Finite Group Databasebeta

← #29688614

Subgroups — #29688614

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×#16839875C211yesmaximal, characteristic, complemented
3C2×#16839864C211yesmaximal, characteristic, complemented
4C2×C4×#11758190C211yesmaximal, characteristic, complemented
5C2×C4×D488C211yesmaximal, characteristic, complemented
6#21448544C211yesmaximal, characteristic, complemented
7#21448530C211yesmaximal, characteristic, complemented
8#21446700C211yesmaximal, characteristic, complemented
9#21446695C211yesmaximal, characteristic, complemented
10#21446685C211yesmaximal, characteristic, complemented
11#21446673C211yesmaximal, characteristic, complemented
12#21446630C211yesmaximal, characteristic, complemented
13#21446625C211yesmaximal, characteristic, complemented
14#21445987C211yesmaximal, characteristic, complemented
15C61×#1726C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C22×D488C2211yescharacteristic
19C22×D488C2211yescharacteristic
20C2×#14519774C2211yescharacteristic
21C2×#14519765C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26#16840151C418yescomplemented
27#16839875—24no—
28#16839864—24no—
29C4×#11758190—24no—
30C4×D488—24no—
31C2×#14519711C2211yescharacteristic
32C2×#14519711C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519709C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519699C2211yescharacteristic
39C2×#14519699C2211yescharacteristic
40C2×#14519699C2211yescharacteristic
41C2×#14519699C2211yescharacteristic
42C2×#14519699C2211yescharacteristic
43#16839668C2211yescharacteristic
44#16839664C2211yescharacteristic
45#16839664C2211yescharacteristic
46#16839664C2211yescharacteristic
47#16839654—22no—
48#16839640—22no—
49#16839640—22no—
50#16839636—22no—
51#16839632C2211yescharacteristic
52#16839628C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#376C2211yescharacteristic
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
80#14519774—22no—
81#14519774C2×C414yescomplemented
82#14519765—22no—
83#14519765C2×C414yescomplemented
84C4×Dic122—22no—
85C4×Dic122—22no—
86C4×D244—22no—
87C4×D244—22no—
88C4×D244—22no—
89C4×D244—22no—
90C4×D244—22no—
91C4×D244—22no—
92C2×C4⋊Dic61—22no—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758198—22no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—22no—
103C2×#11758198—22no—
104C2×#11758198—22no—
105C2×C4×Dic61—21no—
106#14519711—22no—
107#14519711—22no—
108#14519709—22no—
109#14519709—22no—
110#14519702—22no—
111#14519702—22no—
112#14519699—22no—
113#14519699—22no—
114#14519699—22no—
115#14519699—22no—
116#14519699—22no—
117#14519699—22no—
118#14519684—22no—
119#14519684—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#97×C61—22no—
128#97×C61C2×C414yescomplemented
129#97×C61D414yescomplemented
130C4×C8×C61—22no—
131C4×C8×C61—22no—
132C4×C8×C61D414yescomplemented
133#95×C61D412yes—
134#95×C61D412yes—
135C23×D61—21no—
136C23×D61—21no—
137C23×D61—21no—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
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×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C2×C8×C61C22×C412yes—
187C2×C8×C61C2×D412yes—
188C2×C8×C61C2×D412yes—
189C2×C8×C61C2×D412yes—
190C2×C8×C61C2×D412yes—
191C2×C8×C61#4112yes—
192C2×C8×C61#4112yes—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61C22×C412yes—
196C42×C61C2×D412yes—
197C42×C61C2×D412yes—
198C42×C61#4112yes—
199#11758198—28no—
200#11758198—42no—
201#11758198—42no—
202#11758198—42no—
203#11758198—82no—
204#11758198—82no—
205#11758198—82no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—82no—
215#11758196—42no—
216C4×Dic61—42no—
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—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61C4×D414yes—
245C2×C4×C61#12712yes—
246C2×C4×C61#12712yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
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×C61—22no—
285C8×C61—22no—
286C8×C61C4×D414yes—
287C8×C61C4×D414yes—
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—22no—
310C4×C61—22no—
311C4×C61—24no—
312C4×C61—42no—
313C4×C61C2×C4×D411yescharacteristic
314C4×C61C2×C4×D411yescharacteristic
315C4×C61#53211yescharacteristic
316C4×C61#53211yescharacteristic
317Dic61—42no—
318Dic61—81no—
319Dic61—81no—
320Dic61—81no—
321Dic61—81no—
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#1847—611no—
336#1726D6111yescharacteristic, complemented
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×#107—611no—
361C2×#107—611no—
362C2×#106—611no—
363C2×#106—611no—
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×#97C2×D6111yescharacteristic
373C2×#97C2×D6111yescharacteristic
374C2×#97C2×D6111yescharacteristic
375C2×C4×C8C2×D6111yescharacteristic
376C2×C4×C8C2×D6111yescharacteristic
377C2×C4×C8C2×D6111yescharacteristic
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#384—611no—
387#382—611no—
388#376C2×D6111yescharacteristic
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×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
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—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1221no—
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—1221no—
440C2×C42C22×D6111yescharacteristic
441C2×C42C22×D6111yescharacteristic
442C2×C42C22×D6111yescharacteristic
443#107—1222no—
444#107—1222no—
445#106—1222no—
446#106—1222no—
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—22no—
456#97C4×D6114yescomplemented
457#97D24414yescomplemented
458C4×C8—22no—
459C4×C8—22no—
460C4×C8D24414yescomplemented
461#95—1222no—
462#95—1222no—
463#95D24412yes—
464#95D24412yes—
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—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—4882no—
501C22×C4—4882no—
502C22×C4—4882no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4C2×D24412yes—
507C22×C4C2×D24412yes—
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—1228no—
512SD16—1228no—
513D8—1228no—
514D8—1228no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8C2×C4×D6112yes—
520C2×C8C2×C4×D6112yes—
521C2×C8C2×D24412yes—
522C2×C8C2×D24412yes—
523C2×C8C2×D24412yes—
524C2×C8C2×D24412yes—
525C2×C8#1175822212yes—
526C2×C8#1175822212yes—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—4882no—
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—21no—
545C42—21no—
546C42—2442no—
547C42C2×C4×D6112yes—
548C42C2×D24412yes—
549C42C2×D24412yes—
550C42#1175822212yes—
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—
566C23C22×D24411yescharacteristic
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—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—1222no—
592C2×C4—4884no—
593C2×C4—4884no—
594C2×C4—4884no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4C4×D24414yes—
606C2×C4#1451977412yes—
607C2×C4#1451977412yes—
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C22×D24411yescharacteristic
610C2×C4C22×D24411yescharacteristic
611C2×C4C2×#1175822211yescharacteristic
612C2×C4C2×#1175822211yescharacteristic
613C2×C4C2×#1175822211yescharacteristic
614C8—22no—
615C8—22no—
616C8C4×D24414yes—
617C8C4×D24414yes—
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×C4×D24411yescharacteristic
634C22C2×#1451977411yescharacteristic
635C22#1684031211yescharacteristic
636C22#1684056712yes—
637C22#1684057412yes—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—4881no—
643C4—4881no—
644C4—4881no—
645C4—4881no—
646C4—2442no—
647C4C2×C4×D24411yescharacteristic
648C4C2×C4×D24411yescharacteristic
649C4#1684031211yescharacteristic
650C4#1684031211yescharacteristic
651C2—4882no—
652C2—4882no—
653C2—4882no—
654C2C4×#1451977411yescharacteristic
655C2#2145465312yes—
656C2#2145629511yescharacteristic
657C2#2145630511yescharacteristic
658C2#2145650212yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1726C2
socleC23×C61C22×D4
prime radicalC2×#16840151C2
cosocleC22×C4×C61C24
hypercenter#1726D61
hypocenterC61#21704
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. C2×#16839875C2semidirect
3. C2×#16839864C2semidirect
4. C2×C4×#11758190C2semidirect
5. C2×C4×D488C2semidirect
6. #21448544C2semidirect
7. #21448530C2semidirect
8. #21446700C2semidirect
9. #21446695C2semidirect
10. #21446685C2semidirect
11. #21446673C2semidirect
12. #21446630C2semidirect
13. #21446625C2semidirect
14. #21445987C2semidirect
15. C61×#1726C2semidirect
26. #16840151C4semidirect
72. C2×#11758190C2×C4semidirect
73. C2×#11758190C2×C4semidirect
78. C2×D488C2×C4semidirect
79. C2×D488C2×C4semidirect
81. #14519774C2×C4semidirect
83. #14519765C2×C4semidirect
128. #97×C61C2×C4semidirect
129. #97×C61D4semidirect
132. C4×C8×C61D4semidirect
336. #1726D61semidirect
389. C61#21704semidirect
456. #97C4×D61semidirect
457. #97D244semidirect
460. C4×C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168401512
C2×#168398752
C2×#168398642
C2×C4×#117581902
C2×C4×D4882
#214485442
#214485302
#214467002
#214466952
#214466852
#214466732
#214466302
#214466252
#214459872
C61×#17262
#2170461