Finite Group Databasebeta

← #29721291

Subgroups — #29721291

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×#16840069C211yesmaximal, characteristic, complemented
2C2×#16839819C211yesmaximal, characteristic
3#21454129C211yesmaximal, characteristic
4#21447256C211yesmaximal, characteristic
5#21447250C211yesmaximal, characteristic
6#21447019C211yesmaximal, characteristic
7#21447012C211yesmaximal, characteristic
8#21446776C211yesmaximal, characteristic, complemented
9#21446587C211yesmaximal, characteristic, complemented
10#21446559C211yesmaximal, characteristic, complemented
11#21446554C211yesmaximal, characteristic, complemented
12#21446512C211yesmaximal, characteristic, complemented
13#21446512C211yesmaximal, characteristic, complemented
14#21445834C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C61C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17#16840069—24no—
18#16839819—24no—
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839646—21no—
36#16839646—21no—
37#16839635—21no—
38#16839635—21no—
39#16839628C2211yescharacteristic
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×C22⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C22×C42×C61C2211yescharacteristic
53C61×#391C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#387—21no—
56C61×#385—21no—
57C61×#380C2211yescharacteristic
58C4×C4⋊C4×C61—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C22⋊C4×C61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519769—22no—
66#14519769—22no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77#14519710—22no—
78#14519710—22no—
79#14519708—22no—
80#14519708—22no—
81C4⋊#92418—22no—
82C4⋊#92418D414yes—
83#14519701—22no—
84#14519701—22no—
85#14519701—22no—
86#14519701—22no—
87C22⋊#92418—22no—
88C22⋊#92418—22no—
89#14519688—22no—
90#14519688—22no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61C2311yescharacteristic
98C23×C4×C61D411yescharacteristic
99C23×C4×C61D411yescharacteristic
100C23×C4×C61C2311yescharacteristic
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C4×Q8×C61—22no—
104C4×Q8×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—22no—
114C2×C42×C61—22no—
115C2×C42×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61D411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
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—22no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—41no—
147C22×C4×C61—41no—
148C22×C4×C61—41no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—42no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—42no—
175C42×C61—42no—
176C42×C61C2×D412yes—
177C42×C61#4112yes—
178C22⋊Dic61—42no—
179C22⋊Dic61—42no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D412yes—
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—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
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×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×C4×C612−1+411yescharacteristic
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—41no—
266C2×Dic61—41no—
267#92418—22no—
268#92418—22no—
269#92418—41no—
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—22no—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61#54711yescharacteristic
296C4×C61#54811yescharacteristic
297C4×C61#55211yescharacteristic
298C4×C61#55611yescharacteristic
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302Dic61—81no—
303C2×#479—611no—
304C2×#432—611no—
305#2851—611no—
306C4×C22⋊Q8D6111yescharacteristic
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—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×Q8C2×D6111yescharacteristic
329C22×C42C2×D6111yescharacteristic
330#479—1224no—
331#432—1224no—
332C2×#107—611no—
333C2×#107—611no—
334C2×#106—611no—
335C2×#106—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×#97—611no—
345C2×#97—611no—
346#391C2×D6111yescharacteristic
347#389—611no—
348#389—611no—
349#389C2×D6111yescharacteristic
350#387—21no—
351#387—1221no—
352#387—1221no—
353#385—21no—
354#385—1221no—
355#385—1221no—
356#382—611no—
357#380—611no—
358#380—611no—
359#380C2×D6111yescharacteristic
360C4×C4⋊C4—21no—
361C4×C4⋊C4C2×D6111yescharacteristic
362C4×C22⋊C4—21no—
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×C4C22⋊D6111yescharacteristic
374C23×C4C22⋊D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380#125—1222no—
381#125—1222no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
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—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C22⋊D6111yescharacteristic
409C2×C42C22⋊D6111yescharacteristic
410C2×C42C22×D6111yescharacteristic
411C2×C42C22×D6111yescharacteristic
412#107—1222no—
413#107—1222no—
414#106—1222no—
415#106—1222no—
416C4⋊C8—614no—
417C4⋊C8—1222no—
418#103—1222no—
419#103—1222no—
420#103—1222no—
421#103—1222no—
422C22⋊C8—1222no—
423C22⋊C8—1222no—
424#97—1222no—
425#97—1222no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437C24C2×C22⋊D6111yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1175822212yes—
444C22×C4—22no—
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—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C2×C8—612no—
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4#1175822212yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42D4×D6112yes—
514C42#1175822412yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1451980812yes—
522C23C22×C22⋊D6111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
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#1451974912yes—
571C2×C4#1451974912yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—1222no—
579C8—1222no—
580C8—2441no—
581C8—2441no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1451974911yescharacteristic
590C22#1684027311yescharacteristic
591C22#1684033311yescharacteristic
592C22C2×#1451980811yescharacteristic
593C22#1684056011yescharacteristic
594C22#1684056011yescharacteristic
595C22#1684056011yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—4881no—
606C4—4881no—
607C4—4881no—
608C4—4881no—
609C4#1684041711yescharacteristic
610C4#1684042111yescharacteristic
611C4#1684045211yescharacteristic
612C4#1684048211yescharacteristic
613C2—42no—
614C2#2144924811yescharacteristic
615C2#2145542511yescharacteristic
616C2#2145544511yescharacteristic
617C2#2145592811yescharacteristic
618C2#2145599011yescharacteristic
619C2#2145702111yescharacteristic
620C2#2145705511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC4×C22⋊Q8D61
hypocenterC61#24707
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840069C2semidirect
8. #21446776C2semidirect
9. #21446587C2semidirect
10. #21446559C2semidirect
11. #21446554C2semidirect
12. #21446512C2semidirect
13. #21446512C2semidirect
14. #21445834C2semidirect
371. C61#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#168400692
C2×#168398192
#214541292
#214472562
#214472502
#214470192
#214470122
#214467762
#214465872
#214465592
#214465542
#214465122
#214458342
C4×C22⋊Q8×C612
#2470761