Finite Group Databasebeta

← #29752699

Subgroups — #29752699

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#21448835C211yesmaximal, characteristic
3#21447226C211yesmaximal, characteristic, complemented
4#21447194C211yesmaximal, characteristic, complemented
5#21447009C211yesmaximal, characteristic
6#21447008C211yesmaximal, characteristic
7#21446581C211yesmaximal, characteristic, complemented
8#21446546C211yesmaximal, characteristic, complemented
9#21446422C211yesmaximal, characteristic, complemented
10#21446402C211yesmaximal, characteristic, complemented
11#21446396C211yesmaximal, characteristic, complemented
12#21446081C211yesmaximal, characteristic, complemented
13#21446012C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1823C211yesmaximal, characteristic, complemented
16C2×#14519751C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×C4×C22⋊D61—21no—
19C2×#14519709C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839702—21no—
31#16839685—21no—
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839652—21no—
37#16839650—21no—
38#16839644—21no—
39#16839641—21no—
40#16839633—21no—
41#16839631C2211yescharacteristic
42#16839622C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C22⋊Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C61×#380C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C22⋊D61—21no—
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519751—22no—
65#14519751—22no—
66#14519739—22no—
67#14519739—41no—
68C4×C22⋊D61—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61D411yescharacteristic
79C2×C22⋊Dic61D411yescharacteristic
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519709—22no—
89#14519708—22no—
90#14519708—22no—
91#14519708—41no—
92#14519706—22no—
93#14519699—22no—
94#14519699—41no—
95#14519697—22no—
96#14519697—22no—
97C22⋊#92418—22no—
98C8×Dic61—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—41no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C22⋊C8×C61—41no—
120#95×C61—21no—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61—81no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244#4112yes—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—81no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—81no—
173C22⋊Dic61—41no—
174C22⋊Dic61—81no—
175C22⋊Dic61SD1614yes—
176#11758198—24no—
177#11758198—42no—
178#11758198—42no—
179#11758198—81no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—81no—
186C4⋊Dic61#4112yes—
187#11758196—42no—
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#4112yes—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61C2×SD1612yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
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—84no—
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C22⋊D61—48no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
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×Dic61C2×SD1612yes—
265C2×Dic61C2×SD1612yes—
266D244—22no—
267D244—22no—
268D244—41no—
269D244—41no—
270C4×D61—84no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—22no—
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#52511yescharacteristic
284C22×C61#54611yescharacteristic
285C22×C61#54711yescharacteristic
286C22×C61C22×SD1611yescharacteristic
287C22×C61#57511yescharacteristic
288C22×C61#57611yescharacteristic
289C22×C61#57711yescharacteristic
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61#52911yescharacteristic
303C4×C61#53911yescharacteristic
304C4×C61#55311yescharacteristic
305C4×C61#55611yescharacteristic
306Dic61—22no—
307Dic61—22no—
308Dic61—41no—
309Dic61—42no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313#2497—611no—
314#2361—611no—
315#2071—611no—
316#2065—611no—
317#2034—611no—
318#2034—611no—
319#1948—611no—
320#1941—611no—
321#1923—611no—
322#1920—611no—
323#1919—611no—
324#1864—611no—
325#1853—611no—
326#1823D6111yescharacteristic, complemented
327C4×#102—611no—
328C2×C61—42no—
329C2×C61#246611yescharacteristic
330C2×C61#310111yescharacteristic
331C2×C61#311911yescharacteristic
332C2×C61#329511yescharacteristic
333C2×C61#332111yescharacteristic
334C2×C61#333511yescharacteristic
335C2×C61#335511yescharacteristic
336D61—82no—
337C2×#121—611no—
338C2×C4×D4—611no—
339C2×C4×D4—1221no—
340C22×C4⋊C4C2×D6111yescharacteristic
341C2×#106—611no—
342C2×#106—611no—
343C2×#106—611no—
344C2×#106—611no—
345C2×C4⋊C8—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#102—611no—
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#340C2×D6111yescharacteristic
375#340C2×D6111yescharacteristic
376#340C2×D6111yescharacteristic
377#340C2×D6111yescharacteristic
378#336—611no—
379#336—611no—
380C61#2703211yescharacteristic, complemented, Sylow
381C22×D4—611no—
382C22×D4—1221no—
383C23×C4—21no—
384C23×C4—1221no—
385C23×C4C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390#121—1222no—
391#121—1222no—
392C4×D4—1222no—
393C4×D4—2441no—
394C4×D4—2444no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—611no—
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—
422#106—1222no—
423#106—1222no—
424#106—1222no—
425#106—2441no—
426C4⋊C8—1222no—
427#102—1222no—
428#102—1222no—
429#102—1222no—
430#102—2441no—
431C22⋊C8—41no—
432C22⋊C8—1222no—
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—1221no—
444C24D4×D6111yescharacteristic
445C2×D4—612no—
446C2×D4—1224no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—2442no—
453C2×D4—2442no—
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—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—4881no—
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—21no—
485C2×C8—612no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—612no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—4881no—
506C4⋊C4—4881no—
507C4⋊C4—4881no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4#1175822212yes—
511C22⋊C4—81no—
512C22⋊C4—614no—
513C22⋊C4—1224no—
514C22⋊C4—4881no—
515C22⋊C4—4881no—
516C22⋊C4—2442no—
517C22⋊C4—2442no—
518C22⋊C4—2441no—
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—1222no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23—2441no—
538C23SD16×D6112yes—
539C23C2×D4×D6111yescharacteristic
540D4—1222no—
541D4—1222no—
542D4—2448no—
543D4—2441no—
544D4—2441no—
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—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—4884no—
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×C4#41×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C2×C4C2×#1175822211yescharacteristic
591C2×C4#1451982211yescharacteristic
592C2×C4#1451982411yescharacteristic
593C8—41no—
594C8—41no—
595C8—1222no—
596C8—2441no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22#1684027011yescharacteristic
612C22#1684040211yescharacteristic
613C22#1684041011yescharacteristic
614C22C2×SD16×D6111yescharacteristic
615C22#1684056111yescharacteristic
616C22#1684056311yescharacteristic
617C22#1684057111yescharacteristic
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—1222no—
624C4—1222no—
625C4—4881no—
626C4—4881no—
627C4—2441no—
628C4—2442no—
629C4—2441no—
630C4#1684029011yescharacteristic
631C4#1684035511yescharacteristic
632C4#1684045711yescharacteristic
633C4#1684048011yescharacteristic
634C2—42no—
635C2—4882no—
636C2#2144985411yescharacteristic
637C2#2145582411yescharacteristic
638C2#2145602111yescharacteristic
639C2#2145763911yescharacteristic
640C2#2145800611yescharacteristic
641C2#2145816911yescharacteristic
642C2#2145842511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1823C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1823D61
hypocenterC61#27032
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
1. #21450248C2semidirect
3. #21447226C2semidirect
4. #21447194C2semidirect
7. #21446581C2semidirect
8. #21446546C2semidirect
9. #21446422C2semidirect
10. #21446402C2semidirect
11. #21446396C2semidirect
12. #21446081C2semidirect
13. #21446012C2semidirect
14. #21445633C2semidirect
15. C61×#1823C2semidirect
326. #1823D61semidirect
380. C61#27032semidirect

Maximal subgroups

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

subgroupindex
#214502482
#214488352
#214472262
#214471942
#214470092
#214470082
#214465812
#214465462
#214464222
#214464022
#214463962
#214460812
#214460122
#214456332
C61×#18232
#2703261