Finite Group Databasebeta

← #29753276

Subgroups — #29753276

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450428C211yesmaximal, characteristic, complemented
2#21448895C211yesmaximal, characteristic
3#21447337C211yesmaximal, characteristic
4#21447133C211yesmaximal, characteristic
5#21447057C211yesmaximal, characteristic
6#21446910C211yesmaximal, characteristic
7#21446456C211yesmaximal, characteristic, complemented
8#21446450C211yesmaximal, characteristic, complemented
9#21446413C211yesmaximal, characteristic, complemented
10#21446376C211yesmaximal, characteristic, complemented
11#21446162C211yesmaximal, characteristic, complemented
12#21446156C211yesmaximal, characteristic
13#21445832C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1969C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C61⋊#401—21no—
29#16839704—21no—
30#16839669C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839654—21no—
35#16839651—21no—
36#16839647—21no—
37#16839636—21no—
38#16839629—21no—
39#16839627C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839609—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C4×D61—21no—
61C2×#11758201—22no—
62C2×#11758201—41no—
63C2×#11758201—41no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519766—22no—
67#14519766—22no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70C4×D244—22no—
71C4×D244—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196D411yescharacteristic
78C2×#11758196D411yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519708—41no—
90#14519706—41no—
91#14519701—22no—
92#14519701—22no—
93#14519701—41no—
94#14519699—22no—
95#14519699—22no—
96#14519697—41no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C2×Q16×C61—22no—
111C2×Q16×C61—22no—
112C2×Q16×C61—41no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#103×C61—41no—
120#103×C61—41no—
121C4×C8×C61—41no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C2×Q8×C61—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—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
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×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244#4112yes—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—81no—
162Q16×C61—44no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61#4112yes—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175#11758201—44no—
176#11758198—24no—
177#11758198—24no—
178#11758198—42no—
179#11758198—42no—
180#11758198—81no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61#4112yes—
189#11758196—22no—
190#11758196—41no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
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×C61C22×D411yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
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#13512yes—
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265D244—22no—
266D244—22no—
267D244—41no—
268D244—41no—
269C4×D61—84no—
270C8×C61—22no—
271C8×C61—41no—
272#92418—41no—
273#92418—41no—
274#27070—611nomaximal, Sylow
275C22×C61#52511yescharacteristic
276C22×C61#54611yescharacteristic
277C22×C61#55011yescharacteristic
278C22×C61C2×#13511yescharacteristic
279C22×C61#57511yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61#52911yescharacteristic
296C4×C61#53811yescharacteristic
297C4×C61#55111yescharacteristic
298C4×C61#55311yescharacteristic
299Dic61—41no—
300Dic61—41no—
301Dic61—42no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306#2514—611no—
307#2367—611no—
308#2092—611no—
309#2055—611no—
310#2042—611no—
311#2015—611no—
312#1969D6111yescharacteristic, complemented
313#1927—611no—
314#1927—611no—
315#1921—611no—
316#1916—611no—
317#1878—611no—
318#1877—611no—
319#1805—611no—
320C4×#102—611no—
321C2×C61#246711yescharacteristic
322C2×C61#310211yescharacteristic
323C2×C61#312011yescharacteristic
324C2×C61#329611yescharacteristic
325C2×C61#331311yescharacteristic
326C2×C61#339311yescharacteristic
327C2×C61#340111yescharacteristic
328D61—82no—
329C22×Q16C2×D6111yescharacteristic
330C22×SD16—611no—
331C2×#124—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×#107C2×D6111yescharacteristic
335C2×#106—611no—
336C2×C4⋊C8—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×#102—611no—
343C2×#102—611no—
344C2×#102—611no—
345C2×C4×C8C2×D6111yescharacteristic
346#401—1221no—
347#400—1221no—
348#391—611no—
349#391—611no—
350#391C2×D6111yescharacteristic
351#391C2×D6111yescharacteristic
352#390—611no—
353#389—611no—
354#388—1221no—
355#388—1221no—
356#387—1221no—
357#385—1221no—
358#383—1221no—
359#382—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—611no—
363C4×C22⋊C4—1221no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#336—611no—
369#336—611no—
370#336—611no—
371#336—611no—
372C61#2707011yescharacteristic, complemented, Sylow
373C22×Q8C22×D6111yescharacteristic
374C22×Q8C22×D6111yescharacteristic
375C22×D4—611no—
376C23×C4—1221no—
377C2×Q16—22no—
378C2×Q16—22no—
379C2×Q16—41no—
380C2×SD16—1222no—
381C2×SD16—2441no—
382C2×SD16—2441no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387#124—1222no—
388#124—1222no—
389C4×Q8—1222no—
390C4×Q8—2441no—
391C4×D4—1222no—
392C4×D4—2441no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413#107—22no—
414#106—2441no—
415C4⋊C8—2441no—
416#103—41no—
417#103—41no—
418#103—1222no—
419#103—1222no—
420#103—2441no—
421#102—1222no—
422#102—1222no—
423#102—2441no—
424C4×C8—41no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—611no—
430#95—611no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C24—1221no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#1175822212yes—
453C2×D4—612no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
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—4881no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476Q16—44no—
477SD16—2444no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8#1175822412yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—2441no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—4881no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—21no—
510C42—21no—
511C42—1221no—
512C42—1221no—
513C42—1221no—
514C42—1221no—
515C42—4881no—
516C42—2442no—
517C42—2442no—
518C23—1222no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23—2441no—
525C23C2×D4×D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534D4—1222no—
535D4—1222no—
536D4—2441no—
537D4—2441no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—4884no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4#1451980012yes—
578C2×C4#41×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4C2×#1175822411yescharacteristic
581C2×C4C2×#1175822211yescharacteristic
582C2×C4#1451982211yescharacteristic
583C2×C4#1451982211yescharacteristic
584C8—22no—
585C8—41no—
586C8—2441no—
587C8—2441no—
588C22—4881no—
589C22—4881no—
590C22—4881no—
591C22—4881no—
592C22—2442no—
593C22—2442no—
594C22—2442no—
595C22#1684027111yescharacteristic
596C22#1684040111yescharacteristic
597C22#1684043211yescharacteristic
598C22C2×#1451980011yescharacteristic
599C22#1684055911yescharacteristic
600C22#1684056911yescharacteristic
601C22#1684057511yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—4881no—
609C4—4881no—
610C4—2441no—
611C4—2441no—
612C4—2441no—
613C4—2441no—
614C4—2442no—
615C4#1684029111yescharacteristic
616C4#1684034411yescharacteristic
617C4#1684043811yescharacteristic
618C4#1684046111yescharacteristic
619C2—4882no—
620C2#2144987611yescharacteristic
621C2#2145584311yescharacteristic
622C2#2145603911yescharacteristic
623C2#2145766011yescharacteristic
624C2#2145789111yescharacteristic
625C2#2145880911yescharacteristic
626C2#2145889111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1969C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1969D61
hypocenterC61#27070
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1969

Complements

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

NH ≅ G/Ntype
1. #21450428C2semidirect
7. #21446456C2semidirect
8. #21446450C2semidirect
9. #21446413C2semidirect
10. #21446376C2semidirect
11. #21446162C2semidirect
13. #21445832C2semidirect
15. C61×#1969C2semidirect
312. #1969D61semidirect
372. C61#27070semidirect

Maximal subgroups

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

subgroupindex
#214504282
#214488952
#214473372
#214471332
#214470572
#214469102
#214464562
#214464502
#214464132
#214463762
#214461622
#214461562
#214458322
C4×#145196992
C61×#19692
#2707061