Finite Group Databasebeta

← #29670081

Subgroups — #29670081

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
2SD16×D244C211yesmaximal, characteristic, complemented
3#21457419C211yesmaximal, characteristic
4#21445443C211yesmaximal, characteristic, complemented
5#21445330C211yesmaximal, characteristic, complemented
6#21445239C211yesmaximal, characteristic, complemented
7#21445169C211yesmaximal, characteristic, complemented
8#21445160C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445081C211yesmaximal, characteristic, complemented
11#21444996C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3298C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17C2×SD16×D61—21no—
18Q8×D244C2211yescharacteristic
19D4×D244C2211yescharacteristic
20#16840165—21no—
21#16840165C2211yescharacteristic, complemented
22#16840158C2211yescharacteristic, complemented
23#16840136C2211yescharacteristic
24#16840135C2211yescharacteristic
25#16840110C2211yescharacteristic
26#16840042—21no—
27#16840042C2211yescharacteristic, complemented
28#16840039—21no—
29#16840039C2211yescharacteristic
30#16840010—21no—
31#16839992—21no—
32#16839967C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839914—21no—
37#16839912—21no—
38#16839907—21no—
39#16839902—21no—
40Q8×#92418C2211yescharacteristic, complemented
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43#16839840C2211yescharacteristic
44C8×D244C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56D4×Q8×C61C2211yescharacteristic, complemented
57C61×#547C2211yescharacteristic
58C61×#487C2211yescharacteristic, complemented
59C61×#483—21no—
60C61×#475C2211yescharacteristic, complemented
61C61×#465—21no—
62C61×#461C2211yescharacteristic
63C61×#448—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic, complemented
66C2×#103×C61—21no—
67C2×Q8×D61—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519802—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78SD16×D61—44no—
79C2×#11758199—21no—
80C2×#11758199D411yescharacteristic
81C2×#11758194—21no—
82C2×C8×D61—21no—
83#14519777—21no—
84#14519777C2311yescharacteristic
85#14519774C2311yescharacteristic
86#14519757—21no—
87#14519757—41no—
88#14519752—21no—
89#14519751—21no—
90#14519747—41no—
91C4×D244—21no—
92C4×D244C2311yescharacteristic
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710D411yescharacteristic
98#14519708—41no—
99C4⋊#92418—21no—
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519701—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
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
117C2×#41×C61—21no—
118C22×Q8×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61D411yescharacteristic
121C22×C8×C61—21no—
122C4⋊Q8×C61—21no—
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
125#123×C61—41no—
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61—21no—
128C22⋊Q8×C61—41no—
129#121×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C2×C4⋊C4×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61—21no—
142#103×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×Q8×C61—41no—
152C2×Q8×C61—42no—
153C2×Q8×C61—81no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61—41no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C2×C22⋊D61—41no—
163Q8×D61—42no—
164Q8×D61—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×C4×D61—41no—
176C2×Dic122—21no—
177C2×Dic122—41no—
178SD16×C61—22no—
179SD16×C61—81no—
180C2×C8×C61—21no—
181C2×C8×C61—81no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61—41no—
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—41no—
197C22⋊C4×C61—41no—
198C42×C61—21no—
199C42×C61C2411yescharacteristic
200#11758201—42no—
201#11758201—42no—
202#11758201—42no—
203#11758201—81no—
204#11758200—42no—
205#11758199—42no—
206#11758199SD1614yescomplemented
207#11758198—21no—
208#11758198—41no—
209#11758198—41no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—41no—
213C4⋊Dic61C2×D411yescharacteristic
214#11758194—42no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221#11758190—42no—
222#11758190—81no—
223C8×D61—42no—
224C23×C61—21no—
225C23×C61—21no—
226C22×D61—21no—
227C22×D61—41no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—161no—
231Q8×C61—21no—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—82no—
237Q8×C61AGL(1,ℤ/8)12yescomplemented
238D4×C61—21no—
239D4×C61—21no—
240D4×C61—21no—
241D4×C61—82no—
242D4×C61C2×SD1612yescomplemented
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—41no—
252C2×C4×C61—41no—
253C2×C4×C61—82no—
254C2×C4×C61—82no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C22⋊D61—82no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261D244—21no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—81no—
267D244—82no—
268D244C2×SD1612yes—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—82no—
272Dic122—41no—
273Dic122—41no—
274Dic122—82no—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—41no—
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418C2×SD1612yescomplemented
284#19803—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61D4211yescharacteristic
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
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—81no—
305C4×C61—81no—
306C4×C61C22×SD1611yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61#57611yescharacteristic
309C4×C61#57711yescharacteristic
310Dic61—41no—
311Dic61—81no—
312Q8×D8—611no—
313#3298D6111yescharacteristic, complemented
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#1627—611no—
325#1583—611no—
326#1580—611no—
327C2×C61—42no—
328C2×C61—81no—
329C2×C61#326411yescharacteristic
330C2×C61D4×SD1611yescharacteristic
331C2×C61#328911yescharacteristic
332D61—42no—
333D61—161no—
334C2×AGL(1,ℤ/8)—1221no—
335C22×SD16—1221no—
336D4×Q8—611no—
337D4×Q8C2×D6111yescharacteristic, complemented
338#547C2×D6111yescharacteristic
339D42—611no—
340#500—611no—
341#500—1221no—
342#498—611no—
343#492—611no—
344#492—611no—
345#487C2×D6111yescharacteristic, complemented
346#486—611no—
347#483—21no—
348#475C2×D6111yescharacteristic, complemented
349#474—611no—
350#474—611no—
351#474—1221no—
352#474—1221no—
353#468—1221no—
354#465—21no—
355#465—1221no—
356#461—611no—
357#461—611no—
358#461C2×D6111yescharacteristic
359#460—611no—
360#459—611no—
361#450—1221no—
362#450—1221no—
363#449—1221no—
364#448—21no—
365#448—1221no—
366C8×Q8—611no—
367C4×SD16C2×D6111yescharacteristic
368C4×D8—611no—
369C4×D8—1221no—
370#436—611no—
371C8×D4—611no—
372C8×D4C2×D6111yescharacteristic, complemented
373C2×#103—21no—
374#334—611no—
375#334—611no—
376#332—611no—
377#332—611no—
378#332—611no—
379#331—611no—
380Q8⋊C8—611no—
381D4⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384#322—611no—
385C61#1980311yescharacteristic, complemented, Sylow
386C2×#41—21no—
387C22×Q8—21no—
388C22×Q8—1221no—
389C22×D4—1221no—
390C22×D4—1221no—
391AGL(1,ℤ/8)—2444no—
392C2×SD16—21no—
393C2×SD16—1221no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—2444no—
401C2×SD16C22⋊D6111yescharacteristic
402C2×D8—611no—
403C2×D8—1221no—
404C2×M16—1221no—
405C22×C8—21no—
406C22×C8—1221no—
407C4⋊Q8—21no—
408C4⋊Q8—611no—
409C4⋊Q8—1221no—
410C4⋊Q8C22×D6111yescharacteristic
411C4⋊Q8C22×D6111yescharacteristic
412#127—611no—
413#123—41no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—41no—
417C22⋊Q8—1221no—
418C22⋊Q8—2441no—
419#121—21no—
420#121—1221no—
421#121—1221no—
422#120—2441no—
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429C2×C4⋊C4—21no—
430#107—611no—
431#107—1221no—
432#107—1221no—
433#107—1221no—
434#107—1221no—
435#106—21no—
436#106—2441no—
437#106C22⋊D6111yescharacteristic
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—1221no—
442C4⋊C8C22×D6111yescharacteristic
443#103—21no—
444#103—21no—
445#103—21no—
446#103—21no—
447#103—1221no—
448#103—1221no—
449#103C22⋊D6111yescharacteristic
450#102—611no—
451#102—611no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—2441no—
457#102C22⋊D6111yescharacteristic
458C22⋊C8—21no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C24—2441no—
466#41—42no—
467#41—81no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—41no—
471C2×Q8—42no—
472C2×Q8—81no—
473C2×Q8—1221no—
474C2×Q8—4881no—
475C2×Q8—2441no—
476C2×Q8—2442no—
477C2×Q8C2×C22⋊D6111yescharacteristic
478C2×D4—41no—
479C2×D4—611no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—4881no—
484C2×D4—4881no—
485C2×D4—2441no—
486C2×D4—2442no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4D4×D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—1221no—
496C22×C4—2441no—
497SD16—22no—
498SD16—81no—
499SD16—4881no—
500SD16—4881no—
501SD16—2442no—
502SD16—2442no—
503SD16—2442no—
504SD16—2442no—
505SD16—2442no—
506D8—614no—
507D8—2442no—
508M16—2442no—
509C2×C8—21no—
510C2×C8—81no—
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—41no—
526C4⋊C4—41no—
527C4⋊C4—611no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—2441no—
531C4⋊C4D4×D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—1221no—
539C22⋊C4—2441no—
540C22⋊C4—2441no—
541C42—21no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—21no—
545C23—1221no—
546C23—9761no—
547C23—4882no—
548C23—4882no—
549C23—2441no—
550Q8—21no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—82no—
556Q8—4882no—
557Q8—2441no—
558Q8—2441no—
559Q8#1451980212yescomplemented
560D4—21no—
561D4—21no—
562D4—21no—
563D4—82no—
564D4—612no—
565D4—1221no—
566D4—4881no—
567D4—4882no—
568D4—4882no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4SD16×D6112yescomplemented
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—41no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—1221no—
587C2×C4—4882no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—21no—
595C8—21no—
596C8—41no—
597C8—612no—
598C8—1221no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—1222no—
610C22—9764no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—2441no—
615C22—2441no—
616C22D4×C22⋊D6111yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—4881no—
625C4—2441no—
626C4C2×SD16×D6111yescharacteristic
627C4C2×#1451980211yescharacteristic
628C4#1684056511yescharacteristic
629C4#1684056811yescharacteristic
630C2—42no—
631C2—81no—
632C2—9761no—
633C2—2442no—
634C2#2145730011yescharacteristic
635C2SD16×C22⋊D6111yescharacteristic
636C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3298C2
socleC22×C61D42
prime radical#21457419C2
hypercenter#3298D61
hypocenterC61#19803
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3298

Complements

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

NH ≅ G/Ntype
1. Q8×#11758199C2semidirect
2. SD16×D244C2semidirect
4. #21445443C2semidirect
5. #21445330C2semidirect
6. #21445239C2semidirect
7. #21445169C2semidirect
8. #21445160C2semidirect
9. #21445103C2semidirect
10. #21445081C2semidirect
11. #21444996C2semidirect
12. #21444922C2semidirect
13. #21444663C2semidirect
14. #21444641C2semidirect
15. C61×#3298C2semidirect
21. #16840165C22semidirect
22. #16840158C22semidirect
27. #16840042C22semidirect
40. Q8×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
56. D4×Q8×C61C22semidirect
58. C61×#487C22semidirect
60. C61×#475C22semidirect
65. C8×D4×C61C22semidirect
206. #11758199SD16semidirect
237. Q8×C61AGL(1,ℤ/8)semidirect
242. D4×C61C2×SD16semidirect
283. #92418C2×SD16semidirect
313. #3298D61semidirect
337. D4×Q8C2×D61semidirect
345. #487C2×D61semidirect
348. #475C2×D61semidirect
372. C8×D4C2×D61semidirect
385. C61#19803semidirect
559. Q8#14519802semidirect
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
SD16×D2442
#214574192
#214454432
#214453302
#214452392
#214451692
#214451602
#214451032
#214450812
#214449962
#214449222
#214446632
#214446412
C61×#32982
#1980361