Finite Group Databasebeta

← #29687787

Subgroups — #29687787

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840050C211yesmaximal, characteristic
2#21448563C211yesmaximal, characteristic, complemented
3#21448180C211yesmaximal, characteristic
4#21446769C211yesmaximal, characteristic, complemented
5#21446590C211yesmaximal, characteristic, complemented
6#21446407C211yesmaximal, characteristic, complemented
7#21446396C211yesmaximal, characteristic, complemented
8#21446078C211yesmaximal, characteristic
9#21446068C211yesmaximal, characteristic
10#21445827C211yesmaximal, characteristic
11#21445716C211yesmaximal, characteristic, complemented
12C4×#14519708C211yesmaximal, characteristic, complemented
13#21445679C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16840050C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839668—21no—
30#16839666—21no—
31#16839661C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839649—21no—
34#16839646—21no—
35#16839627C2211yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C22⋊C4×Dic61—21no—
41C22⋊C4×Dic61—21no—
42C4×C22⋊Dic61—21no—
43#16839608C2211yescharacteristic
44C2×#14519684—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519776—22no—
66#14519776C2×C414yescomplemented
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61C2×C414yes—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519709C2×C414yescomplemented
88#14519708—22no—
89#14519708C2×C414yescomplemented
90#14519706—22no—
91#14519706C2×C414yescomplemented
92#14519698—22no—
93#14519698C2×C414yes—
94C4×#92418—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#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107C22×D4×C61C2311yescharacteristic
108C23×C4×C61—21no—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C4×D4×C61—22no—
112C4×D4×C61Q814yescomplemented
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#102×C61—22no—
120#102×C61C2×C414yes—
121#97×C61—22no—
122#95×C61—21no—
123C24×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61C22×C412yes—
128C2×D4×C61C2×Q812yes—
129C2×D4×C61#4112yes—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61—21no—
134C22×C4×C61—81no—
135C22×C4×C61—81no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—24no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—81no—
146C22×Dic61—81no—
147C22×Dic61—81no—
148C22×Dic61—81no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—24no—
154C2×Dic122—81no—
155C2×Dic122—81no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C22×C412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61C2×Q812yes—
168C22⋊C4×C61—24no—
169C42×C61—81no—
170C42×C61C2×Q812yes—
171C42×C61#4112yes—
172C22⋊Dic61—24no—
173C22⋊Dic61—24no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—81no—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61#4112yes—
183#11758196—24no—
184#11758196—81no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—81no—
188C4×Dic61—81no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C4×Dic61C22×C412yes—
192C4×Dic61#4112yes—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C412yes—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—22no—
210D4×C61C4×Q814yes—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—84no—
225C2×C4×C61C22⋊Q812yes—
226C2×C4×C61C22⋊Q812yes—
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×Q811yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—84no—
256C2×Dic61—84no—
257C2×Dic61C4×D412yes—
258C2×Dic61C4×D412yes—
259Dic122—84no—
260C8×C61—22no—
261C8×C61—41no—
262#92418—22no—
263#92418—41no—
264#21623—611nomaximal, Sylow
265C22×C61—24no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61C2×C4×D411yescharacteristic
273C22×C61C2×C22⋊Q811yescharacteristic
274C22×C61#52511yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57711yescharacteristic
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61C2×C4×Q811yescharacteristic
285C4×C61C4×#4111yescharacteristic
286C4×C61#52911yescharacteristic
287C4×C61#53311yescharacteristic
288Dic61—22no—
289Dic61—22no—
290Dic61—41no—
291Dic61—42no—
292Dic61—41no—
293Dic61—42no—
294Dic61—42no—
295Dic61—81no—
296C2×#476—611no—
297#2331—611no—
298#2275—611no—
299#1988—611no—
300#1949—611no—
301#1920—611no—
302#1919—611no—
303#1864—611no—
304#1862—611no—
305#1804—611no—
306#1776—611no—
307C4×#106—611no—
308#1768—611no—
309#1762D6111yescharacteristic
310C4×#102—611no—
311C2×C61—44no—
312C2×C61C4×C22⋊Q811yescharacteristic
313C2×C61#294611yescharacteristic
314C2×C61#294911yescharacteristic
315C2×C61#308111yescharacteristic
316C2×C61#308411yescharacteristic
317C2×C61#311811yescharacteristic
318C2×C61#312511yescharacteristic
319C2×C4⋊Q8—611no—
320C2×C4×D4—611no—
321C2×C4×D4—611no—
322C2×C4×D4C2×D6111yescharacteristic
323#476—618no—
324C2×#107—611no—
325C2×#106—611no—
326C2×#106—611no—
327C2×#106—611no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102C2×D6111yescharacteristic
333C2×#102C2×D6111yescharacteristic
334C2×#97C2×D6111yescharacteristic
335C2×C4×C8—611no—
336#391—1221no—
337#391—1221no—
338#389—611no—
339#389—611no—
340#387—1221no—
341#387—1221no—
342#382—611no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4C2×D6111yescharacteristic
348C4×C22⋊C4—1221no—
349C4×C22⋊C4—1221no—
350C4×C22⋊C4—1221no—
351#376—611no—
352C2×#95—1221no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340C2×D6111yescharacteristic
360#340C2×D6111yescharacteristic
361#336—611no—
362#336—611no—
363C61#2162311yescharacteristic, complemented, Sylow
364C22×Q8—1221no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—21no—
367C23×C4—1221no—
368C23×C4—1221no—
369C22×C8—611no—
370C22×C8—611no—
371C22×C8C22×D6111yescharacteristic
372C22×C8C22×D6111yescharacteristic
373C4⋊Q8—614no—
374C4⋊Q8—1222no—
375C4×D4—22no—
376C4×D4—614no—
377C4×D4—1222no—
378C4×D4—1222no—
379C4×D4—1222no—
380C4×D4Dic12214yescomplemented
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—21no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402#107—1222no—
403#106—614no—
404#106—614no—
405#106—1222no—
406C4⋊C8—614no—
407C4⋊C8—1222no—
408#102—22no—
409#102—614no—
410#102—1222no—
411#102C4×D6114yes—
412#97—22no—
413C4×C8—1222no—
414#95—21no—
415#95—611no—
416#95—611no—
417#95—611no—
418#95—611no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—21no—
428C2×Q8—1224no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic12212yes—
435C2×D4C2×C4×D6112yes—
436C2×D4#1175822212yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—4881no—
454C22×C4—4881no—
455C22×C4—4881no—
456C22×C4—4881no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—612no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×C4×D6112yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—612no—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—4881no—
486C4⋊C4—4881no—
487C4⋊C4C2×Dic12212yes—
488C4⋊C4C2×C4×D6112yes—
489C22⋊C4—24no—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C42—81no—
493C42—612no—
494C42—612no—
495C42—1221no—
496C42—1221no—
497C42—4881no—
498C42—4881no—
499C42—4881no—
500C42—4881no—
501C42C2×Dic12212yes—
502C42#1175822412yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D6111yescharacteristic
511Q8—4884no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C4×Dic12214yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—4884no—
546C2×C4—4884no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#1451975412yes—
557C2×C4#1451975412yes—
558C2×C4C22×C4×D6111yescharacteristic
559C2×C4C22×Dic12211yescharacteristic
560C2×C4#41×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4C2×#1175822411yescharacteristic
563C2×C4C2×#1175822211yescharacteristic
564C8—22no—
565C8—41no—
566C8—1222no—
567C8—2441no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C4×D4×D6111yescharacteristic
576C22C2×#1451975411yescharacteristic
577C22#1684027111yescharacteristic
578C22#1684056611yescharacteristic
579C22#1684057111yescharacteristic
580C22#1684057211yescharacteristic
581C22#1684057211yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—1222no—
588C4—1222no—
589C4—4881no—
590C4—2442no—
591C4—2442no—
592C4—2441no—
593C4—2442no—
594C4—2441no—
595C4C2×C4×Dic12211yescharacteristic
596C4C4×#1175822411yescharacteristic
597C4#1684029211yescharacteristic
598C4#1684031511yescharacteristic
599C2—44no—
600C2C4×#1451975411yescharacteristic
601C2#2145464811yescharacteristic
602C2#2145467511yescharacteristic
603C2#2145565211yescharacteristic
604C2#2145568511yescharacteristic
605C2#2145600711yescharacteristic
606C2#2145610011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#21623
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #21448563C2semidirect
4. #21446769C2semidirect
5. #21446590C2semidirect
6. #21446407C2semidirect
7. #21446396C2semidirect
11. #21445716C2semidirect
12. C4×#14519708C2semidirect
13. #21445679C2semidirect
19. #16840050C4semidirect
66. #14519776C2×C4semidirect
87. #14519709C2×C4semidirect
89. #14519708C2×C4semidirect
91. #14519706C2×C4semidirect
112. C4×D4×C61Q8semidirect
363. C61#21623semidirect
380. C4×D4Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168400502
#214485632
#214481802
#214467692
#214465902
#214464072
#214463962
#214460782
#214460682
#214458272
#214457162
C4×#145197082
#214456792
C4×#145196982
C61×#17622
#2162361