Finite Group Databasebeta

← #29662820

Subgroups — #29662820

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456190C211yesmaximal, characteristic, complemented
2#21455829C211yesmaximal, characteristic, complemented
3#459×D61C211yesmaximal, characteristic
4#21454555C211yesmaximal, characteristic, complemented
5#21445380C211yesmaximal, characteristic, complemented
6#21445378C211yesmaximal, characteristic, complemented
7#21445354C211yesmaximal, characteristic, complemented
8#21445257C211yesmaximal, characteristic, complemented
9#21445255C211yesmaximal, characteristic, complemented
10#21445231C211yesmaximal, characteristic, complemented
11#21445225C211yesmaximal, characteristic, complemented
12#21444570C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1697C211yesmaximal, characteristic, complemented
16C2×D8×D61—21no—
17C2×M16×D61—21no—
18#16840317C2211yescharacteristic
19#127×D61C2211yescharacteristic
20C4×D4×D61C2211yescharacteristic
21#16840125—21no—
22#16840125—21no—
23#16840110C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic, complemented
25#16840108C2211yescharacteristic, complemented
26#16840094—21no—
27#16840049C2211yescharacteristic, complemented
28#16839968C2211yescharacteristic, complemented
29#16839967C2211yescharacteristic, complemented
30#16839958C2211yescharacteristic
31#16839957C2211yescharacteristic
32#16839956—21no—
33#16839955—21no—
34#16839954C2211yescharacteristic
35#16839914—21no—
36#16839853—21no—
37#16839849—21no—
38#16839840C2211yescharacteristic, complemented
39#16839836—21no—
40#16839831—21no—
41C8×D244C2211yescharacteristic, complemented
42#16839827C2211yescharacteristic, complemented
43#16839814C2211yescharacteristic
44#16839784C2211yescharacteristic
45C4⋊C8×D61C2211yescharacteristic
46#16839765—21no—
47#16839765—21no—
48#102×D61—21no—
49#16839724—21no—
50C4×#11758189C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839513C2211yescharacteristic, 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
63C61×#436C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, 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×#11758200—21no—
74C2×D488—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77D8×D61—44no—
78C2×#11758194—21no—
79C2×#11758189—21no—
80C2×#11758189D411yescharacteristic
81C2×#11758189D411yescharacteristic
82M16×D61—44no—
83C2×C8×D61—21no—
84C2×C8×D61—21no—
85C2×C8×D61—21no—
86#14519777C2311yescharacteristic
87#14519774C2311yescharacteristic
88#14519773—21no—
89#14519773—21no—
90#14519773—21no—
91#14519768—21no—
92#14519757—21no—
93#14519753—21no—
94Q8×Dic61—21no—
95#14519742—21no—
96C4×D244C2311yescharacteristic
97D4×Dic61D411yescharacteristic
98#14519739D411yescharacteristic
99C4×C22⋊D61—21no—
100#14519737—21no—
101#14519735—21no—
102C4⋊C4×D61D411yescharacteristic
103C22⋊C4×D61—21no—
104C42×D61C2311yescharacteristic
105#14519708—21no—
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519702—21no—
112#14519700—21no—
113#14519699—21no—
114#14519699—21no—
115#14519699—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122#14519697—21no—
123C22⋊#92418—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689—21no—
127#14519687D411yescharacteristic
128#14519687D411yescharacteristic
129C4×#92418C2311yescharacteristic
130C8×Dic61—21no—
131C2×D8×C61—21no—
132C2×M16×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134#127×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#106×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—21no—
147C23×D61—41no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C22×Dic61—21no—
159D4×D61—22no—
160D4×D61—41no—
161D4×D61—42no—
162D4×D61—42no—
163D4×D61—42no—
164D4×D61—42no—
165D4×D61—82no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61#4111yescharacteristic
178C2×Dic122—21no—
179D8×C61—42no—
180M16×C61—42no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—21no—
192#11758200—42no—
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
208#11758194—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214D488—42no—
215#11758190—42no—
216#11758189—22no—
217#11758189—22no—
218#11758189—42no—
219C8×D61—42no—
220C8×D61—42no—
221C8×D61—42no—
222C23×C61—21no—
223C23×C61—41no—
224C22×D61—21no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—41no—
228C22×D61—81no—
229C22×D61—82no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61#12111yescharacteristic
233Q8×C61—81no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—81no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—42no—
249C22⋊D61—82no—
250C22⋊D61—82no—
251C22⋊D61—82no—
252C2×Dic61—21no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61#12111yescharacteristic
256C2×Dic61#12111yescharacteristic
257C2×Dic61#12111yescharacteristic
258D244—21no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—81no—
264D244AGL(1,ℤ/8)12yes—
265C4×D61—21no—
266C4×D61—21no—
267C4×D61—41no—
268C4×D61—42no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—42no—
272C4×D61#13512yes—
273C4×D61AGL(1,ℤ/8)12yes—
274Dic122—41no—
275Dic122—41no—
276C8×C61—21no—
277C8×C61—21no—
278C8×C61—41no—
279C8×C61—41no—
280C8×C61—41no—
281#92418—21no—
282#92418—21no—
283#92418—41no—
284#92418—41no—
285#92418—41no—
286#19166—611nomaximal, Sylow
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—22no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—84no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—164no—
304C2×D61#46312yes—
305C4×C61—21no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61C2×#13511yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C4×C61#57511yescharacteristic
312Dic61—21no—
313Dic61—21no—
314Dic61—41no—
315Dic61—81no—
316Dic61#46312yes—
317#3138—611no—
318#3101—611no—
319C2×#459—611no—
320D4×M16—611no—
321#1701—611no—
322#1701—611no—
323#1697—611no—
324#1697D6111yescharacteristic, complemented
325#1681—611no—
326#1681—611no—
327#1677—611no—
328#1677—611no—
329#1565—611no—
330#1550—611no—
331#1545—611no—
332C2×C61—81no—
333C2×C61—161no—
334C2×C61C2×#46311yescharacteristic
335C2×C61#305511yescharacteristic
336C2×C61#306811yescharacteristic
337D61—22no—
338D61—42no—
339D61—161no—
340C22×D8—1221no—
341C22×M16—1221no—
342#533—611no—
343C2×#127—611no—
344C2×C4×D4—611no—
345#490—1221no—
346#490—1221no—
347#486—611no—
348#486—611no—
349#486—611no—
350#486C2×D6111yescharacteristic, complemented
351#483—1221no—
352#476—611no—
353#476C2×D6111yescharacteristic, complemented
354#461—611no—
355#461—611no—
356#459—611no—
357#459—611no—
358#459—611no—
359#459—1221no—
360#459—1221no—
361#459C2×D6111yescharacteristic
362#450—1221no—
363C4×SD16—1221no—
364C4×SD16—1221no—
365#436—611no—
366#436C2×D6111yescharacteristic, complemented
367#435—1221no—
368#435—1221no—
369C8×D4—611no—
370C8×D4—611no—
371#431—611no—
372#423—611no—
373C2×C4⋊C8—611no—
374#418—1221no—
375#418—1221no—
376C2×#102—1221no—
377#407—1221no—
378C4×M16—611no—
379#335—611no—
380#335C2×D6111yescharacteristic, complemented
381#334—611no—
382#334—611no—
383#333—611no—
384#333C2×D6111yescharacteristic, complemented
385#332—611no—
386#332—611no—
387#331—611no—
388#331—611no—
389#331—611no—
390#331C2×D6111yescharacteristic, complemented
391C61#1916611yescharacteristic, complemented, Sylow
392C22×D4—611no—
393C22×D4—1221no—
394C22×D4—1221no—
395C22×D4—1221no—
396C23×C4—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×D8—21no—
400C2×D8—1221no—
401C2×D8—1221no—
402C2×D8—1221no—
403C2×D8—2444no—
404C2×M16—21no—
405C2×M16—611no—
406C2×M16—611no—
407C2×M16—1221no—
408C2×M16—1221no—
409C2×M16—2444no—
410C22×C8—1221no—
411C22×C8—1221no—
412C22×C8—1221no—
413C4⋊Q8—611no—
414C4⋊Q8C22×D6111yescharacteristic
415#127—611no—
416#127—1221no—
417#127—1221no—
418#127—1221no—
419#127C22×D6111yescharacteristic
420#125—1221no—
421C22⋊Q8—1221no—
422C22⋊Q8—1221no—
423C4×Q8—1221no—
424C4×Q8—1221no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—611no—
428C4×D4—1221no—
429C4×D4—1221no—
430C4×D4C22×D6111yescharacteristic
431#117—1221no—
432C2×C4⋊C4—611no—
433C2×C22⋊C4—1221no—
434C2×C42—611no—
435#106—21no—
436#106—1221no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—1221no—
441C4⋊C8—1221no—
442C4⋊C8C22×D6111yescharacteristic
443C4⋊C8C22×D6111yescharacteristic
444C4⋊C8C22×D6111yescharacteristic
445#103—1221no—
446#103—1221no—
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—
459#102—1221no—
460#102—1221no—
461C22⋊C8—21no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464C22⋊C8—1221no—
465C22⋊C8—1221no—
466#97—611no—
467#97—611no—
468C4×C8—611no—
469C4×C8—1221no—
470C4×C8C22×D6111yescharacteristic
471C24—1221no—
472C24—2441no—
473C2×Q8—21no—
474C2×Q8—1221no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—611no—
479C2×D4—1221no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—4882no—
485C2×D4—2442no—
486C2×D4—2442no—
487C2×D4—2441no—
488C2×D4—2441no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4—2442no—
492C2×D4—2442no—
493C2×D4D4×D6111yescharacteristic
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—2442no—
503C22×C4—2442no—
504SD16—2442no—
505SD16—2442no—
506D8—42no—
507D8—2442no—
508D8—2442no—
509D8—2442no—
510M16—42no—
511M16—1222no—
512M16—1222no—
513M16—2442no—
514M16—2442no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—611no—
519C2×C8—611no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—2442no—
524C2×C8—2442no—
525C2×C8—2442no—
526C2×C8D4×D6111yescharacteristic
527C2×C8D4×D6111yescharacteristic
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—611no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2441no—
535C4⋊C4—2441no—
536C4⋊C4D4×D6111yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—2441no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—2441no—
546C42C23×D6111yescharacteristic
547C23—21no—
548C23—41no—
549C23—611no—
550C23—1221no—
551C23—4881no—
552C23—4882no—
553C23—4882no—
554C23—4882no—
555C23—2442no—
556C23—2442no—
557C23—2441no—
558Q8—81no—
559Q8—2441no—
560Q8—2441no—
561D4—21no—
562D4—21no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—81no—
568D4—612no—
569D4—1221no—
570D4—4881no—
571D4—4882no—
572D4—4882no—
573D4—4882no—
574D4—2442no—
575D4—2441no—
576D4—2441no—
577D4—2441no—
578D4—2441no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—611no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C8—21no—
602C8—21no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8—1221no—
607C8—1221no—
608C8—2441no—
609C8—2441no—
610C8—2441no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—612no—
618C22—1221no—
619C22—1221no—
620C22—1222no—
621C22—9764no—
622C22—4884no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2441no—
627C22—2441no—
628C22#121×D6111yescharacteristic
629C4—21no—
630C4—41no—
631C4—81no—
632C4—612no—
633C4—1221no—
634C4—1221no—
635C4—4881no—
636C4—2441no—
637C4#135×D6111yescharacteristic
638C4AGL(1,ℤ/8)×D6111yescharacteristic
639C4AGL(1,ℤ/8)×D6111yescharacteristic
640C4#1684056111yescharacteristic
641C2—81no—
642C2—161no—
643C2—1222no—
644C2—9761no—
645C2—2442no—
646C2#463×D6111yescharacteristic
647C2#2145543311yescharacteristic
648C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1697C2
socleC22×C61C2×#121
prime radical#459×D61C2
hypercenter#1697D61
hypocenterC61#19166
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1697

Complements

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

NH ≅ G/Ntype
1. #21456190C2semidirect
2. #21455829C2semidirect
4. #21454555C2semidirect
5. #21445380C2semidirect
6. #21445378C2semidirect
7. #21445354C2semidirect
8. #21445257C2semidirect
9. #21445255C2semidirect
10. #21445231C2semidirect
11. #21445225C2semidirect
12. #21444570C2semidirect
13. #21444506C2semidirect
14. #21444485C2semidirect
15. C61×#1697C2semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
25. #16840108C22semidirect
27. #16840049C22semidirect
28. #16839968C22semidirect
29. #16839967C22semidirect
38. #16839840C22semidirect
41. C8×D244C22semidirect
42. #16839827C22semidirect
51. #16839514C22semidirect
52. #16839513C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839507C22semidirect
57. #16839506C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#486C22semidirect
61. C61×#476C22semidirect
63. C61×#436C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#331C22semidirect
324. #1697D61semidirect
350. #486C2×D61semidirect
353. #476C2×D61semidirect
366. #436C2×D61semidirect
380. #335C2×D61semidirect
384. #333C2×D61semidirect
390. #331C2×D61semidirect
391. C61#19166semidirect

Maximal subgroups

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

subgroupindex
#214561902
#214558292
#459×D612
#214545552
#214453802
#214453782
#214453542
#214452572
#214452552
#214452312
#214452252
#214445702
#214445062
#214444852
C61×#16972
#1916661