Finite Group Databasebeta

← #29671383

Subgroups — #29671383

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#21458798C211yesmaximal, characteristic, complemented
2#21458740C211yesmaximal, characteristic, complemented
3#21457435C211yesmaximal, characteristic
4#21445409C211yesmaximal, characteristic, complemented
5#21445330C211yesmaximal, characteristic, complemented
6#21445301C211yesmaximal, characteristic, complemented
7#21445164C211yesmaximal, characteristic, complemented
8#21445160C211yesmaximal, characteristic, complemented
9#21445017C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444922C211yesmaximal, characteristic, complemented
12#21444913C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14C8×#11758200C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17#16840490C2211yescharacteristic, complemented
18#16840433C2211yescharacteristic
19#16840168—21no—
20#16840165C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840135C2211yescharacteristic, complemented
23#16840133—21no—
24#16840110C2211yescharacteristic, complemented
25#16840108C2211yescharacteristic
26#16840062—21no—
27#16840054—21no—
28#16840044C2211yescharacteristic, complemented
29#16840042C2211yescharacteristic
30#16840013—21no—
31#16839967C2211yescharacteristic
32#16839966—21no—
33#16839962C2211yescharacteristic, complemented
34#16839961—21no—
35#16839959C2211yescharacteristic
36#16839957C2211yescharacteristic
37#16839905—21no—
38#16839896—21no—
39#16839893C2211yescharacteristic, complemented
40C8×Dic122C2211yescharacteristic, complemented
41#16839862—21no—
42#16839849—21no—
43C4×#11758200C2211yescharacteristic
44#16839840C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic, 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×#11758190—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C2×#11758201—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758194—21no—
79#14519777C2311yescharacteristic
80#14519775—21no—
81#14519774C2311yescharacteristic
82#14519773—21no—
83#14519769—21no—
84#14519767—41no—
85#14519751—21no—
86#14519750—41no—
87C4×Dic122—21no—
88C4×Dic122C2311yescharacteristic
89#14519742—21no—
90#14519739—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700D411yescharacteristic
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697—41no—
111C22⋊#92418—21no—
112#14519688—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×Dic122C2×D411yescharacteristic
176SD16×C61—22no—
177SD16×C61—42no—
178SD16×C61—42no—
179SD16×C61—42no—
180SD16×C61—81no—
181C2×C8×C61—21no—
182C2×C8×C61—42no—
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—41no—
196C42×C61—21no—
197C42×C61C2411yescharacteristic
198#11758201—81no—
199#11758200—42no—
200#11758200—81no—
201#11758200SD1614yescomplemented
202#11758199—42no—
203#11758199—42no—
204#11758199—81no—
205#11758198—41no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—21no—
210#11758196—41no—
211#11758196—41no—
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
222#11758190—42no—
223#11758190—81no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—82no—
228C23×C61—82no—
229C23×C61—161no—
230C22×D61—41no—
231Q8×C61—21no—
232Q8×C61—21no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—82no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—81no—
244D4×C61—82no—
245D4×C61—82no—
246D4×C61C2×SD1612yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—82no—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C22⋊D61—82no—
261C22⋊D61—161no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265D244—41no—
266D244—41no—
267D244—41no—
268D244—41no—
269D244—81no—
270C4×D61—82no—
271Dic122—21no—
272Dic122—41no—
273Dic122—41no—
274Dic122C2×SD1612yescomplemented
275C8×C61—21no—
276C8×C61—41no—
277C8×C61AGL(1,ℤ/8)12yescomplemented
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418C2×SD1612yescomplemented
284#19892—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—164no—
295C22×C61D4211yescharacteristic
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61C22×SD1611yescharacteristic
306C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C61#57611yescharacteristic
308C4×C61#57711yescharacteristic
309Dic61—22no—
310Dic61—41no—
311Dic61—81no—
312#3392—611no—
313#3386—611no—
314D4×SD16D6111yescharacteristic, complemented
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—81no—
329C2×C61—161no—
330C2×C61#327411yescharacteristic
331C2×C61D4×SD1611yescharacteristic
332C2×C61#329011yescharacteristic
333D61—161no—
334C2×AGL(1,ℤ/8)—1221no—
335C22×SD16—21no—
336#557—611no—
337#550—611no—
338D4×Q8C2×D6111yescharacteristic
339D42C2×D6111yescharacteristic
340#501—1221no—
341#500—611no—
342#493—611no—
343#492—611no—
344#492—1221no—
345#492C2×D6111yescharacteristic
346#486—611no—
347#486—611no—
348#478—1221no—
349#477—1221no—
350#475—611no—
351#474—611no—
352#469—1221no—
353#465—21no—
354#461—611no—
355#461—1221no—
356#461C2×D6111yescharacteristic
357#460—611no—
358#460—611no—
359#460—1221no—
360#460C2×D6111yescharacteristic
361#459—611no—
362#450—21no—
363#449—1221no—
364#448—21no—
365#447—1221no—
366#446—611no—
367C8×Q8—611no—
368#440—1221no—
369C4×SD16—611no—
370C4×SD16—1221no—
371C4×SD16C2×D6111yescharacteristic
372#436—611no—
373C8×D4C2×D6111yescharacteristic
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—21no—
389C22×D4—21no—
390AGL(1,ℤ/8)—2444no—
391C2×SD16—21no—
392C2×SD16—21no—
393C2×SD16—21no—
394C2×SD16—21no—
395C2×SD16—44no—
396C2×SD16—611no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16C22⋊D6111yescharacteristic
403C2×D8—1221no—
404C2×D8—1221no—
405C2×D8—1221no—
406C2×M16—1221no—
407C22×C8—21no—
408C4⋊Q8—21no—
409C4⋊Q8—611no—
410C4⋊Q8—1221no—
411C4⋊Q8C22×D6111yescharacteristic
412#127—611no—
413#127—1221no—
414#127C22×D6111yescharacteristic
415#125—1221no—
416#125—2441no—
417C22⋊Q8—21no—
418C22⋊Q8—41no—
419#121—21no—
420#121—21no—
421#121—1221no—
422#121—2441no—
423#120—41no—
424C4×Q8—611no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×Q8C22×D6111yescharacteristic
428C4×D4—21no—
429C4×D4—1221no—
430C4×D4C22×D6111yescharacteristic
431C4×D4C22×D6111yescharacteristic
432#107—1221no—
433#107—1221no—
434#107—1221no—
435#106—611no—
436#106—1221no—
437#106C22⋊D6111yescharacteristic
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—1221no—
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—21no—
459C22⋊C8—1221no—
460#97—1221no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C24—41no—
466#41—4881no—
467#41—2442no—
468C2×Q8—21no—
469C2×Q8—41no—
470C2×Q8—42no—
471C2×Q8—81no—
472C2×Q8—611no—
473C2×Q8—1221no—
474C2×Q8C2×C22⋊D6111yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—81no—
483C2×D4—81no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—2441no—
488C2×D4—2441no—
489C2×D4D4×D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—2441no—
495SD16—22no—
496SD16—42no—
497SD16—42no—
498SD16—42no—
499SD16—81no—
500SD16—614no—
501SD16—4881no—
502SD16—4881no—
503SD16—4881no—
504SD16—2442no—
505SD16—2442no—
506D8—4881no—
507D8—2442no—
508D8—2442no—
509M16—2442no—
510C2×C8—21no—
511C2×C8—42no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8C2×C22⋊D6111yescharacteristic
519C2×C8C2×C22⋊D6111yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—611no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—2441no—
529C4⋊C4—2441no—
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—2441no—
539C42—21no—
540C42—1221no—
541C42—2441no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—21no—
545C23—41no—
546C23—82no—
547C23—82no—
548C23—161no—
549C23—2441no—
550Q8—21no—
551Q8—21no—
552Q8—41no—
553Q8—41no—
554Q8—82no—
555Q8—612no—
556Q8—1221no—
557Q8—2441no—
558Q8—2441no—
559D4—21no—
560D4—21no—
561D4—21no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—81no—
567D4—82no—
568D4—82no—
569D4—9761no—
570D4—4881no—
571D4—4882no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575D4—2441no—
576D4SD16×D6112yes—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—4882no—
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—
602C8#1451980312yescomplemented
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—164no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22D4×C22⋊D6111yescharacteristic
617C4—21no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—1222no—
624C4—4881no—
625C4—2441no—
626C4C2×SD16×D6111yescharacteristic
627C4C2×#1451980311yescharacteristic
628C4#1684056511yescharacteristic
629C4#1684056811yescharacteristic
630C2—42no—
631C2—81no—
632C2—161no—
633C2—9761no—
634C2#2145743511yescharacteristic
635C2SD16×C22⋊D6111yescharacteristic
636C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radical#21457435C2
hypercenterD4×SD16D61
hypocenterC61#19892
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. #21458798C2semidirect
2. #21458740C2semidirect
4. #21445409C2semidirect
5. #21445330C2semidirect
6. #21445301C2semidirect
7. #21445164C2semidirect
8. #21445160C2semidirect
9. #21445017C2semidirect
10. #21444998C2semidirect
11. #21444922C2semidirect
12. #21444913C2semidirect
13. #21444710C2semidirect
14. C8×#11758200C2semidirect
15. D4×SD16×C61C2semidirect
17. #16840490C22semidirect
20. #16840165C22semidirect
22. #16840135C22semidirect
24. #16840110C22semidirect
28. #16840044C22semidirect
33. #16839962C22semidirect
39. #16839893C22semidirect
40. C8×Dic122C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
201. #11758200SD16semidirect
274. Dic122C2×SD16semidirect
277. C8×C61AGL(1,ℤ/8)semidirect
283. #92418C2×SD16semidirect
314. D4×SD16D61semidirect
385. C61#19892semidirect
602. C8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214587982
#214587402
#214574352
#214454092
#214453302
#214453012
#214451642
#214451602
#214450172
#214449982
#214449222
#214449132
#214447102
C8×#117582002
D4×SD16×C612
#1989261