Finite Group Databasebeta

← #29699277

Subgroups — #29699277

Order 15616.

870 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×#16839847C211yesmaximal, characteristic
2#21448783C211yesmaximal, characteristic, complemented
3#21448728C211yesmaximal, characteristic
4#21446497C211yesmaximal, characteristic, complemented
5#21446407C211yesmaximal, characteristic, complemented
6#21446401C211yesmaximal, characteristic, complemented
7#21446156C211yesmaximal, characteristic
8#21446150C211yesmaximal, characteristic
9#21445948C211yesmaximal, characteristic, complemented
10#21445832C211yesmaximal, characteristic, complemented
11#21445828C211yesmaximal, characteristic
12C8×#11758196C211yesmaximal, characteristic, complemented
13C4×#14519699C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#1933C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21#16839847C418yescomplemented
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839647—21no—
33#16839645—21no—
34#16839636—21no—
35#16839633—21no—
36#16839629—21no—
37#16839627C2211yescharacteristic
38#16839627C2211yescharacteristic
39#16839619—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839607—21no—
44#95×D61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69Q8×Dic61—22no—
70Q8×Dic61C2×C414yescomplemented
71C4×D244—22no—
72C4×D244—41no—
73#14519739—22no—
74#14519739—22no—
75#14519739—41no—
76#14519739C2×C414yes—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519708C2×C414yescomplemented
93#14519706—22no—
94#14519701—41no—
95#14519699—22no—
96#14519699C2×C414yes—
97#14519697—22no—
98C8×Dic61—41no—
99C8×Dic61C2×C414yescomplemented
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—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#103×C61—41no—
120#103×C61C2×C414yescomplemented
121C4×C8×C61—41no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125C23×D61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244C22×C412yes—
150C2×D244#4112yes—
151C2×D244#4112yes—
152C2×C4×D61—24no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C412yes—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—81no—
173C4⋊C4×C61C22×C412yes—
174C42×C61—21no—
175C42×C61—21no—
176#11758201—28no—
177#11758198—24no—
178#11758198—42no—
179#11758198—42no—
180C4⋊Dic61—21no—
181C4⋊Dic61—24no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—81no—
187C4⋊Dic61C22×C412yes—
188#11758196—22no—
189#11758196—22no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—24no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C4×Dic61C22×C412yes—
196C4×Dic61#4112yes—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C22×C412yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—41no—
214Q8×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
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#13512yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61C4×D412yes—
258C2×Dic61C4×D412yes—
259C2×Dic61#13512yes—
260D244—22no—
261D244—22no—
262D244—22no—
263D244#11714yes—
264C4×D61—84no—
265C4×D61—84no—
266C4×D61—84no—
267C8×C61—22no—
268C8×C61—41no—
269#92418—22no—
270#92418—41no—
271#22768—611nomaximal, Sylow
272C22×C61C2×C4×D411yescharacteristic
273C22×C61#54611yescharacteristic
274C22×C61#54811yescharacteristic
275C22×C61C2×#13511yescharacteristic
276C22×C61C2×#13511yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57711yescharacteristic
279C2×D61—24no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61C2×#11711yescharacteristic
293C4×C61#52011yescharacteristic
294C4×C61#55211yescharacteristic
295C4×C61#55311yescharacteristic
296Dic61—22no—
297Dic61—22no—
298Dic61—22no—
299Dic61—42no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303C2×C4×SD16—611no—
304#2356—611no—
305#2349—611no—
306#1933—611no—
307#1933D6111yescharacteristic, complemented
308#1920—611no—
309#1920—611no—
310#1877—611no—
311#1877—611no—
312#1835—611no—
313#1805—611no—
314#1804—611no—
315C8×C4⋊C4—611no—
316C4×#102—611no—
317C4×#102—611no—
318C2×C61#235211yescharacteristic
319C2×C61C4×#13511yescharacteristic
320C2×C61#294911yescharacteristic
321C2×C61#329611yescharacteristic
322C2×C61#329611yescharacteristic
323C2×C61#336011yescharacteristic
324C2×C61#336011yescharacteristic
325D61—44no—
326C22×SD16—611no—
327C2×C4×Q8—611no—
328C2×C4×D4—611no—
329C2×C4×D4—611no—
330C2×C4×D4—611no—
331C4×SD16—618no—
332C2×#106—611no—
333C2×C4⋊C8—611no—
334C2×#103—611no—
335C2×#103C2×D6111yescharacteristic
336C2×#103C2×D6111yescharacteristic
337C2×#102—611no—
338C2×#102—611no—
339C2×#102—611no—
340C2×C4×C8—611no—
341C2×C4×C8—611no—
342C2×C4×C8C2×D6111yescharacteristic
343#391—611no—
344#391C2×D6111yescharacteristic
345#389—611no—
346#389C2×D6111yescharacteristic
347#387—1221no—
348#387—1221no—
349#385—1221no—
350#385—1221no—
351#383—1221no—
352#382—611no—
353#382—611no—
354#380—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358#376—1221no—
359C2×#95—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#340C2×D6111yescharacteristic
368#336—611no—
369#336—611no—
370C61#2276811yescharacteristic, complemented, Sylow
371C22×Q8C22×D6111yescharacteristic
372C22×D4—611no—
373C23×C4—1221no—
374C23×C4—1221no—
375C23×C4—1221no—
376C2×SD16—614no—
377C2×SD16—1222no—
378C2×SD16—1222no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8C22×D6111yescharacteristic
382C22×C8C22×D6111yescharacteristic
383C4×Q8—614no—
384C4×Q8—1222no—
385C4×D4—614no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—2441no—
390C4×D4—2441no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42C22×D6111yescharacteristic
411#106—614no—
412C4⋊C8—1222no—
413#103—41no—
414#103—2441no—
415#103C4×D6114yescomplemented
416#102—614no—
417#102—1222no—
418#102—1222no—
419C4×C8—41no—
420C4×C8—614no—
421C4×C8—2441no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—611no—
426#95—611no—
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—
436C24—1221no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D6112yes—
443C2×D4—612no—
444C2×D4—612no—
445C2×D4—612no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—1224no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—4881no—
462C22×C4—4881no—
463C22×C4—4881no—
464C22×C4—4881no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—1228no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—612no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8C2×C4×D6112yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—81no—
491C4⋊C4—612no—
492C4⋊C4—1222no—
493C4⋊C4—1224no—
494C4⋊C4—1221no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—4881no—
501C4⋊C4C2×C4×D6112yes—
502C22⋊C4—1224no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C42—21no—
506C42—21no—
507C42—612no—
508C42—612no—
509C42—1224no—
510C42—1221no—
511C42—1221no—
512C42—2442no—
513C42—2442no—
514C23—1222no—
515C23—1222no—
516C23—1222no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23C2×D4×D6111yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—614no—
527D4—1222no—
528D4—1222no—
529D4—1222no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
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—41no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4884no—
561C2×C4—4884no—
562C2×C4—4884no—
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#1451980012yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#1451982211yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—22no—
579C8—41no—
580C8—1222no—
581C8—2441no—
582C22—1224no—
583C22—2442no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22C4×D4×D6111yescharacteristic
590C22#1684040111yescharacteristic
591C22#1684042011yescharacteristic
592C22#135×D6111yescharacteristic
593C22C2×#1451980011yescharacteristic
594C22#1684056811yescharacteristic
595C22#1684056911yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—1222no—
603C4—1222no—
604C4—1222no—
605C4—4881no—
606C4—4881no—
607C4—2441no—
608C4—2442no—
609C4#117×D6111yescharacteristic
610C4#1684024111yescharacteristic
611C4#1684044811yescharacteristic
612C4#1684045711yescharacteristic
613C2—2444no—
614C2#2144874711yescharacteristic
615C2C4×#1451980011yescharacteristic
616C2#2145467311yescharacteristic
617C2#2145764911yescharacteristic
618C2#2145766011yescharacteristic
619C2#2145846511yescharacteristic
620C2#2145847311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1933C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1933D61
hypocenterC61#22768
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. #21448783C2semidirect
4. #21446497C2semidirect
5. #21446407C2semidirect
6. #21446401C2semidirect
9. #21445948C2semidirect
10. #21445832C2semidirect
12. C8×#11758196C2semidirect
15. C61×#1933C2semidirect
21. #16839847C4semidirect
70. Q8×Dic61C2×C4semidirect
92. #14519708C2×C4semidirect
99. C8×Dic61C2×C4semidirect
120. #103×C61C2×C4semidirect
307. #1933D61semidirect
370. C61#22768semidirect
415. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398472
#214487832
#214487282
#214464972
#214464072
#214464012
#214461562
#214461502
#214459482
#214458322
#214458282
C8×#117581962
C4×#145196992
#214456332
C61×#19332
#2276861