Finite Group Databasebeta

← #29683935

Subgroups — #29683935

Order 15616.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839768C211yesmaximal, characteristic, complemented
5#21448309C211yesmaximal, characteristic, complemented
6C2×#16839664C211yesmaximal, characteristic, complemented
7#21446700C212yesmaximal, complemented
8#21446695C212yesmaximal, complemented
9#21446685C212yesmaximal, complemented
10#21446673C212yesmaximal, complemented
11C2×C4×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×D244C2211yescharacteristic
14C2×#14519805—24no—
15C2×#14519805—24no—
16C2×#14519805C418yescomplemented
17C22×#11758190C2212yes—
18C22×D488C2212yes—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758198—22no—
22#16839804—24no—
23C2×#14519711C2212yescomplemented
24C2×#14519709C2212yescomplemented
25#16839768—24no—
26#16839764—24no—
27C2×#14519702C2212yescomplemented
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yescomplemented
31#16839668C2211yescharacteristic, complemented
32#16839668C2211yescharacteristic, complemented
33#16839664—22no—
34#16839664C2212yes—
35#16839664C2212yes—
36#16839664C2211yescharacteristic, complemented
37#16839664C2211yescharacteristic, complemented
38#16839641—22no—
39C4×#11758198—22no—
40C22×M16×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C22×C42×C61C2211yescharacteristic
43C4×M16×C61—24no—
44C2×#97×C61C2212yescomplemented
45C2×C4×C8×C61C2212yescomplemented
46C24×D61—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222C2×C414yes—
53C22×Dic122C2311yescharacteristic
54C22×D244—22no—
55C22×D244—22no—
56C22×D244—22no—
57C22×D244—22no—
58C22×D244—21no—
59C22×D244—22no—
60C22×D244C2×C414yes—
61C22×D244C2312yes—
62C22×D244C2311yescharacteristic
63C22×C22⋊D61—21no—
64C23×Dic61—21no—
65C22×C4×D61—21no—
66#14519805—232no—
67C2×#11758190—24no—
68C2×#11758190—24no—
69C2×#11758190C2×C418yescomplemented
70C2×D488—24no—
71C2×D488—24no—
72C2×D488C2×C418yescomplemented
73#14519736—22no—
74#14519736—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×#11758198—24no—
83C2×#11758198—24no—
84C2×#11758198—22no—
85C2×#11758198—22no—
86C2×#11758198—48no—
87C2×C4×Dic61—21no—
88#14519711—24no—
89#14519709—24no—
90#14519702—24no—
91#14519699—24no—
92#14519699—24no—
93#14519699—24no—
94#14519684—22no—
95#14519684—22no—
96C23×C4×C61D412yes—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—22no—
99C2×M16×C61—22no—
100C2×M16×C61C2×C414yes—
101C2×M16×C61D414yes—
102C2×M16×C61D414yes—
103C2×M16×C61D414yes—
104C22×C8×C61C2312yes—
105C22×C8×C61C2312yes—
106C2×C42×C61—22no—
107C2×C42×C61—22no—
108C2×C42×C61D412yes—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111#97×C61—24no—
112C4×C8×C61—24no—
113C24×C61C2×D411yescharacteristic
114C23×D61—24no—
115C23×D61—21no—
116C23×D61—22no—
117C23×D61—44no—
118C23×D61—44no—
119C23×D61—42no—
120C23×D61—42no—
121C22×C4×C61—24no—
122C22×C4×C61—24no—
123C22×C4×C61—42no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—44no—
142C22×Dic61—42no—
143#11758222—28no—
144#11758222—28no—
145#11758222—44no—
146#11758222—44no—
147C2×D244—28no—
148C2×D244—28no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—24no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244—44no—
161C2×D244—42no—
162C2×D244—44no—
163C2×D244—42no—
164C2×D244C22×C414yes—
165C2×D244C22×C412yes—
166C2×C4×D61—24no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122C22×C412yes—
175M16×C61—28no—
176M16×C61C22⋊C4116yescomplemented
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61C22×C414yes—
180C2×C8×C61C2×D414yes—
181C2×C8×C61C2×D414yes—
182C2×C8×C61C2×D414yes—
183C42×C61—24no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—42no—
189#11758198—44no—
190#11758198—48no—
191#11758198—44no—
192#11758198—48no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4×Dic61—42no—
201D488—216no—
202#11758190—216no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61C4×D412yes—
206C23×C61#12112yes—
207C23×C61#12712yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—24no—
211C22×D61—416no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—41no—
215C22×D61—42no—
216C22×D61—44no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—44no—
220C22×D61—48no—
221C22×D61—84no—
222C22×D61—84no—
223C22×D61—88no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—48no—
230C2×C4×C61C2×C22⋊C414yes—
231C2×C4×C61C4×D414yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61#12114yes—
234C2×C4×C61#12112yes—
235C2×C4×C61#12414yes—
236C2×C4×C61#12712yes—
237C2×C4×C61AGL(1,ℤ/8)14yescomplemented
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C22⋊D61—48no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—44no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—84no—
255D244—24no—
256D244—22no—
257D244—24no—
258D244—22no—
259D244—42no—
260D244—41no—
261D244—42no—
262D244—41no—
263D244—48no—
264C4×D61—48no—
265Dic122—22no—
266Dic122—22no—
267Dic122—41no—
268Dic122—41no—
269C8×C61—24no—
270C8×C61C2×C22⋊C418yescomplemented
271#21236—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—24no—
275C22×C61—24no—
276C22×C61#39014yes—
277C22×C61C2×C4×D411yescharacteristic
278C22×C61C2×#12111yescharacteristic
279C22×C61C2×#12711yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)12yes—
281C22×C61#57512yes—
282C2×D61—44no—
283C2×D61—48no—
284C2×D61—42no—
285C2×D61—44no—
286C2×D61—816no—
287C2×D61—81no—
288C2×D61—82no—
289C2×D61—81no—
290C2×D61—82no—
291C2×D61—82no—
292C2×D61—84no—
293C4×C61—24no—
294C4×C61—42no—
295C4×C61#39014yes—
296C4×C61C22×C22⋊C411yescharacteristic
297C4×C61C2×C4×D411yescharacteristic
298C4×C61C2×#12111yescharacteristic
299C4×C61C2×#12411yescharacteristic
300Dic61—42no—
301Dic61—81no—
302Dic61—82no—
303Dic61—81no—
304C22×AGL(1,ℤ/8)—611no—
305C2×#428—611no—
306C2×#417—611no—
307C2×#418—611no—
308C2×C4×M16D6111yescharacteristic, complemented
309#2297—611no—
310C2×#390—611no—
311#1973—612no—
312#1972—612no—
313#1970—612no—
314#1968—612no—
315C2×C61—24no—
316C2×C61C2×#39011yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)12yes—
318C2×C61#305512yes—
319C2×C61#315111yescharacteristic
320C2×C61#315311yescharacteristic
321D61—82no—
322D61—84no—
323C22×#41—611no—
324C23×D4—611no—
325C2×AGL(1,ℤ/8)—618no—
326C2×AGL(1,ℤ/8)—1224no—
327C2×AGL(1,ℤ/8)—1224no—
328C22×SD16—612no—
329C22×D8—612no—
330C22×M16C2×D6111yescharacteristic
331C22×M16C2×D6111yescharacteristic
332C2×#117—611no—
333C22×C4⋊C4—611no—
334C22×C22⋊C4—1222no—
335C22×C42C2×D6111yescharacteristic
336#428—1224no—
337C2×#107—612no—
338C2×#106—612no—
339#418—1224no—
340#417—1224no—
341C2×#103—612no—
342C2×#102—612no—
343C2×#102—612no—
344C2×#102—612no—
345C4×M16—24no—
346C2×#97C2×D6112yescomplemented
347C2×C4×C8C2×D6112yescomplemented
348#391—611no—
349#391—611no—
350#390—612no—
351#390—611no—
352#390—611no—
353#390—612no—
354#390—1222no—
355#386—1222no—
356C4×C22⋊C4—1222no—
357C61#2123611yescharacteristic, complemented, Sylow
358C25—1221no—
359C2×#41—614no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—1222no—
363C2×#41—1222no—
364C2×#41—1222no—
365C22×Q8—611no—
366C22×D4—614no—
367C22×D4—611no—
368C22×D4—612no—
369C22×D4—1221no—
370C22×D4—1222no—
371C22×D4—1222no—
372C22×D4—1221no—
373C22×D4—1222no—
374C22×D4—1222no—
375C22×D4—1222no—
376C23×C4—1221no—
377C23×C4—1221no—
378C23×C4D24412yes—
379C23×C4C22×D6111yescharacteristic
380AGL(1,ℤ/8)—12232no—
381C2×SD16—618no—
382C2×SD16—1224no—
383C2×SD16—1224no—
384C2×D8—618no—
385C2×D8—1224no—
386C2×D8—1224no—
387C2×M16—22no—
388C2×M16—22no—
389C2×M16C4×D6114yes—
390C2×M16D24414yes—
391C2×M16D24414yes—
392C2×M16C22⋊D6114yes—
393C22×C8C22×D6112yes—
394C22×C8C22×D6112yes—
395#117—1222no—
396#117—1222no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—2448no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—1221no—
412C2×C42D24412yes—
413C2×C42C22×D6111yescharacteristic
414C2×C42C22×D6111yescharacteristic
415#107—1224no—
416#106—1224no—
417#103—1224no—
418#102—1224no—
419#102—1224no—
420#102—1224no—
421#97—24no—
422C4×C8—24no—
423#95—1222no—
424#95—1222no—
425C24—1221no—
426C24—1222no—
427C24—1224no—
428C24—2444no—
429C24—2444no—
430C24—2442no—
431C24—2442no—
432C24C2×D24411yescharacteristic
433#41—1228no—
434#41—1228no—
435#41—2444no—
436#41—2444no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×D4—612no—
444C2×D4—614no—
445C2×D4—1224no—
446C2×D4—1221no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1221no—
454C2×D4—1222no—
455C2×D4—1221no—
456C2×D4—1228no—
457C2×D4—1228no—
458C2×D4—1224no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2444no—
462C2×D4—2442no—
463C2×D4—2444no—
464C2×D4—2442no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—1221no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—2444no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4C2×C4×D6112yes—
478C22×C4C2×D24412yes—
479C22×C4C2×D24412yes—
480C22×C4C2×D24412yes—
481C22×C4C2×D24412yes—
482C22×C4C2×D24411yescharacteristic
483C22×C4#1175822212yes—
484C22×C4#1175822212yes—
485C22×C4C2×C22⋊D6112yes—
486C22×C4C23×D6111yescharacteristic
487SD16—12216no—
488D8—12216no—
489M16—28no—
490M16#11758198116yescomplemented
491C2×C8—22no—
492C2×C8—22no—
493C2×C8C2×C4×D6114yes—
494C2×C8C2×D24414yes—
495C2×C8C2×D24414yes—
496C2×C8C2×C22⋊D6114yes—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—2442no—
504C22⋊C4—2448no—
505C22⋊C4—2444no—
506C22⋊C4—2442no—
507C22⋊C4—2448no—
508C22⋊C4—2444no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—24no—
514C42—2442no—
515C23—22no—
516C23—22no—
517C23—1222no—
518C23—1224no—
519C23—4888no—
520C23—4884no—
521C23—4884no—
522C23—24416no—
523C23—2441no—
524C23—2442no—
525C23—2444no—
526C23—2442no—
527C23—2441no—
528C23—2442no—
529C23—2442no—
530C23—2444no—
531C23—2448no—
532C23C4×D24412yes—
533C23#1451975112yes—
534C23#1451977412yes—
535C23C22×D24411yescharacteristic
536Q8—1222no—
537Q8—1222no—
538Q8—2441no—
539Q8—2441no—
540D4—1222no—
541D4—1224no—
542D4—1224no—
543D4—1222no—
544D4—2448no—
545D4—2442no—
546D4—2441no—
547D4—2442no—
548D4—2441no—
549D4—2448no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—1222no—
557C2×C4—4884no—
558C2×C4—2444no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2448no—
567C2×C4C2×#1175819814yes—
568C2×C4C4×D24414yes—
569C2×C4C4×D24412yes—
570C2×C4#1451975114yes—
571C2×C4#1451975112yes—
572C2×C4#1451976514yes—
573C2×C4#1451977412yes—
574C2×C4#1451980514yescomplemented
575C2×C4C22×C4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4C22×D24411yescharacteristic
578C2×C4C22×D24411yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C8—24no—
582C8C2×#1175819818yescomplemented
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—48816no—
588C22—4884no—
589C22—4882no—
590C22—4882no—
591C22—4881no—
592C22—4882no—
593C22—4881no—
594C22—2444no—
595C22—2448no—
596C22—2442no—
597C22—2444no—
598C22#1683966414yes—
599C22C2×C4×D24411yescharacteristic
600C22C2×#1451975111yescharacteristic
601C22C2×#1451977411yescharacteristic
602C22C2×#1451980512yes—
603C22#1684056212yes—
604C4—24no—
605C4—42no—
606C4—4882no—
607C4—4881no—
608C4—4881no—
609C4—2442no—
610C4#1683966414yes—
611C4C22×#1175819811yescharacteristic
612C4C2×C4×D24411yescharacteristic
613C4C2×#1451975111yescharacteristic
614C4C2×#1451976511yescharacteristic
615C2—24no—
616C2—4884no—
617C2—4882no—
618C2C2×#1683966411yescharacteristic
619C2C4×#1451980512yes—
620C2#2145543812yes—
621C2#2145627111yescharacteristic
622C2#2145628511yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×M16×C61C2
socleC23×C61C22×D4
prime radicalC22×#14519805C2
cosocleC22×C4×C61C24
hypercenterC2×C4×M16D61
hypocenterC61#21236
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
3. C2×#16839764C2semidirect
4. C2×#16839768C2semidirect
5. #21448309C2semidirect
6. C2×#16839664C2semidirect
7. #21446700C2semidirect
8. #21446695C2semidirect
9. #21446685C2semidirect
10. #21446673C2semidirect
11. C2×C4×M16×C61C2semidirect
16. C2×#14519805C4semidirect
23. C2×#14519711C22semidirect
24. C2×#14519709C22semidirect
27. C2×#14519702C22semidirect
30. C2×#14519699C22semidirect
31. #16839668C22semidirect
32. #16839668C22semidirect
36. #16839664C22semidirect
37. #16839664C22semidirect
44. C2×#97×C61C22semidirect
45. C2×C4×C8×C61C22semidirect
69. C2×#11758190C2×C4semidirect
72. C2×D488C2×C4semidirect
176. M16×C61C22⋊C4semidirect
237. C2×C4×C61AGL(1,ℤ/8)semidirect
270. C8×C61C2×C22⋊C4semidirect
308. C2×C4×M16D61semidirect
346. C2×#97C2×D61semidirect
347. C2×C4×C8C2×D61semidirect
357. C61#21236semidirect
490. M16#11758198semidirect
574. C2×C4#14519805semidirect
582. C8C2×#11758198semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
C2×#168398042
C2×#168397642
C2×#168397682
#214483092
C2×#168396642
#214467002
#214466952
#214466852
#214466732
C2×C4×M16×C612
#2123661