Finite Group Databasebeta

← #29683684

Subgroups — #29683684

Order 15616.

1638 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840308C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5C2×M16×Dic61C211yesmaximal, characteristic, complemented
6C2×#16839627C211yesmaximal, characteristic, complemented
7#21446641C212yesmaximal, complemented
8#21446637C212yesmaximal, complemented
9#21446629C212yesmaximal, complemented
10#21446627C212yesmaximal, complemented
11C2×C61×#417C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840308—24no—
15#16840308C418yescomplemented
16C2×#14519775C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519749—22no—
21C2×C4×C22⋊D61—22no—
22C22×C4⋊Dic61C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839764—24no—
25#16839763—24no—
26#16839761—24no—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yes—
33M16×Dic61—24no—
34C2×#14519687C2212yes—
35C2×C8×Dic61C2212yes—
36#16839627—22no—
37#16839627C2212yes—
38#16839627C2212yes—
39C2×#14519684—22no—
40C22×#41×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#417—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C22⋊D61—22no—
56C23×Dic61—21no—
57C23×Dic61D412yes—
58C22×C4×D61—21no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194D414yescomplemented
62C22×#92418C2312yes—
63#14519775—22no—
64#14519775C2×C414yescomplemented
65#14519773—22no—
66#14519773C2×C414yescomplemented
67#14519766—22no—
68#14519766C2×C414yescomplemented
69#14519764—22no—
70#14519764C2×C414yescomplemented
71#14519749—216no—
72C4×C22⋊D61—216no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C2×C22⋊Dic61—22no—
79C2×#11758198—22no—
80C2×C4×Dic61C2×C414yes—
81C2×C4×Dic61D412yes—
82C2×C4×Dic61D414yes—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519702—24no—
86#14519701—24no—
87#14519700—24no—
88#14519699—24no—
89#14519698—24no—
90#14519697—24no—
91#14519687—24no—
92C8×Dic61—24no—
93#14519684—24no—
94#14519684—24no—
95C2×#41×C61—22no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C2×#41×C61—22no—
99C2×#41×C61—22no—
100C2×#41×C61C2×C414yes—
101C22×Q8×C61C2311yescharacteristic
102C22×D4×C61—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61—21no—
106C23×C4×C61C2311yescharacteristic
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61D414yescomplemented
110C22×C8×C61C2312yes—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2312yes—
115#103×C61—24no—
116#102×C61—24no—
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120#41×C61—28no—
121#41×C61—28no—
122#41×C61—44no—
123#41×C61—44no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C22×C412yes—
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—22no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—44no—
145C22×C4×C61—42no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61#4112yes—
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—24no—
155C2×C22⋊D61—28no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—42no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—44no—
165C22×Dic61—42no—
166C22×Dic61C22⋊C418yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169#11758222—28no—
170#11758222—28no—
171#11758222—44no—
172#11758222—44no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244C22×C412yes—
179C2×C4×D61—24no—
180C2×C4×D61—42no—
181C2×C4×D61—42no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122C22×C412yes—
188M16×C61—28no—
189C2×C8×C61—22no—
190C2×C8×C61—22no—
191C2×C8×C61C2×D414yes—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—42no—
195C22⋊Dic61—28no—
196#11758198—28no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—42no—
200#11758196—28no—
201C4×Dic61C22⋊C418yes—
202C4×Dic61C22×C412yes—
203C4×Dic61C22×C412yes—
204C4×Dic61C2×D412yes—
205C4×Dic61C2×D412yes—
206#11758194—28no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418C2×D414yes—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61C2×C22⋊C412yes—
218C23×C61#12412yes—
219C23×C61#12712yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—22no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—42no—
225C22×D61—42no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—48no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—48no—
239C2×C4×C61—42no—
240C2×C4×C61—44no—
241C2×C4×C61—84no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C4×D414yes—
244C2×C4×C61C4×D414yes—
245C2×C4×C61#12114yes—
246C2×C4×C61#12114yes—
247C2×C4×C61#12412yes—
248C2×C4×C61#12712yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C22⋊D61—416no—
256C22⋊D61—48no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—48no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—44no—
267C2×Dic61—84no—
268C2×Dic61C2×C22⋊C414yes—
269C2×Dic61C2×C22⋊C414yes—
270D244—22no—
271D244—22no—
272D244—41no—
273D244—41no—
274C4×D61—48no—
275Dic122—22no—
276Dic122—22no—
277Dic122—41no—
278Dic122—41no—
279C8×C61—24no—
280#92418—24no—
281#21201—611nomaximal, Sylow
282C22×C61—24no—
283C22×C61—24no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—44no—
287C22×C61—42no—
288C22×C61—82no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61#39014yes—
292C22×C61C22×C22⋊C411yescharacteristic
293C22×C61C2×#12411yescharacteristic
294C22×C61C2×#12711yescharacteristic
295C22×C61#57512yes—
296C22×C61#57512yes—
297C2×D61—44no—
298C2×D61—42no—
299C2×D61—82no—
300C2×D61—81no—
301C2×D61—81no—
302C4×C61—42no—
303C4×C61—82no—
304C4×C61#39014yes—
305C4×C61C2×C4×D411yescharacteristic
306C4×C61C2×C4×D411yescharacteristic
307C4×C61C2×#12111yescharacteristic
308C4×C61C2×#12111yescharacteristic
309Dic61—24no—
310Dic61—24no—
311Dic61—42no—
312Dic61—82no—
313C2×#532—611no—
314C2×#417—611no—
315C2×#417—611no—
316C2×#417—611no—
317C2×#417D6111yescharacteristic, complemented
318C2×C4×M16—611no—
319C2×#382—611no—
320#1961—612no—
321#1960—612no—
322#1958—612no—
323#1958—612no—
324C2×C61—24no—
325C2×C61—82no—
326C2×C61C2×#39011yescharacteristic
327C2×C61#289612yes—
328C2×C61#314212yes—
329C2×C61#315311yescharacteristic
330C2×C61#315311yescharacteristic
331D61—82no—
332C22×#41—611no—
333C22×#41C2×D6111yescharacteristic
334C22×M16—611no—
335C22×M16C2×D6111yescharacteristic
336#532—618no—
337#532—1224no—
338C2×C4⋊Q8—611no—
339C2×#127—611no—
340C2×#124—611no—
341C2×#124—611no—
342C2×#121—1222no—
343C2×C4×D4—1222no—
344C22×C4⋊C4—611no—
345C22×C4⋊C4C2×D6111yescharacteristic
346C22×C42—611no—
347#417—24no—
348#417—1224no—
349#417—1224no—
350#417—1224no—
351C2×#103—612no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#103C2×D6112yes—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×#102C2×D6112yes—
359C4×M16—1224no—
360C2×#97—612no—
361C2×C4×C8—612no—
362#382—612no—
363#382—612no—
364#382—1222no—
365C2×#95—1222no—
366C61#2120111yescharacteristic, complemented, Sylow
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—614no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1222no—
377C2×#41—1222no—
378C2×#41C4×D6114yes—
379C22×Q8—611no—
380C22×Q8C22×D6111yescharacteristic
381C22×D4—21no—
382C22×D4—611no—
383C22×D4—1222no—
384C22×D4—1221no—
385C22×D4C22×D6111yescharacteristic
386C23×C4—21no—
387C23×C4—21no—
388C23×C4—612no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C2×M16—22no—
393C2×M16—22no—
394C2×M16—614no—
395C2×M16—1222no—
396C2×M16—1222no—
397C2×M16C22⋊D6114yescomplemented
398C22×C8—612no—
399C22×C8C22×D6112yes—
400C4⋊Q8—614no—
401C4⋊Q8—1222no—
402#127—614no—
403#127—1222no—
404#124—614no—
405#124—614no—
406#124—1222no—
407#124—1222no—
408#121—12216no—
409C4×D4—12216no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22×D6112yes—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1222no—
421C2×C42—614no—
422C2×C42—612no—
423C2×C42—614no—
424C2×C42—611no—
425C2×C42—611no—
426#103—24no—
427#103—1224no—
428#103—1224no—
429#103—1224no—
430#102—24no—
431#102—1224no—
432#102—1224no—
433#102—1224no—
434#97—1224no—
435C4×C8—1224no—
436#95—1224no—
437#95—1224no—
438C24—21no—
439C24—1221no—
440C24D4×D6111yescharacteristic
441#41—28no—
442#41—28no—
443#41—44no—
444#41—44no—
445#41—1228no—
446#41—1228no—
447#41—2444no—
448#41—2444no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8C2×C4×D6112yes—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—612no—
470C2×D4—1228no—
471C2×D4—1224no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4—2442no—
480C2×D4—2442no—
481C2×D4C2×C4×D6112yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—618no—
492C22×C4—1221no—
493C22×C4—1224no—
494C22×C4—1222no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2444no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4C2×C4×D6112yes—
502C22×C4#1175822212yes—
503C22×C4D4×D6112yes—
504C22×C4D4×D6112yes—
505C22×C4D4×D6111yescharacteristic
506C22×C4#1175822412yes—
507C22×C4C2×C22⋊D6112yes—
508C22×C4C23×D6111yescharacteristic
509M16—28no—
510M16—1228no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—614no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8C2×C22⋊D6114yes—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—42no—
520C4⋊C4—1228no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—2442no—
524C22⋊C4—1228no—
525C22⋊C4—1228no—
526C42—612no—
527C42—612no—
528C42—612no—
529C42—612no—
530C42—618no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—1222no—
539C23—2441no—
540C23—2441no—
541C23—2442no—
542C23—2442no—
543C23C22⋊C4×D6112yes—
544C23#1451976312yes—
545C23#1451977312yes—
546C23C2×D4×D6111yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—1222no—
552Q8—1222no—
553Q8—2441no—
554Q8—2441no—
555D4—22no—
556D4—22no—
557D4—48no—
558D4—41no—
559D4—41no—
560D4—1222no—
561D4—1222no—
562D4—24416no—
563D4—2448no—
564D4—2441no—
565D4—2441no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—84no—
573C2×C4—614no—
574C2×C4—614no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—4884no—
581C2×C4—2448no—
582C2×C4—2442no—
583C2×C4—2444no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2448no—
587C2×C4C22⋊C4×D6112yes—
588C2×C4#1451973714yes—
589C2×C4C4×C22⋊D6114yes—
590C2×C4#1451974914yes—
591C2×C4#1451975014yes—
592C2×C4#1451976312yes—
593C2×C4#1451977312yes—
594C2×C4C22×C4×D6111yescharacteristic
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C2×#1175822411yescharacteristic
599C2×C4C2×#1175822211yescharacteristic
600C8—24no—
601C8—1224no—
602C22—24no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—42no—
607C22—44no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—4882no—
612C22—4881no—
613C22—4881no—
614C22—2442no—
615C22—2444no—
616C22#1683966214yes—
617C22C2×C22⋊C4×D6111yescharacteristic
618C22C2×#1451976311yescharacteristic
619C22C2×#1451977311yescharacteristic
620C22#1684055812yes—
621C22#1684056112yes—
622C4—42no—
623C4—82no—
624C4—1224no—
625C4—1224no—
626C4—4882no—
627C4—2442no—
628C4#1683966214yes—
629C4C2×#1451973711yescharacteristic
630C4C2×C4×C22⋊D6111yescharacteristic
631C4C2×#1451974911yescharacteristic
632C4C2×#1451975011yescharacteristic
633C2—24no—
634C2—82no—
635C2—4882no—
636C2C2×#1683966211yescharacteristic
637C2#2145433512yes—
638C2#2145621712yes—
639C2#2145628111yescharacteristic
640C2#2145628611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC2×#16840308C2
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61#21201
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C2×#16839763C2semidirect
4. C2×#16839761C2semidirect
5. C2×M16×Dic61C2semidirect
6. C2×#16839627C2semidirect
7. #21446641C2semidirect
8. #21446637C2semidirect
9. #21446629C2semidirect
10. #21446627C2semidirect
11. C2×C61×#417C2semidirect
15. #16840308C4semidirect
61. C2×#11758194D4semidirect
64. #14519775C2×C4semidirect
66. #14519773C2×C4semidirect
68. #14519766C2×C4semidirect
70. #14519764C2×C4semidirect
109. C2×M16×C61D4semidirect
317. C2×#417D61semidirect
366. C61#21201semidirect
397. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403082
C2×#168397642
C2×#168397632
C2×#168397612
C2×M16×Dic612
C2×#168396272
#214466412
#214466372
#214466292
#214466272
C2×C61×#4172
#2120161