Finite Group Databasebeta

← #29721292

Subgroups — #29721292

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840071C211yesmaximal, characteristic, complemented
2C2×#16839816C211yesmaximal, characteristic
3#21454127C211yesmaximal, characteristic
4C22⋊Q8×Dic61C211yesmaximal, characteristic
5#21447255C211yesmaximal, characteristic
6#21447247C211yesmaximal, characteristic
7#21447019C211yesmaximal, characteristic
8#21446773C211yesmaximal, characteristic, complemented
9#21446588C211yesmaximal, characteristic, complemented
10#21446563C211yesmaximal, characteristic, complemented
11#21446556C211yesmaximal, characteristic, complemented
12#21446511C211yesmaximal, characteristic, complemented
13#21446506C211yesmaximal, characteristic, complemented
14#21445832C211yesmaximal, characteristic, complemented
15C61×#2035C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C22×C4×Dic61C2211yescharacteristic
19#16840071—24no—
20#16839816—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839649—21no—
36#16839648—21no—
37#16839645—21no—
38#16839637—21no—
39#16839637—21no—
40#16839634—21no—
41#16839627C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839622C2211yescharacteristic
44C4⋊C4×Dic61—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C22⋊C4×Dic61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C23×Dic61D411yescharacteristic
62C23×Dic61D411yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519768—22no—
66#14519768—22no—
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519710—22no—
91#14519708—22no—
92#14519706—22no—
93#14519706D414yes—
94#14519701—22no—
95#14519701—22no—
96C22⋊#92418—22no—
97#14519687—22no—
98#14519687—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C22×Q8×C61C2311yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—22no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121#106×C61—22no—
122#103×C61—22no—
123#103×C61—22no—
124C22⋊C8×C61—22no—
125#95×C61—21no—
126C24×C61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—41no—
153C22×Dic61—41no—
154C22×Dic61—41no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61C2×D412yes—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188#11758196—42no—
189#11758196—42no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C4×Dic61C2×D412yes—
197C4×Dic61#4112yes—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C2×D412yes—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61#13712yes—
211C23×C61C22×D411yescharacteristic
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—44no—
252C2×Dic61—44no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61#12112yes—
266C2×Dic61#12112yes—
267C8×C61—22no—
268C8×C61—41no—
269#92418—22no—
270#92418—41no—
271#24707—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×#12111yescharacteristic
280C22×C61#52511yescharacteristic
281C22×C61#53611yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57511yescharacteristic
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#54711yescharacteristic
292C4×C61#54811yescharacteristic
293C4×C61#55211yescharacteristic
294C4×C61#55611yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—41no—
298Dic61—41no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302Dic61—81no—
303C2×#479—611no—
304C2×#432—611no—
305#2851—611no—
306C4×C22⋊Q8—611no—
307#2077—611no—
308#2076—611no—
309#2036—611no—
310#2035D6111yescharacteristic
311#1989—611no—
312#1949—611no—
313#1944—611no—
314#1943—611no—
315#1935—611no—
316#1935—611no—
317#1805—611no—
318C2×C61—42no—
319C2×C61#241511yescharacteristic
320C2×C61#305411yescharacteristic
321C2×C61#305611yescharacteristic
322C2×C61#311111yescharacteristic
323C2×C61#311711yescharacteristic
324C2×C61#323611yescharacteristic
325C2×C61#324011yescharacteristic
326C2×#125—611no—
327C2×C22⋊Q8C2×D6111yescharacteristic
328C2×C4×Q8—611no—
329C22×C42—611no—
330#479—1224no—
331#432—1224no—
332C2×#107—611no—
333C2×#107—611no—
334C2×#106—611no—
335C2×#106C2×D6111yescharacteristic
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×C22⋊C8—611no—
343C2×C22⋊C8C2×D6111yescharacteristic
344C2×#97—611no—
345C2×#97—611no—
346#391—611no—
347#389—611no—
348#389—611no—
349#389—611no—
350#387—1221no—
351#387—1221no—
352#387—1221no—
353#385—1221no—
354#385—1221no—
355#385—1221no—
356#382—611no—
357#380—611no—
358#380—611no—
359#380C2×D6111yescharacteristic
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340C2×D6111yescharacteristic
371C61#2470711yescharacteristic, complemented, Sylow
372C22×Q8C22×D6111yescharacteristic
373C23×C4—611no—
374C23×C4—611no—
375C23×C4C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380#125—1222no—
381#125—1222no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1222no—
409C2×C42—1222no—
410C2×C42—1221no—
411C2×C42—1221no—
412#107—1222no—
413#107—1222no—
414#106—22no—
415#106—1222no—
416C4⋊C8—614no—
417C4⋊C8—1222no—
418#103—22no—
419#103—22no—
420#103—1222no—
421#103—1222no—
422C22⋊C8—22no—
423C22⋊C8—1222no—
424#97—1222no—
425#97—1222no—
426#95—21no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437C24D4×D6111yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1175822412yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4—2441no—
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—612no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8D4×D6112yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4#1175822412yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—612no—
508C42—612no—
509C42—1222no—
510C42—1222no—
511C42—1221no—
512C42—1221no—
513C42—2442no—
514C42—2442no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1451980612yes—
522C23C2×D4×D6111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—2444no—
559C2×C4—2444no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—22no—
579C8—41no—
580C8—1222no—
581C8—2441no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22#121×D6111yescharacteristic
590C22#1684026811yescharacteristic
591C22#1684033411yescharacteristic
592C22C2×#1451980611yescharacteristic
593C22#1684055911yescharacteristic
594C22#1684056111yescharacteristic
595C22#1684056111yescharacteristic
596C4—41no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—1222no—
602C4—1222no—
603C4—4881no—
604C4—4881no—
605C4—4881no—
606C4—4881no—
607C4—2441no—
608C4—2441no—
609C4#1684041211yescharacteristic
610C4#1684041811yescharacteristic
611C4#1684044911yescharacteristic
612C4#1684048111yescharacteristic
613C2—42no—
614C2#2144924111yescharacteristic
615C2#2145543011yescharacteristic
616C2#2145544211yescharacteristic
617C2#2145592911yescharacteristic
618C2#2145598511yescharacteristic
619C2#2145702211yescharacteristic
620C2#2145706311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2035C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2035D61
hypocenterC61#24707
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
1. C2×#16840071C2semidirect
8. #21446773C2semidirect
9. #21446588C2semidirect
10. #21446563C2semidirect
11. #21446556C2semidirect
12. #21446511C2semidirect
13. #21446506C2semidirect
14. #21445832C2semidirect
371. C61#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#168400712
C2×#168398162
#214541272
C22⋊Q8×Dic612
#214472552
#214472472
#214470192
#214467732
#214465882
#214465632
#214465562
#214465112
#214465062
#214458322
C61×#20352
#2470761