Finite Group Databasebeta

← #29669849

Subgroups — #29669849

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457637C211yesmaximal, characteristic, complemented
2#21457634C211yesmaximal, characteristic, complemented
3#21457549C211yesmaximal, characteristic, complemented
4#21445455C211yesmaximal, characteristic, complemented
5#21445387C211yesmaximal, characteristic, complemented
6#21445230C211yesmaximal, characteristic, complemented
7#21445225C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445033C211yesmaximal, characteristic, complemented
10#21444969C211yesmaximal, characteristic, complemented
11#21444884C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic, complemented
13#21444653C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3366C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840417C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840138C2211yescharacteristic
20#16840134C2211yescharacteristic, complemented
21#16840112C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840080—21no—
25#16840077—21no—
26#16840049C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16839997—21no—
29#16839993—21no—
30#16839989—21no—
31#16839972C2211yescharacteristic, complemented
32#16839962C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839924—21no—
36#16839912—21no—
37#16839905—21no—
38#16839894—21no—
39C4×#11758201C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519697—21no—
44#102×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic
57C61×#550C2211yescharacteristic, complemented
58C61×#490—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#478—21no—
61C61×#476C2211yescharacteristic, complemented
62C61×#459—21no—
63C61×#459C2211yescharacteristic
64C61×#446C2211yescharacteristic, complemented
65C61×#442—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519797—44no—
71C2×#11758202—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201D411yescharacteristic
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758199—41no—
79C22×#92418—21no—
80C22×#92418—21no—
81C2×C8×D61—21no—
82#14519777C2311yescharacteristic
83#14519774C2311yescharacteristic
84#14519759—41no—
85#14519756—21no—
86#14519752—21no—
87#14519751—21no—
88#14519748—21no—
89#14519747—41no—
90C4×D244C2311yescharacteristic
91C4⋊C4×D61—21no—
92#14519710—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—41no—
99#14519701D411yescharacteristic
100#14519700—21no—
101#14519699—21no—
102#14519699—21no—
103#14519698—21no—
104#14519698—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×D8×C61—21no—
118C2×D8×C61D411yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61—41no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163D4×D61—42no—
164D4×D61—81no—
165#11758222—42no—
166#11758222—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×C4×D61—41no—
176C2×Dic122—21no—
177D8×C61—22no—
178D8×C61—42no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—41no—
193C22⋊C4×C61—41no—
194C42×C61—21no—
195C42×C61C2411yescharacteristic
196C22⋊Dic61—41no—
197#11758202—42no—
198#11758201—22no—
199#11758201—81no—
200#11758200—42no—
201#11758200—81no—
202#11758199—42no—
203#11758199—81no—
204#11758199—82no—
205#11758198—21no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61C2×D411yescharacteristic
210#11758196—41no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—42no—
216C2×#92418—81no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C8×D61—81no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—41no—
223C22×D61—21no—
224C22×D61—41no—
225C22×D61—82no—
226C22×D61—82no—
227C22×D61—161no—
228Q8×C61—21no—
229Q8×C61—21no—
230Q8×C61—81no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61—82no—
240D4×C61—161no—
241D4×C61#13512yescomplemented
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—82no—
252C2×C4×C61—161no—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C22⋊D61—82no—
257C22⋊D61—82no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260D244—21no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—81no—
266D244—82no—
267D244AGL(1,ℤ/8)12yes—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—82no—
271Dic122—41no—
272Dic122—41no—
273C8×C61—21no—
274C8×C61—21no—
275C8×C61—41no—
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#19788—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61D4211yescharacteristic
293C2×D61—22no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—164no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61C2×#13511yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61#57611yescharacteristic
309C4×C61#57711yescharacteristic
310Dic61—41no—
311Dic61—81no—
312#3366D6111yescharacteristic, complemented
313#3295—611no—
314#3294—611no—
315#3284—611no—
316#1717—611no—
317#1702—611no—
318#1677—611no—
319#1677—611no—
320#1665—611no—
321#1644—611no—
322#1635—611no—
323#1621—611no—
324#1619—611no—
325#1582—611no—
326#1580—611no—
327C2×C61—42no—
328C2×C61—81no—
329C2×C61—161no—
330C2×C61#326411yescharacteristic
331C2×C61#328211yescharacteristic
332C2×C61#328811yescharacteristic
333D61—42no—
334D61—161no—
335C2×#135—1221no—
336#554C2×D6111yescharacteristic
337#550C2×D6111yescharacteristic, complemented
338#547—611no—
339D42—611no—
340#493—611no—
341#492—611no—
342#490—21no—
343#487—611no—
344#486—611no—
345#486—611no—
346#486C2×D6111yescharacteristic, complemented
347#481—1221no—
348#480—1221no—
349#478—21no—
350#476—611no—
351#476C2×D6111yescharacteristic, complemented
352#474—611no—
353#466—1221no—
354#465—1221no—
355#465—1221no—
356#462—611no—
357#460—611no—
358#459—21no—
359#459—611no—
360#459—611no—
361#459C2×D6111yescharacteristic
362#452—1221no—
363#450—1221no—
364#449—1221no—
365#447—1221no—
366#446C2×D6111yescharacteristic, complemented
367#442—21no—
368C4×SD16—611no—
369C4×D8C2×D6111yescharacteristic
370C8×D4—611no—
371C8×D4—611no—
372C8×D4—611no—
373C2×#102—1221no—
374C2×#102—1221no—
375#334—611no—
376#334—611no—
377#333—611no—
378#332—611no—
379#331—611no—
380#331—611no—
381Q8⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384D4⋊C8—611no—
385#322—611no—
386C61#1978811yescharacteristic, complemented, Sylow
387C2×#41—21no—
388C2×#41—1221no—
389C22×D4—1221no—
390C22×D4—1221no—
391#135—2444no—
392C2×Q16—1221no—
393C2×SD16—611no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×D8—21no—
398C2×D8—1221no—
399C2×D8—1221no—
400C2×D8—2441no—
401C2×D8C22⋊D6111yescharacteristic
402C22×C8—1221no—
403C22×C8—1221no—
404C22×C8—1221no—
405C4⋊Q8—611no—
406C4⋊Q8C22×D6111yescharacteristic
407#127—21no—
408#127—611no—
409#127C22×D6111yescharacteristic
410#125—21no—
411#123—2441no—
412C22⋊Q8—21no—
413C22⋊Q8—41no—
414C22⋊Q8—1221no—
415#121—21no—
416#121—41no—
417#121—1221no—
418#121—1221no—
419#121—1221no—
420#120—2441no—
421C4×Q8C22×D6111yescharacteristic
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—611no—
425C4×D4C22×D6111yescharacteristic
426C4×D4C22×D6111yescharacteristic
427C2×C4⋊C4—21no—
428C2×C4⋊C4—1221no—
429#107—1221no—
430#107C22⋊D6111yescharacteristic
431#106—21no—
432#106—611no—
433#106—1221no—
434C4⋊C8—21no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8C22×D6111yescharacteristic
439#103—611no—
440#103—1221no—
441#103—2441no—
442#102—21no—
443#102—21no—
444#102—21no—
445#102—21no—
446#102—611no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102C22⋊D6111yescharacteristic
457#102C22⋊D6111yescharacteristic
458C22⋊C8—1221no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461#97—21no—
462C4×C8—611no—
463C4×C8—611no—
464C4×C8C22×D6111yescharacteristic
465C24—2441no—
466#41—42no—
467#41—81no—
468#41—4881no—
469#41—2442no—
470C2×Q8—21no—
471C2×Q8—1221no—
472C2×Q8D4×D6111yescharacteristic
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—611no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—4881no—
483C2×D4—4881no—
484C2×D4—2441no—
485C2×D4—2442no—
486C2×D4—2441no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4C2×C22⋊D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—1221no—
496C22×C4—2441no—
497Q16—2442no—
498SD16—1222no—
499SD16—4881no—
500SD16—4881no—
501SD16—2442no—
502D8—22no—
503D8—42no—
504D8—4881no—
505D8—4882no—
506D8—2442no—
507C2×C8—21no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—4881no—
515C2×C8—4881no—
516C2×C8—2442no—
517C2×C8C2×C22⋊D6111yescharacteristic
518C2×C8C2×C22⋊D6111yescharacteristic
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—611no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—2441no—
528C4⋊C4D4×D6111yescharacteristic
529C4⋊C4C2×C22⋊D6111yescharacteristic
530C4⋊C4C2×C22⋊D6111yescharacteristic
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—1221no—
536C22⋊C4—2441no—
537C22⋊C4—2441no—
538C42—21no—
539C42C23×D6111yescharacteristic
540C23—21no—
541C23—21no—
542C23—41no—
543C23—1221no—
544C23—9761no—
545C23—4882no—
546C23—4882no—
547C23—2441no—
548Q8—21no—
549Q8—21no—
550Q8—81no—
551Q8—2441no—
552Q8—2441no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—81no—
561D4—82no—
562D4—161no—
563D4—612no—
564D4—1221no—
565D4—4881no—
566D4—4882no—
567D4—4882no—
568D4—4882no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4#1451979712yescomplemented
574C2×C4—21no—
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—82no—
584C2×C4—161no—
585C2×C4—1221no—
586C2×C4—4882no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4C22×C22⋊D6111yescharacteristic
592C2×C4#1451982211yescharacteristic
593C8—21no—
594C8—21no—
595C8—41no—
596C8—1221no—
597C8—1221no—
598C8—2441no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—1222no—
612C22—9764no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—2441no—
617C22—2441no—
618C22D4×C22⋊D6111yescharacteristic
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—4881no—
626C4—2441no—
627C4C2×#1451979711yescharacteristic
628C4AGL(1,ℤ/8)×D6111yescharacteristic
629C4#1684056611yescharacteristic
630C4#1684057011yescharacteristic
631C2—42no—
632C2—81no—
633C2—161no—
634C2—9761no—
635C2—2442no—
636C2#2145729311yescharacteristic
637C2#2145752711yescharacteristic
638C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3366C2
socleC22×C61D42
hypercenter#3366D61
hypocenterC61#19788
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3366

Complements

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

NH ≅ G/Ntype
1. #21457637C2semidirect
2. #21457634C2semidirect
3. #21457549C2semidirect
4. #21445455C2semidirect
5. #21445387C2semidirect
6. #21445230C2semidirect
7. #21445225C2semidirect
8. #21445158C2semidirect
9. #21445033C2semidirect
10. #21444969C2semidirect
11. #21444884C2semidirect
12. #21444872C2semidirect
13. #21444653C2semidirect
14. #21444643C2semidirect
15. C61×#3366C2semidirect
20. #16840134C22semidirect
21. #16840112C22semidirect
31. #16839972C22semidirect
42. D4×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839483C22semidirect
57. C61×#550C22semidirect
59. C61×#486C22semidirect
61. C61×#476C22semidirect
64. C61×#446C22semidirect
241. D4×C61#135semidirect
312. #3366D61semidirect
337. #550C2×D61semidirect
346. #486C2×D61semidirect
351. #476C2×D61semidirect
366. #446C2×D61semidirect
386. C61#19788semidirect
573. D4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214576372
#214576342
#214575492
#214454552
#214453872
#214452302
#214452252
#214451582
#214450332
#214449692
#214448842
#214448722
#214446532
#214446432
C61×#33662
#1978861