Finite Group Databasebeta

← #29671381

Subgroups — #29671381

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458796C211yesmaximal, characteristic, complemented
2D4×#11758201C211yesmaximal, characteristic, complemented
3#21457440C211yesmaximal, characteristic
4#21445412C211yesmaximal, characteristic, complemented
5#21445329C211yesmaximal, characteristic, complemented
6#21445303C211yesmaximal, characteristic, complemented
7#21445169C211yesmaximal, characteristic, complemented
8#21445162C211yesmaximal, characteristic, complemented
9#21445018C211yesmaximal, characteristic, complemented
10#21444997C211yesmaximal, characteristic, complemented
11#21444922C211yesmaximal, characteristic, complemented
12#21444915C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3386C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C22×#11758201—21no—
18#16840434C2211yescharacteristic
19D4×D244C2211yescharacteristic
20#16840139C2211yescharacteristic
21#16840136—21no—
22#16840136C2211yescharacteristic
23#16840136C2211yescharacteristic, complemented
24#16840110C2211yescharacteristic
25#16840110C2211yescharacteristic, complemented
26#16840016—21no—
27#16839993—21no—
28#16839964—21no—
29#16839964C2211yescharacteristic
30#16839964C2211yescharacteristic
31#16839962—21no—
32#16839962C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic, complemented
35#16839957C2211yescharacteristic
36#16839912—21no—
37#16839910—21no—
38#16839901—21no—
39#16839897—21no—
40Q8×#92418C2211yescharacteristic, complemented
41C4×#11758201—21no—
42C4×#11758201C2211yescharacteristic
43#16839841C2211yescharacteristic
44D4×#92418C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839505C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#557C2211yescharacteristic, complemented
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#500C2211yescharacteristic, complemented
60C61×#478—21no—
61C61×#477—21no—
62C61×#475C2211yescharacteristic, complemented
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic, complemented
65C61×#440—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758226—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519804—44no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—21no—
80C2×#11758201—44no—
81C2×#11758201D411yescharacteristic
82C2×D488—21no—
83C2×D488—21no—
84C2×#11758199—21no—
85C2×#11758189—21no—
86C22×#92418—21no—
87#14519774—21no—
88#14519774C2311yescharacteristic
89#14519774C2311yescharacteristic
90#14519752—21no—
91#14519752—21no—
92#14519752—41no—
93#14519751—21no—
94#14519747—41no—
95C4×D244—21no—
96C4×D244C2311yescharacteristic
97#14519708D411yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519701—21no—
104#14519701D411yescharacteristic
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697—41no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C22×Q8×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#125×C61—21no—
124#125×C61—41no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#106×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61—21no—
143#102×C61D411yescharacteristic
144#97×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C2×C22⋊D61—41no—
158#11758226—42no—
159#11758226—81no—
160D4×D61—42no—
161D4×D61—81no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—42no—
169C2×D244—41no—
170C2×D244—41no—
171C2×D244—41no—
172C2×D244—81no—
173C2×D244C2×D411yescharacteristic
174C2×C4×D61—21no—
175C2×C4×D61—41no—
176SD16×C61SD1614yescomplemented
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—41no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—41no—
198C42×C61C2411yescharacteristic
199#11758201—22no—
200#11758201—42no—
201#11758201—42no—
202#11758201—42no—
203#11758201—42no—
204#11758201—42no—
205#11758201—81no—
206#11758201—81no—
207#11758201—81no—
208#11758201—81no—
209#11758199—42no—
210#11758198—21no—
211#11758198—41no—
212#11758198—41no—
213C4⋊Dic61C2×D411yescharacteristic
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—42no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221D488—42no—
222D488—81no—
223#11758189—42no—
224C23×C61—21no—
225C22×D61—21no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—161no—
231Q8×C61—21no—
232Q8×C61—21no—
233Q8×C61—21no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61—82no—
239Q8×C61C2×SD1612yescomplemented
240D4×C61—21no—
241D4×C61C2×SD1612yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—82no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C22⋊D61—82no—
256C2×Dic61—21no—
257D244—21no—
258D244—21no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—41no—
267D244—82no—
268D244—81no—
269D244—81no—
270D244—82no—
271D244—161no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—82no—
275C8×C61—21no—
276C8×C61—41no—
277C8×C61C2×SD1612yescomplemented
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418AGL(1,ℤ/8)12yescomplemented
284#19892—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61D4211yescharacteristic
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—164no—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61C22×SD1611yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61#57611yescharacteristic
310C4×C61#57711yescharacteristic
311Dic61—41no—
312#3392—611no—
313#3386D6111yescharacteristic, complemented
314D4×SD16—611no—
315#3274—611no—
316#1706—611no—
317#1693—611no—
318#1688—611no—
319#1666—611no—
320#1665—611no—
321#1642—611no—
322#1639—611no—
323#1627—611no—
324#1626—611no—
325#1590—611no—
326C8×SD16—611no—
327C2×C61—42no—
328C2×C61#327411yescharacteristic
329C2×C61D4×SD1611yescharacteristic
330C2×C61#329011yescharacteristic
331D61—81no—
332D61—161no—
333D61—161no—
334C2×AGL(1,ℤ/8)—1221no—
335C22×SD16—1221no—
336#557C2×D6111yescharacteristic, complemented
337#550—611no—
338D4×Q8C2×D6111yescharacteristic
339D42—611no—
340#501—21no—
341#500C2×D6111yescharacteristic, complemented
342#493—611no—
343#492—611no—
344#492—611no—
345#492—1221no—
346#486—611no—
347#486—611no—
348#478—21no—
349#477—21no—
350#475C2×D6111yescharacteristic, complemented
351#474C2×D6111yescharacteristic
352#469—1221no—
353#465—1221no—
354#461—611no—
355#461—611no—
356#461—1221no—
357#460—611no—
358#460—611no—
359#460—611no—
360#460—1221no—
361#459—611no—
362#450—1221no—
363#449—1221no—
364#448—1221no—
365#447—1221no—
366#446C2×D6111yescharacteristic, complemented
367C8×Q8—611no—
368#440—21no—
369C4×SD16—611no—
370C4×SD16—1221no—
371C4×SD16C2×D6111yescharacteristic
372#436—611no—
373C8×D4—611no—
374#334—611no—
375#334—611no—
376#332—611no—
377#332—611no—
378#331—611no—
379#331—611no—
380Q8⋊C8—611no—
381Q8⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384C82—611no—
385C61#1989211yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8—21no—
388C22×D4—1221no—
389C22×D4—1221no—
390AGL(1,ℤ/8)—2444no—
391C2×SD16—611no—
392C2×SD16—1221no—
393C2×SD16—1221no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—2444no—
402C2×SD16C22⋊D6111yescharacteristic
403C2×D8—1221no—
404C2×D8—1221no—
405C2×D8—1221no—
406C2×M16—1221no—
407C22×C8—1221no—
408C4⋊Q8—21no—
409C4⋊Q8—21no—
410C4⋊Q8C22×D6111yescharacteristic
411C4⋊Q8C22×D6111yescharacteristic
412#127—611no—
413#127—611no—
414#127—1221no—
415#125—21no—
416#125—41no—
417C22⋊Q8—21no—
418C22⋊Q8—41no—
419#121—1221no—
420#121—1221no—
421#121—1221no—
422#121—2441no—
423#120—2441no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8C22×D6111yescharacteristic
427C4×Q8C22×D6111yescharacteristic
428C4×D4—21no—
429C4×D4—611no—
430C4×D4—1221no—
431C4×D4C22×D6111yescharacteristic
432#107—21no—
433#107—21no—
434#107—21no—
435#106—21no—
436#106—611no—
437#106C22⋊D6111yescharacteristic
438C4⋊C8—21no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—1221no—
443C4⋊C8C22×D6111yescharacteristic
444#103—21no—
445#103—611no—
446#103—1221no—
447#103—1221no—
448#103C22⋊D6111yescharacteristic
449#102—21no—
450#102—611no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—2441no—
456#102—2441no—
457#102C22⋊D6111yescharacteristic
458C22⋊C8—1221no—
459C22⋊C8—1221no—
460#97—21no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C24—2441no—
466#41—4881no—
467#41—2442no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—41no—
471C2×Q8—42no—
472C2×Q8—81no—
473C2×Q8D4×D6111yescharacteristic
474C2×Q8C2×C22⋊D6111yescharacteristic
475C2×D4—611no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—4881no—
483C2×D4—4881no—
484C2×D4—2441no—
485C2×D4—2441no—
486C2×D4—2441no—
487C2×D4—2442no—
488C2×D4—2441no—
489C2×D4—2442no—
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—1221no—
494C22×C4—2441no—
495SD16—1222no—
496SD16—4881no—
497SD16—4881no—
498SD16—4881no—
499SD16—4881no—
500SD16—2442no—
501SD16—2442no—
502SD16—2442no—
503SD16—2442no—
504SD16—2442no—
505SD16#1175820114yescomplemented
506D8—4881no—
507D8—2442no—
508D8—2442no—
509M16—2442no—
510C2×C8—21no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—2442no—
518C2×C8C2×C22⋊D6111yescharacteristic
519C2×C8C2×C22⋊D6111yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—611no—
531C4⋊C4D4×D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C22⋊C4—21no—
535C22⋊C4—41no—
536C22⋊C4—1221no—
537C22⋊C4—2441no—
538C22⋊C4—2441no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—1221no—
545C23—9761no—
546C23—4882no—
547C23—4882no—
548C23—2441no—
549C23—2441no—
550Q8—21no—
551Q8—21no—
552Q8—21no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—82no—
558Q8C2×#1175820112yescomplemented
559D4—21no—
560D4—1221no—
561D4—1221no—
562D4—9761no—
563D4—4881no—
564D4—4881no—
565D4—4882no—
566D4—4882no—
567D4—4882no—
568D4—2441no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575D4—2441no—
576D4C2×#1175820112yes—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
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×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—21no—
595C8—41no—
596C8—612no—
597C8—1221no—
598C8—2441no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C8C2×#1175820112yescomplemented
603C22—22no—
604C22—41no—
605C22—41no—
606C22—9764no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22D4×C22⋊D6111yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—2441no—
626C4C22×#1175820111yescharacteristic
627C4AGL(1,ℤ/8)×D6111yescharacteristic
628C4#1684056611yescharacteristic
629C4#1684057011yescharacteristic
630C2—42no—
631C2—9761no—
632C2—9761no—
633C2—4881no—
634C2#2145743311yescharacteristic
635C2D4×#1175820111yescharacteristic
636C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3386C2
socleC22×C61D42
prime radical#21457440C2
hypercenter#3386D61
hypocenterC61#19892
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3386

Complements

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

NH ≅ G/Ntype
1. #21458796C2semidirect
2. D4×#11758201C2semidirect
4. #21445412C2semidirect
5. #21445329C2semidirect
6. #21445303C2semidirect
7. #21445169C2semidirect
8. #21445162C2semidirect
9. #21445018C2semidirect
10. #21444997C2semidirect
11. #21444922C2semidirect
12. #21444915C2semidirect
13. #21444709C2semidirect
14. SD16×#92418C2semidirect
15. C61×#3386C2semidirect
23. #16840136C22semidirect
25. #16840110C22semidirect
34. #16839962C22semidirect
40. Q8×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. C61×#557C22semidirect
59. C61×#500C22semidirect
62. C61×#475C22semidirect
64. C61×#446C22semidirect
176. SD16×C61SD16semidirect
239. Q8×C61C2×SD16semidirect
277. C8×C61C2×SD16semidirect
283. #92418AGL(1,ℤ/8)semidirect
313. #3386D61semidirect
336. #557C2×D61semidirect
341. #500C2×D61semidirect
350. #475C2×D61semidirect
366. #446C2×D61semidirect
385. C61#19892semidirect
505. SD16#11758201semidirect
558. Q8C2×#11758201semidirect
602. C8C2×#11758201semidirect

Maximal subgroups

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

subgroupindex
#214587962
D4×#117582012
#214574402
#214454122
#214453292
#214453032
#214451692
#214451622
#214450182
#214449972
#214449222
#214449152
#214447092
SD16×#924182
C61×#33862
#1989261