Finite Group Databasebeta

← #29669481

Subgroups — #29669481

Order 15616.

756 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458718C211yesmaximal, characteristic, complemented
2SD16×D244C211yesmaximal, characteristic, complemented
3D4×#11758199C211yesmaximal, characteristic
4#21445449C211yesmaximal, characteristic, complemented
5#21445301C211yesmaximal, characteristic, complemented
6#21445225C211yesmaximal, characteristic, complemented
7#21445196C211yesmaximal, characteristic, complemented
8#21445160C211yesmaximal, characteristic, complemented
9#21444998C211yesmaximal, characteristic, complemented
10#21444934C211yesmaximal, characteristic, complemented
11#21444920C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3295C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17C22×#11758199—21no—
18Q8×D244C2211yescharacteristic, complemented
19D4×D244C2211yescharacteristic
20#16840172—21no—
21#16840158C2211yescharacteristic, complemented
22#16840138C2211yescharacteristic
23#16840136C2211yescharacteristic, complemented
24#16840135C2211yescharacteristic, complemented
25#16840108C2211yescharacteristic, complemented
26#16840059—21no—
27#16840052—21no—
28#16840049C2211yescharacteristic, complemented
29#16840042C2211yescharacteristic, complemented
30#16839996—21no—
31#16839992—21no—
32#16839967C2211yescharacteristic
33#16839962C2211yescharacteristic, complemented
34#16839957C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839914—21no—
37#16839902—21no—
38#16839896—21no—
39#16839895—21no—
40#16839893C2211yescharacteristic, complemented
41#16839872—21no—
42C4×#11758199C2211yescharacteristic
43C8×D244C2211yescharacteristic, complemented
44D4×#92418C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic, complemented
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic, complemented
59C61×#481—21no—
60C61×#476C2211yescharacteristic
61C61×#465—21no—
62C61×#460C2211yescharacteristic, complemented
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic, complemented
66C2×#102×C61—21no—
67C2×Q8×D61—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74C2×#11758201—41no—
75SD16×D61—44no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—44no—
81C2×#11758199D411yescharacteristic
82C22×#92418—21no—
83C2×C8×D61—21no—
84#14519777—21no—
85#14519777C2311yescharacteristic
86#14519774C2311yescharacteristic
87#14519757—21no—
88#14519757—41no—
89#14519752—21no—
90#14519751—21no—
91#14519747—41no—
92C4×D244—21no—
93C4×D244C2311yescharacteristic
94#14519710—21no—
95#14519710D411yescharacteristic
96#14519708—21no—
97#14519708—21no—
98#14519708—21no—
99C4⋊#92418—21no—
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519700—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697—21no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113#14519689—21no—
114#14519688—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61D411yescharacteristic
121C22×C8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61C2311yescharacteristic
124#125×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127#121×C61—21no—
128#121×C61—21no—
129#120×C61—41no—
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×C61D411yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147C23×D61—41no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
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—81no—
158C2×D4×C61—81no—
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—41no—
164C2×C22⋊D61—41no—
165Q8×D61—42no—
166Q8×D61—81no—
167D4×D61—42no—
168D4×D61—81no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—42no—
173C2×D244—41no—
174C2×D244—81no—
175C2×D244C2×D411yescharacteristic
176C2×C4×D61—21no—
177C2×C4×D61—41no—
178C2×Dic122—21no—
179C2×Dic122—41no—
180SD16×C61—22no—
181SD16×C61—81no—
182C2×C8×C61—21no—
183C2×C8×C61—81no—
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×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—82no—
202#11758200—42no—
203#11758199—42no—
204#11758199—42no—
205#11758199—42no—
206#11758199—81no—
207#11758199SD1614yescomplemented
208#11758198—21no—
209#11758198—41no—
210#11758198—41no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—41no—
214C4⋊Dic61C2×D411yescharacteristic
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—42no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222#11758190—42no—
223#11758190—81no—
224C8×D61—42no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—41no—
228C23×C61—82no—
229C23×C61—82no—
230C23×C61—161no—
231C22×D61—21no—
232C22×D61—41no—
233C22×D61—82no—
234C22×D61—82no—
235C22×D61—161no—
236Q8×C61—21no—
237Q8×C61—21no—
238Q8×C61—81no—
239D4×C61—21no—
240D4×C61—21no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—81no—
246D4×C61—82no—
247D4×C61—82no—
248D4×C61C2×SD1612yescomplemented
249D4×C61AGL(1,ℤ/8)12yes—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—161no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C22⋊D61—82no—
264C2×Dic61—21no—
265C2×Dic61—41no—
266D244—21no—
267D244—41no—
268D244—41no—
269D244—41no—
270D244—41no—
271D244—81no—
272D244—82no—
273D244C2×SD1612yes—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—82no—
277Dic122—41no—
278Dic122—41no—
279Dic122—82no—
280C8×C61—21no—
281C8×C61—21no—
282C8×C61—41no—
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418C2×SD1612yescomplemented
289#19763—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—164no—
300C22×C61D4211yescharacteristic
301C2×D61—22no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—164no—
308C4×C61—21no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61C22×SD1611yescharacteristic
315C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
316C4×C61#57611yescharacteristic
317C4×C61#57711yescharacteristic
318Dic61—41no—
319Dic61—81no—
320#3384—611no—
321#3295D6111yescharacteristic, complemented
322D4×SD16—611no—
323D4×D8—611no—
324#1715—611no—
325#1688—611no—
326#1677—611no—
327#1671—611no—
328#1665—611no—
329#1639—611no—
330#1629—611no—
331#1627—611no—
332#1622—611no—
333#1582—611no—
334#1580—611no—
335C2×C61—42no—
336C2×C61—42no—
337C2×C61—161no—
338C2×C61#326411yescharacteristic
339C2×C61D4×SD1611yescharacteristic
340C2×C61#328811yescharacteristic
341D61—42no—
342D61—161no—
343C22×SD16—1221no—
344C22×D8—1221no—
345#554C2×D6111yescharacteristic, complemented
346D4×Q8—611no—
347D42—611no—
348D42C2×D6111yescharacteristic
349#501—1221no—
350#498—611no—
351#493—611no—
352#492—611no—
353#492—611no—
354#486—611no—
355#486C2×D6111yescharacteristic, complemented
356#481—21no—
357#478—1221no—
358#476—611no—
359#476—1221no—
360#476C2×D6111yescharacteristic
361#474—611no—
362#466—1221no—
363#465—21no—
364#465—1221no—
365#461—611no—
366#460—611no—
367#460C2×D6111yescharacteristic, complemented
368#459—611no—
369#459—611no—
370#450—21no—
371#450—1221no—
372#448—1221no—
373#447—1221no—
374#447—1221no—
375#446—611no—
376#442—1221no—
377C4×SD16C2×D6111yescharacteristic
378C4×D8—611no—
379C8×D4—611no—
380C8×D4—611no—
381C8×D4C2×D6111yescharacteristic, complemented
382C2×#102—21no—
383#334—611no—
384#334—611no—
385#333—611no—
386#332—611no—
387#331—611no—
388#331—611no—
389Q8⋊C8—611no—
390D4⋊C8—611no—
391D4⋊C8—611no—
392D4⋊C8—611no—
393#322—611no—
394C61#1976311yescharacteristic, complemented, Sylow
395C22×Q8—1221no—
396C22×D4—21no—
397C22×D4—21no—
398C22×D4—1221no—
399C22×D4—1221no—
400C2×SD16—21no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—2441no—
406C2×SD16—2444no—
407C2×SD16C22⋊D6111yescharacteristic
408C2×D8—611no—
409C2×D8—1221no—
410C2×D8—1221no—
411C2×D8—1221no—
412C2×D8—1221no—
413C2×D8—2444no—
414C22×C8—21no—
415C22×C8—1221no—
416C22×C8—1221no—
417C4⋊Q8—611no—
418C4⋊Q8—1221no—
419C4⋊Q8C22×D6111yescharacteristic
420#127—611no—
421#127C22×D6111yescharacteristic
422#125—21no—
423C22⋊Q8—21no—
424C22⋊Q8—41no—
425C22⋊Q8—1221no—
426C22⋊Q8—2441no—
427#121—21no—
428#121—21no—
429#121—1221no—
430#121—1221no—
431#120—41no—
432#120—2441no—
433C4×Q8C22×D6111yescharacteristic
434C4×D4—21no—
435C4×D4—611no—
436C4×D4—1221no—
437C4×D4C22×D6111yescharacteristic
438C4×D4C22×D6111yescharacteristic
439C2×C4⋊C4—21no—
440#107—611no—
441#107—1221no—
442#106—21no—
443#106—1221no—
444#106—1221no—
445#106—1221no—
446#106C22⋊D6111yescharacteristic
447C4⋊C8—611no—
448C4⋊C8—611no—
449C4⋊C8—611no—
450C4⋊C8—1221no—
451C4⋊C8C22×D6111yescharacteristic
452#103—1221no—
453#103C22⋊D6111yescharacteristic
454#102—21no—
455#102—21no—
456#102—21no—
457#102—21no—
458#102—611no—
459#102—611no—
460#102—1221no—
461#102—1221no—
462#102—1221no—
463#102—1221no—
464#102—1221no—
465#102—2441no—
466#102C22⋊D6111yescharacteristic
467C22⋊C8—21no—
468C22⋊C8—1221no—
469C22⋊C8—1221no—
470#97—1221no—
471C4×C8—611no—
472C4×C8—611no—
473C4×C8C22×D6111yescharacteristic
474C24—41no—
475C24—2441no—
476C2×Q8—21no—
477C2×Q8—1221no—
478C2×Q8—4881no—
479C2×Q8—2441no—
480C2×Q8—2442no—
481C2×Q8C2×C22⋊D6111yescharacteristic
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—81no—
490C2×D4—81no—
491C2×D4—611no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—4881no—
496C2×D4—4881no—
497C2×D4—2441no—
498C2×D4—2442no—
499C2×D4—2441no—
500C2×D4—2442no—
501C2×D4D4×D6111yescharacteristic
502C2×D4C2×C22⋊D6111yescharacteristic
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—41no—
506C22×C4—1221no—
507C22×C4—2441no—
508SD16—22no—
509SD16—81no—
510SD16—4881no—
511SD16—4882no—
512SD16—2442no—
513SD16—2442no—
514SD16—2442no—
515D8—614no—
516D8—4881no—
517D8—2442no—
518D8—2442no—
519D8—2442no—
520C2×C8—21no—
521C2×C8—81no—
522C2×C8—611no—
523C2×C8—611no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—2442no—
529C2×C8—2442no—
530C2×C8C2×C22⋊D6111yescharacteristic
531C2×C8C2×C22⋊D6111yescharacteristic
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—611no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2441no—
541C4⋊C4D4×D6111yescharacteristic
542C4⋊C4C2×C22⋊D6111yescharacteristic
543C4⋊C4C2×C22⋊D6111yescharacteristic
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—1221no—
549C22⋊C4—2441no—
550C22⋊C4—2441no—
551C42—21no—
552C42C23×D6111yescharacteristic
553C23—21no—
554C23—21no—
555C23—41no—
556C23—82no—
557C23—82no—
558C23—161no—
559C23—1221no—
560C23—9761no—
561C23—4882no—
562C23—4882no—
563C23—2441no—
564Q8—21no—
565Q8—21no—
566Q8—81no—
567Q8—4882no—
568Q8—2441no—
569Q8—2441no—
570D4—21no—
571D4—21no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—81no—
577D4—82no—
578D4—82no—
579D4—612no—
580D4—1221no—
581D4—4881no—
582D4—4882no—
583D4—4882no—
584D4—2441no—
585D4—2441no—
586D4—2441no—
587D4—2441no—
588D4SD16×D6112yescomplemented
589D4#1451980212yes—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—161no—
600C2×C4—1221no—
601C2×C4—4882no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4#1451982211yescharacteristic
608C8—21no—
609C8—21no—
610C8—41no—
611C8—612no—
612C8—1221no—
613C8—2441no—
614C8—2441no—
615C8—2441no—
616C8—2441no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—164no—
627C22—1222no—
628C22—9764no—
629C22—4881no—
630C22—4881no—
631C22—4881no—
632C22—2441no—
633C22—2441no—
634C22D4×C22⋊D6111yescharacteristic
635C4—21no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—4881no—
642C4—2441no—
643C4C2×SD16×D6111yescharacteristic
644C4C2×#1451980211yescharacteristic
645C4#1684056511yescharacteristic
646C4#1684056811yescharacteristic
647C2—42no—
648C2—42no—
649C2—161no—
650C2—9761no—
651C2—2442no—
652C2#2145730011yescharacteristic
653C2SD16×C22⋊D6111yescharacteristic
654C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3295C2
socleC22×C61D42
prime radicalD4×#11758199C2
hypercenter#3295D61
hypocenterC61#19763
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3295

Complements

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

NH ≅ G/Ntype
1. #21458718C2semidirect
2. SD16×D244C2semidirect
4. #21445449C2semidirect
5. #21445301C2semidirect
6. #21445225C2semidirect
7. #21445196C2semidirect
8. #21445160C2semidirect
9. #21444998C2semidirect
10. #21444934C2semidirect
11. #21444920C2semidirect
12. #21444891C2semidirect
13. #21444656C2semidirect
14. #21444641C2semidirect
15. C61×#3295C2semidirect
18. Q8×D244C22semidirect
21. #16840158C22semidirect
23. #16840136C22semidirect
24. #16840135C22semidirect
25. #16840108C22semidirect
28. #16840049C22semidirect
29. #16840042C22semidirect
33. #16839962C22semidirect
40. #16839893C22semidirect
43. C8×D244C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
49. #16839506C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
56. C61×#554C22semidirect
58. C61×#486C22semidirect
62. C61×#460C22semidirect
65. C8×D4×C61C22semidirect
207. #11758199SD16semidirect
248. D4×C61C2×SD16semidirect
288. #92418C2×SD16semidirect
321. #3295D61semidirect
345. #554C2×D61semidirect
355. #486C2×D61semidirect
367. #460C2×D61semidirect
381. C8×D4C2×D61semidirect
394. C61#19763semidirect
588. D4SD16×D61semidirect

Maximal subgroups

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

subgroupindex
#214587182
SD16×D2442
D4×#117581992
#214454492
#214453012
#214452252
#214451962
#214451602
#214449982
#214449342
#214449202
#214448912
#214446562
#214446412
C61×#32952
#1976361