Finite Group Databasebeta

← #29650762

Subgroups — #29650762

Order 15616.

726 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#21456395C211yesmaximal, characteristic
2#21454528C211yesmaximal, characteristic, complemented
3#21454526C211yesmaximal, characteristic, complemented
4#21444686C211yesmaximal, characteristic, complemented
5#21444685C211yesmaximal, characteristic, complemented
6#21444684C211yesmaximal, characteristic, complemented
7#21444683C211yesmaximal, characteristic, complemented
8#21444682C211yesmaximal, characteristic, complemented
9#21444681C211yesmaximal, characteristic, complemented
10#21444680C211yesmaximal, characteristic, complemented
11#21444573C211yesmaximal, characteristic, complemented
12#21444268C211yesmaximal, characteristic, complemented
13#21444268C211yesmaximal, characteristic, complemented
14#322×D61C211yesmaximal, characteristic, complemented
15C61×#1586C211yesmaximal, characteristic, complemented
16C2×#14519785—21no—
17C2×#14519783—21no—
18#16840309C2211yescharacteristic
19#16840309C2211yescharacteristic
20#16840166C411yescharacteristic
21#16840137C411yescharacteristic
22#16840107C411yescharacteristic
23#16840107C411yescharacteristic
24#16839999C411yescharacteristic
25#16839997C411yescharacteristic
26#16839997C411yescharacteristic
27#16839995C411yescharacteristic
28C4×D488C2211yescharacteristic
29C4×#11758199C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31#16839841C2211yescharacteristic
32#16839840C2211yescharacteristic
33#16839839C2211yescharacteristic
34#16839838—21no—
35#16839837—21no—
36C8×D244C2211yescharacteristic
37D4×#92418C2211yescharacteristic
38#16839827C2211yescharacteristic
39C8×C22⋊D61—21no—
40#16839825—21no—
41#16839784C2211yescharacteristic
42#16839784C2211yescharacteristic
43#16839727—21no—
44#16839727—21no—
45C4×C8×D61C2211yescharacteristic
46C4×C8×D61C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839513C2211yescharacteristic
49#16839512C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839491C2211yescharacteristic
52#16839491C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839489C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839484C2211yescharacteristic
59#16839483C2211yescharacteristic
60C4×D8×C61C2211yescharacteristic
61C61×#436C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#334C2211yescharacteristic
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222—21no—
69C2×D488C2×C411yescharacteristic
70C2×#11758199C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72#14519785—44no—
73#14519783—44no—
74C2×#11758194—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C2×#11758189—21no—
79C22×#92418—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83C2×C8×D61C2×C411yescharacteristic
84C2×C8×D61C2×C411yescharacteristic
85C2×C8×D61D411yescharacteristic
86C2×C8×D61D411yescharacteristic
87#14519776C2×C411yescharacteristic
88#14519776C2×C411yescharacteristic
89#14519773C2×C411yescharacteristic
90#14519773C2×C411yescharacteristic
91#14519763—21no—
92#14519763—21no—
93#14519752C2×C411yescharacteristic
94#14519752C2×C411yescharacteristic
95#14519750—21no—
96#14519750—21no—
97#14519748C2×C411yescharacteristic
98#14519748C2×C411yescharacteristic
99C4×D244C2311yescharacteristic
100C4×D244C2311yescharacteristic
101C4×C22⋊D61—21no—
102C4×C22⋊D61—21no—
103C42×D61C2311yescharacteristic
104#14519711C2×C411yescharacteristic
105#14519710C2×C411yescharacteristic
106#14519710C2×C411yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519699C2×C411yescharacteristic
112#14519699C2×C411yescharacteristic
113#14519698C2×C411yescharacteristic
114#14519698C2×C411yescharacteristic
115#14519697C2×C411yescharacteristic
116#14519697C2×C411yescharacteristic
117C22⋊#92418—21no—
118C22⋊#92418—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689—21no—
122#14519689—21no—
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61C2×C411yescharacteristic
128C8×Dic61C2×C411yescharacteristic
129C8×Dic61D411yescharacteristic
130C8×Dic61D411yescharacteristic
131C2×D8×C61C2×C411yescharacteristic
132C2×M16×C61—21no—
133C22×C8×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#107×C61C2×C411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61C2×C411yescharacteristic
140#102×C61C2×C411yescharacteristic
141C22⋊C8×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C4×C8×C61C2311yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—21no—
156#11758222—42no—
157#11758222—42no—
158#11758222—42no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C411yescharacteristic
163C2×D244C22×C411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61C22×C411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61#4111yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171D8×C61—41no—
172M16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C4⋊C4×C61C22×C411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188#11758199—41no—
189#11758199—41no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196#11758196—21no—
197#11758196—21no—
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201#11758194—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—42no—
206C2×#92418C22×C411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210D488—41no—
211#11758189—42no—
212#11758189—42no—
213#11758189—42no—
214#11758189—42no—
215C8×D61—24no—
216C8×D61—22no—
217C8×D61—22no—
218C8×D61—41no—
219C8×D61—41no—
220C8×D61—42no—
221C8×D61—42no—
222C23×C61—21no—
223C23×C61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61C4×D411yescharacteristic
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—42no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61C4×D411yescharacteristic
249C2×Dic61C4×D411yescharacteristic
250C2×Dic61C4×D411yescharacteristic
251D244—21no—
252D244—21no—
253D244—21no—
254D244—21no—
255D244—42no—
256D244—42no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—22no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61AGL(1,ℤ/8)12yes—
265C4×D61AGL(1,ℤ/8)12yes—
266Dic122—42no—
267Dic122—42no—
268C8×C61—22no—
269C8×C61—21no—
270C8×C61—21no—
271C8×C61—21no—
272C8×C61—21no—
273C8×C61—41no—
274C8×C61—41no—
275#92418—22no—
276#92418—21no—
277#92418—21no—
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#92418—41no—
282#17981—611nomaximal, Sylow
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61C2×C4×D411yescharacteristic
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61#44212yes—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61#56811yescharacteristic
303C4×C61#56811yescharacteristic
304C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
306Dic61—21no—
307Dic61—41no—
308Dic61—41no—
309Dic61#44212yes—
310Dic61#44312yes—
311#3169—611no—
312#2931—611no—
313#2931—611no—
314#1586—611no—
315#1586—611no—
316#1586—611no—
317#1586—611no—
318#1586—611no—
319#1586—611no—
320#1586—611no—
321#1586D6111yescharacteristic, complemented
322#1566—611no—
323#1492—611no—
324#1492—611no—
325C2×#322—611no—
326C2×C61—81no—
327C2×C61—81no—
328C2×C61#293211yescharacteristic
329C2×C61C2×#44211yescharacteristic
330C2×C61C2×#44311yescharacteristic
331D61—22no—
332D61—81no—
333D61—81no—
334C2×#131—1221no—
335C2×#131—1221no—
336#532—611no—
337#532—611no—
338#500—611no—
339#493—611no—
340#486—611no—
341#486—611no—
342#466—611no—
343#466—611no—
344#466—611no—
345#466—611no—
346C4×D8—611no—
347C4×D8—611no—
348C4×D8—611no—
349C4×D8C2×D6111yescharacteristic
350#436—611no—
351#436—611no—
352#436—611no—
353#436—1221no—
354#436—1221no—
355#436C2×D6111yescharacteristic
356C8×D4—611no—
357C8×D4—611no—
358C8×D4—611no—
359C8×D4—1221no—
360C8×D4—1221no—
361C8×D4C2×D6111yescharacteristic
362#423—611no—
363#423—611no—
364#408—1221no—
365#408—1221no—
366C2×C4×C8—611no—
367C2×C4×C8—611no—
368C2×C4×C8—611no—
369#334—611no—
370#334—611no—
371#334—611no—
372#334C2×D6111yescharacteristic
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8C2×D6111yescharacteristic
380D4⋊C8C2×D6111yescharacteristic
381#322—611no—
382#322—611no—
383#322—611no—
384#322C2×D6111yescharacteristic
385C61#1798111yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C2×#41—1221no—
388C2×D8—611no—
389C2×D8—611no—
390C2×D8—611no—
391C2×D8C4×D6111yescharacteristic
392#131—2444no—
393#131—2444no—
394C2×M16—21no—
395C2×M16—1221no—
396C2×M16—1221no—
397C2×M16—1221no—
398C2×M16—1221no—
399C2×M16—1221no—
400C22×C8—21no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—1221no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8—1221no—
409C4⋊Q8—611no—
410C4⋊Q8—611no—
411#127—611no—
412#127—611no—
413#124—1221no—
414#124—1221no—
415#121—611no—
416#121—611no—
417#121—611no—
418#121—611no—
419#121—1221no—
420#121—1221no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4C22×D6111yescharacteristic
426C4×D4C22×D6111yescharacteristic
427C2×C42—611no—
428#107—611no—
429#107—611no—
430#107—611no—
431#107C4×D6111yescharacteristic
432C4⋊C8—611no—
433C4⋊C8—611no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C4⋊C8C22×D6111yescharacteristic
437C4⋊C8C22×D6111yescharacteristic
438#102—611no—
439#102—611no—
440#102—611no—
441#102—611no—
442#102—611no—
443#102—611no—
444#102C4×D6111yescharacteristic
445#102C4×D6111yescharacteristic
446C22⋊C8—21no—
447C22⋊C8—21no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—611no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8—1221no—
462C4×C8C22×D6111yescharacteristic
463C4×C8C22×D6111yescharacteristic
464C4×C8C22×D6111yescharacteristic
465#41—2442no—
466#41—2442no—
467#41—2442no—
468#41—2442no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×D4—611no—
472C2×D4—611no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4C2×C4×D6111yescharacteristic
482C2×D4C2×C4×D6111yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—1221no—
489C22×C4—1221no—
490D8—41no—
491D8—2441no—
492D8—2441no—
493D8—2441no—
494M16—42no—
495M16—2442no—
496M16—2442no—
497M16—2442no—
498M16—2442no—
499M16—2442no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—42no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1224no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—2441no—
515C2×C8—2441no—
516C2×C8—2442no—
517C2×C8—2442no—
518C2×C8—2442no—
519C2×C8C2×C4×D6111yescharacteristic
520C2×C8C2×C4×D6111yescharacteristic
521C2×C8D4×D6111yescharacteristic
522C2×C8D4×D6111yescharacteristic
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4C2×C4×D6111yescharacteristic
528C4⋊C4C2×C4×D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42C23×D6111yescharacteristic
541C23—21no—
542C23—21no—
543C23—611no—
544C23—1221no—
545C23—1221no—
546Q8—2442no—
547Q8—2442no—
548D4—21no—
549D4—21no—
550D4—21no—
551D4—21no—
552D4—1221no—
553D4—1221no—
554D4—1221no—
555D4—1221no—
556D4—2442no—
557D4—2442no—
558D4—2442no—
559D4—2442no—
560D4—2442no—
561D4—2442no—
562D4—2442no—
563D4—2442no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—611no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C22×C4×D6111yescharacteristic
585C2×C4C2×D4×D6111yescharacteristic
586C2×C4#41×D6111yescharacteristic
587C8—22no—
588C8—21no—
589C8—21no—
590C8—21no—
591C8—21no—
592C8—41no—
593C8—41no—
594C8—1222no—
595C8—1221no—
596C8—1221no—
597C8—1221no—
598C8—1221no—
599C8—2441no—
600C8—2441no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—612no—
608C22—1221no—
609C22—1221no—
610C22—2441no—
611C22—2441no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22C4×D4×D6111yescharacteristic
617C4—21no—
618C4—41no—
619C4—41no—
620C4—612no—
621C4—612no—
622C4—1221no—
623C4—2441no—
624C4—2441no—
625C4#1684052711yescharacteristic
626C4#1684052811yescharacteristic
627C4AGL(1,ℤ/8)×D6111yescharacteristic
628C4AGL(1,ℤ/8)×D6111yescharacteristic
629C2—81no—
630C2—81no—
631C2—1222no—
632C2—4881no—
633C2—4881no—
634C2#2145453311yescharacteristic
635C2#442×D6111yescharacteristic
636C2#443×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1586C2
socleC22×C61C2×C4×D4
prime radical#21456395C2
cosocleC42×C61C24
hypercenter#1586D61
hypocenterC61#17981
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #21454528C2semidirect
3. #21454526C2semidirect
4. #21444686C2semidirect
5. #21444685C2semidirect
6. #21444684C2semidirect
7. #21444683C2semidirect
8. #21444682C2semidirect
9. #21444681C2semidirect
10. #21444680C2semidirect
11. #21444573C2semidirect
12. #21444268C2semidirect
13. #21444268C2semidirect
14. #322×D61C2semidirect
15. C61×#1586C2semidirect
321. #1586D61semidirect
385. C61#17981semidirect

Maximal subgroups

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

subgroupindex
#214563952
#214545282
#214545262
#214446862
#214446852
#214446842
#214446832
#214446822
#214446812
#214446802
#214445732
#214442682
#322×D612
C61×#15862
#1798161