Finite Group Databasebeta

← #29650717

Subgroups — #29650717

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#21444651C211yesmaximal, characteristic, complemented
5#21444650C211yesmaximal, characteristic, complemented
6#21444649C211yesmaximal, characteristic, complemented
7#21444648C211yesmaximal, characteristic, complemented
8#21444647C211yesmaximal, characteristic, complemented
9#21444646C211yesmaximal, characteristic, complemented
10#21444645C211yesmaximal, characteristic, complemented
11#21444576C211yesmaximal, characteristic, complemented
12#21444268C211yesmaximal, characteristic, complemented
13#21444268C211yesmaximal, characteristic, complemented
14#322×D61C211yesmaximal, characteristic, complemented
15C61×#1581C211yesmaximal, 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#16839515C2211yescharacteristic
49#16839514C2211yescharacteristic
50#16839514C2211yescharacteristic
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×#335C2211yescharacteristic
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—22no—
216C8×D61—41no—
217C8×D61—41no—
218C8×D61—42no—
219C8×D61—42no—
220C8×D61—42no—
221C8×D61D814yescomplemented
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×D61C2×D812yes—
265C4×D61#13512yes—
266Dic122—42no—
267Dic122—42no—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—21no—
271C8×C61—41no—
272C8×C61—41no—
273C8×C61—41no—
274C8×C61C2×D812yes—
275#92418—21no—
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#92418C2×D812yes—
282#17975—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×D61C4×D812yes—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61#56811yescharacteristic
303C4×C61#56811yescharacteristic
304C4×C61C22×D811yescharacteristic
305C4×C61C2×#13511yescharacteristic
306Dic61—21no—
307Dic61—41no—
308Dic61—41no—
309Dic61C4×D812yes—
310Dic61#44412yes—
311#3169—611no—
312#2931—611no—
313#2931—611no—
314#1581—611no—
315#1581—611no—
316#1581—611no—
317#1581—611no—
318#1581—611no—
319#1581—611no—
320#1581—611no—
321#1581D6111yescharacteristic, complemented
322#1567—611no—
323#1492—611no—
324#1492—611no—
325C2×#322—611no—
326C2×C61—81no—
327C2×C61—81no—
328C2×C61#293211yescharacteristic
329C2×C61C2×C4×D811yescharacteristic
330C2×C61C2×#44411yescharacteristic
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#335—611no—
370#335—611no—
371#335—611no—
372#335C2×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#1797511yescharacteristic, 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—614no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—1222no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—2441no—
514C2×C8—2441no—
515C2×C8—2442no—
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—21no—
588C8—21no—
589C8—21no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—612no—
594C8—1221no—
595C8—1221no—
596C8—1221no—
597C8—2441no—
598C8—2441no—
599C8—2441no—
600C8D8×D6112yes—
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
627C4C2×D8×D6111yescharacteristic
628C4#135×D6111yescharacteristic
629C2—81no—
630C2—81no—
631C2—1222no—
632C2—4881no—
633C2—4881no—
634C2#2145453311yescharacteristic
635C2C4×D8×D6111yescharacteristic
636C2#444×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1581C2
socleC22×C61C2×C4×D4
prime radical#21456395C2
cosocleC42×C61C24
hypercenter#1581D61
hypocenterC61#17975
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #21454528C2semidirect
3. #21454526C2semidirect
4. #21444651C2semidirect
5. #21444650C2semidirect
6. #21444649C2semidirect
7. #21444648C2semidirect
8. #21444647C2semidirect
9. #21444646C2semidirect
10. #21444645C2semidirect
11. #21444576C2semidirect
12. #21444268C2semidirect
13. #21444268C2semidirect
14. #322×D61C2semidirect
15. C61×#1581C2semidirect
221. C8×D61D8semidirect
321. #1581D61semidirect
385. C61#17975semidirect

Maximal subgroups

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

subgroupindex
#214563952
#214545282
#214545262
#214446512
#214446502
#214446492
#214446482
#214446472
#214446462
#214446452
#214445762
#214442682
#322×D612
C61×#15812
#1797561