Finite Group Databasebeta

← #29752695

Subgroups — #29752695

Order 15616.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450248C211yesmaximal, characteristic, complemented
2#21448839C211yesmaximal, characteristic
3#21447225C211yesmaximal, characteristic, complemented
4#21447191C211yesmaximal, characteristic, complemented
5#21447005C211yesmaximal, characteristic
6#21446579C211yesmaximal, characteristic, complemented
7#21446548C211yesmaximal, characteristic, complemented
8#21446428C211yesmaximal, characteristic, complemented
9#21446403C211yesmaximal, characteristic, complemented
10#21446400C211yesmaximal, characteristic, complemented
11#21446082C211yesmaximal, characteristic, complemented
12#21446014C211yesmaximal, characteristic, complemented
13#21445903C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2034C211yesmaximal, characteristic
16C2×D4×Dic61—21no—
17C2×D4×Dic61C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839702—21no—
28#16839685—21no—
29#16839665C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839655—21no—
34#16839650—21no—
35#16839644—21no—
36#16839642—21no—
37#16839637—21no—
38#16839631C2211yescharacteristic
39#16839622C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#121×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63D4×Dic61—22no—
64D4×Dic61—41no—
65D4×Dic61—44no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—41no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61D411yescharacteristic
83C2×C22⋊Dic61D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519709—22no—
89#14519709—41no—
90#14519708—22no—
91#14519706—22no—
92#14519698—22no—
93#14519698—41no—
94C22⋊#92418—41no—
95C8×Dic61—41no—
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×C61C2311yescharacteristic
106C23×C4×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#121×C61—22no—
110#121×C61—22no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116#106×C61—22no—
117#102×C61—22no—
118#102×C61—22no—
119C22⋊C8×C61—22no—
120#95×C61—21no—
121C24×C61—21no—
122C24×C61C2×D411yescharacteristic
123C2×D4×C61—24no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61—81no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61#4112yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61#4112yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—81no—
171C22⋊Dic61—41no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C22⋊Dic61—81no—
176C22⋊Dic61SD1614yes—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187C4⋊Dic61#4112yes—
188#11758196—42no—
189#11758196—81no—
190#11758196—81no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
196C4×Dic61—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61C2×SD1612yes—
217C23×C61C22×D411yescharacteristic
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—48no—
221D4×C61—41no—
222D4×C61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—84no—
268C2×Dic61—84no—
269C2×Dic61C2×SD1612yes—
270C2×Dic61C2×SD1612yes—
271C8×C61—22no—
272C8×C61—41no—
273#92418—41no—
274#92418—41no—
275#27032—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61#52511yescharacteristic
291C22×C61#54611yescharacteristic
292C22×C61#54711yescharacteristic
293C22×C61C22×SD1611yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57711yescharacteristic
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61#52911yescharacteristic
302C4×C61#53911yescharacteristic
303C4×C61#55311yescharacteristic
304C4×C61#55611yescharacteristic
305Dic61—22no—
306Dic61—22no—
307Dic61—41no—
308Dic61—42no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312Dic61—81no—
313#2497—611no—
314#2361—611no—
315#2071—611no—
316#2065—611no—
317#2034—611no—
318#2034D6111yescharacteristic
319#1948—611no—
320#1941—611no—
321#1923—611no—
322#1920—611no—
323#1919—611no—
324#1864—611no—
325#1853—611no—
326#1823—611no—
327C4×#102—611no—
328C2×C61—42no—
329C2×C61—82no—
330C2×C61#246611yescharacteristic
331C2×C61#310111yescharacteristic
332C2×C61#311911yescharacteristic
333C2×C61#329511yescharacteristic
334C2×C61#332111yescharacteristic
335C2×C61#333511yescharacteristic
336C2×C61#335511yescharacteristic
337C2×#121C2×D6111yescharacteristic
338C2×C4×D4—611no—
339C2×C4×D4—1221no—
340C22×C4⋊C4—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×#106—611no—
344C2×#106C2×D6111yescharacteristic
345C2×C4⋊C8—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102C2×D6111yescharacteristic
349C2×#102C2×D6111yescharacteristic
350C2×C22⋊C8—611no—
351C2×C22⋊C8C2×D6111yescharacteristic
352C2×C4×C8—611no—
353#400—1221no—
354#396—1221no—
355#390—611no—
356#389—611no—
357#389—611no—
358#389—611no—
359#388—1221no—
360#388—1221no—
361#387—1221no—
362#386—1221no—
363#385—1221no—
364#384—611no—
365#380—611no—
366#380C2×D6111yescharacteristic
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C22⋊C4—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340C2×D6111yescharacteristic
378#336—611no—
379#336—611no—
380C61#2703211yescharacteristic, complemented, Sylow
381C22×D4—21no—
382C22×D4C22×D6111yescharacteristic
383C23×C4—1221no—
384C23×C4—1221no—
385C23×C4C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390#121—22no—
391#121—22no—
392C4×D4—1222no—
393C4×D4—2441no—
394C4×D4—2444no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—2441no—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—611no—
413C2×C22⋊C4—611no—
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—
422#106—22no—
423#106—1222no—
424#106—1222no—
425#106—2441no—
426C4⋊C8—1222no—
427#102—22no—
428#102—22no—
429#102—1222no—
430#102—2441no—
431C22⋊C8—22no—
432C22⋊C8—2441no—
433C4×C8—2441no—
434#95—21no—
435#95—611no—
436#95—611no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443C24—21no—
444C24D4×D6111yescharacteristic
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4#1175822412yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—4881no—
472C22×C4—4881no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4D4×D6111yescharacteristic
478C22×C4C23×D6111yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8#1175822412yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—612no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—4881no—
505C4⋊C4—4881no—
506C4⋊C4—4881no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4#1175822412yes—
511C22⋊C4—24no—
512C22⋊C4—81no—
513C22⋊C4—614no—
514C22⋊C4—4881no—
515C22⋊C4—4881no—
516C22⋊C4—2442no—
517C22⋊C4—2441no—
518C22⋊C4—2442no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42—1221no—
523C42—4881no—
524C42—2442no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23SD16×D6112yes—
539C23C2×D4×D6111yescharacteristic
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1222no—
573C2×C4—1221no—
574C2×C4—4884no—
575C2×C4—4884no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4#41×D6111yescharacteristic
588C2×C4C2×#1175822411yescharacteristic
589C2×C4C2×#1175822411yescharacteristic
590C2×C4C2×#1175822411yescharacteristic
591C2×C4#1451982311yescharacteristic
592C2×C4#1451982411yescharacteristic
593C8—22no—
594C8—41no—
595C8—2441no—
596C8—2441no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1684026811yescharacteristic
612C22#1684040411yescharacteristic
613C22#1684041211yescharacteristic
614C22C2×SD16×D6111yescharacteristic
615C22#1684055811yescharacteristic
616C22#1684056611yescharacteristic
617C22#1684056911yescharacteristic
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—1222no—
623C4—1222no—
624C4—4881no—
625C4—4881no—
626C4—4881no—
627C4—2441no—
628C4—2442no—
629C4—2441no—
630C4#1684029311yescharacteristic
631C4#1684035611yescharacteristic
632C4#1684046411yescharacteristic
633C4#1684048111yescharacteristic
634C2—42no—
635C2—82no—
636C2#2144985011yescharacteristic
637C2#2145582211yescharacteristic
638C2#2145601911yescharacteristic
639C2#2145764111yescharacteristic
640C2#2145800811yescharacteristic
641C2#2145817111yescharacteristic
642C2#2145842011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2034C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2034D61
hypocenterC61#27032
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
1. #21450248C2semidirect
3. #21447225C2semidirect
4. #21447191C2semidirect
6. #21446579C2semidirect
7. #21446548C2semidirect
8. #21446428C2semidirect
9. #21446403C2semidirect
10. #21446400C2semidirect
11. #21446082C2semidirect
12. #21446014C2semidirect
13. #21445903C2semidirect
14. #102×Dic61C2semidirect
380. C61#27032semidirect

Maximal subgroups

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

subgroupindex
#214502482
#214488392
#214472252
#214471912
#214470052
#214465792
#214465482
#214464282
#214464032
#214464002
#214460822
#214460142
#214459032
#102×Dic612
C61×#20342
#2703261