Finite Group Databasebeta

← #29699845

Subgroups — #29699845

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839874C211yesmaximal, characteristic
2#21448791C211yesmaximal, characteristic, complemented
3#21448767C211yesmaximal, characteristic, complemented
4#21446696C211yesmaximal, characteristic
5#21446449C211yesmaximal, characteristic, complemented
6#21446443C211yesmaximal, characteristic, complemented
7#21446442C211yesmaximal, characteristic, complemented
8#21446436C211yesmaximal, characteristic, complemented
9#21445986C211yesmaximal, characteristic, complemented
10#21445827C211yesmaximal, characteristic, complemented
11#21445817C211yesmaximal, characteristic, complemented
12#21445718C211yesmaximal, characteristic, complemented
13C4×#14519709C211yesmaximal, characteristic, complemented
14#21445693C211yesmaximal, characteristic, complemented
15C61×#1970C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519733—21no—
19#16839874C418yescomplemented
20C2×#14519710C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839656—21no—
34#16839656—21no—
35#16839650—21no—
36#16839650—21no—
37#16839631C2211yescharacteristic
38#16839627C2211yescharacteristic
39#16839623—21no—
40#16839620—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46C22⋊C4×Dic61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67D4×Dic61C2×C414yes—
68#14519733—44no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196D411yescharacteristic
76C2×#11758196D411yescharacteristic
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519710—41no—
90#14519709—22no—
91#14519709C2×C414yes—
92#14519708—41no—
93#14519706—22no—
94#14519698—41no—
95#14519698—41no—
96#14519698C2×C414yes—
97#14519698C2×C414yes—
98#14519687—22no—
99#14519687C2×C414yes—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C2×D8×C61—22no—
111C2×D8×C61—22no—
112C2×D8×C61C2×C414yes—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123#102×C61—41no—
124#102×C61—41no—
125C4×C8×C61—22no—
126C24×C61—21no—
127C24×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C2×D4×C61C22×C412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—24no—
148C22×Dic61—24no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—81no—
154C22×Dic61—81no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159D8×C61—28no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C22×C412yes—
164C2×C8×C61#4112yes—
165C2×C8×C61#4112yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C22⋊C4×C61—42no—
171C22⋊C4×C61—42no—
172C22⋊C4×C61—81no—
173C22⋊C4×C61—81no—
174C42×C61—21no—
175C42×C61—21no—
176C22⋊Dic61—24no—
177C22⋊Dic61—24no—
178C22⋊Dic61—42no—
179C22⋊Dic61—42no—
180C22⋊Dic61—81no—
181C22⋊Dic61—81no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22×C412yes—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191#11758196—41no—
192#11758196—42no—
193#11758196—81no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61—81no—
199C4×Dic61—81no—
200C4×Dic61C22×C412yes—
201C4×Dic61#4112yes—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C412yes—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×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×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61AGL(1,ℤ/8)12yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—48no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—84no—
273C2×Dic61C4×D412yes—
274C2×Dic61C4×D412yes—
275C8×C61—41no—
276C8×C61#11714yescomplemented
277#92418—22no—
278#92418—41no—
279#22817—611nomaximal, Sylow
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61C2×C4×D411yescharacteristic
295C22×C61#54711yescharacteristic
296C22×C61#55011yescharacteristic
297C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57711yescharacteristic
301C4×C61—22no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C2×#11711yescharacteristic
306C4×C61#52011yescharacteristic
307C4×C61#55511yescharacteristic
308C4×C61#55611yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—42no—
314Dic61—42no—
315Dic61—81no—
316Dic61—81no—
317C2×#442—611no—
318#2357—611no—
319#2354—611no—
320#1972—611no—
321#1970D6111yescharacteristic
322#1926—611no—
323#1926—611no—
324#1925—611no—
325#1925—611no—
326#1847—611no—
327#1804—611no—
328#1802—611no—
329#1776—611no—
330C4×#106—611no—
331#1771—611no—
332C2×C61—82no—
333C2×C61—82no—
334C2×C61#235411yescharacteristic
335C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
336C2×C61#294611yescharacteristic
337C2×C61#334511yescharacteristic
338C2×C61#334711yescharacteristic
339C2×C61#339911yescharacteristic
340C2×C61#340111yescharacteristic
341C22×D8C2×D6111yescharacteristic
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×#117—1221no—
345#442—618no—
346C2×#107—611no—
347C2×#107C2×D6111yescharacteristic
348C2×#106—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×C4⋊C8—611no—
352C2×#102—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102C2×D6111yescharacteristic
357C2×#102C2×D6111yescharacteristic
358C2×#97—611no—
359C2×#97—611no—
360C2×C4×C8C2×D6111yescharacteristic
361#390—611no—
362#390—611no—
363#390C2×D6111yescharacteristic
364#390C2×D6111yescharacteristic
365#388—1221no—
366#388—1221no—
367#388—1221no—
368#388—1221no—
369#384—611no—
370#382—611no—
371#381—1221no—
372#380—1221no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—611no—
378C4×C22⋊C4—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#336—611no—
384#336—611no—
385C61#2281711yescharacteristic, complemented, Sylow
386C22×D4C22×D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—1221no—
389C23×C4—1221no—
390C2×D8—22no—
391C2×D8—22no—
392C2×D8C4×D6114yes—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4×D4—614no—
398C4×D4—614no—
399C4×D4—1222no—
400C4×D4—1222no—
401#117—2444no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—22no—
430#107—2441no—
431#106—614no—
432#106—1222no—
433#106—2441no—
434C4⋊C8—1222no—
435#102—41no—
436#102—41no—
437#102—614no—
438#102—614no—
439#102—2441no—
440#102—2441no—
441#97—614no—
442#97—1222no—
443C4×C8—22no—
444#95—611no—
445#95—611no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452C24—21no—
453C24—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×C4×D6112yes—
465C2×D4C2×C4×D6112yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—4881no—
479C22×C4—4881no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4C23×D6111yescharacteristic
485D8—28no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—612no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8C2×C4×D6112yes—
496C2×C8#1175822412yes—
497C2×C8#1175822412yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—612no—
503C4⋊C4—612no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1222no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—4881no—
512C4⋊C4—2441no—
513C4⋊C4—2442no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—4881no—
521C22⋊C4—4881no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—21no—
525C42—21no—
526C42—612no—
527C42—612no—
528C42—1221no—
529C42—1221no—
530C42—4881no—
531C42—4881no—
532C42—2442no—
533C42—2442no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D6111yescharacteristic
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—4884no—
583C2×C4—2448no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4#1451980112yes—
595C2×C4C22×C4×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822411yescharacteristic
598C2×C4C2×#1175822411yescharacteristic
599C2×C4#1451982211yescharacteristic
600C2×C4#1451982411yescharacteristic
601C8—41no—
602C8—1222no—
603C8—2441no—
604C8#1451973314yescomplemented
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C4×D4×D6111yescharacteristic
620C22#1684041211yescharacteristic
621C22#1684043211yescharacteristic
622C22C2×#1451980111yescharacteristic
623C22#1684055911yescharacteristic
624C22#1684056311yescharacteristic
625C22#1684056911yescharacteristic
626C4—22no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—1222no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—2442no—
635C4—2442no—
636C4—2441no—
637C4—2441no—
638C4C2×#1451973311yescharacteristic
639C4#1684024111yescharacteristic
640C4#1684047611yescharacteristic
641C4#1684048111yescharacteristic
642C2—82no—
643C2—82no—
644C2#2144876711yescharacteristic
645C2C4×#1451980111yescharacteristic
646C2#2145464411yescharacteristic
647C2#2145829511yescharacteristic
648C2#2145832511yescharacteristic
649C2#2145886611yescharacteristic
650C2#2145889111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1970C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
cosocleC22×C4×C61C24
hypercenter#1970D61
hypocenterC61#22817
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. #21448791C2semidirect
3. #21448767C2semidirect
5. #21446449C2semidirect
6. #21446443C2semidirect
7. #21446442C2semidirect
8. #21446436C2semidirect
9. #21445986C2semidirect
10. #21445827C2semidirect
11. #21445817C2semidirect
12. #21445718C2semidirect
13. C4×#14519709C2semidirect
14. #21445693C2semidirect
19. #16839874C4semidirect
276. C8×C61#117semidirect
385. C61#22817semidirect
604. C8#14519733semidirect

Maximal subgroups

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

subgroupindex
C2×#168398742
#214487912
#214487672
#214466962
#214464492
#214464432
#214464422
#214464362
#214459862
#214458272
#214458172
#214457182
C4×#145197092
#214456932
C61×#19702
#2281761