Finite Group Databasebeta

← #29699749

Subgroups — #29699749

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839847C211yesmaximal, characteristic
2#21448788C211yesmaximal, characteristic
3#21448783C211yesmaximal, characteristic, complemented
4#21446676C211yesmaximal, characteristic
5#21446670C211yesmaximal, characteristic
6#21446462C211yesmaximal, characteristic, complemented
7#21446440C211yesmaximal, characteristic
8#21446438C211yesmaximal, characteristic
9#21445982C211yesmaximal, characteristic, complemented
10#21445832C211yesmaximal, characteristic, complemented
11#21445828C211yesmaximal, characteristic
12C4×#14519708C211yesmaximal, characteristic, complemented
13#21445707C211yesmaximal, characteristic, complemented
14#21445680C211yesmaximal, characteristic, complemented
15C61×#1928C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19#16839847C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839653—21no—
36#16839651—21no—
37#16839647—21no—
38#16839645—21no—
39#16839629—21no—
40#16839627C2211yescharacteristic
41#16839627C2211yescharacteristic
42#16839624—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46C4×C4⋊Dic61C2211yescharacteristic
47C4×#11758198—21no—
48#16839607—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67Q8×Dic61—22no—
68Q8×Dic61C2×C414yes—
69#14519739—22no—
70#14519739C2×C414yes—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196D411yescharacteristic
76C2×#11758196D411yescharacteristic
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519711—41no—
89#14519708—22no—
90#14519708—22no—
91#14519708C2×C414yes—
92#14519706—22no—
93#14519701—41no—
94#14519699—41no—
95#14519699C2×C414yes—
96#14519697—41no—
97C4×#92418—22no—
98C8×Dic61—22no—
99C8×Dic61C2×C414yes—
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×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#107×C61—41no—
120#103×C61—41no—
121#103×C61C2×C414yes—
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×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—81no—
154C2×C4×D61—81no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C412yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—81no—
170#11758201—28no—
171#11758198—24no—
172#11758198—42no—
173#11758198—42no—
174#11758198—81no—
175#11758198—81no—
176C4⋊Dic61—21no—
177C4⋊Dic61—24no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C412yes—
183#11758196—22no—
184#11758196—22no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—24no—
188C4×Dic61—42no—
189C4×Dic61—42no—
190C4×Dic61—81no—
191C4×Dic61C22×C412yes—
192C4×Dic61#4112yes—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418C22×C412yes—
197C2×#92418#4112yes—
198C2×#92418#4112yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—41no—
210Q8×C61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
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—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61#13512yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C4×D412yes—
254C2×Dic61C4×D412yes—
255C2×Dic61#13512yes—
256D244—22no—
257D244—22no—
258D244—41no—
259D244—41no—
260C4×D61—84no—
261C8×C61—22no—
262C8×C61—41no—
263#92418—41no—
264#92418#11714yes—
265#22806—611nomaximal, Sylow
266C22×C61C2×C4×D411yescharacteristic
267C22×C61#54711yescharacteristic
268C22×C61#55011yescharacteristic
269C22×C61C2×#13511yescharacteristic
270C22×C61C2×#13511yescharacteristic
271C22×C61#57711yescharacteristic
272C22×C61#57711yescharacteristic
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61C2×#11711yescharacteristic
287C4×C61#52011yescharacteristic
288C4×C61#55511yescharacteristic
289C4×C61#55611yescharacteristic
290Dic61—22no—
291Dic61—22no—
292Dic61—22no—
293Dic61—41no—
294Dic61—42no—
295Dic61—81no—
296Dic61—81no—
297C2×C4×SD16—611no—
298#2357—611no—
299#2356—611no—
300#1968—611no—
301#1968—611no—
302#1928—611no—
303#1928D6111yescharacteristic, complemented
304#1925—611no—
305#1925—611no—
306#1846—611no—
307#1805—611no—
308#1804—611no—
309C4×#106—611no—
310C4×#106—611no—
311C8×C4⋊C4—611no—
312C2×C61#235411yescharacteristic
313C2×C61C4×#13511yescharacteristic
314C2×C61#294911yescharacteristic
315C2×C61#333711yescharacteristic
316C2×C61#333711yescharacteristic
317C2×C61#339411yescharacteristic
318C2×C61#339411yescharacteristic
319D61—82no—
320C22×SD16—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C4×SD16—618no—
324C2×#107—611no—
325C2×#107C2×D6111yescharacteristic
326C2×#106—611no—
327C2×#106—611no—
328C2×#106—611no—
329C2×C4⋊C8—611no—
330C2×#103—611no—
331C2×#103C2×D6111yescharacteristic
332C2×#103C2×D6111yescharacteristic
333C2×#102—611no—
334C2×#102—611no—
335C2×#102—611no—
336C2×C4×C8—611no—
337C2×C4×C8—611no—
338C2×C4×C8—611no—
339#391—611no—
340#391C2×D6111yescharacteristic
341#390—611no—
342#390—611no—
343#388—1221no—
344#388—1221no—
345#387—1221no—
346#387—1221no—
347#383—1221no—
348#382—611no—
349#382—611no—
350#381—1221no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4C2×D6111yescharacteristic
356C4×C22⋊C4—1221no—
357#376—1221no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340C2×D6111yescharacteristic
362#336—611no—
363#336C2×D6111yescharacteristic
364C61#2280611yescharacteristic, complemented, Sylow
365C22×Q8C22×D6111yescharacteristic
366C22×D4—611no—
367C23×C4—1221no—
368C2×SD16—614no—
369C2×SD16—1222no—
370C2×SD16—1222no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—614no—
378C4×D4—1222no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#107—41no—
402#107—2441no—
403#106—614no—
404#106—1222no—
405#106—1222no—
406C4⋊C8—1222no—
407#103—41no—
408#103—2441no—
409#103C4×D6114yes—
410#102—614no—
411#102—2441no—
412#102—2441no—
413C4×C8—614no—
414C4×C8—1222no—
415C4×C8—1222no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—611no—
420#95—611no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430C24—1221no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D6112yes—
437C2×D4—612no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1224no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—4881no—
456C22×C4—4881no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460SD16—1228no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8C2×C4×D6112yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—612no—
479C4⋊C4—1222no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1222no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4C2×C4×D6112yes—
488C22⋊C4—1224no—
489C22⋊C4—4881no—
490C22⋊C4—4881no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—21no—
494C42—21no—
495C42—81no—
496C42—612no—
497C42—612no—
498C42—1224no—
499C42—1221no—
500C42—1221no—
501C42—4881no—
502C42—2442no—
503C42—2442no—
504C23—1222no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23C2×D4×D6111yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—1222no—
517D4—1222no—
518D4—2441no—
519D4—2441no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—4884no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4#1451980012yes—
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#41×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4#1451982211yescharacteristic
565C2×C4#1451982411yescharacteristic
566C8—22no—
567C8—41no—
568C8—614no—
569C8—2441no—
570C22—4881no—
571C22—4881no—
572C22—4881no—
573C22—4881no—
574C22—2442no—
575C22—2442no—
576C22—2442no—
577C22C4×D4×D6111yescharacteristic
578C22#1684041011yescharacteristic
579C22#1684043011yescharacteristic
580C22#135×D6111yescharacteristic
581C22C2×#1451980011yescharacteristic
582C22#1684057111yescharacteristic
583C22#1684057211yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—1222no—
591C4—1222no—
592C4—1222no—
593C4—4881no—
594C4—4881no—
595C4—2441no—
596C4—2442no—
597C4#117×D6111yescharacteristic
598C4#1684024111yescharacteristic
599C4#1684047411yescharacteristic
600C4#1684047911yescharacteristic
601C2—4882no—
602C2#2144876511yescharacteristic
603C2C4×#1451980011yescharacteristic
604C2#2145467511yescharacteristic
605C2#2145819611yescharacteristic
606C2#2145820111yescharacteristic
607C2#2145881411yescharacteristic
608C2#2145881811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1928C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1928D61
hypocenterC61#22806
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
3. #21448783C2semidirect
6. #21446462C2semidirect
9. #21445982C2semidirect
10. #21445832C2semidirect
12. C4×#14519708C2semidirect
13. #21445707C2semidirect
14. #21445680C2semidirect
15. C61×#1928C2semidirect
19. #16839847C4semidirect
303. #1928D61semidirect
364. C61#22806semidirect

Maximal subgroups

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

subgroupindex
C2×#168398472
#214487882
#214487832
#214466762
#214466702
#214464622
#214464402
#214464382
#214459822
#214458322
#214458282
C4×#145197082
#214457072
#214456802
C61×#19282
#2280661