Finite Group Databasebeta

← #29683933

Subgroups — #29683933

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×#14519803C211yesmaximal, characteristic
2C2×#16839802C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C2×#16839767C211yesmaximal, characteristic, complemented
5C2×C4×#11758194C211yesmaximal, characteristic, complemented
6#21448309C211yesmaximal, characteristic, complemented
7#21446699C212yesmaximal, complemented
8#21446694C212yesmaximal, complemented
9#21446684C212yesmaximal, complemented
10#21446672C212yesmaximal, complemented
11C2×C61×#390C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519803—24no—
14C2×#14519803—24no—
15C2×#14519803C418yescomplemented
16C22×#11758200C2212yes—
17C22×#11758199C2212yes—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839802—24no—
22C2×#14519710C2212yescomplemented
23C2×#14519708C2212yescomplemented
24#16839767—24no—
25#16839761—24no—
26C2×#14519700C2212yescomplemented
27C2×#14519698C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yescomplemented
30C4×#11758194—24no—
31C2×#14519688C2212yescomplemented
32C2×C4×#92418C2212yescomplemented
33#16839668C2211yescharacteristic, complemented
34#16839668C2211yescharacteristic, complemented
35#16839664C2211yescharacteristic, complemented
36#16839664C2211yescharacteristic, complemented
37#16839641—22no—
38C4×#11758198—22no—
39C23×D4×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C22×C22⋊C4×C61—22no—
42C22×C42×C61C2211yescharacteristic
43C61×#390—22no—
44C61×#390C2212yes—
45C61×#390C2212yes—
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×C4×D61—21no—
56#14519803—232no—
57C2×#11758200—24no—
58C2×#11758200—24no—
59C2×#11758200C2×C418yescomplemented
60C2×#11758199—24no—
61C2×#11758199—24no—
62C2×#11758199C2×C418yescomplemented
63C2×#11758194—22no—
64C2×#11758194—22no—
65C2×#11758194C2×C414yes—
66C2×#11758194D414yes—
67C2×#11758194D414yes—
68C2×#11758194D414yes—
69C22×#92418C2312yes—
70C22×#92418C2312yes—
71#14519736—22no—
72#14519736—22no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×C4×Dic61—21no—
79#14519710—24no—
80#14519708—24no—
81#14519700—24no—
82#14519698—24no—
83#14519698—24no—
84#14519697—24no—
85#14519688—24no—
86C4×#92418—24no—
87#14519684—22no—
88#14519684—22no—
89C25×C61—21no—
90C22×D4×C61—22no—
91C22×D4×C61—22no—
92C22×D4×C61—22no—
93C22×D4×C61—21no—
94C22×D4×C61—22no—
95C22×D4×C61—22no—
96C22×D4×C61C2×C414yes—
97C22×D4×C61C2312yes—
98C23×C4×C61—21no—
99C23×C4×C61D412yes—
100C23×C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—24no—
106C2×C22⋊C4×C61—24no—
107C2×C22⋊C4×C61—48no—
108C2×C42×C61—22no—
109C2×C42×C61—22no—
110C2×C42×C61D412yes—
111C2×C42×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C24×C61—24no—
114C24×C61—22no—
115C24×C61—44no—
116C24×C61—44no—
117C24×C61—42no—
118C24×C61—42no—
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C2×D4×C61—28no—
122C2×D4×C61—28no—
123C2×D4×C61—24no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—44no—
130C2×D4×C61—42no—
131C2×D4×C61—44no—
132C2×D4×C61—42no—
133C2×D4×C61C22×C414yes—
134C22×C4×C61—24no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—42no—
138C22×C4×C61—44no—
139C22×C4×C61—42no—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4112yes—
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156#11758222—28no—
157#11758222—28no—
158#11758222—44no—
159#11758222—44no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
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—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—42no—
178C22⋊C4×C61—48no—
179C22⋊C4×C61—48no—
180C42×C61—24no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—42no—
186#11758200—216no—
187#11758199—216no—
188#11758198—44no—
189#11758198—44no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4×Dic61—42no—
195#11758194—28no—
196#11758194C22⋊C4116yescomplemented
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418C22×C414yes—
200C2×#92418C2×D414yes—
201C2×#92418C2×D414yes—
202C2×#92418C2×D414yes—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—24no—
206C23×C61—416no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—44no—
210C23×C61—44no—
211C23×C61—48no—
212C23×C61—84no—
213C23×C61—84no—
214C23×C61—88no—
215C23×C61C4×D412yes—
216C23×C61#12112yes—
217C23×C61#12712yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—42no—
223C22×D61—42no—
224D4×C61—24no—
225D4×C61—24no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—48no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—48no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61—84no—
238C2×C4×C61C2×C22⋊C414yes—
239C2×C4×C61C4×D414yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61#12114yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12414yes—
244C2×C4×C61#12712yes—
245C2×C4×C61AGL(1,ℤ/8)14yescomplemented
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—48no—
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260D244—22no—
261D244—22no—
262D244—41no—
263D244—41no—
264C4×D61—48no—
265Dic122—22no—
266Dic122—22no—
267Dic122—41no—
268Dic122—41no—
269#92418—24no—
270#92418C2×C22⋊C418yescomplemented
271#21236—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—24no—
275C22×C61—24no—
276C22×C61—48no—
277C22×C61—44no—
278C22×C61—816no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61—84no—
282C22×C61#39014yes—
283C22×C61C2×C4×D411yescharacteristic
284C22×C61C2×#12111yescharacteristic
285C22×C61C2×#12711yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)12yes—
287C22×C61#57512yes—
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—82no—
293C4×C61—24no—
294C4×C61—42no—
295C4×C61—82no—
296C4×C61#39014yes—
297C4×C61C22×C22⋊C411yescharacteristic
298C4×C61C2×C4×D411yescharacteristic
299C4×C61C2×#12111yescharacteristic
300C4×C61C2×#12411yescharacteristic
301Dic61—42no—
302Dic61—81no—
303Dic61—81no—
304C22×AGL(1,ℤ/8)—611no—
305C2×#428—611no—
306C2×#417—611no—
307C2×#418—611no—
308C2×C4×M16—611no—
309#2297—611no—
310C2×#390D6111yescharacteristic, complemented
311#1973—612no—
312#1972—612no—
313#1970—612no—
314#1968—612no—
315C2×C61—24no—
316C2×C61—84no—
317C2×C61C2×#39011yescharacteristic
318C2×C61C4×AGL(1,ℤ/8)12yes—
319C2×C61#305512yes—
320C2×C61#315111yescharacteristic
321C2×C61#315311yescharacteristic
322D61—82no—
323C22×#41—611no—
324C23×D4C2×D6111yescharacteristic
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×M16—611no—
331C22×M16—611no—
332C2×#117—611no—
333C22×C4⋊C4C2×D6111yescharacteristic
334C22×C22⋊C4—22no—
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—1224no—
346C2×#97—612no—
347C2×C4×C8—612no—
348#391—611no—
349#391—611no—
350#390—22no—
351#390—611no—
352#390—611no—
353#390C2×D6112yes—
354#390C2×D6112yes—
355#386—1222no—
356C4×C22⋊C4—1222no—
357C61#2123611yescharacteristic, complemented, Sylow
358C25—21no—
359C2×#41—614no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—1222no—
363C2×#41—1222no—
364C2×#41—1222no—
365C22×Q8—611no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—22no—
370C22×D4—22no—
371C22×D4—21no—
372C22×D4—611no—
373C22×D4—1221no—
374C22×D4C4×D6114yes—
375C22×D4C22×D6112yes—
376C23×C4—21no—
377C23×C4—1221no—
378C23×C4C22⋊D6112yes—
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—614no—
388C2×M16—614no—
389C2×M16—614no—
390C2×M16—614no—
391C2×M16—1222no—
392C2×M16—1222no—
393C22×C8—612no—
394C22×C8—612no—
395#117—1222no—
396#117—1222no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—48no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—1222no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—1221no—
412C2×C42C22⋊D6112yes—
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—1224no—
422C4×C8—1224no—
423#95—1222no—
424#95—1222no—
425C24—22no—
426C24—24no—
427C24—44no—
428C24—44no—
429C24—42no—
430C24—42no—
431C24—1221no—
432C24C2×C22⋊D6111yescharacteristic
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—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—24no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—612no—
456C2×D4—1224no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—2442no—
463C2×D4—2442no—
464C2×D4C2×C4×D6114yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4C2×C4×D6112yes—
478C22×C4#1175822212yes—
479C22×C4D4×D6112yes—
480C22×C4D4×D6112yes—
481C22×C4D4×D6112yes—
482C22×C4#1175822412yes—
483C22×C4C2×C22⋊D6112yes—
484C22×C4C2×C22⋊D6112yes—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487SD16—12216no—
488D8—12216no—
489M16—6116no—
490M16—1228no—
491C2×C8—614no—
492C2×C8—614no—
493C2×C8—614no—
494C2×C8—614no—
495C2×C8—1222no—
496C2×C8—1222no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—2444no—
507C22⋊C4—2442no—
508C22⋊C4—2444no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—24no—
514C42—2442no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—416no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—48no—
524C23—88no—
525C23—84no—
526C23—84no—
527C23—1222no—
528C23—2441no—
529C23—2442no—
530C23—2441no—
531C23—2442no—
532C23C4×C22⋊D6112yes—
533C23#1451974912yes—
534C23#1451977312yes—
535C23C22×C22⋊D6111yescharacteristic
536Q8—1222no—
537Q8—1222no—
538Q8—2441no—
539Q8—2441no—
540D4—24no—
541D4—24no—
542D4—42no—
543D4—42no—
544D4—48no—
545D4—1222no—
546D4—1222no—
547D4—2448no—
548D4—2441no—
549D4—2441no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—84no—
559C2×C4—1222no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2442no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2448no—
567C2×C4C22⋊C4×D6114yes—
568C2×C4#1451973714yes—
569C2×C4C4×C22⋊D6112yes—
570C2×C4#1451974912yes—
571C2×C4#1451975014yes—
572C2×C4#1451976314yes—
573C2×C4#1451977312yes—
574C2×C4#1451980314yescomplemented
575C2×C4C22×C4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C8—618no—
582C8—1224no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—48no—
588C22—44no—
589C22—816no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—4882no—
594C22—4881no—
595C22—4881no—
596C22—2444no—
597C22—2442no—
598C22#1683966214yes—
599C22C2×C4×C22⋊D6111yescharacteristic
600C22C2×#1451974911yescharacteristic
601C22C2×#1451977311yescharacteristic
602C22C2×#1451980312yes—
603C22#1684056012yes—
604C4—24no—
605C4—42no—
606C4—82no—
607C4—4881no—
608C4—4881no—
609C4—2442no—
610C4#1683966214yes—
611C4C2×C22⋊C4×D6111yescharacteristic
612C4C2×#1451973711yescharacteristic
613C4C2×#1451975011yescharacteristic
614C4C2×#1451976311yescharacteristic
615C2—24no—
616C2—84no—
617C2—4882no—
618C2C2×#1683966211yescharacteristic
619C2C4×#1451980312yes—
620C2#2145543412yes—
621C2#2145626911yescharacteristic
622C2#2145628311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#390C2
socleC23×C61C22×D4
prime radicalC22×#14519803C2
cosocleC22×C4×C61C24
hypercenterC2×#390D61
hypocenterC61#21236
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
3. C2×#16839761C2semidirect
4. C2×#16839767C2semidirect
5. C2×C4×#11758194C2semidirect
6. #21448309C2semidirect
7. #21446699C2semidirect
8. #21446694C2semidirect
9. #21446684C2semidirect
10. #21446672C2semidirect
11. C2×C61×#390C2semidirect
15. C2×#14519803C4semidirect
22. C2×#14519710C22semidirect
23. C2×#14519708C22semidirect
26. C2×#14519700C22semidirect
29. C2×#14519697C22semidirect
31. C2×#14519688C22semidirect
32. C2×C4×#92418C22semidirect
33. #16839668C22semidirect
34. #16839668C22semidirect
35. #16839664C22semidirect
36. #16839664C22semidirect
59. C2×#11758200C2×C4semidirect
62. C2×#11758199C2×C4semidirect
196. #11758194C22⋊C4semidirect
245. C2×C4×C61AGL(1,ℤ/8)semidirect
270. #92418C2×C22⋊C4semidirect
310. C2×#390D61semidirect
357. C61#21236semidirect
574. C2×C4#14519803semidirect

Maximal subgroups

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

subgroupindex
C22×#145198032
C2×#168398022
C2×#168397612
C2×#168397672
C2×C4×#117581942
#214483092
#214466992
#214466942
#214466842
#214466722
C2×C61×#3902
#2123661