Finite Group Databasebeta

← #29688609

Subgroups — #29688609

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×#16840150C211yesmaximal, characteristic
2C2×#16839871C211yesmaximal, characteristic, complemented
3C2×#16839859C211yesmaximal, characteristic, complemented
4C2×#16839849C211yesmaximal, characteristic, complemented
5C2×D8×Dic61C211yesmaximal, characteristic, complemented
6#21448543C211yesmaximal, characteristic, complemented
7#21448529C211yesmaximal, characteristic, complemented
8#21446700C211yesmaximal, characteristic, complemented
9#21446684C211yesmaximal, characteristic, complemented
10#21446672C211yesmaximal, characteristic, complemented
11#21446629C211yesmaximal, characteristic, complemented
12#21446623C211yesmaximal, characteristic, complemented
13#21445984C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1972C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758199C2211yescharacteristic
19C2×#14519773C2211yescharacteristic
20C2×#14519764C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25#16840150C418yescomplemented
26#16839871—24no—
27#16839859—24no—
28#16839849—24no—
29D8×Dic61—24no—
30C2×#14519711C2211yescharacteristic
31C2×#14519710C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519698C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×C4×#92418C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44#16839668C2211yescharacteristic
45#16839664C2211yescharacteristic
46#16839656—22no—
47#16839639—22no—
48#16839638—22no—
49#16839634—22no—
50#16839631C2211yescharacteristic
51#16839627C2211yescharacteristic
52#16839608C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—22no—
63C23×Dic61—21no—
64C23×Dic61—21no—
65C22×C4×D61—21no—
66C2×#11758190—22no—
67C2×#11758190—22no—
68C2×#11758190C2×C414yescomplemented
69C2×#11758200—22no—
70C2×#11758200—22no—
71C2×#11758200C2×C414yescomplemented
72C2×#11758199—22no—
73C2×#11758199—22no—
74C2×#11758199C2×C414yescomplemented
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519773—22no—
78#14519773C2×C414yescomplemented
79#14519764—22no—
80#14519764C2×C414yescomplemented
81#14519742—22no—
82#14519742—22no—
83D4×Dic61—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—22no—
87#14519739—22no—
88#14519739—22no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—22no—
93C2×C22⋊Dic61—22no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×#11758198—21no—
97C2×#11758198—22no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711—22no—
102#14519710—22no—
103#14519708—22no—
104#14519702—22no—
105#14519700—22no—
106#14519699—22no—
107#14519698—22no—
108#14519698—22no—
109#14519697—22no—
110#14519687C2×C414yescomplemented
111#14519687D414yescomplemented
112C4×#92418—22no—
113C8×Dic61—22no—
114C8×Dic61D414yescomplemented
115#14519684—22no—
116#14519684—22no—
117#14519684D412yes—
118#14519684D412yes—
119C22×D4×C61C2311yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C2×D8×C61—22no—
122C2×D8×C61—22no—
123C2×D8×C61C2×C414yescomplemented
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—22no—
129C2×C22⋊C4×C61—22no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—22no—
132#102×C61—22no—
133#102×C61—22no—
134#97×C61—22no—
135C24×C61—21no—
136C24×C61—21no—
137C23×D61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—28no—
155C2×C22⋊D61—82no—
156C2×C22⋊D61—82no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—82no—
163C22×Dic61—82no—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244C22×C412yes—
173C2×C4×D61—42no—
174C2×C4×D61—82no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C22×C412yes—
181D8×C61—28no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C22×C412yes—
185C2×C8×C61C2×D412yes—
186C2×C8×C61C2×D412yes—
187C2×C8×C61#4112yes—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C22⋊C4×C61—82no—
193C22⋊C4×C61—82no—
194C42×C61—21no—
195C42×C61—21no—
196C22⋊Dic61—28no—
197C22⋊Dic61—42no—
198C22⋊Dic61—42no—
199#11758200—28no—
200#11758199—28no—
201#11758198—42no—
202#11758198—82no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207#11758196—42no—
208#11758196—82no—
209C4×Dic61—42no—
210C4×Dic61C22×C412yes—
211C4×Dic61C2×D412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C22×C412yes—
217C2×#92418C2×D412yes—
218C2×#92418C2×D412yes—
219C2×#92418#4112yes—
220#11758190—28no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—42no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61—42no—
231C23×C61C22×D411yescharacteristic
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—42no—
236C22×D61—42no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—88no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—84no—
271C2×Dic61—84no—
272C2×Dic61C4×D414yes—
273C2×Dic61#12712yes—
274C2×Dic61#12712yes—
275D244—22no—
276D244—22no—
277D244—41no—
278D244—41no—
279C4×D61—84no—
280Dic122—22no—
281Dic122—22no—
282Dic122—41no—
283Dic122—41no—
284C8×C61—22no—
285C8×C61C4×D414yes—
286#92418—22no—
287#92418C4×D414yes—
288#21704—611nomaximal, Sylow
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—44no—
292C22×C61—44no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—82no—
298C22×C61—82no—
299C22×C61C2×C4×D411yescharacteristic
300C22×C61C2×#12711yescharacteristic
301C22×C61#53211yescharacteristic
302C22×C61#57612yes—
303C22×C61#57712yes—
304C2×D61—42no—
305C2×D61—44no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—82no—
309C4×C61—42no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C2×C4×D411yescharacteristic
313C4×C61C2×C4×D411yescharacteristic
314C4×C61#53211yescharacteristic
315C4×C61#53211yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—24no—
319Dic61—42no—
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#1972D6111yescharacteristic, complemented
331#1970—611no—
332#1968—611no—
333#1958—611no—
334#1957—611no—
335#1847—611no—
336#1726—611no—
337C2×C61—82no—
338C2×C61—82no—
339C2×C61C4×#12711yescharacteristic
340C2×C61#294612yes—
341C2×C61#315411yescharacteristic
342C2×C61#315511yescharacteristic
343C2×C61#317912yes—
344D61—82no—
345C22×SD16—611no—
346C22×SD16—611no—
347C22×D8—611no—
348C22×D8C2×D6111yescharacteristic
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×#106C2×D6111yescharacteristic
364C2×#103—611no—
365C2×#103—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#102C2×D6111yescharacteristic
371C2×#102C2×D6111yescharacteristic
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#390C2×D6111yescharacteristic
381#390C2×D6111yescharacteristic
382#388—1222no—
383#386—1222no—
384#386—1222no—
385#385—1222no—
386#384—611no—
387#382—611no—
388#376—611no—
389C61#2170411yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×D4—611no—
392C22×D4—1222no—
393C22×D4C22×D6111yescharacteristic
394C22×D4C22×D6111yescharacteristic
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—22no—
405C2×D8—22no—
406C2×D8—614no—
407C2×D8—1222no—
408C2×D8—1222no—
409C2×D8C4×D6114yescomplemented
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—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—22no—
433C2×C22⋊C4—22no—
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—1222no—
444#107—1222no—
445#106—22no—
446#106—1222no—
447#103—1222no—
448#103—1222no—
449#102—22no—
450#102—22no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#102—1222no—
455#97—22no—
456#97—614no—
457#97—614no—
458C4×C8—614no—
459C4×C8—1222no—
460C4×C8—1222no—
461#95—612no—
462#95—612no—
463#95—1222no—
464#95—1222no—
465C24—21no—
466C24—21no—
467C24—1221no—
468C2×Q8—612no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—612no—
485C2×D4—1228no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—4882no—
492C2×D4—4882no—
493C2×D4C2×C4×D6112yes—
494C2×D4C2×C4×D6112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—612no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—1221no—
501C22×C4—1221no—
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—28no—
514D8—1228no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—612no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—612no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8C2×C4×D6112yes—
524C2×C8D4×D6112yes—
525C2×C8D4×D6112yes—
526C2×C8#1175822412yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—4882no—
536C4⋊C4—2442no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—1228no—
540C22⋊C4—4882no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C42—21no—
545C42—21no—
546C42—612no—
547C42—612no—
548C42—612no—
549C42—612no—
550C42—2442no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—1222no—
562C23—2441no—
563C23—2441no—
564C23—2442no—
565C23—2442no—
566C23C2×D4×D6111yescharacteristic
567Q8—1222no—
568Q8—1222no—
569Q8—2441no—
570Q8—2441no—
571D4—22no—
572D4—22no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—1222no—
580D4—1222no—
581D4—4888no—
582D4—2441no—
583D4—2441no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—614no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—4884no—
600C2×C4—4884no—
601C2×C4—4884no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4#41×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4C2×#1175822411yescharacteristic
614C8—22no—
615C8—614no—
616C8—1222no—
617C8#1451973714yes—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—4882no—
629C22—4881no—
630C22—4881no—
631C22—2444no—
632C22—2442no—
633C22C4×D4×D6111yescharacteristic
634C22#127×D6111yescharacteristic
635C22#1684030711yescharacteristic
636C22#1684056612yes—
637C22#1684057212yes—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—1224no—
642C4—1222no—
643C4—1222no—
644C4—4881no—
645C4—4881no—
646C4—2442no—
647C4C4×D4×D6111yescharacteristic
648C4C2×#1451973711yescharacteristic
649C4#1684030711yescharacteristic
650C4#1684031111yescharacteristic
651C2—82no—
652C2—82no—
653C2—4882no—
654C2#2144838311yescharacteristic
655C2#2145464812yes—
656C2#2145629211yescharacteristic
657C2#2145630311yescharacteristic
658C2#2145650312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1972C2
socleC23×C61C22×D4
prime radicalC2×#16840150C2
cosocleC22×C4×C61C24
hypercenter#1972D61
hypocenterC61#21704
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1972

Complements

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

NH ≅ G/Ntype
2. C2×#16839871C2semidirect
3. C2×#16839859C2semidirect
4. C2×#16839849C2semidirect
5. C2×D8×Dic61C2semidirect
6. #21448543C2semidirect
7. #21448529C2semidirect
8. #21446700C2semidirect
9. #21446684C2semidirect
10. #21446672C2semidirect
11. #21446629C2semidirect
12. #21446623C2semidirect
13. #21445984C2semidirect
14. #21445507C2semidirect
15. C61×#1972C2semidirect
25. #16840150C4semidirect
68. C2×#11758190C2×C4semidirect
71. C2×#11758200C2×C4semidirect
74. C2×#11758199C2×C4semidirect
78. #14519773C2×C4semidirect
80. #14519764C2×C4semidirect
110. #14519687C2×C4semidirect
111. #14519687D4semidirect
114. C8×Dic61D4semidirect
123. C2×D8×C61C2×C4semidirect
330. #1972D61semidirect
389. C61#21704semidirect
409. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401502
C2×#168398712
C2×#168398592
C2×#168398492
C2×D8×Dic612
#214485432
#214485292
#214467002
#214466842
#214466722
#214466292
#214466232
#214459842
#214455072
C61×#19722
#2170461