Finite Group Databasebeta

← #29670091

Subgroups — #29670091

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
1Q8×#11758200C211yesmaximal, characteristic, complemented
2#21458488C211yesmaximal, characteristic, complemented
3#21457435C211yesmaximal, characteristic
4#21445446C211yesmaximal, characteristic, complemented
5#21445316C211yesmaximal, characteristic, complemented
6#21445228C211yesmaximal, characteristic, complemented
7#21445177C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445017C211yesmaximal, characteristic, complemented
10#21444935C211yesmaximal, characteristic, complemented
11#21444922C211yesmaximal, characteristic, complemented
12#21444896C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17Q8×Dic122C2211yescharacteristic, complemented
18#16840433C2211yescharacteristic
19#16840158—21no—
20#16840158C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840134C2211yescharacteristic
23#16840110C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic, complemented
25#16840107—21no—
26#16840052—21no—
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840051—21no—
30#16840044—21no—
31#16840044C2211yescharacteristic, complemented
32#16840013—21no—
33#16839962C2211yescharacteristic, complemented
34#16839960—21no—
35#16839959C2211yescharacteristic
36#16839957C2211yescharacteristic
37#16839905—21no—
38#16839896—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C8×Dic122C2211yescharacteristic, complemented
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43#16839847—21no—
44#16839840C2211yescharacteristic
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C22×SD16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C61×#448—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519803—44no—
69C2×#11758200—21no—
70C2×#11758200—41no—
71C2×#11758200D411yescharacteristic
72C2×#11758201—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758194—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519776—21no—
80#14519776—41no—
81#14519774C2311yescharacteristic
82#14519773—21no—
83#14519751—21no—
84#14519750—41no—
85Q8×Dic61—21no—
86C4×Dic122—21no—
87C4×Dic122C2311yescharacteristic
88#14519739—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519700—21no—
101#14519700D411yescharacteristic
102#14519699—21no—
103#14519699—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697—41no—
111C22⋊#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×Q8×C61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×SD16×C61—44no—
124C2×SD16×C61D411yescharacteristic
125C22×C8×C61—21no—
126C4⋊Q8×C61—21no—
127C4⋊Q8×C61C2311yescharacteristic
128#127×C61C2311yescharacteristic
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132#121×C61—21no—
133#120×C61—41no—
134C4×Q8×C61C2311yescharacteristic
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137C4×D4×C61C2311yescharacteristic
138#106×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61—21no—
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—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×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—42no—
158C2×D4×C61—42no—
159C2×D4×C61—81no—
160C2×D4×C61—81no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C2×C22⋊D61—41no—
167C2×C22⋊D61—41no—
168#11758222—42no—
169#11758222—81no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×C4×D61—41no—
174C2×Dic122—21no—
175C2×Dic122—41no—
176C2×Dic122C2×D411yescharacteristic
177SD16×C61—22no—
178SD16×C61—42no—
179SD16×C61—42no—
180SD16×C61—42no—
181SD16×C61—81no—
182C2×C8×C61—21no—
183C2×C8×C61—42no—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—41no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61C2411yescharacteristic
199#11758201—42no—
200#11758200—42no—
201#11758200—82no—
202#11758200SD1614yescomplemented
203#11758199—42no—
204#11758199—42no—
205#11758199—81no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—41no—
210C4⋊Dic61C2×D411yescharacteristic
211#11758196—21no—
212#11758196—41no—
213C4×Dic61—21no—
214C4×Dic61—41no—
215#11758194—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—82no—
226C23×C61—82no—
227C23×C61—161no—
228C22×D61—41no—
229Q8×C61—21no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61—82no—
233Q8×C61AGL(1,ℤ/8)12yescomplemented
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—81no—
242D4×C61—82no—
243D4×C61—82no—
244D4×C61C2×SD1612yes—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—82no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C22⋊D61—82no—
259C22⋊D61—161no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—41no—
267D244—81no—
268C4×D61—82no—
269Dic122—21no—
270Dic122—81no—
271Dic122—161no—
272Dic122C2×SD1612yescomplemented
273C8×C61—21no—
274C8×C61—21no—
275C8×C61—41no—
276#92418—21no—
277#92418—41no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#92418C2×SD1612yescomplemented
282#19804—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×C61—164no—
293C22×C61D4211yescharacteristic
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C4×C61—22no—
298C4×C61—21no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—81no—
303C4×C61C22×SD1611yescharacteristic
304C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C4×C61#57711yescharacteristic
306C4×C61#57711yescharacteristic
307Dic61—22no—
308Dic61—41no—
309Dic61—81no—
310Q8×SD16—611no—
311#3362—611no—
312D4×SD16D6111yescharacteristic, complemented
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—81no—
327C2×C61—161no—
328C2×C61#326511yescharacteristic
329C2×C61D4×SD1611yescharacteristic
330C2×C61#328811yescharacteristic
331D61—161no—
332C2×AGL(1,ℤ/8)—1221no—
333C22×SD16—21no—
334Q82—611no—
335#550—611no—
336D4×Q8C2×D6111yescharacteristic
337D42C2×D6111yescharacteristic
338#498—611no—
339#498—1221no—
340#493—611no—
341#492—611no—
342#492C2×D6111yescharacteristic
343#486—611no—
344#486—611no—
345#486—1221no—
346#476—611no—
347#476—611no—
348#476—1221no—
349#476—1221no—
350#475—611no—
351#475—1221no—
352#469—1221no—
353#465—21no—
354#461C2×D6111yescharacteristic
355#460—611no—
356#460—611no—
357#460—1221no—
358#460C2×D6111yescharacteristic
359#459—611no—
360#450—21no—
361#449—1221no—
362#448—21no—
363#447—1221no—
364C8×Q8—611no—
365C8×Q8—611no—
366C4×SD16—611no—
367C4×SD16—1221no—
368C4×SD16—1221no—
369C4×SD16C2×D6111yescharacteristic
370#436—611no—
371C8×D4C2×D6111yescharacteristic
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—21no—
387C22×D4—21no—
388AGL(1,ℤ/8)—2444no—
389C2×SD16—21no—
390C2×SD16—21no—
391C2×SD16—21no—
392C2×SD16—21no—
393C2×SD16—44no—
394C2×SD16—611no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—2441no—
398C2×SD16C22⋊D6111yescharacteristic
399C2×D8—1221no—
400C2×D8—1221no—
401C2×D8—1221no—
402C2×M16—1221no—
403C22×C8—21no—
404C4⋊Q8—21no—
405C4⋊Q8—611no—
406C4⋊Q8—1221no—
407C4⋊Q8—1221no—
408C4⋊Q8—2441no—
409C4⋊Q8C22×D6111yescharacteristic
410#127—611no—
411#127—1221no—
412#127C22×D6111yescharacteristic
413C22⋊Q8—21no—
414C22⋊Q8—41no—
415#121—21no—
416#121—21no—
417#121—1221no—
418#121—2441no—
419#120—41no—
420C4×Q8—611no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—1221no—
426C4×D4C22×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428#106—611no—
429#106—1221no—
430#106—1221no—
431#106—1221no—
432#106—1221no—
433#106—1221no—
434#106C22⋊D6111yescharacteristic
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—1221no—
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—21no—
457C22⋊C8—1221no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—1221no—
461C4×C8—1221no—
462C4×C8C22×D6111yescharacteristic
463C24—41no—
464#41—4881no—
465#41—2442no—
466C2×Q8—21no—
467C2×Q8—41no—
468C2×Q8—42no—
469C2×Q8—81no—
470C2×Q8—611no—
471C2×Q8—1221no—
472C2×Q8—2441no—
473C2×Q8C2×C22⋊D6111yescharacteristic
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—81no—
482C2×D4—81no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—2441no—
487C2×D4—2441no—
488C2×D4D4×D6111yescharacteristic
489C2×D4C2×C22⋊D6111yescharacteristic
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—2441no—
494SD16—22no—
495SD16—42no—
496SD16—42no—
497SD16—42no—
498SD16—81no—
499SD16—614no—
500SD16—4882no—
501SD16—2442no—
502SD16—2442no—
503D8—4881no—
504D8—2442no—
505D8—2442no—
506M16—2442no—
507C2×C8—21no—
508C2×C8—42no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8C2×C22⋊D6111yescharacteristic
516C2×C8C2×C22⋊D6111yescharacteristic
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—41no—
521C4⋊C4—611no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4D4×D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C4⋊C4C2×C22⋊D6111yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—2441no—
535C42—21no—
536C42—1221no—
537C42—2441no—
538C42C23×D6111yescharacteristic
539C23—21no—
540C23—21no—
541C23—41no—
542C23—82no—
543C23—82no—
544C23—161no—
545C23—2441no—
546Q8—21no—
547Q8—41no—
548Q8—41no—
549Q8—82no—
550Q8—612no—
551Q8—1221no—
552Q8—9761no—
553Q8—4881no—
554Q8#1451980312yescomplemented
555D4—21no—
556D4—21no—
557D4—21no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—81no—
563D4—82no—
564D4—82no—
565D4—9761no—
566D4—4881no—
567D4—4882no—
568D4—2441no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4SD16×D6112yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—82no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—4882no—
586C2×C4—2441no—
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C22×C22⋊D6111yescharacteristic
589C2×C4#1451982211yescharacteristic
590C8—21no—
591C8—21no—
592C8—41no—
593C8—612no—
594C8—1221no—
595C8—2441no—
596C8—2441no—
597C8—2441no—
598C8—2441no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—164no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22D4×C22⋊D6111yescharacteristic
613C4—22no—
614C4—21no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—1222no—
620C4—4881no—
621C4—2441no—
622C4C2×SD16×D6111yescharacteristic
623C4C2×#1451980311yescharacteristic
624C4#1684056811yescharacteristic
625C4#1684057311yescharacteristic
626C2—42no—
627C2—81no—
628C2—161no—
629C2—9761no—
630C2#2145731011yescharacteristic
631C2SD16×C22⋊D6111yescharacteristic
632C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radical#21457435C2
hypercenterD4×SD16D61
hypocenterC61#19804
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#11758200C2semidirect
2. #21458488C2semidirect
4. #21445446C2semidirect
5. #21445316C2semidirect
6. #21445228C2semidirect
7. #21445177C2semidirect
8. #21445164C2semidirect
9. #21445017C2semidirect
10. #21444935C2semidirect
11. #21444922C2semidirect
12. #21444896C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
15. D4×SD16×C61C2semidirect
17. Q8×Dic122C22semidirect
20. #16840158C22semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
31. #16840044C22semidirect
33. #16839962C22semidirect
39. Q8×#92418C22semidirect
40. C8×Dic122C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839484C22semidirect
202. #11758200SD16semidirect
233. Q8×C61AGL(1,ℤ/8)semidirect
272. Dic122C2×SD16semidirect
281. #92418C2×SD16semidirect
312. D4×SD16D61semidirect
383. C61#19804semidirect
554. Q8#14519803semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582002
#214584882
#214574352
#214454462
#214453162
#214452282
#214451772
#214451642
#214450172
#214449352
#214449222
#214448962
#214446632
#214446312
D4×SD16×C612
#1980461