Finite Group Databasebeta

← #29650982

Subgroups — #29650982

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455817C211yesmaximal, characteristic
2#21454558C211yesmaximal, characteristic, complemented
3#21454555C211yesmaximal, characteristic, complemented
4#21444733C211yesmaximal, characteristic, complemented
5#21444731C211yesmaximal, characteristic, complemented
6#21444686C211yesmaximal, characteristic, complemented
7#21444686C211yesmaximal, characteristic, complemented
8#21444685C211yesmaximal, characteristic, complemented
9#21444642C211yesmaximal, characteristic, complemented
10#21444641C211yesmaximal, characteristic, complemented
11#21444588C211yesmaximal, characteristic, complemented
12#21444268C211yesmaximal, characteristic, complemented
13D4⋊C8×D61C211yesmaximal, characteristic, complemented
14#21444105C211yesmaximal, characteristic, complemented
15C61×#1586C211yesmaximal, characteristic, complemented
16C2×#14519783—21no—
17C2×M16×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#16839841C2211yescharacteristic, complemented
32#16839840C2211yescharacteristic
33#16839840C2211yescharacteristic, complemented
34#16839836—21no—
35#16839836—21no—
36#16839833—21no—
37#16839831—21no—
38C8×D244C2211yescharacteristic
39#16839827C2211yescharacteristic
40#16839827C2211yescharacteristic, complemented
41#16839813C2211yescharacteristic
42#16839784C2211yescharacteristic
43C4⋊C8×D61C2211yescharacteristic
44#16839727—21no—
45#16839724—21no—
46C4×#11758189C2211yescharacteristic
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#16839483C2211yescharacteristic, complemented
59#16839483C2211yescharacteristic, complemented
60C4×D8×C61C2211yescharacteristic
61C61×#436C2211yescharacteristic
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×#11758194—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C2×#11758189D411yescharacteristic
79C2×#11758189D411yescharacteristic
80M16×D61—44no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84C2×C8×D61—21no—
85C2×C8×D61—21no—
86#14519776C2×C411yescharacteristic
87#14519773C2×C411yescharacteristic
88#14519763—21no—
89#14519752C2×C411yescharacteristic
90#14519750—21no—
91#14519748C2×C411yescharacteristic
92C4×D244C2311yescharacteristic
93C4×D244C2311yescharacteristic
94D4×Dic61C2×C411yescharacteristic
95#14519739C2×C411yescharacteristic
96C4×C22⋊D61—21no—
97C4×C22⋊D61—21no—
98#14519737—21no—
99C4⋊C4×D61C2×C411yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519711C2×C411yescharacteristic
103#14519710C2×C411yescharacteristic
104#14519710C2×C411yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
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#14519689C2×C411yescharacteristic
124#14519689C2×C411yescharacteristic
125#14519687D411yescharacteristic
126#14519687D411yescharacteristic
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C4×#92418C2311yescharacteristic
130C8×Dic61—21no—
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
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—41no—
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—41no—
174M16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
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—41no—
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—
204#11758194—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212D488—22no—
213#11758189—22no—
214#11758189—22no—
215#11758189—42no—
216#11758189—42no—
217C8×D61—42no—
218C8×D61—42no—
219C8×D61—42no—
220C8×D61—42no—
221C8×D61—42no—
222C23×C61—21no—
223C23×C61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—81no—
229C22×D61C4×D411yescharacteristic
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—42no—
246C22⋊D61—81no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—42no—
250C2×Dic61C4×D411yescharacteristic
251C2×Dic61C4×D411yescharacteristic
252C2×Dic61C4×D411yescharacteristic
253D244—21no—
254D244—21no—
255D244—21no—
256D244—42no—
257D244#13112yescomplemented
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—41no—
261C4×D61—42no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—42no—
267C4×D61AGL(1,ℤ/8)12yes—
268C4×D61AGL(1,ℤ/8)12yes—
269Dic122—42no—
270C8×C61—22no—
271C8×C61—21no—
272C8×C61—21no—
273C8×C61—21no—
274C8×C61—21no—
275C8×C61—41no—
276C8×C61—41no—
277#92418—21no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#18001—611nomaximal, Sylow
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61C2×C4×D411yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—22no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—84no—
301C2×D61#44212yes—
302C4×C61—21no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61C2×#13111yescharacteristic
306C4×C61#56811yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309Dic61—21no—
310Dic61—21no—
311Dic61—41no—
312Dic61—41no—
313Dic61#44212yes—
314#3100—611no—
315#2935—611no—
316D4×M16—611no—
317#1594—611no—
318#1594—611no—
319#1586—611no—
320#1586—611no—
321#1586—611no—
322#1586D6111yescharacteristic, complemented
323#1580—611no—
324#1580—611no—
325#1570—611no—
326#1492—611no—
327C2×D4⋊C8—611no—
328#1451—611no—
329C2×C61—81no—
330C2×C61—81no—
331C2×C61#293311yescharacteristic
332C2×C61C2×#44211yescharacteristic
333C2×C61#295611yescharacteristic
334D61—22no—
335D61—42no—
336D61—81no—
337C2×#131—1221no—
338C22×M16—1221no—
339#532—611no—
340C2×C4×D4—611no—
341#480—611no—
342#480—611no—
343#474—611no—
344#474—611no—
345#459—611no—
346#459—611no—
347#447—611no—
348#447—611no—
349C4×D8—611no—
350C4×D8—611no—
351C4×D8—611no—
352C4×D8C2×D6111yescharacteristic
353#436—611no—
354#436—611no—
355#436—611no—
356#436C2×D6111yescharacteristic
357#435—1221no—
358#435—1221no—
359#435—1221no—
360#435—1221no—
361C8×D4—611no—
362C8×D4—611no—
363C8×D4—611no—
364C8×D4C2×D6111yescharacteristic, complemented
365#431—611no—
366#423—611no—
367C2×C4⋊C8—611no—
368#408—1221no—
369#407—1221no—
370C4×M16—611no—
371C2×C4×C8—611no—
372#334—611no—
373#334—611no—
374#334—611no—
375#334C2×D6111yescharacteristic, complemented
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381D4⋊C8—611no—
382D4⋊C8C2×D6111yescharacteristic
383D4⋊C8C2×D6111yescharacteristic, complemented
384#322—611no—
385#322—611no—
386#322—611no—
387#322C2×D6111yescharacteristic, complemented
388C61#1800111yescharacteristic, complemented, Sylow
389C2×#41—1221no—
390C22×D4—611no—
391C23×C4—1221no—
392C2×D8—611no—
393C2×D8—611no—
394C2×D8—611no—
395C2×D8C4×D6111yescharacteristic
396#131—2444no—
397C2×M16—21no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—1221no—
401C2×M16—1221no—
402C2×M16—1221no—
403C2×M16—1221no—
404C2×M16—2444no—
405C22×C8—21no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8—1221no—
409C22×C8—1221no—
410C22×C8—1221no—
411C4⋊Q8—611no—
412#127—611no—
413#124—1221no—
414#121—611no—
415#121—611no—
416#121—1221no—
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—611no—
420C4×D4—611no—
421C4×D4—1221no—
422C4×D4—1221no—
423C4×D4—1221no—
424C4×D4C22×D6111yescharacteristic
425C4×D4C22×D6111yescharacteristic
426C2×C4⋊C4—611no—
427C2×C22⋊C4—1221no—
428C2×C42—611no—
429#107—611no—
430#107—611no—
431#107—611no—
432#107C4×D6111yescharacteristic
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—1221no—
438C4⋊C8—1221no—
439C4⋊C8C22×D6111yescharacteristic
440C4⋊C8C22×D6111yescharacteristic
441#102—611no—
442#102—611no—
443#102—611no—
444#102—611no—
445#102—611no—
446#102—611no—
447#102C4×D6111yescharacteristic
448#102C4×D6111yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8—611no—
452C22⋊C8—611no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C22⋊C8—1221no—
459#97—611no—
460#97—611no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—611no—
464C4×C8—1221no—
465C4×C8C22×D6111yescharacteristic
466C4×C8C22×D6111yescharacteristic
467C4×C8C22×D6111yescharacteristic
468C24—1221no—
469#41—2442no—
470#41—2442no—
471C2×Q8—1221no—
472C2×D4—611no—
473C2×D4—611no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—2441no—
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—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—2442no—
494C22×C4—2442no—
495D8—41no—
496D8—1222no—
497D8—1222no—
498D8—2441no—
499M16—42no—
500M16—1222no—
501M16—1222no—
502M16—2442no—
503M16—2442no—
504M16—2442no—
505M16—2442no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—2442no—
518C2×C8—2442no—
519C2×C8—2442no—
520C2×C8—2442no—
521C2×C8—2442no—
522C2×C8C2×C4×D6111yescharacteristic
523C2×C8C2×C4×D6111yescharacteristic
524C2×C8D4×D6111yescharacteristic
525C2×C8D4×D6111yescharacteristic
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4C2×C4×D6111yescharacteristic
531C4⋊C4C2×C4×D6111yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42C23×D6111yescharacteristic
544C23—21no—
545C23—21no—
546C23—611no—
547C23—1221no—
548C23—1221no—
549C23—4881no—
550C23—2442no—
551C23—2442no—
552Q8—2442no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—21no—
557D4—612no—
558D4—1221no—
559D4—1221no—
560D4—1221no—
561D4—4881no—
562D4—2442no—
563D4—2442no—
564D4—2442no—
565D4—2442no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—611no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—2441no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4C22×C4×D6111yescharacteristic
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4#41×D6111yescharacteristic
592C8—22no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—41no—
598C8—41no—
599C8—1221no—
600C8—1221no—
601C8—1221no—
602C8—1221no—
603C8—2441no—
604C8—2441no—
605C8—2441no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—612no—
613C22—1221no—
614C22—1221no—
615C22—1222no—
616C22—4884no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22—2441no—
622C22C4×D4×D6111yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—612no—
627C4—1221no—
628C4—1221no—
629C4—2441no—
630C4—2441no—
631C4#131×D6111yescharacteristic
632C4#1684052811yescharacteristic
633C4AGL(1,ℤ/8)×D6111yescharacteristic
634C4AGL(1,ℤ/8)×D6111yescharacteristic
635C2—81no—
636C2—81no—
637C2—1222no—
638C2—4881no—
639C2—2442no—
640C2#2145454211yescharacteristic
641C2#442×D6111yescharacteristic
642C2#2145472411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1586C2
socleC22×C61C2×C4×D4
prime radical#21455817C2
cosocleC42×C61C24
hypercenter#1586D61
hypocenterC61#18001
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. #21454558C2semidirect
3. #21454555C2semidirect
4. #21444733C2semidirect
5. #21444731C2semidirect
6. #21444686C2semidirect
7. #21444686C2semidirect
8. #21444685C2semidirect
9. #21444642C2semidirect
10. #21444641C2semidirect
11. #21444588C2semidirect
12. #21444268C2semidirect
13. D4⋊C8×D61C2semidirect
14. #21444105C2semidirect
15. C61×#1586C2semidirect
31. #16839841C22semidirect
33. #16839840C22semidirect
40. #16839827C22semidirect
48. #16839513C22semidirect
49. #16839512C22semidirect
50. #16839512C22semidirect
52. #16839491C22semidirect
54. D4⋊#92418C22semidirect
56. #16839489C22semidirect
57. #16839484C22semidirect
58. #16839483C22semidirect
59. #16839483C22semidirect
62. C8×D4×C61C22semidirect
63. C61×#334C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×#322C22semidirect
257. D244#131semidirect
322. #1586D61semidirect
364. C8×D4C2×D61semidirect
375. #334C2×D61semidirect
383. D4⋊C8C2×D61semidirect
387. #322C2×D61semidirect
388. C61#18001semidirect

Maximal subgroups

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

subgroupindex
#214558172
#214545582
#214545552
#214447332
#214447312
#214446862
#214446852
#214446422
#214446412
#214445882
#214442682
D4⋊C8×D612
#214441052
C61×#15862
#1800161