Finite Group Databasebeta

← #29670095

Subgroups — #29670095

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458490C211yesmaximal, characteristic, complemented
2D4×#11758201C211yesmaximal, characteristic, complemented
3#21457440C211yesmaximal, characteristic
4#21445448C211yesmaximal, characteristic, complemented
5#21445315C211yesmaximal, characteristic, complemented
6#21445230C211yesmaximal, characteristic, complemented
7#21445179C211yesmaximal, characteristic, complemented
8#21445169C211yesmaximal, characteristic, complemented
9#21445018C211yesmaximal, characteristic, complemented
10#21444934C211yesmaximal, characteristic, complemented
11#21444922C211yesmaximal, characteristic, complemented
12#21444897C211yesmaximal, characteristic, complemented
13#21444662C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C22×#11758201—21no—
18#16840434C2211yescharacteristic
19D4×D244C2211yescharacteristic
20#16840139C2211yescharacteristic
21#16840136C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840110—21no—
24#16840110C2211yescharacteristic, complemented
25#16840110C2211yescharacteristic, complemented
26#16840049—21no—
27#16840049C2211yescharacteristic
28#16840016—21no—
29#16839993—21no—
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#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839505C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56Q82×C61C2211yescharacteristic, complemented
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic, complemented
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×SD16×C61—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—41no—
78C2×#11758201—44no—
79C2×#11758201D411yescharacteristic
80C2×D488—21no—
81C2×D488—21no—
82C2×#11758199—21no—
83C2×#11758189—21no—
84C22×#92418—21no—
85#14519774—21no—
86#14519774C2311yescharacteristic
87#14519774C2311yescharacteristic
88#14519752—21no—
89#14519752—21no—
90#14519752—41no—
91#14519751—21no—
92#14519747—41no—
93C4×D244—21no—
94C4×D244C2311yescharacteristic
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
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×C61—21no—
122C4⋊Q8×C61—41no—
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
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#106×C61—21no—
134#106×C61—21no—
135#106×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
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—41no—
151C2×Q8×C61—42no—
152C2×Q8×C61—81no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C2×C22⋊D61—41no—
159#11758226—42no—
160#11758226—81no—
161D4×D61—42no—
162D4×D61—81no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—42no—
170C2×D244—41no—
171C2×D244—41no—
172C2×D244—41no—
173C2×D244—81no—
174C2×D244C2×D411yescharacteristic
175C2×C4×D61—21no—
176C2×C4×D61—41no—
177SD16×C61SD1614yescomplemented
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—21no—
186C4⋊C4×C61—21no—
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—82no—
207#11758199—42no—
208#11758198—21no—
209#11758198—41no—
210#11758198—41no—
211C4⋊Dic61C2×D411yescharacteristic
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—42no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219D488—42no—
220D488—81no—
221#11758189—42no—
222C23×C61—21no—
223C22×D61—21no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—82no—
227C22×D61—82no—
228C22×D61—161no—
229Q8×C61—21no—
230Q8×C61—21no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61—82no—
234Q8×C61—81no—
235Q8×C61—161no—
236Q8×C61C2×SD1612yescomplemented
237Q8×C61AGL(1,ℤ/8)12yescomplemented
238D4×C61—21no—
239D4×C61C2×SD1612yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
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—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C2×Dic61—21no—
255D244—21no—
256D244—21no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—82no—
266D244—81no—
267D244—81no—
268D244—82no—
269D244—161no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—82no—
273C8×C61—21no—
274C8×C61—41no—
275C8×C61C2×SD1612yescomplemented
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#19804—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61D4211yescharacteristic
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—164no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C22×SD1611yescharacteristic
306C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C61#57711yescharacteristic
308C4×C61#57711yescharacteristic
309Dic61—41no—
310Q8×SD16D6111yescharacteristic, complemented
311#3362—611no—
312D4×SD16—611no—
313#3274—611no—
314#1715—611no—
315#1691—611no—
316#1677—611no—
317#1668—611no—
318#1666—611no—
319#1642—611no—
320#1629—611no—
321#1627—611no—
322#1623—611no—
323#1583—611no—
324SD16⋊C8—611no—
325C2×C61—42no—
326C2×C61#326511yescharacteristic
327C2×C61D4×SD1611yescharacteristic
328C2×C61#328811yescharacteristic
329D61—81no—
330D61—161no—
331D61—161no—
332C2×AGL(1,ℤ/8)—1221no—
333C22×SD16—1221no—
334Q82C2×D6111yescharacteristic, complemented
335#550—611no—
336D4×Q8C2×D6111yescharacteristic
337D42—611no—
338#498—21no—
339#498C2×D6111yescharacteristic, complemented
340#493—611no—
341#492—611no—
342#492—611no—
343#486—611no—
344#486—611no—
345#486—1221no—
346#476—21no—
347#476—611no—
348#476—1221no—
349#476C2×D6111yescharacteristic
350#475—21no—
351#475C2×D6111yescharacteristic, complemented
352#469—1221no—
353#465—1221no—
354#461—611no—
355#460—611no—
356#460—611no—
357#460—611no—
358#460—1221no—
359#459—611no—
360#450—1221no—
361#449—1221no—
362#448—1221no—
363#447—1221no—
364C8×Q8—611no—
365C8×Q8C2×D6111yescharacteristic, complemented
366C4×SD16—21no—
367C4×SD16—611no—
368C4×SD16—1221no—
369C4×SD16C2×D6111yescharacteristic
370#436—611no—
371C8×D4—611no—
372#335—611no—
373#334—611no—
374#333—611no—
375#332—611no—
376#331—611no—
377#331—611no—
378Q8⋊C8—611no—
379Q8⋊C8—611no—
380D4⋊C8—611no—
381D4⋊C8—611no—
382#322—611no—
383C61#1980411yescharacteristic, complemented, Sylow
384C2×#41—1221no—
385C22×Q8—21no—
386C22×D4—1221no—
387C22×D4—1221no—
388AGL(1,ℤ/8)—2444no—
389C2×SD16—611no—
390C2×SD16—1221no—
391C2×SD16—1221no—
392C2×SD16—1221no—
393C2×SD16—1221no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—2441no—
397C2×SD16—2444no—
398C2×SD16C22⋊D6111yescharacteristic
399C2×D8—1221no—
400C2×D8—1221no—
401C2×D8—1221no—
402C2×M16—1221no—
403C22×C8—1221no—
404C4⋊Q8—21no—
405C4⋊Q8—21no—
406C4⋊Q8—21no—
407C4⋊Q8—41no—
408C4⋊Q8C22×D6111yescharacteristic
409C4⋊Q8C22×D6111yescharacteristic
410#127—611no—
411#127—611no—
412#127—1221no—
413C22⋊Q8—21no—
414C22⋊Q8—41no—
415#121—1221no—
416#121—1221no—
417#121—1221no—
418#121—2441no—
419#120—2441no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8C22×D6111yescharacteristic
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4C22×D6111yescharacteristic
428#106—21no—
429#106—21no—
430#106—21no—
431#106—21no—
432#106—611no—
433#106—1221no—
434#106C22⋊D6111yescharacteristic
435C4⋊C8—21no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—1221no—
440C4⋊C8C22×D6111yescharacteristic
441#103—21no—
442#103—611no—
443#103—1221no—
444#103C22⋊D6111yescharacteristic
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—2441no—
455#102C22⋊D6111yescharacteristic
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C4×C8—21no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8—1221no—
462C4×C8C22×D6111yescharacteristic
463C24—2441no—
464#41—4881no—
465#41—2442no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—41no—
469C2×Q8—41no—
470C2×Q8—42no—
471C2×Q8—81no—
472C2×Q8D4×D6111yescharacteristic
473C2×Q8C2×C22⋊D6111yescharacteristic
474C2×D4—611no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—4881no—
482C2×D4—4881no—
483C2×D4—2441no—
484C2×D4—2441no—
485C2×D4—2441no—
486C2×D4—2442no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4C2×C22⋊D6111yescharacteristic
490C22×C4—21no—
491C22×C4—41no—
492C22×C4—1221no—
493C22×C4—2441no—
494SD16—1222no—
495SD16—4881no—
496SD16—4882no—
497SD16—2442no—
498SD16—2442no—
499SD16—2442no—
500SD16—2442no—
501SD16—2442no—
502SD16#1175820114yescomplemented
503D8—4881no—
504D8—2442no—
505D8—2442no—
506M16—2442no—
507C2×C8—21no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—2442no—
515C2×C8C2×C22⋊D6111yescharacteristic
516C2×C8C2×C22⋊D6111yescharacteristic
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—611no—
527C4⋊C4D4×D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C4⋊C4C2×C22⋊D6111yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—41no—
532C22⋊C4—1221no—
533C22⋊C4—2441no—
534C22⋊C4—2441no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42C23×D6111yescharacteristic
539C23—21no—
540C23—1221no—
541C23—9761no—
542C23—4882no—
543C23—4882no—
544C23—2441no—
545C23—2441no—
546Q8—21no—
547Q8—21no—
548Q8—41no—
549Q8—41no—
550Q8—82no—
551Q8—81no—
552Q8—161no—
553Q8C2×#1175820112yescomplemented
554Q8#1451980412yescomplemented
555D4—21no—
556D4—1221no—
557D4—1221no—
558D4—9761no—
559D4—4881no—
560D4—4881no—
561D4—4882no—
562D4—4882no—
563D4—4882no—
564D4—2441no—
565D4—2441no—
566D4—2441no—
567D4—2441no—
568D4—2441no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4C2×#1175820112yes—
573C2×C4—21no—
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—82no—
583C2×C4—1221no—
584C2×C4—4882no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C22×C22⋊D6111yescharacteristic
589C2×C4#1451982211yescharacteristic
590C8—21no—
591C8—41no—
592C8—1221no—
593C8—1221no—
594C8—2441no—
595C8—2441no—
596C8—2441no—
597C8—2441no—
598C8C2×#1175820112yescomplemented
599C22—22no—
600C22—41no—
601C22—41no—
602C22—9764no—
603C22—4881no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—4881no—
609C22—2441no—
610C22—2441no—
611C22—2441no—
612C22D4×C22⋊D6111yescharacteristic
613C4—22no—
614C4—22no—
615C4—21no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—2441no—
622C4C22×#1175820111yescharacteristic
623C4C2×#1451980411yescharacteristic
624C4#1684056811yescharacteristic
625C4#1684057011yescharacteristic
626C2—42no—
627C2—9761no—
628C2—9761no—
629C2—4881no—
630C2#2145731411yescharacteristic
631C2D4×#1175820111yescharacteristic
632C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radical#21457440C2
hypercenterQ8×SD16D61
hypocenterC61#19804
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. #21458490C2semidirect
2. D4×#11758201C2semidirect
4. #21445448C2semidirect
5. #21445315C2semidirect
6. #21445230C2semidirect
7. #21445179C2semidirect
8. #21445169C2semidirect
9. #21445018C2semidirect
10. #21444934C2semidirect
11. #21444922C2semidirect
12. #21444897C2semidirect
13. #21444662C2semidirect
14. SD16⋊#92418C2semidirect
15. Q8×SD16×C61C2semidirect
24. #16840110C22semidirect
25. #16840110C22semidirect
34. #16839962C22semidirect
40. Q8×#92418C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. #16839483C22semidirect
56. Q82×C61C22semidirect
59. C61×#498C22semidirect
63. C61×#475C22semidirect
64. C8×Q8×C61C22semidirect
177. SD16×C61SD16semidirect
236. Q8×C61C2×SD16semidirect
237. Q8×C61AGL(1,ℤ/8)semidirect
275. C8×C61C2×SD16semidirect
310. Q8×SD16D61semidirect
334. Q82C2×D61semidirect
339. #498C2×D61semidirect
351. #475C2×D61semidirect
365. C8×Q8C2×D61semidirect
383. C61#19804semidirect
502. SD16#11758201semidirect
553. Q8C2×#11758201semidirect
554. Q8#14519804semidirect
598. C8C2×#11758201semidirect

Maximal subgroups

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

subgroupindex
#214584902
D4×#117582012
#214574402
#214454482
#214453152
#214452302
#214451792
#214451692
#214450182
#214449342
#214449222
#214448972
#214446622
SD16⋊#924182
Q8×SD16×C612
#1980461