Finite Group Databasebeta

← #29653859

Subgroups — #29653859

Order 15616.

1142 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840387C211yesmaximal, characteristic, complemented
2#21456576C211yesmaximal, characteristic, complemented
3#21451555C211yesmaximal, characteristic
4#21444808C214yesmaximal, complemented
5#21444740C214yesmaximal, complemented
6#21444392C211yesmaximal, characteristic, complemented
7#21444391C211yesmaximal, characteristic, complemented
8C2×#16839485C211yesmaximal, characteristic, complemented
9C61×#3187C211yesmaximal, characteristic, complemented
10#16840387—22no—
11#16840387—22no—
12#16840387—22no—
13#16840387C2214yes—
14C2×#14519757—21no—
15C2×#14519756C2211yescharacteristic
16C2×#14519755—21no—
17C2×#14519755—21no—
18C2×#14519754—21no—
19C2×#14519753—21no—
20C2×#14519745—21no—
21C22×#11758198C2211yescharacteristic
22C2×C22⋊C4×D61—21no—
23#16840092C2212yescomplemented
24#16840078—21no—
25#16839989—22no—
26#16839913C2212yescomplemented
27#16839900—21no—
28C2×C22⋊#92418C2211yescharacteristic
29#16839688C2212yes—
30#16839678—22no—
31#16839670C2212yes—
32#16839642—21no—
33#16839622C2211yescharacteristic
34#16839500C2214yescomplemented
35#16839499C2214yescomplemented
36#16839485C2214yes—
37C2×C22⋊Q8×C61C2211yescharacteristic
38C2×#121×C61C2211yescharacteristic
39C61×#483C2212yescomplemented
40C61×#481—21no—
41C61×#465—22no—
42C61×#450C2212yescomplemented
43C61×#448—21no—
44C2×C22⋊C8×C61C2211yescharacteristic
45C24×D61D411yescharacteristic
46C22×Dic122—21no—
47C22×Dic122D411yescharacteristic, complemented
48C22×C22⋊D61—22no—
49C23×Dic61D411yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C2×#11758200—22no—
53C22×#92418—21no—
54#14519757—24no—
55#14519757—24no—
56#14519756—21no—
57#14519756—21no—
58#14519756D412yescomplemented
59#14519756C2312yes—
60#14519755—24no—
61#14519755—24no—
62#14519755—24no—
63#14519755—24no—
64#14519754—24no—
65#14519754—24no—
66#14519753—24no—
67#14519753—24no—
68#14519745—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic, complemented
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—22no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—22no—
85C2×#11758198D412yes—
86C2×#11758198C2312yes—
87C22⋊C4×D61—24no—
88C22⋊C4×D61—24no—
89#14519708—22no—
90#14519700—22no—
91#14519698—22no—
92C22⋊#92418—21no—
93C22⋊#92418C2312yes—
94#14519684—22no—
95#14519684C2312yes—
96C22×Q8×C61D411yescharacteristic, complemented
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C2×SD16×C61—22no—
101C22×C8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61D412yescomplemented
105C22⋊Q8×C61C2312yes—
106#121×C61—21no—
107#121×C61—21no—
108#121×C61—21no—
109#121×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61D411yescharacteristic, complemented
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115#106×C61—22no—
116#103×C61—22no—
117#102×C61—22no—
118C22⋊C8×C61—21no—
119C22⋊C8×C61C2312yes—
120C24×C61—21no—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123C23×D61—21no—
124C23×D61—22no—
125C23×D61—81no—
126C23×D61—81no—
127C23×D61SD1614yescomplemented
128C23×D61C2×D412yes—
129C2×Q8×C61—21no—
130C2×Q8×C61—81no—
131C2×Q8×C61C2×D412yes—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—41no—
135C2×D4×C61—41no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—81no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—81no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—44no—
150C2×C22⋊D61—82no—
151C2×C22⋊D61—82no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—22no—
157C22×Dic61—81no—
158C22×Dic61C2×D412yes—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—82no—
162C2×C4×D61—82no—
163C2×Dic122—24no—
164C2×Dic122—21no—
165C2×Dic122—81no—
166C2×Dic122—82no—
167C2×Dic122C2×D412yes—
168SD16×C61—82no—
169C2×C8×C61—22no—
170C2×C8×C61—81no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D412yes—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—42no—
184C22⋊Dic61—82no—
185#11758200—82no—
186#11758198—24no—
187#11758198—24no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—44no—
193C4⋊Dic61—24no—
194C4⋊Dic61—21no—
195C4⋊Dic61—81no—
196C4⋊Dic61C2×D412yes—
197#11758196—24no—
198#11758196—24no—
199#11758196—22no—
200#11758196—22no—
201C2×#92418—22no—
202C2×#92418—81no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61#12011yescharacteristic
211C23×C61#12011yescharacteristic
212C23×C61C2×SD1612yes—
213C23×C61#13712yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—24no—
217C22×D61—22no—
218C22×D61—24no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—84no—
224C22×D61—81no—
225C22×D61—81no—
226C22×D61—84no—
227C22×D61C2×SD1614yes—
228Q8×C61—42no—
229Q8×C61—81no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—82no—
233D4×C61—82no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—88no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—81no—
263C4×D61—84no—
264C4×D61—84no—
265Dic122—42no—
266Dic122—81no—
267Dic122—84no—
268C8×C61—42no—
269#92418—42no—
270#18286—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61#44812yes—
285C22×C61#44812yes—
286C22×C61C2×#12011yescharacteristic
287C22×C61C22×SD1611yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C2×D61—24no—
290C2×D61—48no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—44no—
294C2×D61—82no—
295C2×D61—84no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—81no—
303Dic61—42no—
304Dic61—42no—
305Dic61—42no—
306Dic61—82no—
307C2×#543—611no—
308#3187—611no—
309#3187D6111yescharacteristic, complemented
310#2617—611no—
311#1608—614no—
312#1596—614no—
313#1520—611no—
314#1520—611no—
315C2×#323—611no—
316C2×C61—21no—
317C2×C61—22no—
318C2×C61—21no—
319C2×C61—22no—
320C2×C61—161no—
321C2×C61C2×#44811yescharacteristic
322C2×C61#301911yescharacteristic
323C2×C61#302611yescharacteristic
324D61—44no—
325#543—614no—
326#543—1222no—
327#543—1222no—
328#543—1222no—
329C2×C22⋊Q8—611no—
330C2×C22⋊Q8—1221no—
331C2×C22⋊Q8—1221no—
332C2×C22⋊Q8—1221no—
333C2×C22⋊Q8—1221no—
334C2×C22⋊Q8—1221no—
335C2×C22⋊Q8C2×D6111yescharacteristic
336C2×#121C2×D6111yescharacteristic
337C2×#120—1221no—
338C22×C22⋊C4—611no—
339C22×C22⋊C4—1221no—
340#483—612no—
341#483C2×D6112yescomplemented
342#481—21no—
343#481—1221no—
344#465—22no—
345#465—1222no—
346#450—612no—
347#450C2×D6112yescomplemented
348#448—21no—
349#448—1221no—
350C2×C22⋊C8—611no—
351C2×C22⋊C8C2×D6111yescharacteristic
352#397—612no—
353#394—1222no—
354#392—612no—
355#386—1221no—
356#380—611no—
357#328—614no—
358#328—614no—
359#323—614no—
360C61#1828611yescharacteristic, complemented, Sylow
361C25—611no—
362C22×Q8—611no—
363C22×Q8—1221no—
364C22×Q8C22⋊D6111yescharacteristic, complemented
365C22×D4—21no—
366C22×D4—1222no—
367C22×D4C22×D6111yescharacteristic
368C23×C4—611no—
369C23×C4—1221no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372C2×SD16—22no—
373C2×SD16—1222no—
374C22×C8—21no—
375C22×C8—1221no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—612no—
379C22⋊Q8—612no—
380C22⋊Q8—1224no—
381C22⋊Q8—1224no—
382C22⋊Q8—1224no—
383C22⋊Q8—1224no—
384C22⋊Q8—1224no—
385C22⋊Q8—1224no—
386C22⋊Q8—1224no—
387C22⋊Q8—1224no—
388C22⋊Q8—1224no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391C22⋊Q8—1224no—
392C22⋊Q8C22⋊D6112yescomplemented
393C22⋊Q8C22×D6112yes—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#120—2444no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—612no—
411C2×C22⋊C4—612no—
412C2×C22⋊C4—1224no—
413C2×C22⋊C4—1224no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1221no—
424#106—22no—
425#106—1222no—
426#103—22no—
427#103—1222no—
428#102—22no—
429#102—1222no—
430C22⋊C8—21no—
431C22⋊C8—612no—
432C22⋊C8—1221no—
433C22⋊C8C22×D6112yes—
434#95—612no—
435#95—1222no—
436C24—21no—
437C24—614no—
438C24—612no—
439C24—1221no—
440C24—1222no—
441C24—1221no—
442C24—4881no—
443C24—4881no—
444C24D4×D6111yescharacteristic
445C2×Q8—21no—
446C2×Q8—81no—
447C2×Q8—612no—
448C2×Q8—1224no—
449C2×Q8—1221no—
450C2×Q8—4882no—
451C2×Q8—4881no—
452C2×Q8C2×C22⋊D6112yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—81no—
460C2×D4—4882no—
461C2×D4—4882no—
462C2×D4—2444no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—81no—
467C22×C4—612no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—4882no—
476C22×C4—4882no—
477C22×C4—4881no—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—82no—
485SD16—4882no—
486C2×C8—22no—
487C2×C8—81no—
488C2×C8—1222no—
489C2×C8—4881no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—612no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—1224no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—4881no—
503C4⋊C4C2×C22⋊D6112yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—4882no—
517C22⋊C4—2444no—
518C22⋊C4—2442no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—614no—
527C23—1222no—
528C23—1224no—
529C23—1222no—
530C23—1224no—
531C23—4881no—
532C23—4881no—
533C23—4884no—
534C23—4884no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23#1451974511yescharacteristic
540C23#1451974511yescharacteristic
541C23SD16×D6112yes—
542C23#1451980612yes—
543C23C2×D4×D6111yescharacteristic
544Q8—42no—
545Q8—81no—
546Q8—4881no—
547Q8—4884no—
548Q8—2442no—
549D4—42no—
550D4—42no—
551D4—82no—
552D4—82no—
553D4—4888no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—4884no—
571C2×C4—4884no—
572C2×C4—4881no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4#1451974511yescharacteristic
581C2×C4#1451974511yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C8—42no—
585C8—2442no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—42no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—1224no—
600C22—4884no—
601C22—4882no—
602C22—2444no—
603C22—2448no—
604C22—2442no—
605C22—2442no—
606C22#1683989912yes—
607C22#1683989912yes—
608C22C2×#1451974511yescharacteristic
609C22C2×SD16×D6111yescharacteristic
610C22C2×#1451980611yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—4882no—
619C4—2442no—
620C4—2442no—
621C4—2442no—
622C2—21no—
623C2—21no—
624C2—22no—
625C2—22no—
626C2—161no—
627C2—2444no—
628C2C2×#1683989911yescharacteristic
629C2#2145517411yescharacteristic
630C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3187C2
socleC22×C61C2×#120
prime radical#21451555C2
hypercenter#3187D61
hypocenterC61#18286
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C2×#16840387C2semidirect
2. #21456576C2semidirect
4. #21444808C2semidirect
5. #21444740C2semidirect
6. #21444392C2semidirect
7. #21444391C2semidirect
8. C2×#16839485C2semidirect
9. C61×#3187C2semidirect
23. #16840092C22semidirect
26. #16839913C22semidirect
34. #16839500C22semidirect
35. #16839499C22semidirect
39. C61×#483C22semidirect
42. C61×#450C22semidirect
47. C22×Dic122D4semidirect
58. #14519756D4semidirect
71. C2×C4⋊Dic61D4semidirect
96. C22×Q8×C61D4semidirect
104. C22⋊Q8×C61D4semidirect
111. C2×C4⋊C4×C61D4semidirect
127. C23×D61SD16semidirect
309. #3187D61semidirect
341. #483C2×D61semidirect
347. #450C2×D61semidirect
360. C61#18286semidirect
364. C22×Q8C22⋊D61semidirect
392. C22⋊Q8C22⋊D61semidirect
406. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403872
#214565762
#214515552
#214448082
#214447402
#214443922
#214443912
C2×#168394852
C61×#31872
#1828661