Finite Group Databasebeta

← #29748562

Subgroups — #29748562

Order 15616.

804 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450366C211yesmaximal, characteristic, complemented
2#21448884C211yesmaximal, characteristic
3#21447235C211yesmaximal, characteristic, complemented
4#21447191C211yesmaximal, characteristic, complemented
5#21447050C211yesmaximal, characteristic
6#21446499C211yesmaximal, characteristic, complemented
7#21446448C211yesmaximal, characteristic, complemented
8#21446408C211yesmaximal, characteristic, complemented
9#21446398C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446064C211yesmaximal, characteristic, complemented
12#21446064C211yesmaximal, characteristic, complemented
13#21445827C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2041C211yesmaximal, characteristic
16C2×D4×Dic61—21no—
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839702—21no—
28#16839700—21no—
29#16839665—21no—
30#16839665C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839659—21no—
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839643—21no—
37#16839642—21no—
38#16839642—21no—
39#16839627C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#127×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C23×Dic61—21no—
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63D4×Dic61—22no—
64D4×Dic61—22no—
65D4×Dic61—41no—
66D4×Dic61—41no—
67D4×Dic61—44no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519710—22no—
88#14519709—22no—
89#14519706—41no—
90#14519698—22no—
91#14519698—22no—
92#14519698—22no—
93#14519698—22no—
94C8×Dic61—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×D4×C61—21no—
105C22×D4×C61—21no—
106C22×D4×C61C2311yescharacteristic
107C22×D4×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110#127×C61—22no—
111#127×C61—22no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116#102×C61—22no—
117#102×C61—22no—
118#102×C61—41no—
119#102×C61—41no—
120#95×C61—21no—
121C24×C61—21no—
122C24×C61—21no—
123C2×D4×C61—24no—
124C2×D4×C61—24no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61—81no—
138C2×D4×C61—81no—
139C2×D4×C61#4112yes—
140C2×D4×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61—81no—
155C22×Dic61—81no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—81no—
171C42×C61—21no—
172C42×C61—21no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C22⋊Dic61—42no—
177C22⋊Dic61—81no—
178C22⋊Dic61—81no—
179C4⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—22no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61#4112yes—
188C4⋊Dic61#4112yes—
189#11758196—42no—
190#11758196—42no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—42no—
197C4×Dic61—81no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61C22×D411yescharacteristic
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—22no—
223D4×C61—48no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—84no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61#13512yes—
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
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—84no—
271C2×Dic61—84no—
272C2×Dic61AGL(1,ℤ/8)12yes—
273C8×C61—41no—
274C8×C61—41no—
275#92418—41no—
276#92418—41no—
277#26755—611nomaximal, Sylow
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61#52511yescharacteristic
293C22×C61#54611yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57611yescharacteristic
298C22×C61#57711yescharacteristic
299C4×C61—22no—
300C4×C61—41no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61#52911yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#55211yescharacteristic
306C4×C61#55311yescharacteristic
307Dic61—22no—
308Dic61—42no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315#2509—611no—
316#2366—611no—
317#2073—611no—
318#2065—611no—
319#2041—611no—
320#2041D6111yescharacteristic
321#1934—611no—
322#1926—611no—
323#1921—611no—
324#1919—611no—
325#1916—611no—
326#1862—611no—
327#1862—611no—
328#1804—611no—
329C4×#102—611no—
330C2×C61—82no—
331C2×C61—82no—
332C2×C61#246011yescharacteristic
333C2×C61#310511yescharacteristic
334C2×C61#311811yescharacteristic
335C2×C61#330411yescharacteristic
336C2×C61#331811yescharacteristic
337C2×C61#336111yescharacteristic
338C2×C61#336811yescharacteristic
339C2×#127C2×D6111yescharacteristic
340C2×C4×D4—611no—
341C2×C4×D4—611no—
342C2×C4×D4—1221no—
343C2×#107—611no—
344C2×#106—611no—
345C2×C4⋊C8—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×#102C2×D6111yescharacteristic
351C2×#102C2×D6111yescharacteristic
352C2×#102C2×D6111yescharacteristic
353C2×#102C2×D6111yescharacteristic
354C2×C4×C8—611no—
355#400—1221no—
356#400—1221no—
357#390—611no—
358#390—611no—
359#390—1221no—
360#389—611no—
361#389—611no—
362#389—611no—
363#389—1221no—
364#389C2×D6111yescharacteristic
365#387—1221no—
366#386—1221no—
367#386—1221no—
368#382—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336C2×D6111yescharacteristic
382C61#2675511yescharacteristic, complemented, Sylow
383C22×D4—21no—
384C22×D4—21no—
385C22×D4C22×D6111yescharacteristic
386C22×D4C22×D6111yescharacteristic
387C23×C4—1221no—
388C23×C4—1221no—
389C22×C8—611no—
390C22×C8—611no—
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393#127—22no—
394#127—22no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—2441no—
398C4×D4—2441no—
399C4×D4—2444no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423#107—1222no—
424#106—1222no—
425C4⋊C8—2441no—
426#102—22no—
427#102—22no—
428#102—41no—
429#102—41no—
430#102—1222no—
431#102—1222no—
432#102—1222no—
433#102—1222no—
434C4×C8—2441no—
435#95—21no—
436#95—611no—
437#95—611no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C24—21no—
446C24—21no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4#1175822412yes—
464C2×D4#1175822412yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—4881no—
476C22×C4—4881no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—81no—
501C4⋊C4—612no—
502C4⋊C4—612no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1222no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—4881no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C22⋊C4—4881no—
515C22⋊C4—4881no—
516C22⋊C4—2442no—
517C22⋊C4—2442no—
518C22⋊C4—2442no—
519C22⋊C4—2442no—
520C42—21no—
521C42—21no—
522C42—1221no—
523C42—1221no—
524C42—1221no—
525C42—1221no—
526C42—4881no—
527C42—2442no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C2×D4×D6111yescharacteristic
543D4—22no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—48no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—84no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—612no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—4884no—
579C2×C4—4884no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4#1451979812yes—
591C2×C4#41×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C2×C4C2×#1175822411yescharacteristic
594C2×C4C2×#1175822411yescharacteristic
595C2×C4#1451982211yescharacteristic
596C2×C4#1451982411yescharacteristic
597C8—41no—
598C8—41no—
599C8—2441no—
600C8—2441no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1684026811yescharacteristic
616C22#1684040911yescharacteristic
617C22#1684042011yescharacteristic
618C22C2×#1451979811yescharacteristic
619C22AGL(1,ℤ/8)×D6111yescharacteristic
620C22#1684056311yescharacteristic
621C22#1684057211yescharacteristic
622C4—22no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—1222no—
627C4—4881no—
628C4—4881no—
629C4—2441no—
630C4—2442no—
631C4—2442no—
632C4—2441no—
633C4—2441no—
634C4#1684028911yescharacteristic
635C4#1684035711yescharacteristic
636C4#1684044911yescharacteristic
637C4#1684045511yescharacteristic
638C2—82no—
639C2—82no—
640C2#2144976411yescharacteristic
641C2#2145586711yescharacteristic
642C2#2145599611yescharacteristic
643C2#2145776111yescharacteristic
644C2#2145796711yescharacteristic
645C2#2145848411yescharacteristic
646C2#2145854511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2041D61
hypocenterC61#26755
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
1. #21450366C2semidirect
3. #21447235C2semidirect
4. #21447191C2semidirect
6. #21446499C2semidirect
7. #21446448C2semidirect
8. #21446408C2semidirect
9. #21446398C2semidirect
10. #21446375C2semidirect
11. #21446064C2semidirect
12. #21446064C2semidirect
13. #21445827C2semidirect
14. #102×Dic61C2semidirect
382. C61#26755semidirect

Maximal subgroups

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

subgroupindex
#214503662
#214488842
#214472352
#214471912
#214470502
#214464992
#214464482
#214464082
#214463982
#214463752
#214460642
#214458272
#102×Dic612
C61×#20412
#2675561