Finite Group Databasebeta

← #29753204

Subgroups — #29753204

Order 15616.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450405C211yesmaximal, characteristic
2#21447362C211yesmaximal, characteristic, complemented
3#21447134C211yesmaximal, characteristic
4#21447062C211yesmaximal, characteristic, complemented
5#21446902C211yesmaximal, characteristic
6#21446676C211yesmaximal, characteristic
7#21446450C211yesmaximal, characteristic, complemented
8#21446446C211yesmaximal, characteristic
9#21446397C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446174C211yesmaximal, characteristic
12#21446174C211yesmaximal, characteristic
13#21445824C211yesmaximal, characteristic, complemented
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#2365C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519772—21no—
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839689—21no—
32#16839684—21no—
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839654—21no—
38#16839632C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#391C2211yescharacteristic
53C61×#389—21no—
54C61×#387—21no—
55C61×#387—21no—
56C4×C4⋊C4×C61—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C2×#11758201—22no—
61C2×#11758201—22no—
62C2×#11758201—22no—
63C2×#11758201—41no—
64C2×#11758201—41no—
65C2×#11758201—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519772—44no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84#14519708—22no—
85#14519708—41no—
86C4⋊#92418—41no—
87#14519701—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—41no—
91#14519701—41no—
92#14519701—41no—
93#14519697—22no—
94#14519697—41no—
95C4×#92418—41no—
96#14519684—21no—
97#14519684—21no—
98C22×Q8×C61C2311yescharacteristic
99C22×Q8×C61C2311yescharacteristic
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102C4×Q8×C61—22no—
103C4×Q8×C61—22no—
104C4×Q8×C61—41no—
105C4×Q8×C61—41no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61D411yescharacteristic
126#95×C61D411yescharacteristic
127C23×D61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61#4112yes—
139C2×Q8×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61SD1614yescomplemented
169C4⋊C4×C61#4112yes—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—42no—
177C42×C61—42no—
178C42×C61—42no—
179#11758201—44no—
180#11758201—44no—
181#11758198—42no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—81no—
186#11758198—81no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—81no—
195#11758196—42no—
196C4×Dic61—42no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418#4112yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C2×SD1612yes—
253C2×C4×C61C2×SD1612yes—
254C2×C4×C61C2×SD1612yes—
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
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—
270D244—41no—
271D244—41no—
272D244—41no—
273D244—41no—
274#92418—22no—
275#92418—41no—
276#92418—41no—
277#92418—41no—
278#27066—611nomaximal, Sylow
279C22×C61#52511yescharacteristic
280C22×C61#54611yescharacteristic
281C22×C61#55011yescharacteristic
282C22×C61C22×SD1611yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57711yescharacteristic
285C22×C61#57811yescharacteristic
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—81no—
303C4×C61#52911yescharacteristic
304C4×C61#53811yescharacteristic
305C4×C61#55111yescharacteristic
306C4×C61#55311yescharacteristic
307Dic61—81no—
308Dic61—81no—
309Dic61—81no—
310#2512—611no—
311#2365D6111yescharacteristic, complemented
312#2096—611no—
313#2055—611no—
314#2043—611no—
315#2013—611no—
316#1968—611no—
317#1927—611no—
318#1926—611no—
319#1919—611no—
320#1916—611no—
321#1880—611no—
322#1880—611no—
323#1803—611no—
324C4×#103—611no—
325C2×C61#246711yescharacteristic
326C2×C61#310311yescharacteristic
327C2×C61#312111yescharacteristic
328C2×C61#329811yescharacteristic
329C2×C61#330711yescharacteristic
330C2×C61#339211yescharacteristic
331C2×C61#340211yescharacteristic
332D61—161no—
333C22×SD16—611no—
334C22×SD16—611no—
335C2×C4⋊Q8C2×D6111yescharacteristic
336C2×#126—1221no—
337C2×C4×Q8C2×D6111yescharacteristic
338C2×C4×Q8C2×D6111yescharacteristic
339C2×#106—611no—
340C2×#106—611no—
341C2×C4⋊C8—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×C4×C8—611no—
351#397—1221no—
352#396—1221no—
353#391C2×D6111yescharacteristic
354#391C2×D6111yescharacteristic
355#390—611no—
356#390—611no—
357#390—611no—
358#389—21no—
359#389—611no—
360#388—1221no—
361#387—21no—
362#387—21no—
363#384—611no—
364C4×C4⋊C4—21no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4C2×D6111yescharacteristic
367C4×C4⋊C4C2×D6111yescharacteristic
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376C61#2706611yescharacteristic, complemented, Sylow
377C22×Q8C22×D6111yescharacteristic
378C22×Q8C22×D6111yescharacteristic
379C22×D4—611no—
380C2×SD16—1222no—
381C2×SD16—1222no—
382C2×SD16—1222no—
383C2×SD16—2441no—
384C2×SD16—2441no—
385C2×SD16—2441no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—611no—
390C4⋊Q8—22no—
391C4⋊Q8—22no—
392#126—2444no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—41no—
396C4×Q8—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4C22⋊D6111yescharacteristic
408C2×C4⋊C4C22⋊D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—1221no—
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424#106—1222no—
425#106—2441no—
426C4⋊C8—2441no—
427#103—1222no—
428#103—1222no—
429#103—1222no—
430#103—2441no—
431#103—2441no—
432#103—2441no—
433#102—1222no—
434#102—2441no—
435C4×C8—2441no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1221no—
442#95—1221no—
443#95C22⋊D6111yescharacteristic
444#95C22⋊D6111yescharacteristic
445C24—1221no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8#1175822212yes—
457C2×Q8#1175822212yes—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—2444no—
479SD16—2444no—
480C2×C8—612no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—4881no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4#1175820114yescomplemented
510C4⋊C4#1175822412yes—
511C22⋊C4—4881no—
512C22⋊C4—4881no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C22⋊C4—2442no—
516C22⋊C4—2442no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—2442no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23C22×C22⋊D6111yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4C2×#1175820112yes—
584C2×C4C2×#1175820112yes—
585C2×C4C2×#1175820112yes—
586C2×C4#41×D6111yescharacteristic
587C2×C4C2×#1175822411yescharacteristic
588C2×C4C2×#1175822211yescharacteristic
589C2×C4C2×#1175822211yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982311yescharacteristic
592C8—1222no—
593C8—2441no—
594C8—2441no—
595C8—2441no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—4881no—
603C22#1684027311yescharacteristic
604C22#1684040611yescharacteristic
605C22#1684043111yescharacteristic
606C22C22×#1175820111yescharacteristic
607C22#1684056011yescharacteristic
608C22#1684057011yescharacteristic
609C22#1684057711yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—4881no—
621C4—4881no—
622C4—4881no—
623C4#1684029211yescharacteristic
624C4#1684034511yescharacteristic
625C4#1684044111yescharacteristic
626C4#1684045811yescharacteristic
627C2—9761no—
628C2#2144986411yescharacteristic
629C2#2145585011yescharacteristic
630C2#2145604811yescharacteristic
631C2#2145768411yescharacteristic
632C2#2145779511yescharacteristic
633C2#2145879611yescharacteristic
634C2#2145889411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2365C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2365D61
hypocenterC61#27066
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #21447362C2semidirect
4. #21447062C2semidirect
7. #21446450C2semidirect
9. #21446397C2semidirect
10. #21446378C2semidirect
13. #21445824C2semidirect
14. C4×#14519701C2semidirect
15. C61×#2365C2semidirect
168. C4⋊C4×C61SD16semidirect
311. #2365D61semidirect
376. C61#27066semidirect
509. C4⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214504052
#214473622
#214471342
#214470622
#214469022
#214466762
#214464502
#214464462
#214463972
#214463782
#214461742
#214458242
C4×#145197012
C61×#23652
#2706661