Finite Group Databasebeta

← #29748164

Subgroups — #29748164

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450154C211yesmaximal, characteristic, complemented
2#21448853C211yesmaximal, characteristic
3#21447135C211yesmaximal, characteristic, complemented
4#21447006C211yesmaximal, characteristic
5#21446963C211yesmaximal, characteristic, complemented
6#21446823C211yesmaximal, characteristic
7#21446775C211yesmaximal, characteristic, complemented
8#21446495C211yesmaximal, characteristic, complemented
9#21446451C211yesmaximal, characteristic, complemented
10#21446425C211yesmaximal, characteristic, complemented
11#21446404C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic, complemented
13#21446130C211yesmaximal, characteristic, complemented
14#21445912C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519757C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29#16839702—21no—
30#16839689—21no—
31#16839668C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839662—21no—
35#16839660C2211yescharacteristic
36#16839653—21no—
37#16839650—21no—
38#16839641—21no—
39#16839628—21no—
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758198C2211yescharacteristic
43C4×C22⋊Dic61—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×C22⋊D61—21no—
61C22×C4×D61C2311yescharacteristic
62C2×#11758200—22no—
63C2×#11758200—41no—
64C2×#11758200—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519768—22no—
68#14519768—41no—
69#14519757—22no—
70#14519757—41no—
71#14519748—22no—
72#14519748—22no—
73#14519735—22no—
74#14519735—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198D411yescharacteristic
89C2×#11758198D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519708—22no—
94#14519706—41no—
95#14519702—22no—
96#14519702—41no—
97#14519700—22no—
98#14519698—41no—
99#14519689—41no—
100#14519689—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4×D4×C61—22no—
115C4×D4×C61—41no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#102×C61—22no—
123#102×C61—22no—
124C4×C8×C61—41no—
125#95×C61—21no—
126C24×C61—21no—
127C23×D61C2×D411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61#4112yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—24no—
144C2×C22⋊D61—81no—
145C2×C22⋊D61—81no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61#4112yes—
176C4⋊C4×C61#4112yes—
177C22⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C22⋊Dic61—24no—
182C22⋊Dic61—42no—
183C22⋊Dic61—81no—
184#11758200—44no—
185#11758198—22no—
186#11758198—41no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4⋊Dic61#4112yes—
198#11758196—42no—
199#11758196—81no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—42no—
203C4×Dic61—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—84no—
244C2×C4×C61#13512yes—
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—84no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281C8×C61—41no—
282C8×C61—41no—
283#92418—41no—
284#92418—41no—
285#26728—611nomaximal, Sylow
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61#52511yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54611yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57711yescharacteristic
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61#52911yescharacteristic
313C4×C61#53811yescharacteristic
314C4×C61#55511yescharacteristic
315C4×C61#55911yescharacteristic
316Dic61—42no—
317Dic61—41no—
318Dic61—41no—
319Dic61—41no—
320Dic61—81no—
321Dic61—81no—
322Dic61—81no—
323#2489—611no—
324#2363—611no—
325#2055—611no—
326#2034—611no—
327#2025—611no—
328#2000—611no—
329#1989—611no—
330#1933—611no—
331#1927—611no—
332#1923—611no—
333#1920—611no—
334#1877—611no—
335#1873—611no—
336#1825—611no—
337C4×#102D6111yescharacteristic, complemented
338C2×C61—82no—
339C2×C61#245911yescharacteristic
340C2×C61#310211yescharacteristic
341C2×C61#312311yescharacteristic
342C2×C61#329611yescharacteristic
343C2×C61#330311yescharacteristic
344C2×C61#331711yescharacteristic
345C2×C61#332611yescharacteristic
346D61—81no—
347C22×SD16—611no—
348C2×#125—611no—
349C2×C22⋊Q8—611no—
350C2×#121—611no—
351C2×C4×D4C2×D6111yescharacteristic
352C2×#117—611no—
353C2×#106—611no—
354C2×C4⋊C8—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#102—611no—
359C2×#102C2×D6111yescharacteristic
360C2×#102C2×D6111yescharacteristic
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363C2×C4×C8C2×D6111yescharacteristic
364#400—1221no—
365#397—1221no—
366#391—611no—
367#391—611no—
368#390—611no—
369#390—1221no—
370#389—611no—
371#388—1221no—
372#388—1221no—
373#386—1221no—
374#382—1221no—
375#380—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C22⋊C4—611no—
379C4×C22⋊C4—1221no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#340C2×D6111yescharacteristic
388#336—611no—
389#336—611no—
390C61#2672811yescharacteristic, complemented, Sylow
391C22×Q8—611no—
392C22×D4—1221no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
395C23×C4—611no—
396C2×SD16—1222no—
397C2×SD16—2441no—
398C2×SD16—2441no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403#125—1222no—
404#125—2441no—
405C22⋊Q8—1222no—
406C22⋊Q8—2441no—
407#121—1222no—
408#121—1222no—
409C4×D4—22no—
410C4×D4—41no—
411#117—1222no—
412#117—2441no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—611no—
425C2×C22⋊C4—611no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—611no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437#106—1222no—
438C4⋊C8—2441no—
439#103—1222no—
440#103—1222no—
441#103—2441no—
442#102—22no—
443#102—22no—
444#102—2441no—
445C22⋊C8—2441no—
446C22⋊C8—2441no—
447C4×C8—41no—
448#95—21no—
449#95—611no—
450#95—611no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—21no—
459C24—611no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—1224no—
472C2×D4—4881no—
473C2×D4—4881no—
474C2×D4#1175822212yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—81no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497SD16—2444no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—612no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—4881no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4#1175822212yes—
525C4⋊C4#1175822212yes—
526C22⋊C4—42no—
527C22⋊C4—1224no—
528C22⋊C4—1222no—
529C22⋊C4—4881no—
530C22⋊C4—2441no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—1221no—
539C42—1221no—
540C42—4881no—
541C42—2442no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1221no—
553C23—1221no—
554C23—1221no—
555C23—1221no—
556C23C2×D4×D6111yescharacteristic
557Q8—2441no—
558Q8—2441no—
559Q8—2441no—
560Q8—2441no—
561D4—22no—
562D4—22no—
563D4—41no—
564D4—41no—
565D4—4884no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4#1451979612yes—
607C2×C4#41×D6111yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4#1451982211yescharacteristic
612C2×C4#1451982311yescharacteristic
613C8—41no—
614C8—41no—
615C8—2441no—
616C8—2441no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—2441no—
625C22—2441no—
626C22—2441no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22#1684027111yescharacteristic
632C22#1684040211yescharacteristic
633C22#1684040811yescharacteristic
634C22C2×#1451979611yescharacteristic
635C22#1684055911yescharacteristic
636C22#1684056811yescharacteristic
637C22#1684056911yescharacteristic
638C4—22no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—4881no—
644C4—4881no—
645C4—4881no—
646C4—2441no—
647C4—2441no—
648C4—2441no—
649C4—2442no—
650C4#1684029611yescharacteristic
651C4#1684034511yescharacteristic
652C4#1684047511yescharacteristic
653C4#1684049511yescharacteristic
654C2—82no—
655C2—4881no—
656C2#2144975611yescharacteristic
657C2#2145583311yescharacteristic
658C2#2145606811yescharacteristic
659C2#2145765011yescharacteristic
660C2#2145774911yescharacteristic
661C2#2145794111yescharacteristic
662C2#2145807711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC4×#102D61
hypocenterC61#26728
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #21450154C2semidirect
3. #21447135C2semidirect
5. #21446963C2semidirect
7. #21446775C2semidirect
8. #21446495C2semidirect
9. #21446451C2semidirect
10. #21446425C2semidirect
11. #21446404C2semidirect
12. #21446153C2semidirect
13. #21446130C2semidirect
14. #21445912C2semidirect
15. C4×#102×C61C2semidirect
337. C4×#102D61semidirect
390. C61#26728semidirect

Maximal subgroups

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

subgroupindex
#214501542
#214488532
#214471352
#214470062
#214469632
#214468232
#214467752
#214464952
#214464512
#214464252
#214464042
#214461532
#214461302
#214459122
C4×#102×C612
#2672861