Finite Group Databasebeta

← #29670073

Subgroups — #29670073

Order 15616.

728 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
1Q8×#11758199C211yesmaximal, characteristic, complemented
2#21457684C211yesmaximal, characteristic, complemented
3#21457510C211yesmaximal, characteristic, complemented
4#21457414C211yesmaximal, characteristic
5#21445440C211yesmaximal, characteristic, complemented
6#21445331C211yesmaximal, characteristic, complemented
7#21445241C211yesmaximal, characteristic, complemented
8#21445165C211yesmaximal, characteristic, complemented
9#21445158C211yesmaximal, characteristic, complemented
10#21445106C211yesmaximal, characteristic, complemented
11#21445085C211yesmaximal, characteristic, complemented
12#21444995C211yesmaximal, characteristic, complemented
13#21444659C211yesmaximal, characteristic, complemented
14#21444640C211yesmaximal, characteristic, complemented
15C61×#1627C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C2×SD16×D61—21no—
18Q8×D244C2211yescharacteristic
19#16840416C2211yescharacteristic
20#16840397C2211yescharacteristic
21#16840165—21no—
22#16840165C2211yescharacteristic, complemented
23#16840160C2211yescharacteristic, complemented
24#16840134C2211yescharacteristic
25#16840134C2211yescharacteristic
26#16840112C2211yescharacteristic, complemented
27#16840090—21no—
28#16840047C2211yescharacteristic, complemented
29#16840042—21no—
30#16840042C2211yescharacteristic, complemented
31#16840039—21no—
32#16840039C2211yescharacteristic
33#16840007—21no—
34#16839993—21no—
35#16839990—21no—
36#16839968C2211yescharacteristic
37#16839968C2211yescharacteristic
38#16839964C2211yescharacteristic
39#16839914—21no—
40#16839914—21no—
41#16839904—21no—
42#16839901—21no—
43#16839899—21no—
44Q8×#92418C2211yescharacteristic, complemented
45C4×#11758201C2211yescharacteristic
46C4×#11758199—21no—
47C4×#11758199C2211yescharacteristic
48#16839839C2211yescharacteristic
49D4×#92418C2211yescharacteristic, complemented
50#16839827C2211yescharacteristic
51C2×#14519701—21no—
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839507C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839491C2211yescharacteristic
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic, complemented
60D4×Q8×C61C2211yescharacteristic, complemented
61C61×#486C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#459C2211yescharacteristic
64C61×#332C2211yescharacteristic, complemented
65C61×Q8⋊C8C2211yescharacteristic, complemented
66C61×D4⋊C8C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71#14519801—44no—
72C2×#11758190—21no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×#11758201—21no—
79C2×#11758201D411yescharacteristic
80SD16×D61—44no—
81C2×#11758199D411yescharacteristic
82C2×#11758189—21no—
83C22×#92418—21no—
84C2×C8×D61—21no—
85#14519777—21no—
86#14519777C2311yescharacteristic
87#14519777C2311yescharacteristic
88#14519761—41no—
89#14519757—21no—
90#14519757—21no—
91#14519757—41no—
92#14519751—21no—
93#14519748—21no—
94#14519748—21no—
95#14519745—41no—
96C4×D244—21no—
97C4×D244C2311yescharacteristic
98C4×D244C2311yescharacteristic
99C2×C4⋊Dic61—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710D411yescharacteristic
105#14519709—41no—
106#14519708—21no—
107#14519708D411yescharacteristic
108C4⋊#92418—21no—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519702—21no—
112#14519702—21no—
113#14519701—21no—
114#14519701—21no—
115#14519701—21no—
116#14519701—21no—
117#14519701D411yescharacteristic
118#14519698—21no—
119#14519698D411yescharacteristic
120#14519697—21no—
121#14519697D411yescharacteristic
122#14519697D411yescharacteristic
123C22⋊#92418—21no—
124#14519689—21no—
125#14519689—21no—
126C4×#92418—21no—
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C22×Q8×C61—21no—
130C2×SD16×C61—21no—
131C2×D8×C61—21no—
132C4⋊Q8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134#127×C61C2311yescharacteristic
135C22⋊Q8×C61—21no—
136C22⋊Q8×C61—41no—
137C4×Q8×C61C2311yescharacteristic
138C4×D4×C61—21no—
139C4×D4×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144#102×C61—41no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—41no—
161C22×Dic61—41no—
162Q8×D61—42no—
163Q8×D61—81no—
164#11758224—42no—
165#11758224—81no—
166D4×D61—42no—
167D4×D61—42no—
168D4×D61—81no—
169D4×D61—81no—
170C2×D244C2×D411yescharacteristic
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—41no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—41no—
178SD16×C61—42no—
179D8×C61—42no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758201—22no—
195#11758201—81no—
196#11758200—42no—
197#11758200—42no—
198#11758200—81no—
199#11758200—81no—
200#11758199SD1614yescomplemented
201#11758198—21no—
202#11758198—21no—
203#11758198—41no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—41no—
210C4⋊Dic61—41no—
211C4⋊Dic61C2×D411yescharacteristic
212C4⋊Dic61C2×D411yescharacteristic
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—81no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220#11758190—42no—
221#11758190—42no—
222#11758189—42no—
223C8×D61—42no—
224C23×C61—21no—
225C23×C61—41no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—161no—
231Q8×C61—21no—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—82no—
235Q8×C61AGL(1,ℤ/8)12yescomplemented
236D4×C61—21no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—81no—
242D4×C61C2×SD1612yescomplemented
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—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C22⋊D61—82no—
255C22⋊D61—82no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—82no—
261D244—21no—
262D244—21no—
263D244—21no—
264D244C2×SD1612yes—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—82no—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274Dic122—82no—
275C8×C61—41no—
276C8×C61—41no—
277C8×C61—41no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
281#92418—41no—
282#92418—41no—
283#92418C2×SD1612yescomplemented
284#19803—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61D4211yescharacteristic
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—164no—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61C22×SD1611yescharacteristic
305C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C4×C61#57611yescharacteristic
307C4×C61#57711yescharacteristic
308Dic61—41no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312Q8×D8—611no—
313#3298—611no—
314D4×SD16—611no—
315#3272—611no—
316#1714—611no—
317#1693—611no—
318#1679—611no—
319#1666—611no—
320#1665—611no—
321#1654—611no—
322#1651—611no—
323#1639—611no—
324#1627D6111yescharacteristic, complemented
325#1583—611no—
326#1580—611no—
327C2×C61—42no—
328C2×C61—161no—
329C2×C61#326411yescharacteristic
330C2×C61D4×SD1611yescharacteristic
331C2×C61#328911yescharacteristic
332D61—42no—
333D61—81no—
334C2×AGL(1,ℤ/8)—1221no—
335C22×SD16—1221no—
336D4×Q8—611no—
337D4×Q8C2×D6111yescharacteristic, complemented
338#547—611no—
339D42—611no—
340#500—611no—
341#500—1221no—
342#498—611no—
343#492—611no—
344#492—611no—
345#487—611no—
346#486C2×D6111yescharacteristic
347#483—1221no—
348#475—611no—
349#474—611no—
350#474—611no—
351#474—1221no—
352#474—1221no—
353#468—1221no—
354#465—1221no—
355#465—1221no—
356#461—611no—
357#461—611no—
358#461—611no—
359#460C2×D6111yescharacteristic
360#459C2×D6111yescharacteristic
361#450—1221no—
362#450—1221no—
363#449—1221no—
364#448—1221no—
365#448—1221no—
366C8×Q8—611no—
367C4×SD16—611no—
368C4×D8—611no—
369C4×D8—1221no—
370#436—611no—
371C8×D4—611no—
372C8×D4—611no—
373C2×#103—1221no—
374#334—611no—
375#334—611no—
376#332—611no—
377#332—611no—
378#332C2×D6111yescharacteristic, complemented
379#331—611no—
380Q8⋊C8C2×D6111yescharacteristic, complemented
381D4⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8C2×D6111yescharacteristic, complemented
384#322—611no—
385C61#1980311yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8—21no—
388C22×Q8—1221no—
389C22×D4—1221no—
390C22×D4—1221no—
391AGL(1,ℤ/8)—2444no—
392C2×SD16—21no—
393C2×SD16—611no—
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×D8—21no—
403C2×D8—611no—
404C2×M16—1221no—
405C22×C8—1221no—
406C22×C8—1221no—
407C4⋊Q8—21no—
408C4⋊Q8—611no—
409C4⋊Q8—611no—
410C4⋊Q8—1221no—
411C4⋊Q8C22×D6111yescharacteristic
412#127C22×D6111yescharacteristic
413#123—2441no—
414C22⋊Q8—21no—
415C22⋊Q8—41no—
416C22⋊Q8—1221no—
417C22⋊Q8—1221no—
418C22⋊Q8—2441no—
419#121—1221no—
420#121—1221no—
421#121—1221no—
422#120—2441no—
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—1221no—
428C4×D4C22×D6111yescharacteristic
429C2×C4⋊C4—1221no—
430#107—611no—
431#107—1221no—
432#107—1221no—
433#107—1221no—
434#107—1221no—
435#106—611no—
436#106—1221no—
437#106—2441no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—1221no—
441C4⋊C8C22×D6111yescharacteristic
442C4⋊C8C22×D6111yescharacteristic
443#103—611no—
444#103—1221no—
445#103—1221no—
446#103—1221no—
447#103—1221no—
448#103—1221no—
449#103—1221no—
450#102—21no—
451#102—21no—
452#102—41no—
453#102—611no—
454#102—611no—
455#102—611no—
456#102—1221no—
457#102—1221no—
458C22⋊C8—1221no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
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—41no—
470C2×Q8—42no—
471C2×Q8—81no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—4881no—
475C2×Q8—2441no—
476C2×Q8—2442no—
477C2×Q8D4×D6111yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—611no—
482C2×D4—611no—
483C2×D4—4881no—
484C2×D4—4881no—
485C2×D4—2441no—
486C2×D4—2441no—
487C2×D4—2442no—
488C2×D4—2442no—
489C2×D4—2441no—
490C2×D4D4×D6111yescharacteristic
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497SD16—42no—
498SD16—1222no—
499SD16—4881no—
500SD16—4881no—
501SD16—4881no—
502SD16—2442no—
503SD16—2442no—
504SD16—2442no—
505SD16—2442no—
506D8—42no—
507D8—614no—
508M16—2442no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—4881no—
519C2×C8—2442no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—2441no—
531C4⋊C4—2441no—
532C4⋊C4D4×D6111yescharacteristic
533C4⋊C4D4×D6111yescharacteristic
534C22⋊C4—21no—
535C22⋊C4—41no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—2441no—
539C22⋊C4—2441no—
540C22⋊C4—2441no—
541C42—21no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—41no—
545C23—1221no—
546C23—1221no—
547C23—9761no—
548C23—4882no—
549C23—4882no—
550Q8—21no—
551Q8—41no—
552Q8—41no—
553Q8—82no—
554Q8—4882no—
555Q8—2441no—
556Q8—2441no—
557Q8—2441no—
558Q8—2441no—
559Q8#1451980112yescomplemented
560D4—21no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—81no—
566D4—612no—
567D4—1221no—
568D4—1221no—
569D4—1221no—
570D4—4882no—
571D4—4882no—
572D4—4882no—
573D4SD16×D6112yescomplemented
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—82no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—4882no—
584C2×C4—4882no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—41no—
595C8—41no—
596C8—41no—
597C8—612no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—2441no—
602C8—2441no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—1222no—
610C22—9764no—
611C22—2441no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22#1684039411yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—4881no—
623C4—4881no—
624C4—2441no—
625C4—2441no—
626C4C2×SD16×D6111yescharacteristic
627C4C2×#1451980111yescharacteristic
628C4#1684056611yescharacteristic
629C4#1684057111yescharacteristic
630C2—42no—
631C2—161no—
632C2—4881no—
633C2—2442no—
634C2#2145729211yescharacteristic
635C2#2145750811yescharacteristic
636C2#2145759611yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1627C2
socleC22×C61D42
prime radical#21457414C2
hypercenter#1627D61
hypocenterC61#19803
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1627

Complements

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

NH ≅ G/Ntype
1. Q8×#11758199C2semidirect
2. #21457684C2semidirect
3. #21457510C2semidirect
5. #21445440C2semidirect
6. #21445331C2semidirect
7. #21445241C2semidirect
8. #21445165C2semidirect
9. #21445158C2semidirect
10. #21445106C2semidirect
11. #21445085C2semidirect
12. #21444995C2semidirect
13. #21444659C2semidirect
14. #21444640C2semidirect
15. C61×#1627C2semidirect
22. #16840165C22semidirect
23. #16840160C22semidirect
26. #16840112C22semidirect
28. #16840047C22semidirect
30. #16840042C22semidirect
44. Q8×#92418C22semidirect
49. D4×#92418C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839507C22semidirect
55. #16839507C22semidirect
59. #16839484C22semidirect
60. D4×Q8×C61C22semidirect
64. C61×#332C22semidirect
65. C61×Q8⋊C8C22semidirect
66. C61×D4⋊C8C22semidirect
200. #11758199SD16semidirect
235. Q8×C61AGL(1,ℤ/8)semidirect
242. D4×C61C2×SD16semidirect
283. #92418C2×SD16semidirect
324. #1627D61semidirect
337. D4×Q8C2×D61semidirect
378. #332C2×D61semidirect
380. Q8⋊C8C2×D61semidirect
383. D4⋊C8C2×D61semidirect
385. C61#19803semidirect
559. Q8#14519801semidirect
573. D4SD16×D61semidirect

Maximal subgroups

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

subgroupindex
Q8×#117581992
#214576842
#214575102
#214574142
#214454402
#214453312
#214452412
#214451652
#214451582
#214451062
#214450852
#214449952
#214446592
#214446402
C61×#16272
#1980361