478—1221no—
374#476—1221no—
375#474—1221no—
376#442—1221no—
377#440—1221no—
378C4×SD16—1221no—
379C4×D8—1221no—
380C2×#108—611no—
381#428—611no—
382#428—611no—
383#428—611no—
384#427—611no—
385#424—611no—
386C2×#104—611no—
387#418—611no—
388#418—611no—
389#418C2×D6111yescharacteristic
390#417C2×D6111yescharacteristic
391#416C2×D6111yescharacteristic
392C2×#102—611no—
393C2×#102—611no—
394C2×#102—611no—
395C2×#102C2×D6111yescharacteristic
396C2×#100—611no—
397C2×#100—611no—
398#408—611no—
399#405—611no—
400#405C2×D6111yescharacteristic
401C2×C4×C8—611no—
402#384—611no—
403#343—611no—
404#343—611no—
405#341—611no—
406#339—611no—
407#339—611no—
408#339—611no—
409C61#2685011yescharacteristic, complemented, Sylow
410C2×#41C22×D6111yescharacteristic
411C22×D4—611no—
412C22×D4—2441no—
413C22×D4C22×D6111yescharacteristic
414AGL(1,ℤ/8)—1222no—
415AGL(1,ℤ/8)—1222no—
416C2×SD16—1221no—
417C2×D8—1221no—
418C2×M16—611no—
419C2×M16—611no—
420C2×M16—611no—
421C2×M16C22×D6111yescharacteristic
422C22×C8—611no—
423C22×C8C22×D6111yescharacteristic
424C4⋊Q8—21no—
425C4⋊Q8—21no—
426#127—21no—
427#127—1222no—
428#127—1222no—
429#125—21no—
430#125—21no—
431#124—21no—
432C22⋊Q8—41no—
433#121—21no—
434#121—21no—
435#120—41no—
436C4×Q8—1221no—
437C4×D4—1221no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4—1221no—
442C4×D4—2441no—
443#117—611no—
444#117—1221no—
445#117C22×D6111yescharacteristic
446#117C22×D6111yescharacteristic
447C2×C4⋊C4—611no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—2441no—
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—2441no—
452C2×C42—611no—
453#108—1222no—
454#107—611no—
455#107—611no—
456#107—1221no—
457#107—1221no—
458#107—1221no—
459#106—611no—
460#106—611no—
461#106—1221no—
462#106—1221no—
463#106—1221no—
464C4⋊C8—1221no—
465C4⋊C8—1221no—
466#104—1222no—
467#103—21no—
468#103C22⋊D6111yescharacteristic, complemented
469#103C22⋊D6111yescharacteristic, complemented
470#102—21no—
471#102—21no—
472#102—21no—
473#102—21no—
474#102—21no—
475#102—1221no—
476#102—1221no—
477#102—1221no—
478#102—1221no—
479#102—1221no—
480#102—1221no—
481#102—1221no—
482#102—1221no—
483#102—1221no—
484#102—1221no—
485#102C22⋊D6111yescharacteristic, complemented
486#102C22⋊D6111yescharacteristic, complemented
487#100—1222no—
488#100—1222no—
489C22⋊C8—1221no—
490C22⋊C8—1221no—
491#97—21no—
492#97—1221no—
493C4×C8—21no—
494C4×C8—1222no—
495C4×C8—1221no—
496C24—21no—
497C24—1221no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41—21no—
502C2×Q8—41no—
503C2×Q8C2×C22⋊D6111yescharacteristic
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—41no—
512C2×D4—41no—
513C2×D4—612no—
514C2×D4—1221no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4—4882no—
520C2×D4—4882no—
521C2×D4—2444no—
522C2×D4C2×C22⋊D6111yescharacteristic
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—2441no—
528C22×C4—2441no—
529C22×C4—2441no—
530C22×C4C23×D6111yescharacteristic
531SD16—1224no—
532D8—1224no—
533M16—21no—
534M16—21no—
535M16—612no—
536M16—1221no—
537M16—1221no—
538M16—1221no—
539M16—1221no—
540M16—1221no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—611no—
545C2×C8—611no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8C2×C22⋊D6111yescharacteristic
552C2×C8C2×C22⋊D6111yescharacteristic
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—611no—
559C4⋊C4—611no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—4882no—
563C4⋊C4—2441no—
564C4⋊C4D4×D6111yescharacteristic
565C4⋊C4D4×D6111yescharacteristic
566C4⋊C4C2×C22⋊D6111yescharacteristic
567C4⋊C4C2×C22⋊D6111yescharacteristic
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—1221no—
573C22⋊C4—2441no—
574C22⋊C4—2441no—
575C22⋊C4—2441no—
576C42—21no—
577C42—21no—
578C42—612no—
579C42—1221no—
580C42—1221no—
581C42—2441no—
582C23—21no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—81no—
587C23—81no—
588C23—1222no—
589C23—4881no—
590C23—4881no—
591C23—2441no—
592C23—2441no—
593C23#1451982211yescharacteristic
594Q8—21no—
595Q8—21no—
596Q8—161no—
597D4—21no—
598D4—21no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—161no—
606D4—161no—
607D4—1222no—
608D4—1222no—
609D4—4888no—
610D4—2441no—
611D4—2441no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—81no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—9761no—
627C2×C4—9761no—
628C2×C4—9761no—
629C2×C4—4881no—
630C2×C4—2441no—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C22×C22⋊D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
635C2×C4#41×D6111yescharacteristic
636C2×C4#1451982411yescharacteristic
637C8—21no—
638C8—21no—
639C8—41no—
640C8—1222no—
641C8—1221no—
642C8—1221no—
643C8—2441no—
644C8—2441no—
645C22—21no—
646C22—21no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—82no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—161no—
655C22—9761no—
656C22—4882no—
657C22—4881no—
658C22—4881no—
659C22—2442no—
660C22#1684037611yescharacteristic
661C22D4×C22⋊D6111yescharacteristic
662C22#1684044211yescharacteristic
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—81no—
669C4—1224no—
670C4—4881no—
671C4—2441no—
672C4—2441no—
673C4#1684026911yescharacteristic
674C4#1684031111yescharacteristic
675C4#1684041111yescharacteristic
676C4#1684042011yescharacteristic
677C2—21no—
678C2—21no—
679C2—81no—
680C2—161no—
681C2—4882no—
682C2#2144978211yescharacteristic
683C2#2145761011yescharacteristic
684C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3136C2
socleC22×C61D42
prime radical#21446832C2
hypercenter#3136D61
hypocenterC61#26850
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3136

Complements

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

NH ≅ G/Ntype
1. #21455771C2semidirect
2. #21455644C2semidirect
3. #21454654C2semidirect
4. #21447179C2semidirect
5. #21447161C2semidirect
6. #21447109C2semidirect
8. #21446623C2semidirect
9. #21446525C2semidirect
10. #21446481C2semidirect
11. #21446478C2semidirect
12. #21446214C2semidirect
13. #21446043C2semidirect
14. #21445699C2semidirect
15. C61×#3136C2semidirect
97. #14519710D4semidirect
98. #14519710D4semidirect
102. #14519708D4semidirect
103. #14519708D4semidirect
142. #103×C61D4semidirect
143. #103×C61D4semidirect
149. #102×C61D4semidirect
150. #102×C61D4semidirect
335. #3136D61semidirect
409. C61#26850semidirect
468. #103C22⋊D61semidirect
469. #103C22⋊D61semidirect
485. #102C22⋊D61semidirect
486. #102C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214557712
#214556442
#214546542
#214471792
#214471612
#214471092
#214468322
#214466232
#214465252
#214464812
#214464782
#214462142
#214460432
#214456992
C61×#31362
#2685061