Finite Group Databasebeta

← #29670076

Subgroups — #29670076

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#21457686C211yesmaximal, characteristic, complemented
3#21457415C211yesmaximal, characteristic
4#21445439C211yesmaximal, characteristic, complemented
5#21445330C211yesmaximal, characteristic, complemented
6#21445242C211yesmaximal, characteristic, complemented
7#21445165C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic, complemented
9#21445101C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic, complemented
12#21444920C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17Q8×D244C2211yescharacteristic, complemented
18#16840417C2211yescharacteristic
19#16840165—21no—
20#16840165C2211yescharacteristic, complemented
21#16840158C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic
23#16840113C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840094—21no—
26#16840047C2211yescharacteristic, complemented
27#16840042—21no—
28#16840042C2211yescharacteristic
29#16840039—21no—
30#16840039C2211yescharacteristic, complemented
31#16840006—21no—
32#16839990—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839913—21no—
37#16839905—21no—
38#16839899—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C4×#11758199—21no—
41C4×#11758199C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic, complemented
44#103×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
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—
68C2×Q8×D61—21no—
69#14519803—44no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758199—21no—
75C2×#11758199D411yescharacteristic
76C2×#11758194—21no—
77C2×C8×D61—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519759—41no—
82#14519757—21no—
83#14519757—41no—
84#14519756—21no—
85#14519748—21no—
86C4×D244—21no—
87C4×D244C2311yescharacteristic
88C4⋊C4×D61—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710D411yescharacteristic
94#14519708—21no—
95#14519708—41no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519702—21no—
101#14519702—21no—
102#14519701—21no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
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—
167Q8×D61—42no—
168Q8×D61—81no—
169#11758222—42no—
170#11758222—81no—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—41no—
178SD16×C61—22no—
179SD16×C61—42no—
180SD16×C61—42no—
181SD16×C61—42no—
182SD16×C61—81no—
183C2×C8×C61—21no—
184C2×C8×C61—42no—
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
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
200C22⋊Dic61—41no—
201#11758200—42no—
202#11758200—42no—
203#11758200—81no—
204#11758200—81no—
205#11758199—42no—
206#11758199SD1614yescomplemented
207#11758198—21no—
208#11758198—41no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—41no—
214C4⋊Dic61C2×D411yescharacteristic
215#11758196—41no—
216#11758194—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C2×D411yescharacteristic
222C2×#92418C2×D411yescharacteristic
223C8×D61—81no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—82no—
228C23×C61—82no—
229C23×C61—161no—
230C22×D61—21no—
231Q8×C61—21no—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—82no—
235Q8×C61AGL(1,ℤ/8)12yescomplemented
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—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264D244—21no—
265D244C2×SD1612yescomplemented
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—82no—
269C4×D61—82no—
270Dic122—41no—
271Dic122—41no—
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×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—164no—
295C22×C61D4211yescharacteristic
296C2×D61—22no—
297C2×D61—41no—
298C2×D61—41no—
299C4×C61—22no—
300C4×C61—21no—
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—41no—
310Dic61—81no—
311Dic61—81no—
312Q8×D8—611no—
313#3298—611no—
314D4×SD16D6111yescharacteristic, complemented
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—161no—
330C2×C61#326411yescharacteristic
331C2×C61D4×SD1611yescharacteristic
332C2×C61#328911yescharacteristic
333D61—42no—
334C2×AGL(1,ℤ/8)—1221no—
335C22×SD16—21no—
336D4×Q8—611no—
337D4×Q8C2×D6111yescharacteristic
338#547—611no—
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—1221no—
354#465—21no—
355#465—1221no—
356#461—611no—
357#461—611no—
358#461C2×D6111yescharacteristic
359#460C2×D6111yescharacteristic
360#459—611no—
361#450—21no—
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
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—1221no—
387C22×Q8—21no—
388C22×Q8—1221no—
389C22×D4—21no—
390C22×D4—21no—
391AGL(1,ℤ/8)—2444no—
392C2×SD16—21no—
393C2×SD16—21no—
394C2×SD16—21no—
395C2×SD16—21no—
396C2×SD16—44no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
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—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—21no—
420#121—21no—
421#121—1221no—
422#120—41no—
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429C2×C4⋊C4—1221no—
430#107—611no—
431#107—1221no—
432#107—1221no—
433#107—1221no—
434#107—1221no—
435#106—1221no—
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—1221no—
445#103—1221no—
446#103—1221no—
447#103—1221no—
448#103—1221no—
449#103C22⋊D6111yescharacteristic
450#102—21no—
451#102—611no—
452#102—611no—
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—41no—
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×Q8C2×C22⋊D6111yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—81no—
486C2×D4—81no—
487C2×D4—611no—
488C2×D4—2441no—
489C2×D4D4×D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497SD16—22no—
498SD16—42no—
499SD16—42no—
500SD16—42no—
501SD16—81no—
502SD16—4881no—
503SD16—4881no—
504SD16—2442no—
505SD16—2442no—
506D8—614no—
507D8—2442no—
508M16—2442no—
509C2×C8—21no—
510C2×C8—42no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—4881no—
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—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—41no—
537C22⋊C4—41no—
538C22⋊C4—1221no—
539C22⋊C4—2441no—
540C22⋊C4—2441no—
541C42—21no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—21no—
545C23—41no—
546C23—82no—
547C23—82no—
548C23—161no—
549C23—1221no—
550Q8—21no—
551Q8—41no—
552Q8—41no—
553Q8—82no—
554Q8—4882no—
555Q8—2441no—
556Q8—2441no—
557Q8—2441no—
558Q8—2441no—
559Q8#1451980312yescomplemented
560D4—21no—
561D4—21no—
562D4—21no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—81no—
568D4—82no—
569D4—82no—
570D4—612no—
571D4—1221no—
572D4—4882no—
573D4SD16×D6112yes—
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—82no—
584C2×C4—1221no—
585C2×C4—4882no—
586C2×C4—4882no—
587C2×C4—2441no—
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—81no—
610C22—81no—
611C22—81no—
612C22—164no—
613C22—1222no—
614C22—2441no—
615C22—2441no—
616C22D4×C22⋊D6111yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—4881no—
624C4—4881no—
625C4—2441no—
626C4C2×SD16×D6111yescharacteristic
627C4C2×#1451980311yescharacteristic
628C4#1684056511yescharacteristic
629C4#1684056811yescharacteristic
630C2—42no—
631C2—81no—
632C2—161no—
633C2—2442no—
634C2#2145729511yescharacteristic
635C2SD16×C22⋊D6111yescharacteristic
636C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radical#21457415C2
hypercenterD4×SD16D61
hypocenterC61#19803
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×#11758199C2semidirect
2. #21457686C2semidirect
4. #21445439C2semidirect
5. #21445330C2semidirect
6. #21445242C2semidirect
7. #21445165C2semidirect
8. #21445157C2semidirect
9. #21445101C2semidirect
10. #21445082C2semidirect
11. #21444998C2semidirect
12. #21444920C2semidirect
13. #21444663C2semidirect
14. #21444641C2semidirect
15. D4×SD16×C61C2semidirect
17. Q8×D244C22semidirect
20. #16840165C22semidirect
21. #16840158C22semidirect
23. #16840113C22semidirect
26. #16840047C22semidirect
30. #16840039C22semidirect
39. Q8×#92418C22semidirect
43. C8×D244C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
206. #11758199SD16semidirect
235. Q8×C61AGL(1,ℤ/8)semidirect
265. D244C2×SD16semidirect
283. #92418C2×SD16semidirect
314. D4×SD16D61semidirect
385. C61#19803semidirect
559. Q8#14519803semidirect

Maximal subgroups

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

subgroupindex
Q8×#117581992
#214576862
#214574152
#214454392
#214453302
#214452422
#214451652
#214451572
#214451012
#214450822
#214449982
#214449202
#214446632
#214446412
D4×SD16×C612
#1980361