Finite Group Databasebeta

← #29670074

Subgroups — #29670074

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
1D8×Dic122C211yesmaximal, characteristic, complemented
2#21457678C211yesmaximal, characteristic, complemented
3D4×#11758200C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic, complemented
5#21445332C211yesmaximal, characteristic, complemented
6#21445244C211yesmaximal, characteristic, complemented
7#21445164C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic, complemented
12#21444921C211yesmaximal, characteristic, complemented
13#21444660C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic
16C22×#11758200—21no—
17D4×Dic122C2211yescharacteristic
18D4×Dic122C2211yescharacteristic, complemented
19#16840167C2211yescharacteristic, complemented
20#16840166—21no—
21#16840158C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic
23#16840114C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840092—21no—
26#16840044C2211yescharacteristic, complemented
27#16840042C2211yescharacteristic
28#16840042C2211yescharacteristic, complemented
29#16840040—21no—
30#16840040—21no—
31#16839989—21no—
32#16839989—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839913—21no—
37#16839900—21no—
38#16839900—21no—
39C8×Dic122C2211yescharacteristic, complemented
40C4×#11758200C2211yescharacteristic
41D8×Dic61—21no—
42D4×#92418C2211yescharacteristic
43D4×#92418C2211yescharacteristic, complemented
44C2×#14519700—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55#16839483C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#547C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#450—21no—
64C61×#449—21no—
65C4×D8×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C22×Dic122—21no—
68C22×Dic122—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—44no—
75C2×#11758200D411yescharacteristic
76C22×#92418—21no—
77C22×#92418—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519756—21no—
83#14519756—21no—
84#14519754—41no—
85#14519754—41no—
86C4×Dic122C2311yescharacteristic
87D4×Dic61—21no—
88D4×Dic61—21no—
89#14519711—21no—
90#14519711—21no—
91#14519710—21no—
92#14519710—21no—
93#14519708—21no—
94#14519708—41no—
95#14519708D411yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700D411yescharacteristic
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698—41no—
110#14519698D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C2×#41×C61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119AGL(1,ℤ/8)×C61—44no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×D8×C61—21no—
124C2×D8×C61D411yescharacteristic
125C2×M16×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#123×C61—41no—
129C22⋊Q8×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133#120×C61—41no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137C2×C4⋊C4×C61—21no—
138#107×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147#41×C61—42no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61—42no—
156C2×D4×C61—42no—
157C2×D4×C61—41no—
158C2×D4×C61—81no—
159C2×D4×C61—81no—
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—41no—
167C22×Dic61—41no—
168C22×Dic61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—41no—
174C2×Dic122—41no—
175C2×Dic122—81no—
176C2×Dic122—81no—
177C2×Dic122C2×D411yescharacteristic
178SD16×C61—42no—
179SD16×C61—42no—
180SD16×C61—81no—
181D8×C61—42no—
182D8×C61SD1614yescomplemented
183M16×C61—42no—
184C2×C8×C61—21no—
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
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—21no—
196C22⋊C4×C61—41no—
197C22⋊C4×C61—41no—
198C42×C61C2411yescharacteristic
199C22⋊Dic61—41no—
200C22⋊Dic61—41no—
201#11758200—22no—
202#11758200—42no—
203#11758200—42no—
204#11758200—42no—
205#11758200—81no—
206#11758200—81no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61C2×D411yescharacteristic
212#11758196—21no—
213#11758196—41no—
214#11758196—41no—
215C4×Dic61—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—42no—
221C2×#92418—81no—
222C2×#92418C2×D411yescharacteristic
223C2×#92418C2×D411yescharacteristic
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—41no—
228C23×C61—82no—
229C23×C61—82no—
230C23×C61—161no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—21no—
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×C61—82no—
245D4×C61C2×SD1612yes—
246D4×C61C2×SD1612yescomplemented
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—82no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—82no—
266C2×Dic61—82no—
267Dic122—21no—
268Dic122—41no—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272Dic122—82no—
273Dic122—82no—
274Dic122AGL(1,ℤ/8)12yescomplemented
275C8×C61—21no—
276C8×C61—41no—
277C8×C61C2×SD1612yescomplemented
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#19803—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—164no—
298C22×C61D4211yescharacteristic
299C4×C61—21no—
300C4×C61—41no—
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—22no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312Q8×D8—611no—
313#3298—611no—
314D4×SD16—611no—
315#3272D6111yescharacteristic
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—42no—
329C2×C61—81no—
330C2×C61—161no—
331C2×C61#326411yescharacteristic
332C2×C61D4×SD1611yescharacteristic
333C2×C61#328911yescharacteristic
334C2×AGL(1,ℤ/8)—21no—
335C22×SD16—1221no—
336D4×Q8—611no—
337D4×Q8—611no—
338#547C2×D6111yescharacteristic
339D42C2×D6111yescharacteristic
340#500—611no—
341#500—1221no—
342#498—611no—
343#492—611no—
344#492C2×D6111yescharacteristic
345#487—611no—
346#486—611no—
347#483—1221no—
348#475—611no—
349#474—611no—
350#474—611no—
351#474—1221no—
352#474—1221no—
353#468—21no—
354#465—1221no—
355#465—1221no—
356#461—611no—
357#461—611no—
358#461C2×D6111yescharacteristic
359#460—611no—
360#459C2×D6111yescharacteristic
361#450—21no—
362#450—1221no—
363#449—21no—
364#448—1221no—
365#448—1221no—
366C8×Q8—611no—
367C4×SD16—611no—
368C4×D8—1221no—
369C4×D8C2×D6111yescharacteristic
370#436C2×D6111yescharacteristic
371C8×D4—611no—
372C8×D4—611no—
373C2×#103—1221no—
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—1221no—
388C22×Q8—1221no—
389C22×D4—21no—
390C22×D4—21no—
391AGL(1,ℤ/8)—44no—
392C2×SD16—21no—
393C2×SD16—21no—
394C2×SD16—21no—
395C2×SD16—611no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—2444no—
402C2×D8—21no—
403C2×D8C22⋊D6111yescharacteristic
404C2×M16—21no—
405C22×C8—1221no—
406C22×C8—1221no—
407C4⋊Q8—611no—
408C4⋊Q8—611no—
409C4⋊Q8—1221no—
410C4⋊Q8—1221no—
411C4⋊Q8C22×D6111yescharacteristic
412#127C22×D6111yescharacteristic
413#123—41no—
414C22⋊Q8—21no—
415C22⋊Q8—1221no—
416C22⋊Q8—1221no—
417C22⋊Q8—2441no—
418C22⋊Q8—2441no—
419#121—21no—
420#121—21no—
421#121—21no—
422#120—41no—
423C4×Q8—611no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4C22×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429C2×C4⋊C4—21no—
430#107—1221no—
431#107—1221no—
432#107—1221no—
433#107—1221no—
434#107C22⋊D6111yescharacteristic
435#106—611no—
436#106—1221no—
437#106—2441no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—1221no—
442C4⋊C8C22×D6111yescharacteristic
443#103—21no—
444#103—611no—
445#103—1221no—
446#103—1221no—
447#103—1221no—
448#103—1221no—
449#103—1221no—
450#102—21no—
451#102—611no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—2441no—
456#102C22⋊D6111yescharacteristic
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—41no—
466#41—42no—
467#41—81no—
468C2×Q8—21no—
469C2×Q8—611no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—4881no—
473C2×Q8—4881no—
474C2×Q8—2441no—
475C2×Q8—2441no—
476C2×Q8—2442no—
477C2×Q8—2442no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—41no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4D4×D6111yescharacteristic
489C2×D4C2×C22⋊D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4—2441no—
496C22×C4—2441no—
497SD16—42no—
498SD16—42no—
499SD16—81no—
500SD16—1222no—
501SD16—4881no—
502SD16—4881no—
503SD16—2442no—
504SD16—2442no—
505SD16—2442no—
506D8—42no—
507D8#1175820014yescomplemented
508M16—42no—
509C2×C8—21no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—4881no—
517C2×C8—2442no—
518C2×C8C2×C22⋊D6111yescharacteristic
519C2×C8C2×C22⋊D6111yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—41no—
523C4⋊C4—611no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
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—21no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—2441no—
540C22⋊C4—2441no—
541C42—1221no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—82no—
548C23—82no—
549C23—161no—
550Q8—41no—
551Q8—41no—
552Q8—612no—
553Q8—1221no—
554Q8—4882no—
555Q8—4882no—
556Q8—2441no—
557Q8—2441no—
558Q8—2441no—
559Q8—2441no—
560D4—21no—
561D4—21no—
562D4—21no—
563D4—21no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4C2×#1175820012yes—
573D4C2×#1175820012yescomplemented
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—41no—
583C2×C4—41no—
584C2×C4—82no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—4882no—
588C2×C4—4882no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—21no—
595C8—41no—
596C8—1221no—
597C8—1221no—
598C8—2441no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C8C2×#1175820012yescomplemented
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—164no—
616C22D4×C22⋊D6111yescharacteristic
617C4—21no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—1222no—
623C4—4881no—
624C4—4881no—
625C4—2441no—
626C4C22×#1175820011yescharacteristic
627C4AGL(1,ℤ/8)×D6111yescharacteristic
628C4#1684056611yescharacteristic
629C4#1684057311yescharacteristic
630C2—42no—
631C2—42no—
632C2—81no—
633C2—161no—
634C2#2145729311yescharacteristic
635C2D4×#1175820011yescharacteristic
636C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#19803
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3272

Complements

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

NH ≅ G/Ntype
1. D8×Dic122C2semidirect
2. #21457678C2semidirect
3. D4×#11758200C2semidirect
4. #21445438C2semidirect
5. #21445332C2semidirect
6. #21445244C2semidirect
7. #21445164C2semidirect
8. #21445158C2semidirect
9. #21445103C2semidirect
10. #21445080C2semidirect
11. #21444998C2semidirect
12. #21444921C2semidirect
13. #21444660C2semidirect
14. #21444643C2semidirect
18. D4×Dic122C22semidirect
19. #16840167C22semidirect
21. #16840158C22semidirect
23. #16840114C22semidirect
26. #16840044C22semidirect
28. #16840042C22semidirect
39. C8×Dic122C22semidirect
43. D4×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839507C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. #16839483C22semidirect
182. D8×C61SD16semidirect
246. D4×C61C2×SD16semidirect
274. Dic122AGL(1,ℤ/8)semidirect
277. C8×C61C2×SD16semidirect
385. C61#19803semidirect
507. D8#11758200semidirect
573. D4C2×#11758200semidirect
602. C8C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
D8×Dic1222
#214576782
D4×#117582002
#214454382
#214453322
#214452442
#214451642
#214451582
#214451032
#214450802
#214449982
#214449212
#214446602
#214446432
C61×#32722
#1980361