Finite Group Databasebeta

← #29688607

Subgroups — #29688607

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840151C211yesmaximal, characteristic
2C2×#16839873C211yesmaximal, characteristic, complemented
3C2×#16839861C211yesmaximal, characteristic, complemented
4C2×#16839849C211yesmaximal, characteristic, complemented
5C2×#16839844C211yesmaximal, characteristic, complemented
6#21448542C211yesmaximal, characteristic, complemented
7#21448527C211yesmaximal, characteristic, complemented
8#21446698C211yesmaximal, characteristic, complemented
9#21446693C211yesmaximal, characteristic, complemented
10#21446683C211yesmaximal, characteristic, complemented
11#21446671C211yesmaximal, characteristic, complemented
12#21446628C211yesmaximal, characteristic, complemented
13#21446624C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1847C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C22×D488C2211yescharacteristic
19C22×D488C2211yescharacteristic
20C2×#14519774C2211yescharacteristic
21C2×#14519765C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×#14519739C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25C2×#14519739C2211yescharacteristic
26#16840151C418yescomplemented
27#16839873—24no—
28#16839861—24no—
29#16839849—24no—
30#16839844—24no—
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×C4×#92418C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44#16839667C2211yescharacteristic
45#16839663C2211yescharacteristic
46#16839663C2211yescharacteristic
47#16839663C2211yescharacteristic
48#16839652—22no—
49#16839640—22no—
50#16839640—22no—
51#16839633—22no—
52#16839608C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244—22no—
62C22×D244C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C2×#11758190—22no—
69C2×#11758190—22no—
70C2×#11758190—22no—
71C2×#11758190—22no—
72C2×#11758190C2×C414yescomplemented
73C2×#11758190C2×C414yescomplemented
74C2×D488—22no—
75C2×D488—22no—
76C2×D488—22no—
77C2×D488—22no—
78C2×D488C2×C414yescomplemented
79C2×D488C2×C414yescomplemented
80C22×#92418C2311yescharacteristic
81C22×#92418C2311yescharacteristic
82#14519774—22no—
83#14519774C2×C414yescomplemented
84#14519765—22no—
85#14519765C2×C414yescomplemented
86#14519742—22no—
87#14519742—22no—
88#14519739—22no—
89#14519739—22no—
90#14519739—22no—
91#14519739—22no—
92#14519739—22no—
93#14519739—22no—
94C2×#11758196—21no—
95C2×#11758198—22no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—22no—
100C2×#11758198—22no—
101C2×#11758198—22no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105#14519700—22no—
106#14519700—22no—
107#14519697—22no—
108#14519697—22no—
109#14519697—22no—
110#14519697—22no—
111#14519697—22no—
112#14519697—22no—
113#14519687—22no—
114#14519687D414yescomplemented
115C4×#92418—22no—
116C8×Dic61—22no—
117C8×Dic61D414yescomplemented
118#14519684—22no—
119#14519684D412yes—
120#14519684D412yes—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#107×C61—22no—
130#107×C61—22no—
131#106×C61—22no—
132#106×C61—22no—
133#97×C61C2×C414yescomplemented
134#95×C61—22no—
135C23×D61—21no—
136C23×D61—21no—
137C23×D61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—28no—
164C2×D244—82no—
165C2×D244—82no—
166C2×D244C22×C412yes—
167C2×D244C22×C412yes—
168C2×D244C22×C412yes—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—82no—
173C2×C4×D61—82no—
174C2×C4×D61—82no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C22×C412yes—
181C2×C8×C61C22×C412yes—
182C2×C8×C61C22×C412yes—
183C2×C8×C61C2×D412yes—
184C2×C8×C61C2×D412yes—
185C2×C8×C61#4112yes—
186C2×C8×C61#4112yes—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—82no—
196C42×C61C22×C412yes—
197C42×C61#4112yes—
198#11758198—28no—
199#11758198—42no—
200#11758198—42no—
201#11758198—42no—
202#11758198—82no—
203#11758198—82no—
204#11758198—82no—
205#11758196—42no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C4×Dic61C2×D412yes—
210C4×Dic61C2×D412yes—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217D488—28no—
218D488—28no—
219#11758190—28no—
220#11758190—28no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—42no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61C4×D414yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61#12712yes—
263C2×Dic61#12712yes—
264D244—22no—
265D244—22no—
266D244—22no—
267D244—22no—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272D244—41no—
273D244—41no—
274D244—41no—
275D244—41no—
276D244—88no—
277C4×D61—84no—
278C4×D61—84no—
279C4×D61—84no—
280Dic122—22no—
281Dic122—22no—
282Dic122—41no—
283Dic122—41no—
284C8×C61C4×D414yes—
285C8×C61C4×D414yes—
286#92418—22no—
287#92418—22no—
288#21704—611nomaximal, Sylow
289C22×C61C2×C4×D411yescharacteristic
290C22×C61C2×#12711yescharacteristic
291C22×C61#53211yescharacteristic
292C22×C61#57612yes—
293C22×C61#57712yes—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61—82no—
309C4×C61—24no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C2×C4×D411yescharacteristic
315C4×C61C2×C4×D411yescharacteristic
316C4×C61#53211yescharacteristic
317C4×C61#53211yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—42no—
322C2×#496—611no—
323C2×#442—611no—
324C2×#440—611no—
325C2×C4×SD16—611no—
326C2×C4×D8—611no—
327#2328—611no—
328#2326—611no—
329#1973—611no—
330#1972—611no—
331#1970—611no—
332#1968—611no—
333#1958—611no—
334#1957—611no—
335#1847D6111yescharacteristic, complemented
336#1726—611no—
337C2×C61C4×#12711yescharacteristic
338C2×C61#294612yes—
339C2×C61#315411yescharacteristic
340C2×C61#315511yescharacteristic
341C2×C61#317912yes—
342D61—82no—
343D61—82no—
344D61—82no—
345C22×SD16—611no—
346C22×SD16—611no—
347C22×D8—611no—
348C22×D8—611no—
349C2×#127—611no—
350C2×#124—611no—
351C2×C4×Q8—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355#496—618no—
356#442—1224no—
357#440—1224no—
358C4×SD16—1224no—
359C4×D8—1224no—
360C2×#107C2×D6111yescharacteristic
361C2×#107C2×D6111yescharacteristic
362C2×#106C2×D6111yescharacteristic
363C2×#106C2×D6111yescharacteristic
364C2×#103—611no—
365C2×#103—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#97—611no—
373C2×#97—611no—
374C2×#97C2×D6111yescharacteristic
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378#391—611no—
379#390—611no—
380#390—611no—
381#390—611no—
382#388—1222no—
383#386—1222no—
384#386—1222no—
385#385—1222no—
386#384C2×D6111yescharacteristic
387#382C2×D6111yescharacteristic
388#376—611no—
389C61#2170411yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×D4—611no—
392C22×D4—611no—
393C22×D4—611no—
394C22×D4—1222no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×SD16—614no—
399C2×SD16—614no—
400C2×SD16—1222no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C2×SD16—1222no—
404C2×D8—614no—
405C2×D8—614no—
406C2×D8—1222no—
407C2×D8—1222no—
408C2×D8—1222no—
409C2×D8—1222no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414#127—614no—
415#127—1222no—
416#124—614no—
417#124—1222no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1222no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443#107—22no—
444#107—22no—
445#106—22no—
446#106—22no—
447#103—1222no—
448#103—1222no—
449#102—1222no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#102—1222no—
455#97—614no—
456#97—1222no—
457#97C4×D6114yescomplemented
458C4×C8—614no—
459C4×C8—1222no—
460C4×C8—1222no—
461#95—22no—
462#95—612no—
463#95—612no—
464#95—1222no—
465C24—1221no—
466C24—1221no—
467C24—1221no—
468C2×Q8—612no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×D4—612no—
475C2×D4—612no—
476C2×D4—612no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1228no—
493C2×D4—4882no—
494C2×D4—4882no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—612no—
502C22×C4—1221no—
503C22×C4—4882no—
504C22×C4—4882no—
505C22×C4—4882no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4D4×D6112yes—
510C22×C4C23×D6111yescharacteristic
511SD16—1228no—
512SD16—1228no—
513D8—1228no—
514D8—1228no—
515C2×C8—612no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8C2×C4×D6112yes—
522C2×C8C2×C4×D6112yes—
523C2×C8D4×D6112yes—
524C2×C8D4×D6112yes—
525C2×C8#1175822612yes—
526C2×C8#1175822612yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—82no—
536C4⋊C4—2442no—
537C22⋊C4—1228no—
538C22⋊C4—4882no—
539C22⋊C4—4882no—
540C22⋊C4—4882no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C42—612no—
545C42—612no—
546C42—1221no—
547C42—1221no—
548C42—2442no—
549C42C2×C4×D6112yes—
550C42#1175822612yes—
551C23—1222no—
552C23—1222no—
553C23—1222no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2442no—
558C23—2442no—
559C23—2442no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2442no—
564C23—2442no—
565C23—2442no—
566C23C2×D4×D6111yescharacteristic
567Q8—1222no—
568Q8—1222no—
569Q8—2441no—
570Q8—2441no—
571D4—1222no—
572D4—1222no—
573D4—1222no—
574D4—1222no—
575D4—1222no—
576D4—1222no—
577D4—4888no—
578D4—2441no—
579D4—2441no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—4884no—
603C2×C4—4884no—
604C2×C4—4884no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4#1451973914yes—
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×#1175822611yescharacteristic
612C2×C4C2×#1175822611yescharacteristic
613C2×C4C2×#1175822611yescharacteristic
614C8—1222no—
615C8—1222no—
616C8#1451973914yes—
617C8#1451973914yes—
618C22—4882no—
619C22—4882no—
620C22—4882no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—2444no—
628C22—2444no—
629C22—2444no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22C2×#1451973911yescharacteristic
634C22#127×D6111yescharacteristic
635C22#1684031311yescharacteristic
636C22#1684056612yes—
637C22#1684057112yes—
638C4—24no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—1222no—
644C4—1222no—
645C4—2442no—
646C4—2442no—
647C4C2×#1451973911yescharacteristic
648C4C2×#1451973911yescharacteristic
649C4#1684031311yescharacteristic
650C4#1684031311yescharacteristic
651C2—4882no—
652C2—4882no—
653C2—4882no—
654C2#2144838111yescharacteristic
655C2#2145464912yes—
656C2#2145629411yescharacteristic
657C2#2145630311yescharacteristic
658C2#2145650712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC23×C61C22×D4
prime radicalC2×#16840151C2
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#21704
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
2. C2×#16839873C2semidirect
3. C2×#16839861C2semidirect
4. C2×#16839849C2semidirect
5. C2×#16839844C2semidirect
6. #21448542C2semidirect
7. #21448527C2semidirect
8. #21446698C2semidirect
9. #21446693C2semidirect
10. #21446683C2semidirect
11. #21446671C2semidirect
12. #21446628C2semidirect
13. #21446624C2semidirect
14. #21445507C2semidirect
15. C61×#1847C2semidirect
26. #16840151C4semidirect
72. C2×#11758190C2×C4semidirect
73. C2×#11758190C2×C4semidirect
78. C2×D488C2×C4semidirect
79. C2×D488C2×C4semidirect
83. #14519774C2×C4semidirect
85. #14519765C2×C4semidirect
114. #14519687D4semidirect
117. C8×Dic61D4semidirect
133. #97×C61C2×C4semidirect
335. #1847D61semidirect
389. C61#21704semidirect
457. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401512
C2×#168398732
C2×#168398612
C2×#168398492
C2×#168398442
#214485422
#214485272
#214466982
#214466932
#214466832
#214466712
#214466282
#214466242
#214455072
C61×#18472
#2170461