Finite Group Databasebeta

← #29649945

Subgroups — #29649945

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455817C211yesmaximal, characteristic
2#21454526C211yesmaximal, characteristic, complemented
3C8×D4×D61C211yesmaximal, characteristic, complemented
4#21444686C211yesmaximal, characteristic, complemented
5#21444684C211yesmaximal, characteristic, complemented
6#21444682C211yesmaximal, characteristic, complemented
7#21444680C211yesmaximal, characteristic, complemented
8#21444644C211yesmaximal, characteristic, complemented
9#21444641C211yesmaximal, characteristic, complemented
10#21444641C211yesmaximal, characteristic, complemented
11#21444588C211yesmaximal, characteristic, complemented
12#21444242C211yesmaximal, characteristic, complemented
13#21444216C211yesmaximal, characteristic, complemented
14#21444110C211yesmaximal, characteristic, complemented
15C61×#1580C211yesmaximal, characteristic, complemented
16C2×#14519783—21no—
17C22×C8×D61—21no—
18#16840309C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840077C411yescharacteristic
21#16840074C411yescharacteristic
22#16840041C411yescharacteristic
23#16840040C411yescharacteristic
24#16839956C411yescharacteristic
25#16839955C411yescharacteristic
26#16839897C411yescharacteristic
27#16839894C411yescharacteristic
28C4×D488C2211yescharacteristic
29C4×#11758199C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31#16839840C2211yescharacteristic, complemented
32#16839839C2211yescharacteristic, complemented
33#16839837—21no—
34C8×D244C2211yescharacteristic
35C8×D244C2211yescharacteristic, complemented
36D4×#92418C2211yescharacteristic
37#16839827C2211yescharacteristic
38C8×C22⋊D61—21no—
39C8×C22⋊D61—21no—
40#16839825—21no—
41#16839813C2211yescharacteristic
42#16839813C2211yescharacteristic
43#16839784C2211yescharacteristic
44C4⋊C8×D61C2211yescharacteristic
45#16839727—21no—
46C22⋊C8×D61—21no—
47C4×C8×D61C2211yescharacteristic
48#16839513C2211yescharacteristic, complemented
49#16839512C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839491C2211yescharacteristic
52#16839491C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55#16839489C2211yescharacteristic
56#16839489C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59#16839483C2211yescharacteristic, complemented
60C4×D8×C61C2211yescharacteristic
61C8×D4×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic, complemented
63C61×#334C2211yescharacteristic, complemented
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic, complemented
66C61×#322C2211yescharacteristic, complemented
67C2×#11758222—21no—
68C2×D4×D61C2×C411yescharacteristic
69C22×C4×D61—21no—
70C2×D488C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758199C2×C411yescharacteristic
73#14519783—44no—
74C2×#11758194—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418—21no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—44no—
83C2×C8×D61D411yescharacteristic
84C2×C8×D61D411yescharacteristic
85#14519776C2×C411yescharacteristic
86#14519773C2×C411yescharacteristic
87#14519763—21no—
88#14519752C2×C411yescharacteristic
89#14519750—21no—
90#14519748C2×C411yescharacteristic
91C4×D244C2311yescharacteristic
92C4×D244C2311yescharacteristic
93D4×Dic61C2×C411yescharacteristic
94#14519739C2×C411yescharacteristic
95C4×C22⋊D61—21no—
96C4×C22⋊D61—21no—
97#14519737—21no—
98C4⋊C4×D61C2×C411yescharacteristic
99C22⋊C4×D61—21no—
100C42×D61C2311yescharacteristic
101#14519711C2×C411yescharacteristic
102#14519710C2×C411yescharacteristic
103#14519710C2×C411yescharacteristic
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706C2×C411yescharacteristic
110#14519706C2×C411yescharacteristic
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—
123#14519689—21no—
124#14519689C2×C411yescharacteristic
125#14519689C2×C411yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61D411yescharacteristic
130C8×Dic61D411yescharacteristic
131C2×D8×C61C2×C411yescharacteristic
132C22×C8×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
146C23×D61—21no—
147C2×D4×C61C22×C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C22×Dic61—21no—
156D4×D61—22no—
157D4×D61—22no—
158#11758222—42no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244C22×C411yescharacteristic
162C2×D244C22×C411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61C22×C411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61#4111yescharacteristic
172C2×Dic122—21no—
173D8×C61—22no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—42no—
178C2×C8×C61—42no—
179C2×C8×C61C22×C411yescharacteristic
180C2×C8×C61C22×C411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—21no—
189C22⋊Dic61—21no—
190#11758199—22no—
191#11758199—22no—
192#11758198—21no—
193#11758198—21no—
194#11758198—21no—
195#11758198—21no—
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200C4×Dic61C22×C411yescharacteristic
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61#4111yescharacteristic
203#11758194—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212D488—22no—
213#11758189—42no—
214#11758189—42no—
215C8×D61—22no—
216C8×D61—22no—
217C8×D61—42no—
218C8×D61—42no—
219C8×D61—42no—
220C8×D61—42no—
221C23×C61—21no—
222C23×C61—21no—
223C22×D61—24no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61C4×D411yescharacteristic
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61#13112yescomplemented
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—24no—
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—42no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—42no—
249C2×Dic61C4×D411yescharacteristic
250C2×Dic61C4×D411yescharacteristic
251C2×Dic61C4×D411yescharacteristic
252D244—21no—
253D244—21no—
254D244—21no—
255D244—42no—
256D244#13112yescomplemented
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—41no—
260C4×D61—42no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—42no—
266C4×D61AGL(1,ℤ/8)12yes—
267C4×D61AGL(1,ℤ/8)12yes—
268Dic122—42no—
269C8×C61—21no—
270C8×C61—21no—
271C8×C61—21no—
272C8×C61—21no—
273C8×C61—41no—
274C8×C61—41no—
275C8×C61—41no—
276#92418—21no—
277#92418—21no—
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#17907—611nomaximal, Sylow
284C22×C61—22no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61C2×C4×D411yescharacteristic
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—22no—
294C2×D61—48no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61#44212yes—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61C2×#13111yescharacteristic
305C4×C61C2×#13111yescharacteristic
306C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308Dic61—21no—
309Dic61—21no—
310Dic61—41no—
311Dic61—41no—
312Dic61#44212yes—
313#3100—611no—
314#2931—611no—
315C2×C8×D4—611no—
316#1586—611no—
317#1586—611no—
318#1586—611no—
319#1586—611no—
320#1580—611no—
321#1580—611no—
322#1580—611no—
323#1580D6111yescharacteristic, complemented
324#1570—611no—
325#1486—611no—
326#1480—611no—
327#1452—611no—
328C2×C61—42no—
329C2×C61—81no—
330C2×C61#292911yescharacteristic
331C2×C61C2×#44211yescharacteristic
332C2×C61#295511yescharacteristic
333D61—22no—
334D61—42no—
335D61—81no—
336C2×#131—1221no—
337C23×C8—1221no—
338#532—611no—
339C2×C4×D4—611no—
340#480—611no—
341#480—611no—
342#474—611no—
343#474—611no—
344#459—611no—
345#459—611no—
346#447—611no—
347#447—611no—
348C4×D8—611no—
349C4×D8—611no—
350C4×D8—611no—
351C4×D8C2×D6111yescharacteristic
352#436—611no—
353#436—611no—
354#436—1221no—
355C8×D4—611no—
356C8×D4—611no—
357C8×D4—611no—
358C8×D4—611no—
359C8×D4—1221no—
360C8×D4—1221no—
361C8×D4—1221no—
362C8×D4C2×D6111yescharacteristic
363C8×D4C2×D6111yescharacteristic, complemented
364#431—611no—
365#431—611no—
366#423—611no—
367C2×C4⋊C8—611no—
368#408—1221no—
369C2×C22⋊C8—1221no—
370C2×C4×C8—611no—
371#334—611no—
372#334—611no—
373#334—611no—
374#334C2×D6111yescharacteristic, complemented
375D4⋊C8—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381D4⋊C8C2×D6111yescharacteristic
382D4⋊C8C2×D6111yescharacteristic, complemented
383#322—611no—
384#322—611no—
385#322—611no—
386#322C2×D6111yescharacteristic, complemented
387C61#1790711yescharacteristic, complemented, Sylow
388C2×#41—1221no—
389C22×D4—611no—
390C23×C4—1221no—
391C2×D8—611no—
392C2×D8—611no—
393C2×D8—611no—
394C2×D8C4×D6111yescharacteristic
395#131—2444no—
396C2×M16—1221no—
397C2×M16—1221no—
398C2×M16—1221no—
399C22×C8—21no—
400C22×C8—21no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—1221no—
404C22×C8—1221no—
405C22×C8—1221no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8—2444no—
409C4⋊Q8—611no—
410#127—611no—
411#124—1221no—
412#121—611no—
413#121—611no—
414#121—1221no—
415C4×D4—611no—
416C4×D4—611no—
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4C22×D6111yescharacteristic
423C4×D4C22×D6111yescharacteristic
424C2×C4⋊C4—611no—
425C2×C22⋊C4—1221no—
426C2×C42—611no—
427#107—611no—
428#107—611no—
429#107—611no—
430#107C4×D6111yescharacteristic
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438C4⋊C8C22×D6111yescharacteristic
439C4⋊C8C22×D6111yescharacteristic
440#102—611no—
441#102—611no—
442#102—611no—
443#102—611no—
444#102—611no—
445#102—611no—
446#102C4×D6111yescharacteristic
447#102C4×D6111yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—611no—
451C22⋊C8—611no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C22⋊C8—1221no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—611no—
464C4×C8C22×D6111yescharacteristic
465C4×C8C22×D6111yescharacteristic
466C4×C8C22×D6111yescharacteristic
467C24—1221no—
468#41—2442no—
469#41—2442no—
470C2×Q8—1221no—
471C2×D4—611no—
472C2×D4—611no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4C2×C4×D6111yescharacteristic
481C2×D4C2×C4×D6111yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494D8—22no—
495D8—1222no—
496D8—1222no—
497D8—1222no—
498M16—2442no—
499M16—2442no—
500M16—2442no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—42no—
505C2×C8—42no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—2442no—
516C2×C8—2442no—
517C2×C8—2442no—
518C2×C8—2442no—
519C2×C8—2442no—
520C2×C8C2×C4×D6111yescharacteristic
521C2×C8C2×C4×D6111yescharacteristic
522C2×C8D4×D6111yescharacteristic
523C2×C8D4×D6111yescharacteristic
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4C2×C4×D6111yescharacteristic
529C4⋊C4C2×C4×D6111yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—1221no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42C23×D6111yescharacteristic
542C23—21no—
543C23—21no—
544C23—611no—
545C23—1224no—
546C23—1221no—
547C23—1221no—
548C23—2442no—
549C23—2442no—
550Q8—2442no—
551D4—21no—
552D4—21no—
553D4—21no—
554D4—612no—
555D4—1224no—
556D4—1221no—
557D4—1221no—
558D4—1221no—
559D4—2442no—
560D4—2442no—
561D4—2442no—
562D4—2442no—
563D4#1451978312yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—611no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—2441no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4C22×C4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C8—21no—
591C8—21no—
592C8—21no—
593C8—21no—
594C8—41no—
595C8—41no—
596C8—41no—
597C8—1221no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—2441no—
602C8—2441no—
603C8—2441no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—612no—
611C22—1221no—
612C22—1221no—
613C22—1222no—
614C22—2448no—
615C22—2441no—
616C22—2441no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22C4×D4×D6111yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—612no—
625C4—1221no—
626C4—1221no—
627C4—2441no—
628C4—2441no—
629C4#131×D6111yescharacteristic
630C4C2×#1451978311yescharacteristic
631C4AGL(1,ℤ/8)×D6111yescharacteristic
632C4AGL(1,ℤ/8)×D6111yescharacteristic
633C2—42no—
634C2—81no—
635C2—1222no—
636C2—4881no—
637C2—2442no—
638C2#2145450811yescharacteristic
639C2#442×D6111yescharacteristic
640C2#2145471911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1580C2
socleC22×C61C2×C4×D4
prime radical#21455817C2
cosocleC42×C61C24
hypercenter#1580D61
hypocenterC61#17907
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #21454526C2semidirect
3. C8×D4×D61C2semidirect
4. #21444686C2semidirect
5. #21444684C2semidirect
6. #21444682C2semidirect
7. #21444680C2semidirect
8. #21444644C2semidirect
9. #21444641C2semidirect
10. #21444641C2semidirect
11. #21444588C2semidirect
12. #21444242C2semidirect
13. #21444216C2semidirect
14. #21444110C2semidirect
15. C61×#1580C2semidirect
31. #16839840C22semidirect
32. #16839839C22semidirect
35. C8×D244C22semidirect
48. #16839513C22semidirect
49. #16839512C22semidirect
50. #16839512C22semidirect
52. #16839491C22semidirect
54. D4⋊#92418C22semidirect
56. #16839489C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. #16839483C22semidirect
62. C8×D4×C61C22semidirect
63. C61×#334C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×#322C22semidirect
232. D4×C61#131semidirect
256. D244#131semidirect
323. #1580D61semidirect
363. C8×D4C2×D61semidirect
374. #334C2×D61semidirect
382. D4⋊C8C2×D61semidirect
386. #322C2×D61semidirect
387. C61#17907semidirect
563. D4#14519783semidirect

Maximal subgroups

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

subgroupindex
#214558172
#214545262
C8×D4×D612
#214446862
#214446842
#214446822
#214446802
#214446442
#214446412
#214445882
#214442422
#214442162
#214441102
C61×#15802
#1790761