Finite Group Databasebeta

← #29662822

Subgroups — #29662822

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#21455826C211yesmaximal, characteristic, complemented
3#459×D61C211yesmaximal, characteristic
4D4×#11758189C211yesmaximal, characteristic, complemented
5#21445383C211yesmaximal, characteristic, complemented
6#21445377C211yesmaximal, characteristic, complemented
7#21445353C211yesmaximal, characteristic, complemented
8#21445351C211yesmaximal, characteristic, complemented
9#21445258C211yesmaximal, characteristic, complemented
10#21445228C211yesmaximal, characteristic, complemented
11#21445226C211yesmaximal, characteristic, complemented
12#21444570C211yesmaximal, characteristic, complemented
13#21444508C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1681C211yesmaximal, characteristic, complemented
16C2×D8×D61—21no—
17C22×#11758189—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#16840091—21no—
27#16840053C2211yescharacteristic, complemented
28#16840052C2211yescharacteristic, complemented
29#16839964C2211yescharacteristic, complemented
30#16839958C2211yescharacteristic
31#16839957C2211yescharacteristic
32#16839956—21no—
33#16839955—21no—
34#16839954C2211yescharacteristic
35#16839911—21no—
36SD16×Dic61—21no—
37#16839847—21no—
38#16839841C2211yescharacteristic, complemented
39#16839839C2211yescharacteristic, complemented
40#16839833—21no—
41#16839830—21no—
42D4×#92418C2211yescharacteristic, 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#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#461C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61D411yescharacteristic
71C22×C4×D61—21no—
72C2×#11758201—21no—
73C2×D488—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76D8×D61—44no—
77C2×#11758189—21no—
78C2×#11758189—21no—
79C2×#11758189—21no—
80C2×#11758189—44no—
81C2×#11758189D411yescharacteristic
82C2×#11758189D411yescharacteristic
83C22×#92418—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#14519709—21no—
105#14519708—21no—
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519701—21no—
112#14519699—21no—
113#14519699—21no—
114#14519699—21no—
115#14519699—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698—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×SD16×C61—21no—
132C2×D8×C61—21no—
133C22×C8×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×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—
179SD16×C61—42no—
180D8×C61—42no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—42no—
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—21no—
193#11758201—42no—
194#11758199—42no—
195#11758199—42no—
196#11758198—21no—
197#11758198—21no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205C4×Dic61—41no—
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61#4111yescharacteristic
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—42no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215D488—42no—
216#11758189—22no—
217#11758189—22no—
218#11758189—42no—
219#11758189—42no—
220#11758189—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—41no—
234Q8×C61—41no—
235D4×C61—21no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—81no—
241D4×C61AGL(1,ℤ/8)12yes—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C22⋊D61—42no—
250C22⋊D61—82no—
251C22⋊D61—82no—
252C22⋊D61—82no—
253C2×Dic61—21no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61#12111yescharacteristic
257C2×Dic61#12111yescharacteristic
258C2×Dic61#12111yescharacteristic
259D244—21no—
260D244—21no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—81no—
266C4×D61—21no—
267C4×D61—21no—
268C4×D61—41no—
269C4×D61—42no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—42no—
273C4×D61#13512yes—
274C4×D61AGL(1,ℤ/8)12yes—
275Dic122—81no—
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—22no—
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—41no—
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#1697—611no—
325#1681—611no—
326#1681D6111yescharacteristic, complemented
327#1677—611no—
328#1677—611no—
329#1565—611no—
330#1550—611no—
331#1545—611no—
332C2×C61—42no—
333C2×C61—161no—
334C2×C61C2×#46311yescharacteristic
335C2×C61#305511yescharacteristic
336C2×C61#306811yescharacteristic
337D61—22no—
338D61—81no—
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#476—611no—
354#461—611no—
355#461C2×D6111yescharacteristic, complemented
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#436—611no—
367#435—1221no—
368#435—1221no—
369C8×D4—611no—
370C8×D4C2×D6111yescharacteristic, complemented
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#333—611no—
385#332—611no—
386#332C2×D6111yescharacteristic, complemented
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—21no—
398C2×SD16—1221no—
399C2×D8—21no—
400C2×D8—1221no—
401C2×D8—1221no—
402C2×D8—1221no—
403C2×D8—2444no—
404C2×M16—611no—
405C2×M16—611no—
406C2×M16—1221no—
407C2×M16—1221no—
408C2×M16—1221no—
409C2×M16—2444no—
410C22×C8—21no—
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—1221no—
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—21no—
446#103—1221no—
447#102—21no—
448#102—21no—
449#102—21no—
450#102—1221no—
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—42no—
505SD16—2442no—
506D8—42no—
507D8—2442no—
508D8—2442no—
509D8—2442no—
510M16—1222no—
511M16—1222no—
512M16—2442no—
513M16—2442no—
514M16—2442no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—42no—
519C2×C8—611no—
520C2×C8—611no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
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—41no—
559Q8—41no—
560Q8—4881no—
561D4—21no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—81no—
567D4—1221no—
568D4—1221no—
569D4—4881no—
570D4—4882no—
571D4—4882no—
572D4—4882no—
573D4—2442no—
574D4—2441no—
575D4—2441no—
576D4—2441no—
577D4—2441no—
578D4#1451980112yes—
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—22no—
612C22—41no—
613C22—41no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—612no—
618C22—1221no—
619C22—1221no—
620C22—9764no—
621C22—4884no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—2441no—
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
639C4C2×#1451980111yescharacteristic
640C4#1684055911yescharacteristic
641C2—42no—
642C2—161no—
643C2—1222no—
644C2—9761no—
645C2—4881no—
646C2#463×D6111yescharacteristic
647C2#2145543511yescharacteristic
648C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1681C2
socleC22×C61C2×#121
prime radical#459×D61C2
hypercenter#1681D61
hypocenterC61#19166
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1681

Complements

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

NH ≅ G/Ntype
1. #21456190C2semidirect
2. #21455826C2semidirect
4. D4×#11758189C2semidirect
5. #21445383C2semidirect
6. #21445377C2semidirect
7. #21445353C2semidirect
8. #21445351C2semidirect
9. #21445258C2semidirect
10. #21445228C2semidirect
11. #21445226C2semidirect
12. #21444570C2semidirect
13. #21444508C2semidirect
14. #21444485C2semidirect
15. C61×#1681C2semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
25. #16840108C22semidirect
27. #16840053C22semidirect
28. #16840052C22semidirect
29. #16839964C22semidirect
38. #16839841C22semidirect
39. #16839839C22semidirect
42. D4×#92418C22semidirect
51. #16839514C22semidirect
52. #16839513C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839508C22semidirect
57. #16839506C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#486C22semidirect
61. C61×#461C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#335C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
326. #1681D61semidirect
350. #486C2×D61semidirect
355. #461C2×D61semidirect
370. C8×D4C2×D61semidirect
380. #335C2×D61semidirect
386. #332C2×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
#214558262
#459×D612
D4×#117581892
#214453832
#214453772
#214453532
#214453512
#214452582
#214452282
#214452262
#214445702
#214445082
#214444852
C61×#16812
#1916661