Finite Group Databasebeta

← #29721287

Subgroups — #29721287

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×#16839818C211yesmaximal, characteristic
2#21454124C211yesmaximal, characteristic
3C4×#14519753C211yesmaximal, characteristic
4#21447253C211yesmaximal, characteristic
5#21447248C211yesmaximal, characteristic
6#21447021C211yesmaximal, characteristic
7#21447014C211yesmaximal, characteristic
8#21446777C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446560C211yesmaximal, characteristic, complemented
11#21446555C211yesmaximal, characteristic, complemented
12#21446506C211yesmaximal, characteristic, complemented
13#21446506C211yesmaximal, characteristic, complemented
14#21445834C211yesmaximal, characteristic, complemented
15C2×C61×#479C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839818—24no—
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28#16839669C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839643—21no—
33#16839643—21no—
34#16839643—21no—
35#16839633—21no—
36#16839633—21no—
37#16839633—21no—
38#16839628C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42C4×#11758196—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44C4×#11758198—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#479—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519753—22no—
67#14519753—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61—22no—
85C42×D61—22no—
86#14519711—22no—
87#14519709—22no—
88C4⋊#92418—22no—
89#14519701—22no—
90#14519701—22no—
91#14519689—22no—
92#14519689—22no—
93#14519687—22no—
94#14519687—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106C22×Q8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#125×C61—22no—
110#125×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122#106×C61—22no—
123C4⋊C8×C61D414yes—
124#103×C61—22no—
125#103×C61—22no—
126C23×D61C2×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×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D412yes—
164C2×C8×C61C2×D412yes—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61—42no—
174C42×C61—42no—
175C42×C61C2×D412yes—
176C42×C61#4112yes—
177#11758198—24no—
178#11758198—42no—
179#11758198—42no—
180#11758198—42no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61#4112yes—
189#11758196—24no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C23×C61C22×D411yescharacteristic
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61#13712yes—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
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#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C4×D61—44no—
262C4×D61—44no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C8×C61—22no—
268C8×C61—22no—
269#92418—41no—
270#92418—41no—
271#24707—611nomaximal, Sylow
272C22×C61C2×#12111yescharacteristic
273C22×C61#52511yescharacteristic
274C22×C61#53611yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57511yescharacteristic
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61#54711yescharacteristic
293C4×C61#54811yescharacteristic
294C4×C61#55211yescharacteristic
295C4×C61#55611yescharacteristic
296Dic61—42no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—81no—
301Dic61—81no—
302Dic61—81no—
303C2×#479D6111yescharacteristic, complemented
304C2×#432—611no—
305#2851—611no—
306C4×C22⋊Q8—611no—
307#2077—611no—
308#2076—611no—
309#2036—611no—
310#2035—611no—
311#1989—611no—
312#1949—611no—
313#1944—611no—
314#1943—611no—
315#1935—611no—
316#1935—611no—
317#1805—611no—
318C2×C61#241511yescharacteristic
319C2×C61#305411yescharacteristic
320C2×C61#305611yescharacteristic
321C2×C61#311111yescharacteristic
322C2×C61#311711yescharacteristic
323C2×C61#323611yescharacteristic
324C2×C61#324011yescharacteristic
325D61—42no—
326C2×#125C2×D6111yescharacteristic
327C2×C22⋊Q8—611no—
328C2×C4×Q8C2×D6111yescharacteristic
329C22×C42—611no—
330#479—24no—
331#432—1224no—
332C2×#107—611no—
333C2×#107C2×D6111yescharacteristic
334C2×#106—611no—
335C2×#106C2×D6111yescharacteristic
336C2×C4⋊C8—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#103C2×D6111yescharacteristic
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
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#380—611no—
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#340—611no—
371C61#2470711yescharacteristic, complemented, Sylow
372C22×Q8C22×D6111yescharacteristic
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—611no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380#125—22no—
381#125—22no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C4×Q8—22no—
385C4×Q8—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
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—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—21no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1222no—
409C2×C42—1222no—
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412#107—22no—
413#107—1222no—
414#106—22no—
415#106—1222no—
416C4⋊C8—1222no—
417C4⋊C8C22⋊D6114yes—
418#103—22no—
419#103—22no—
420#103—1222no—
421#103—1222no—
422C22⋊C8—1222no—
423C22⋊C8—1222no—
424#97—1222no—
425#97—1222no—
426#95—1221no—
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—
437C24—611no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1175822612yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8C2×C22⋊D6112yes—
481C2×C8C2×C22⋊D6112yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—612no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C22⋊C4—1224no—
504C22⋊C4—2442no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—42no—
508C42—1221no—
509C42—1221no—
510C42—1222no—
511C42—1222no—
512C42—2442no—
513C42#1175822412yes—
514C42C2×C22⋊D6112yes—
515C23—612no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23—1221no—
520C23—1221no—
521C23—1221no—
522C23C2×D4×D6111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
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—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
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#1451974812yes—
571C2×C4#1451974812yes—
572C2×C4C22×C22⋊D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822611yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982511yescharacteristic
578C8—22no—
579C8—22no—
580C8—2441no—
581C8—2441no—
582C22—1222no—
583C22—1222no—
584C22—1222no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22—2441no—
589C22C2×#1451974811yescharacteristic
590C22#1684027511yescharacteristic
591C22#1684033511yescharacteristic
592C22#137×D6111yescharacteristic
593C22#1684055911yescharacteristic
594C22#1684055911yescharacteristic
595C22#1684056111yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—4881no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4—2442no—
607C4—2441no—
608C4—2441no—
609C4#1684041111yescharacteristic
610C4#1684042311yescharacteristic
611C4#1684045411yescharacteristic
612C4#1684048311yescharacteristic
613C2—2442no—
614C2#2144924411yescharacteristic
615C2#2145542411yescharacteristic
616C2#2145544311yescharacteristic
617C2#2145593511yescharacteristic
618C2#2145599511yescharacteristic
619C2#2145702311yescharacteristic
620C2#2145706411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#479C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#479D61
hypocenterC61#24707
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
8. #21446777C2semidirect
9. #21446589C2semidirect
10. #21446560C2semidirect
11. #21446555C2semidirect
12. #21446506C2semidirect
13. #21446506C2semidirect
14. #21445834C2semidirect
15. C2×C61×#479C2semidirect
303. C2×#479D61semidirect
371. C61#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#168398182
#214541242
C4×#145197532
#214472532
#214472482
#214470212
#214470142
#214467772
#214465892
#214465602
#214465552
#214465062
#214458342
C2×C61×#4792
#2470761