Finite Group Databasebeta

← #29748549

Subgroups — #29748549

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450349C211yesmaximal, characteristic, complemented
2#21448884C211yesmaximal, characteristic
3#21447225C211yesmaximal, characteristic, complemented
4#21447189C211yesmaximal, characteristic, complemented
5#21447050C211yesmaximal, characteristic
6#21446504C211yesmaximal, characteristic, complemented
7#21446449C211yesmaximal, characteristic, complemented
8#21446405C211yesmaximal, characteristic, complemented
9#21446401C211yesmaximal, characteristic, complemented
10#21446377C211yesmaximal, characteristic, complemented
11#21446082C211yesmaximal, characteristic, complemented
12#21446082C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2041C211yesmaximal, characteristic
16C2×D4×Dic61—21no—
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839695—21no—
28#16839694—21no—
29#16839665—21no—
30#16839665C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839658—21no—
36#16839645—21no—
37#16839642—21no—
38#16839642—21no—
39#16839631C2211yescharacteristic
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⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519709—22no—
90#14519708—22no—
91#14519706—41no—
92#14519698—22no—
93#14519698—22no—
94#14519698—41no—
95#14519698—41no—
96C8×Dic61—41no—
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—22no—
119#102×C61—22no—
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—81no—
170C4⋊C4×C61#4112yes—
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—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61—42no—
188C4⋊Dic61—81no—
189C4⋊Dic61SD1614yes—
190C4⋊Dic61#4112yes—
191#11758196—81no—
192#11758196—81no—
193#11758196—81no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—42no—
199C4×Dic61—81no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C23×C61—22no—
207C23×C61—22no—
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×C61—41no—
219C23×C61—41no—
220C23×C61C22×D411yescharacteristic
221D4×C61—22no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—48no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—84no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61C2×SD1612yes—
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
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—41no—
271C2×Dic61—84no—
272C2×Dic61—84no—
273C2×Dic61C2×SD1612yes—
274C2×Dic61C2×SD1612yes—
275C8×C61—41no—
276C8×C61—41no—
277#92418—41no—
278#92418—41no—
279#26754—611nomaximal, Sylow
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61#52511yescharacteristic
295C22×C61#54611yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61C22×SD1611yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57611yescharacteristic
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61#52911yescharacteristic
306C4×C61#53911yescharacteristic
307C4×C61#55211yescharacteristic
308C4×C61#55311yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—42no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317#2507—611no—
318#2366—611no—
319#2071—611no—
320#2065—611no—
321#2041—611no—
322#2041D6111yescharacteristic
323#1934—611no—
324#1926—611no—
325#1920—611no—
326#1920—611no—
327#1916—611no—
328#1864—611no—
329#1864—611no—
330#1802—611no—
331C4×#102—611no—
332C2×C61—82no—
333C2×C61—82no—
334C2×C61#246011yescharacteristic
335C2×C61#310111yescharacteristic
336C2×C61#312211yescharacteristic
337C2×C61#329511yescharacteristic
338C2×C61#330911yescharacteristic
339C2×C61#336211yescharacteristic
340C2×C61#337011yescharacteristic
341C2×#127C2×D6111yescharacteristic
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×C4×D4—1221no—
345C2×#106—611no—
346C2×#106—611no—
347C2×C4⋊C8—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×#102—611no—
351C2×#102—611no—
352C2×#102C2×D6111yescharacteristic
353C2×#102C2×D6111yescharacteristic
354C2×#102C2×D6111yescharacteristic
355C2×#102C2×D6111yescharacteristic
356C2×C4×C8—611no—
357#398—1221no—
358#398—1221no—
359#390—611no—
360#390—611no—
361#390—1221no—
362#389—611no—
363#389—611no—
364#389—611no—
365#389—1221no—
366#389C2×D6111yescharacteristic
367#387—1221no—
368#386—1221no—
369#386—1221no—
370#384—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#336—611no—
381#336—611no—
382#336—611no—
383#336C2×D6111yescharacteristic
384C61#2675411yescharacteristic, complemented, Sylow
385C22×D4—21no—
386C22×D4—21no—
387C22×D4C22×D6111yescharacteristic
388C22×D4C22×D6111yescharacteristic
389C23×C4—1221no—
390C23×C4—1221no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395#127—22no—
396#127—22no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—2441no—
400C4×D4—2441no—
401C4×D4—2444no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427#106—1222no—
428#106—1222no—
429C4⋊C8—2441no—
430#102—22no—
431#102—22no—
432#102—22no—
433#102—22no—
434#102—1222no—
435#102—1222no—
436#102—2441no—
437#102—2441no—
438C4×C8—2441no—
439#95—21no—
440#95—611no—
441#95—611no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447C24—21no—
448C24—21no—
449C2×D4—24no—
450C2×D4—24no—
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—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4#1175822412yes—
466C2×D4#1175822412yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—4881no—
478C22×C4—4881no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—81no—
502C4⋊C4—614no—
503C4⋊C4—612no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—4881no—
512C4⋊C4—4881no—
513C4⋊C4—4881no—
514C4⋊C4—4881no—
515C4⋊C4—2441no—
516C4⋊C4—2442no—
517C4⋊C4#1175822412yes—
518C22⋊C4—4881no—
519C22⋊C4—4881no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—21no—
525C42—21no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—4881no—
531C42—2442no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×D4×D6111yescharacteristic
547D4—22no—
548D4—22no—
549D4—22no—
550D4—22no—
551D4—48no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—4884no—
582C2×C4—4884no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4SD16×D6112yes—
595C2×C4#41×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822411yescharacteristic
598C2×C4C2×#1175822411yescharacteristic
599C2×C4#1451982311yescharacteristic
600C2×C4#1451982411yescharacteristic
601C8—41no—
602C8—41no—
603C8—2441no—
604C8—2441no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#1684026811yescharacteristic
620C22#1684040511yescharacteristic
621C22#1684041811yescharacteristic
622C22C2×SD16×D6111yescharacteristic
623C22#1684055811yescharacteristic
624C22#1684056311yescharacteristic
625C22#1684056611yescharacteristic
626C4—22no—
627C4—42no—
628C4—41no—
629C4—81no—
630C4—1222no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—4881no—
635C4—2441no—
636C4—2441no—
637C4—2442no—
638C4#1684029311yescharacteristic
639C4#1684035611yescharacteristic
640C4#1684044911yescharacteristic
641C4#1684046411yescharacteristic
642C2—82no—
643C2—82no—
644C2#2144976611yescharacteristic
645C2#2145582211yescharacteristic
646C2#2145605611yescharacteristic
647C2#2145764211yescharacteristic
648C2#2145782411yescharacteristic
649C2#2145848511yescharacteristic
650C2#2145856511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2041D61
hypocenterC61#26754
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. #21450349C2semidirect
3. #21447225C2semidirect
4. #21447189C2semidirect
6. #21446504C2semidirect
7. #21446449C2semidirect
8. #21446405C2semidirect
9. #21446401C2semidirect
10. #21446377C2semidirect
11. #21446082C2semidirect
12. #21446082C2semidirect
13. #21445817C2semidirect
14. #102×Dic61C2semidirect
384. C61#26754semidirect

Maximal subgroups

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

subgroupindex
#214503492
#214488842
#214472252
#214471892
#214470502
#214465042
#214464492
#214464052
#214464012
#214463772
#214460822
#214458172
#102×Dic612
C61×#20412
#2675461