Finite Group Databasebeta

← #29665171

Subgroups — #29665171

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456761C211yesmaximal, characteristic, complemented
2#21455617C211yesmaximal, characteristic, complemented
3#459×D61C211yesmaximal, characteristic
4#21454840C211yesmaximal, characteristic, complemented
5#21445231C211yesmaximal, characteristic, complemented
6#21445230C211yesmaximal, characteristic, complemented
7#21445225C211yesmaximal, characteristic, complemented
8#21445161C211yesmaximal, characteristic, complemented
9#21445160C211yesmaximal, characteristic, complemented
10#21445159C211yesmaximal, characteristic, complemented
11#21445158C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444485C211yesmaximal, characteristic, complemented
14#21444473C211yesmaximal, characteristic, complemented
15C61×#1677C211yesmaximal, characteristic, complemented
16C2×D8×D61—21no—
17#16840317C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840135C2211yescharacteristic, complemented
21#16840134C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic, complemented
23#16840061—21no—
24#16840060—21no—
25#16840049C2211yescharacteristic, complemented
26#16840043C2211yescharacteristic, complemented
27#16840042C2211yescharacteristic, complemented
28#16839961—21no—
29#16839961—21no—
30#16839958C2211yescharacteristic
31#16839957C2211yescharacteristic
32#16839956—21no—
33#16839955—21no—
34#16839954C2211yescharacteristic
35#16839914—21no—
36#16839914—21no—
37C8×D244C2211yescharacteristic, complemented
38D4×#92418C2211yescharacteristic, complemented
39#16839827C2211yescharacteristic, complemented
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839814C2211yescharacteristic
43#16839813C2211yescharacteristic
44#16839801—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839765—21no—
48#102×D61—21no—
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61D411yescharacteristic
71C22×C4×D61—21no—
72C2×#11758190—21no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×D488—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79D8×D61—44no—
80C2×#11758189—21no—
81C22×#92418—21no—
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84C2×C8×D61—21no—
85#14519777C2311yescharacteristic
86#14519774C2311yescharacteristic
87#14519773—21no—
88#14519773—21no—
89#14519773—21no—
90#14519768—21no—
91#14519757—21no—
92#14519753—21no—
93Q8×Dic61—21no—
94#14519742—21no—
95C4×D244C2311yescharacteristic
96D4×Dic61D411yescharacteristic
97#14519739D411yescharacteristic
98C4×C22⋊D61—21no—
99#14519737—21no—
100#14519735—21no—
101C4⋊C4×D61D411yescharacteristic
102C22⋊C4×D61—21no—
103C42×D61C2311yescharacteristic
104#14519711—21no—
105#14519710—21no—
106#14519708—21no—
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109C4⋊#92418C2311yescharacteristic
110#14519706—21no—
111#14519706—21no—
112#14519706D411yescharacteristic
113#14519706D411yescharacteristic
114#14519699—21no—
115#14519699—21no—
116#14519698—21no—
117#14519698—21no—
118#14519697—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689D411yescharacteristic
127#14519689D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×D8×C61—21no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C22×Dic61—21no—
158D4×D61—22no—
159D4×D61—42no—
160D4×D61—42no—
161D4×D61—81no—
162D4×D61—81no—
163D4×D61—82no—
164D4×D61SD1614yescomplemented
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—81no—
173C2×C4×D61—81no—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61#4111yescharacteristic
177C2×Dic122—21no—
178D8×C61—42no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—81no—
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C2×D411yescharacteristic
188C22⋊C4×C61—21no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—21no—
191#11758200—81no—
192#11758200—81no—
193#11758199—42no—
194#11758199—42no—
195#11758198—21no—
196#11758198—21no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—41no—
204C4×Dic61—41no—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61#4111yescharacteristic
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—81no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214D488—42no—
215#11758190—81no—
216#11758190—81no—
217#11758189—42no—
218C8×D61—42no—
219C8×D61—81no—
220C8×D61—81no—
221C23×C61—21no—
222C23×C61—41no—
223C22×D61—21no—
224C22×D61—24no—
225C22×D61—41no—
226C22×D61—81no—
227C22×D61—81no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—82no—
231C22×D61#12111yescharacteristic
232Q8×C61—81no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61C2×SD1612yes—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C22⋊D61—24no—
248C22⋊D61—82no—
249C22⋊D61—82no—
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—81no—
254C2×Dic61#12111yescharacteristic
255C2×Dic61#12111yescharacteristic
256C2×Dic61#12111yescharacteristic
257D244—21no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—81no—
263D244C2×SD1612yes—
264C4×D61—21no—
265C4×D61—21no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—81no—
270C4×D61—81no—
271C4×D61C2×SD1612yes—
272C4×D61#13512yes—
273Dic122—41no—
274Dic122—41no—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—41no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—21no—
281#92418—21no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#19384—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—22no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—84no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—164no—
303C2×D61#46512yes—
304C4×C61—21no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61C22×SD1611yescharacteristic
308C4×C61C2×#13511yescharacteristic
309C4×C61#57611yescharacteristic
310C4×C61#57611yescharacteristic
311Dic61—21no—
312Dic61—21no—
313Dic61—41no—
314Dic61—81no—
315Dic61#46512yes—
316#3208—611no—
317#3077—611no—
318C2×#459—611no—
319#2972—611no—
320#1677—611no—
321#1677—611no—
322#1677—611no—
323#1677D6111yescharacteristic, complemented
324#1665—611no—
325#1665—611no—
326#1665—611no—
327#1665—611no—
328#1570—611no—
329#1545—611no—
330#1542—611no—
331C2×C61—42no—
332C2×C61—161no—
333C2×C61#304111yescharacteristic
334C2×C61C2×#46511yescharacteristic
335C2×C61#306811yescharacteristic
336D61—22no—
337D61—42no—
338D61—161no—
339C22×D8—1221no—
340#533—611no—
341C2×#127—611no—
342C2×C4×D4—611no—
343#492—611no—
344#492—611no—
345#486—611no—
346#486C2×D6111yescharacteristic, complemented
347#478—1221no—
348#478—1221no—
349#476—611no—
350#476C2×D6111yescharacteristic, complemented
351#474—611no—
352#474—611no—
353#460—1221no—
354#460—1221no—
355#459—611no—
356#459—611no—
357#459—611no—
358#459—1221no—
359#459—1221no—
360#459C2×D6111yescharacteristic
361#450—1221no—
362#450—1221no—
363C8×D4—611no—
364C8×D4—611no—
365C8×D4—611no—
366C8×D4—1221no—
367C8×D4—1221no—
368C8×D4C2×D6111yescharacteristic, complemented
369#431—611no—
370#431—611no—
371#428—1221no—
372#423—611no—
373C2×C4⋊C8—611no—
374#418—1221no—
375C2×#102—1221no—
376C2×C22⋊C8—1221no—
377C2×C22⋊C8—1221no—
378#334—611no—
379#334—611no—
380#334—611no—
381#334C2×D6111yescharacteristic, complemented
382#333—611no—
383#333C2×D6111yescharacteristic, complemented
384#332—611no—
385#332—611no—
386#331—611no—
387#331—611no—
388#331—611no—
389#331C2×D6111yescharacteristic, complemented
390C61#1938411yescharacteristic, complemented, Sylow
391C22×D4—611no—
392C22×D4—1221no—
393C22×D4—1221no—
394C22×D4—1221no—
395C23×C4—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×D8—21no—
401C2×D8—1221no—
402C2×D8—1221no—
403C2×D8—1221no—
404C2×D8—2444no—
405C2×M16—1221no—
406C22×C8—21no—
407C22×C8—1221no—
408C22×C8—1221no—
409C22×C8—1221no—
410C22×C8—1221no—
411C4⋊Q8—611no—
412C4⋊Q8C22×D6111yescharacteristic
413#127—611no—
414#127—1221no—
415#127—1221no—
416#127—1221no—
417#127C22×D6111yescharacteristic
418#125—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8—1221no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×D4—611no—
424C4×D4—611no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4C22×D6111yescharacteristic
429#117—1221no—
430C2×C4⋊C4—611no—
431C2×C22⋊C4—1221no—
432C2×C42—611no—
433#107—1221no—
434#107—1221no—
435#106—21no—
436#106—1221no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8C22×D6111yescharacteristic
445C4⋊C8C22×D6111yescharacteristic
446C4⋊C8C22×D6111yescharacteristic
447#102—21no—
448#102—21no—
449#102—21no—
450#102—21no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459C22⋊C8—21no—
460C22⋊C8—611no—
461C22⋊C8—611no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464C22⋊C8—1221no—
465C22⋊C8—1221no—
466C4×C8—611no—
467C4×C8—1221no—
468C4×C8C22×D6111yescharacteristic
469C24—1221no—
470C24—2441no—
471C2×Q8—21no—
472C2×Q8—1221no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—614no—
477C2×D4—611no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—4881no—
484C2×D4—4881no—
485C2×D4—4882no—
486C2×D4—2442no—
487C2×D4—2442no—
488C2×D4—2441no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4D4×D6111yescharacteristic
492C22×C4—21no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—4881no—
501C22×C4—4881no—
502SD16—4881no—
503SD16—4881no—
504SD16—4881no—
505SD16—4881no—
506D8—42no—
507D8—2442no—
508D8—2442no—
509D8—2442no—
510M16—2442no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—81no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—4881no—
521C2×C8—4881no—
522C2×C8—4881no—
523C2×C8—2442no—
524C2×C8D4×D6111yescharacteristic
525C2×C8D4×D6111yescharacteristic
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2441no—
533C4⋊C4—2441no—
534C4⋊C4D4×D6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—2441no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—2441no—
544C42C23×D6111yescharacteristic
545C23—21no—
546C23—41no—
547C23—611no—
548C23—1224no—
549C23—1221no—
550C23—4881no—
551C23—4881no—
552C23—4882no—
553C23—4882no—
554C23—4882no—
555C23—2441no—
556Q8—81no—
557Q8—2441no—
558Q8—2441no—
559D4—21no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—81no—
565D4—612no—
566D4—1224no—
567D4—1221no—
568D4—4881no—
569D4—4882no—
570D4—4882no—
571D4—4882no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575D4—2441no—
576D4SD16×D6112yes—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—611no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—4881no—
590C2×C4—4881no—
591C2×C4—4881no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C8—21no—
600C8—21no—
601C8—41no—
602C8—41no—
603C8—41no—
604C8—1221no—
605C8—1221no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—612no—
616C22—1221no—
617C22—1221no—
618C22—1222no—
619C22—9764no—
620C22—4884no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—2441no—
625C22—2441no—
626C22#121×D6111yescharacteristic
627C4—21no—
628C4—41no—
629C4—81no—
630C4—612no—
631C4—1221no—
632C4—1221no—
633C4—4881no—
634C4—2441no—
635C4C2×SD16×D6111yescharacteristic
636C4#135×D6111yescharacteristic
637C4#1684056411yescharacteristic
638C4#1684056611yescharacteristic
639C2—42no—
640C2—161no—
641C2—1222no—
642C2—9761no—
643C2—2442no—
644C2#2145532011yescharacteristic
645C2#465×D6111yescharacteristic
646C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1677C2
socleC22×C61C2×#121
prime radical#459×D61C2
hypercenter#1677D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1677

Complements

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

NH ≅ G/Ntype
1. #21456761C2semidirect
2. #21455617C2semidirect
4. #21454840C2semidirect
5. #21445231C2semidirect
6. #21445230C2semidirect
7. #21445225C2semidirect
8. #21445161C2semidirect
9. #21445160C2semidirect
10. #21445159C2semidirect
11. #21445158C2semidirect
12. #21444588C2semidirect
13. #21444485C2semidirect
14. #21444473C2semidirect
15. C61×#1677C2semidirect
20. #16840135C22semidirect
21. #16840134C22semidirect
22. #16840110C22semidirect
25. #16840049C22semidirect
26. #16840043C22semidirect
27. #16840042C22semidirect
37. C8×D244C22semidirect
38. D4×#92418C22semidirect
39. #16839827C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839507C22semidirect
57. #16839506C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#486C22semidirect
61. C61×#476C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
66. C61×#331C22semidirect
164. D4×D61SD16semidirect
323. #1677D61semidirect
346. #486C2×D61semidirect
350. #476C2×D61semidirect
368. C8×D4C2×D61semidirect
381. #334C2×D61semidirect
383. #333C2×D61semidirect
389. #331C2×D61semidirect
390. C61#19384semidirect

Maximal subgroups

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

subgroupindex
#214567612
#214556172
#459×D612
#214548402
#214452312
#214452302
#214452252
#214451612
#214451602
#214451592
#214451582
#214445882
#214444852
#214444732
C61×#16772
#1938461