Finite Group Databasebeta

← #29677204

Subgroups — #29677204

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458408C211yesmaximal, characteristic, complemented
2#21457420C211yesmaximal, characteristic, complemented
3#21457419C211yesmaximal, characteristic, complemented
4#21445407C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445315C211yesmaximal, characteristic, complemented
7#21445286C211yesmaximal, characteristic, complemented
8#21445284C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445102C211yesmaximal, characteristic, complemented
11#21444934C211yesmaximal, characteristic, complemented
12#21444902C211yesmaximal, characteristic, complemented
13#21444704C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3354C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17C2×#14519801—21no—
18#16840416C2211yescharacteristic
19D4×D244C2211yescharacteristic
20#16840158C2211yescharacteristic, complemented
21#16840158C2211yescharacteristic, complemented
22#16840136C2211yescharacteristic
23#16840136C2211yescharacteristic
24#16840135C2211yescharacteristic
25#16840086—21no—
26#16840078—21no—
27#16840049C2211yescharacteristic
28#16840042C2211yescharacteristic, complemented
29#16840013—21no—
30#16840010—21no—
31#16840009—21no—
32#16839967C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839914—21no—
37#16839912—21no—
38#16839908—21no—
39#16839907—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic
42#16839840C2211yescharacteristic
43#16839840C2211yescharacteristic, complemented
44#16839805—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#555C2211yescharacteristic, complemented
57C61×#547C2211yescharacteristic
58C61×#498C2211yescharacteristic, complemented
59C61×#482—21no—
60C61×#481—21no—
61C61×#476C2211yescharacteristic
62C61×#474C2211yescharacteristic, complemented
63C61×#469—21no—
64C4×D8×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic, complemented
66C61×#428—21no—
67C2×#11758224—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519802—44no—
71#14519801—44no—
72C2×#11758190—21no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201—41no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199D411yescharacteristic
82C2×#11758194—21no—
83C2×#11758194—21no—
84C2×#11758189—21no—
85#14519777C2311yescharacteristic
86#14519774C2311yescharacteristic
87#14519761—41no—
88#14519757—21no—
89#14519752—21no—
90#14519751—21no—
91#14519751—21no—
92#14519747—41no—
93C4×D244C2311yescharacteristic
94C2×C4⋊Dic61—21no—
95#14519710—21no—
96#14519710D411yescharacteristic
97#14519708—21no—
98#14519708—21no—
99#14519708—21no—
100#14519708—41no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519701—21no—
105#14519700—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697—21no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113C22⋊#92418—21no—
114#14519689—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C2×#41×C61—21no—
118C2×D8×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#124×C61—41no—
123#123×C61—41no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133C2×C4⋊C4×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136#106×C61—21no—
137#106×C61—21no—
138#106×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147#41×C61—42no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C2×C22⋊D61—41no—
161C2×C22⋊D61—41no—
162C22×Dic61—41no—
163#11758224—42no—
164#11758224—81no—
165D4×D61—42no—
166D4×D61—81no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—42no—
171C2×D244—41no—
172C2×D244—81no—
173C2×D244C2×D411yescharacteristic
174C2×C4×D61—21no—
175C2×Dic122—21no—
176D8×C61—22no—
177M16×C61—42no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—41no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—41no—
195C42×C61—41no—
196C42×C61C2411yescharacteristic
197#11758201—42no—
198#11758201—42no—
199#11758201—81no—
200#11758201—82no—
201#11758200—42no—
202#11758199—22no—
203#11758199—42no—
204#11758199—42no—
205#11758198—21no—
206#11758198—41no—
207#11758198—41no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—41no—
211C4⋊Dic61C2×D411yescharacteristic
212#11758194—42no—
213#11758194—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220#11758190—42no—
221#11758190—81no—
222#11758189—42no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C22×D61—21no—
227C22×D61—41no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—161no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61—81no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—21no—
240D4×C61—82no—
241D4×C61—161no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
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×C61—41no—
253C2×C4×C61—82no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C22⋊D61—82no—
258C22⋊D61—82no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—82no—
262D244—21no—
263D244—21no—
264D244—41no—
265D244—41no—
266D244—41no—
267D244—41no—
268D244—81no—
269D244—82no—
270C4×D61—41no—
271C4×D61—41no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—21no—
275C8×C61—41no—
276C8×C61AGL(1,ℤ/8)12yescomplemented
277#92418—22no—
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418AGL(1,ℤ/8)12yescomplemented
283#20314—611nomaximal, Sylow
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61D4211yescharacteristic
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—164no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61#57711yescharacteristic
310C4×C61#57711yescharacteristic
311Dic61—41no—
312Dic61—81no—
313#3354—611no—
314#3354D6111yescharacteristic, complemented
315#3272—611no—
316#3272—611no—
317#1705—611no—
318#1691—611no—
319#1691—611no—
320#1685—611no—
321#1685—611no—
322#1654—611no—
323#1654—611no—
324#1629—611no—
325#1624—611no—
326#1589—611no—
327#1589—611no—
328C2×C61—81no—
329C2×C61—81no—
330C2×C61—81no—
331C2×C61#327211yescharacteristic
332C2×C61#327211yescharacteristic
333C2×C61#329111yescharacteristic
334D61—81no—
335D61—161no—
336C2×AGL(1,ℤ/8)—1221no—
337C2×AGL(1,ℤ/8)—1221no—
338#555C2×D6111yescharacteristic, complemented
339#547—611no—
340#547C2×D6111yescharacteristic
341D42—611no—
342#498—611no—
343#498—611no—
344#498C2×D6111yescharacteristic, complemented
345#492—611no—
346#492—611no—
347#492—611no—
348#482—21no—
349#482—1221no—
350#481—21no—
351#481—1221no—
352#476—611no—
353#476C2×D6111yescharacteristic
354#474—611no—
355#474C2×D6111yescharacteristic, complemented
356#469—21no—
357#469—1221no—
358#468—1221no—
359#468—1221no—
360#461—611no—
361#461—611no—
362#459—611no—
363#459—611no—
364#450—1221no—
365#450—1221no—
366#449—1221no—
367#449—1221no—
368C4×D8—611no—
369C4×D8C2×D6111yescharacteristic
370#436—611no—
371#436—611no—
372#436—611no—
373#436C2×D6111yescharacteristic, complemented
374#428—21no—
375#428—1221no—
376#335—611no—
377#335—611no—
378#333—611no—
379#332—611no—
380#332—611no—
381#331—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386C82—611no—
387C61#2031411yescharacteristic, complemented, Sylow
388C2×#41—21no—
389C2×#41—1221no—
390C22×D4—1221no—
391C22×D4—1221no—
392AGL(1,ℤ/8)—2444no—
393AGL(1,ℤ/8)—2444no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—2441no—
401C2×D8—611no—
402C2×D8—1221no—
403C2×D8—1221no—
404C2×D8C22⋊D6111yescharacteristic
405C2×M16—21no—
406C2×M16—1221no—
407C2×M16—1221no—
408C2×M16—1221no—
409C4⋊Q8—611no—
410C4⋊Q8C22×D6111yescharacteristic
411C4⋊Q8C22×D6111yescharacteristic
412#127—611no—
413#124—41no—
414#123—41no—
415#123—2441no—
416C22⋊Q8—21no—
417C22⋊Q8—1221no—
418#121—21no—
419#121—21no—
420#121—21no—
421#121—1221no—
422#121—1221no—
423#121—1221no—
424#120—2441no—
425C4×D4—611no—
426C4×D4C22×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429#117—21no—
430#117—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—1221no—
433#107—21no—
434#107—611no—
435#107—1221no—
436#107C22⋊D6111yescharacteristic
437#106—21no—
438#106—21no—
439#106—21no—
440#106—1221no—
441#106—1221no—
442#106—1221no—
443#106—2441no—
444C4⋊C8—611no—
445C4⋊C8—611no—
446C4⋊C8—611no—
447C4⋊C8C22×D6111yescharacteristic
448#103—1221no—
449#103—1221no—
450#102—21no—
451#102—21no—
452#102—611no—
453#102—611no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102C22⋊D6111yescharacteristic
459#102C22⋊D6111yescharacteristic
460C22⋊C8—21no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8C22×D6111yescharacteristic
467C24—2441no—
468#41—42no—
469#41—81no—
470#41—4881no—
471#41—2442no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—1221no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—611no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—4881no—
482C2×D4—4881no—
483C2×D4—2441no—
484C2×D4—2442no—
485C2×D4—2441no—
486C2×D4—2442no—
487C2×D4—2441no—
488C2×D4D4×D6111yescharacteristic
489C2×D4C2×C22⋊D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4—1221no—
496C22×C4—2441no—
497SD16—4881no—
498SD16—4882no—
499SD16—4881no—
500SD16—2442no—
501SD16—2442no—
502SD16—2442no—
503SD16—2442no—
504D8—22no—
505D8—1222no—
506D8—2442no—
507D8—2442no—
508M16—42no—
509M16—2442no—
510M16—2442no—
511M16—2442no—
512C2×C8—21no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8C2×C22⋊D6111yescharacteristic
520C2×C8C2×C22⋊D6111yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—611no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—2441no—
530C4⋊C4D4×D6111yescharacteristic
531C4⋊C4C2×C22⋊D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—1221no—
540C22⋊C4—2441no—
541C22⋊C4—2441no—
542C42—41no—
543C42C23×D6111yescharacteristic
544C23—21no—
545C23—21no—
546C23—21no—
547C23—1221no—
548C23—9761no—
549C23—4882no—
550C23—4882no—
551C23—2441no—
552Q8—41no—
553Q8—41no—
554Q8—81no—
555Q8—2441no—
556Q8—2441no—
557D4—21no—
558D4—21no—
559D4—21no—
560D4—21no—
561D4—21no—
562D4—21no—
563D4—82no—
564D4—161no—
565D4—1221no—
566D4—1221no—
567D4—4881no—
568D4—4882no—
569D4—4882no—
570D4—4882no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—1221no—
588C2×C4—4882no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C22×C22⋊D6111yescharacteristic
594C2×C4#1451982211yescharacteristic
595C8—21no—
596C8—41no—
597C8—612no—
598C8—1222no—
599C8—1221no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C8#1451980212yescomplemented
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—9764no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22D4×C22⋊D6111yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—4881no—
628C4—2441no—
629C4AGL(1,ℤ/8)×D6111yescharacteristic
630C4C2×#1451980211yescharacteristic
631C4#1684057011yescharacteristic
632C4#1684057211yescharacteristic
633C2—81no—
634C2—81no—
635C2—81no—
636C2—9761no—
637C2—4881no—
638C2#2145741311yescharacteristic
639C2#2145741911yescharacteristic
640C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3354C2
socleC22×C61D42
hypercenter#3354D61
hypocenterC61#20314
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3354

Complements

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

NH ≅ G/Ntype
1. #21458408C2semidirect
2. #21457420C2semidirect
3. #21457419C2semidirect
4. #21445407C2semidirect
5. #21445320C2semidirect
6. #21445315C2semidirect
7. #21445286C2semidirect
8. #21445284C2semidirect
9. #21445103C2semidirect
10. #21445102C2semidirect
11. #21444934C2semidirect
12. #21444902C2semidirect
13. #21444704C2semidirect
14. #21444702C2semidirect
15. C61×#3354C2semidirect
20. #16840158C22semidirect
21. #16840158C22semidirect
28. #16840042C22semidirect
43. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
49. #16839508C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. C61×#555C22semidirect
58. C61×#498C22semidirect
62. C61×#474C22semidirect
65. C61×#436C22semidirect
276. C8×C61AGL(1,ℤ/8)semidirect
282. #92418AGL(1,ℤ/8)semidirect
314. #3354D61semidirect
338. #555C2×D61semidirect
344. #498C2×D61semidirect
355. #474C2×D61semidirect
373. #436C2×D61semidirect
387. C61#20314semidirect
603. C8#14519802semidirect

Maximal subgroups

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

subgroupindex
#214584082
#214574202
#214574192
#214454072
#214453202
#214453152
#214452862
#214452842
#214451032
#214451022
#214449342
#214449022
#214447042
#214447022
C61×#33542
#2031461