Finite Group Databasebeta

← #29709207

Subgroups — #29709207

Order 15616.

880 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×Dic61C211yesmaximal, characteristic, complemented
2C2×SD16×Dic61C211yesmaximal, characteristic
3#124×Dic61C211yesmaximal, characteristic
4#21448294C211yesmaximal, characteristic, complemented
5#21447302C211yesmaximal, characteristic, complemented
6#21447237C211yesmaximal, characteristic, complemented
7#21447205C211yesmaximal, characteristic
8#21447198C211yesmaximal, characteristic, complemented
9#21447139C211yesmaximal, characteristic, complemented
10#21447132C211yesmaximal, characteristic
11#21447055C211yesmaximal, characteristic
12#21445667C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#2015C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19Q16×Dic61—24no—
20SD16×Dic61—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839655—21no—
39#16839655—21no—
40#16839647—21no—
41#16839645—21no—
42#16839628C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44#16839613—21no—
45C22⋊C4×Dic61—21no—
46C22⋊C4×Dic61—21no—
47#16839607—21no—
48C42×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C22×SD16×C61C2211yescharacteristic
56C2×#124×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61D414yes—
68D4×Dic61—22no—
69D4×Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519711—22no—
91#14519709—22no—
92#14519706—22no—
93#14519706—22no—
94#14519701—22no—
95#14519701—22no—
96#14519701—22no—
97#14519701—22no—
98#14519698—22no—
99#14519698—22no—
100C8×Dic61—22no—
101C8×Dic61—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×Q8×C61C2311yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C2×Q16×C61—22no—
115C2×Q16×C61—22no—
116C2×Q16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122#124×C61—22no—
123#124×C61—22no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#103×C61—22no—
129#102×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C2×D412yes—
144C2×Q8×C61#4112yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168Q16×C61—28no—
169SD16×C61—28no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61#4112yes—
175C2×C8×C61#4112yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C42×C61—21no—
181C42×C61—21no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C4⋊Dic61—21no—
186C4⋊Dic61—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61C2×D412yes—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—24no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—41no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61C2×D412yes—
207C4×Dic61#4112yes—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—84no—
275C2×Dic61#12112yes—
276C2×Dic61#12112yes—
277C2×Dic61#13512yes—
278C2×Dic61#13512yes—
279C2×Dic61#13712yes—
280C8×C61—22no—
281C8×C61—22no—
282#92418—41no—
283#92418—41no—
284#23681—611nomaximal, Sylow
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C22×C61#52511yescharacteristic
294C22×C61#53211yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C22×C61#57511yescharacteristic
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61—81no—
304C4×C61#52511yescharacteristic
305C4×C61#52511yescharacteristic
306C4×C61#52911yescharacteristic
307C4×C61#52911yescharacteristic
308Dic61—22no—
309Dic61—22no—
310Dic61—41no—
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315Dic61#46414yes—
316C2×C4×Q16—611no—
317C2×C4×SD16—611no—
318C4×#124—611no—
319#2295—611no—
320#2085—611no—
321#2074—611no—
322#2067—611no—
323#2066—611no—
324#2056—611no—
325#2055—611no—
326#2042—611no—
327#2015D6111yescharacteristic
328C4×C4⋊C8—611no—
329C4×#103—611no—
330C4×#102—611no—
331C2×C61—82no—
332C2×C61#239211yescharacteristic
333C2×C61C2×#46411yescharacteristic
334C2×C61#303511yescharacteristic
335C2×C61#310211yescharacteristic
336C2×C61#310711yescharacteristic
337C2×C61#316811yescharacteristic
338C2×C61#317211yescharacteristic
339C22×Q16C2×D6111yescharacteristic
340C22×SD16C2×D6111yescharacteristic
341C2×#124C2×D6111yescharacteristic
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×D4—611no—
345C4×Q16—1224no—
346C4×SD16—1224no—
347C2×#107—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103C2×D6111yescharacteristic
356C2×#102—611no—
357C2×#102—611no—
358C2×#102C2×D6111yescharacteristic
359C2×C4×C8—611no—
360C2×C4×C8—611no—
361#391—611no—
362#391—611no—
363#391C2×D6111yescharacteristic
364#390—611no—
365#389—611no—
366#389—611no—
367#388—1221no—
368#388—1221no—
369#387—1221no—
370#387—1221no—
371#382—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374C4×C22⋊C4—1221no—
375C4×C22⋊C4—1221no—
376#376—1221no—
377C43—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#336—611no—
383#336C2×D6111yescharacteristic
384C61#2368111yescharacteristic, complemented, Sylow
385C22×Q8C22×D6111yescharacteristic
386C22×Q8C22×D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—1221no—
389C2×Q16—22no—
390C2×Q16—22no—
391C2×Q16—22no—
392C2×SD16—22no—
393C2×SD16—22no—
394C2×SD16—22no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#124—22no—
400#124—22no—
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#106—1222no—
433C4⋊C8—1222no—
434C4⋊C8—1222no—
435#103—22no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—22no—
441#102—1222no—
442#102—1222no—
443C4×C8—1222no—
444C4×C8—1222no—
445#95—21no—
446#95—21no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8D4×D6112yes—
468C2×Q8#1175822412yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4#1175822412yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492Q16—28no—
493SD16—28no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8#1175822412yes—
505C2×C8#1175822412yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—612no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C42—21no—
526C42—21no—
527C42—612no—
528C42—612no—
529C42—1222no—
530C42—1222no—
531C42—1222no—
532C42—1224no—
533C42—1221no—
534C42—1221no—
535C42—2441no—
536C42—2442no—
537C42—2442no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D4×D6111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554D4—22no—
555D4—22no—
556D4—41no—
557D4—41no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—4884no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822411yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C8—22no—
605C8—22no—
606C8—2441no—
607C8—2441no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#121×D6111yescharacteristic
616C22#1684026811yescharacteristic
617C22#1684031011yescharacteristic
618C22#135×D6111yescharacteristic
619C22#135×D6111yescharacteristic
620C22#137×D6111yescharacteristic
621C22#1684056111yescharacteristic
622C4—41no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—614no—
628C4—1222no—
629C4—1222no—
630C4—4881no—
631C4—4881no—
632C4—4881no—
633C4—2441no—
634C4—2442no—
635C4#1684026811yescharacteristic
636C4#1684026811yescharacteristic
637C4#1684029311yescharacteristic
638C4#1684029311yescharacteristic
639C2—82no—
640C2#2144905411yescharacteristic
641C2#464×D6111yescharacteristic
642C2#2145527311yescharacteristic
643C2#2145583811yescharacteristic
644C2#2145588311yescharacteristic
645C2#2145638611yescharacteristic
646C2#2145642211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2015C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2015D61
hypocenterC61#23681
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2015

Complements

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

NH ≅ G/Ntype
1. C2×Q16×Dic61C2semidirect
4. #21448294C2semidirect
5. #21447302C2semidirect
6. #21447237C2semidirect
8. #21447198C2semidirect
9. #21447139C2semidirect
12. #21445667C2semidirect
13. #103×Dic61C2semidirect
384. C61#23681semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×Dic612
C2×SD16×Dic612
#124×Dic612
#214482942
#214473022
#214472372
#214472052
#214471982
#214471392
#214471322
#214470552
#214456672
#103×Dic612
#102×Dic612
C61×#20152
#2368161