Finite Group Databasebeta

← C4×#16839929 #29685885

Subgroups — C4×#16839929 #29685885

Order 15616.

1578 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
1C4×#14519822C211yesmaximal, characteristic, complemented
2C2×#16839929C211yesmaximal, characteristic
3C4×#14519803C212yesmaximal, complemented
4#21446218C211yesmaximal, characteristic, complemented
5#21446198C212yesmaximal, complemented
6#21446044C212yesmaximal, complemented
7#21446028C211yesmaximal, characteristic, complemented
8#21445772C211yesmaximal, characteristic, complemented
9C4×#14519705C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C4×#127×C61C211yesmaximal, characteristic, complemented
12C2×#14519822C2211yescharacteristic
13C2×#14519803C2212yes—
14#16840243C2212yes—
15#16840241—22no—
16#16840237C2211yescharacteristic
17#16840235—22no—
18#16840234—22no—
19C4×#11758222C2212yes—
20C4×#11758224—22no—
21C2×C4×C22⋊D61—22no—
22C4×D4×D61—22no—
23#16839929C418yesdirect product factor
24#16839872—22no—
25#16839862—22no—
26C4×#11758200—22no—
27C4×#11758199—22no—
28#16839802C2212yescomplemented
29C2×#14519705C2212yes—
30C2×#14519705C2212yes—
31#16839767C2212yescomplemented
32#16839761C2212yes—
33C2×#14519694C2211yescharacteristic
34C2×#14519694C2211yescharacteristic
35C4×#11758194C2212yescomplemented
36#16839531C2212yescomplemented
37#16839522C2212yescomplemented
38#16839522C2212yescomplemented
39C2×#127×C61C2211yescharacteristic
40C2×C4×D4×C61—22no—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic, complemented
43C61×#390—22no—
44C61×#390C2211yescharacteristic
45C61×#390C2211yescharacteristic, complemented
46C61×#384C2211yescharacteristic, complemented
47C43×C61C2211yescharacteristic, complemented
48#14519822—22no—
49#14519822—24no—
50#14519822C2×C414yescomplemented
51C2×#11758222C2312yes—
52C2×#11758224—22no—
53C22×C22⋊D61—22no—
54C2×D4×D61—22no—
55C22×C4×D61—22no—
56#14519803—24no—
57#14519803C2×C418yescomplemented
58C2×#11758200—22no—
59C2×#11758199—22no—
60C2×#11758194C2312yes—
61C2×#11758194C2312yes—
62C4×Dic122D412yescomplemented
63C4×Dic122D412yescomplemented
64#14519742—22no—
65C4×D244D412yescomplemented
66C4×D244D412yescomplemented
67D4×Dic61—22no—
68D4×Dic61—22no—
69#14519739—22no—
70C4×C22⋊D61—24no—
71C4×C22⋊D61—24no—
72C4×C22⋊D61—22no—
73C4×C22⋊D61—24no—
74C4×C22⋊D61—22no—
75#14519737—24no—
76#14519737—24no—
77#14519736C2312yes—
78#14519735—22no—
79#14519734—22no—
80#14519733—24no—
81C2×#11758196—22no—
82C4⋊C4×D61—22no—
83C2×C22⋊Dic61—22no—
84C2×#11758198—22no—
85C22⋊C4×D61—24no—
86C2×C4×Dic61—22no—
87C42×D61—22no—
88#14519710—22no—
89#14519708—22no—
90#14519705—24no—
91#14519705C2×C418yescomplemented
92#14519700—22no—
93#14519698—22no—
94#14519698—22no—
95#14519697—22no—
96#14519694—22no—
97#14519694C2×C414yescomplemented
98#14519688—22no—
99C4×#92418—22no—
100C22×D4×C61—22no—
101C22×D4×C61C2311yescharacteristic
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61D412yes—
105#127×C61—22no—
106#127×C61C2×C414yescomplemented
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—48no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C22⋊C4×C61—22no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—22no—
118C2×C22⋊C4×C61D412yes—
119C2×C42×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C24×C61—21no—
125C24×C61C2×D412yes—
126C23×D61—22no—
127C2×D4×C61—28no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61C22×C412yes—
136C2×D4×C61C22×C412yescomplemented
137C2×D4×C61#4112yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—44no—
144C22×C4×C61—44no—
145C22×C4×C61—41no—
146C22×C4×C61—41no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—24no—
152C2×C22⋊D61—28no—
153C2×C22⋊D61—28no—
154C2×C22⋊D61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158#11758224—28no—
159#11758224—28no—
160D4×D61—28no—
161D4×D61—28no—
162#11758222—22no—
163#11758222—22no—
164#11758222C22×C414yes—
165#11758222#4114yes—
166C2×D244C2×D412yes—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—24no—
170C2×C4×D61—22no—
171C2×C4×D61—84no—
172C2×Dic122C2×D412yes—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—44no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—44no—
178C22⋊C4×C61—44no—
179C22⋊C4×C61—42no—
180C42×C61—24no—
181C42×C61—21no—
182C42×C61—22no—
183C42×C61—42no—
184C42×C61C22×C412yescomplemented
185C42×C61C2×D412yes—
186C42×C61C2×D412yes—
187C22⋊Dic61—24no—
188C22⋊Dic61—22no—
189#11758200—28no—
190#11758199—28no—
191#11758198—24no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61C2×D412yes—
195C4⋊Dic61C2×D412yes—
196#11758196—24no—
197#11758196—22no—
198#11758196—22no—
199C4×Dic61—24no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203#11758194—22no—
204#11758194—22no—
205#11758194—22no—
206#11758194C22×C414yescomplemented
207C2×#92418—22no—
208C2×#92418—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61C4×D414yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—28no—
222C22×D61—82no—
223C22×D61—82no—
224D4×C61—24no—
225D4×C61—24no—
226D4×C61—24no—
227D4×C61—416no—
228D4×C61—42no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—44no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—82no—
243C2×C4×C61—82no—
244C2×C4×C61—82no—
245C2×C4×C61#12012yes—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D412yes—
250C2×C4×C61C2×#4112yes—
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—216no—
253C22⋊D61—24no—
254C22⋊D61—24no—
255C2×Dic61—28no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—82no—
260C2×Dic61—82no—
261D244—22no—
262D244C4×D414yescomplemented
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—44no—
266C4×D61—82no—
267C4×D61—84no—
268Dic122—22no—
269Dic122C4×D414yescomplemented
270#92418—24no—
271#92418—42no—
272C4×#453—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—24no—
276C22×C61—44no—
277C22×C61—44no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C2×#12011yescharacteristic
287C22×C61#52511yescharacteristic
288C2×D61—24no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—88no—
292C4×C61—24no—
293C4×C61—21no—
294C4×C61—22no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—81no—
298C4×C61#45312yescomplemented
299C4×C61C2×C4×D412yes—
300C4×C61#52512yes—
301Dic61—24no—
302Dic61—42no—
303Dic61—42no—
304C4×H(5,𝔽2)—611no—
305C2×#453—611no—
306C4×AGL(1,ℤ/8)—612no—
307C4×#127D6111yescharacteristic, complemented
308#1887—611no—
309#1883—612no—
310#1859—612no—
311#1856—611no—
312#1792—611no—
313C4×#104—612no—
314C4×#100—611no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—44no—
318C2×C61—82no—
319C2×C61C4×#12011yescharacteristic
320C2×C61C2×#45311yescharacteristic
321C2×C61#301811yescharacteristic
322D61—44no—
323C2×H(5,𝔽2)—611no—
324C2×AGL(1,ℤ/8)—612no—
325C2×#127C2×D6111yescharacteristic
326#520—612no—
327#520—1222no—
328#518—611no—
329#518—1222no—
330#518—1222no—
331C4×#41—612no—
332C4×#41—1222no—
333C2×C4×D4—22no—
334C2×C4×D4—1222no—
335C2×C4×D4—1222no—
336C2×C4×D4C2×D6111yescharacteristic
337C2×C4×D4C2×D6111yescharacteristic, complemented
338#453—618no—
339#442—1222no—
340#440—1222no—
341C4×SD16—1222no—
342C4×D8—1222no—
343#428—612no—
344C2×#104—612no—
345C2×#104—612no—
346#418—612no—
347#417—612no—
348C2×#100—611no—
349C2×#100—611no—
350C4×M16—612no—
351#390—22no—
352#390C2×D6111yescharacteristic
353#390C2×D6111yescharacteristic, complemented
354#384C2×D6111yescharacteristic, complemented
355C43C2×D6111yescharacteristic, complemented
356#343—612no—
357#339—612no—
358#339—612no—
359C61C4×#45311yescharacteristic, complemented, Sylow
360H(5,𝔽2)—614no—
361H(5,𝔽2)—1224no—
362H(5,𝔽2)—1222no—
363C2×#41—612no—
364C2×#41—1222no—
365C22×D4—22no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4C22×D6111yescharacteristic
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C23×C4—1222no—
372C23×C4C22⋊D6112yes—
373AGL(1,ℤ/8)—618no—
374AGL(1,ℤ/8)—1224no—
375C2×SD16—1222no—
376C2×D8—1222no—
377C2×M16—612no—
378C2×M16—612no—
379#127—22no—
380#127C4×D6114yescomplemented
381C4×Q8—612no—
382C4×Q8—612no—
383C4×Q8—1222no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—48no—
389C4×D4—612no—
390C4×D4—612no—
391C4×D4—1224no—
392C4×D4—1224no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C4×D4—1224no—
397C4×D4—1224no—
398C4×D4—1222no—
399C4×D4—1224no—
400C4×D4—1222no—
401#117—612no—
402#117—1224no—
403#117—1222no—
404#117—1222no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—22no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1224no—
414C2×C22⋊C4—1222no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4C22⋊D6112yes—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—1222no—
420C2×C42—1222no—
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424#107—1222no—
425#106—1222no—
426#104—618no—
427#104—1224no—
428#103—1222no—
429#102—1222no—
430#102—1222no—
431#102—1222no—
432#100—614no—
433#100—1222no—
434#97—1222no—
435C4×C8—1222no—
436C24—21no—
437C24—1222no—
438C24C2×C22⋊D6112yes—
439#41—614no—
440#41—614no—
441#41—1228no—
442#41—1228no—
443#41—1222no—
444#41—1222no—
445C2×Q8—612no—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—612no—
455C2×D4—1228no—
456C2×D4—1228no—
457C2×D4—1224no—
458C2×D4—1228no—
459C2×D4—1222no—
460C2×D4—1228no—
461C2×D4C2×C4×D6112yes—
462C2×D4C2×C4×D6112yescomplemented
463C2×D4#1175822212yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—1224no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—4884no—
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6112yes—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C23×D6111yescharacteristic
485SD16—1228no—
486D8—1228no—
487M16—614no—
488M16—1222no—
489M16—1222no—
490M16—1222no—
491C2×C8—1222no—
492C2×C8—1222no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—1224no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—24no—
511C42—21no—
512C42—22no—
513C42—42no—
514C42—1224no—
515C42—1222no—
516C42—1222no—
517C42—1222no—
518C42C2×C4×D6112yescomplemented
519C42D4×D6112yes—
520C42D4×D6112yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—1228no—
531C23—1222no—
532C23—4882no—
533C23—4882no—
534C23C4×C22⋊D6114yes—
535C23C22×C22⋊D6111yescharacteristic
536Q8—614no—
537Q8—1222no—
538D4—24no—
539D4—24no—
540D4—24no—
541D4—416no—
542D4—42no—
543D4—614no—
544D4—12216no—
545D4—1224no—
546D4—1224no—
547D4—1222no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—1228no—
565C2×C4—1222no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—4882no—
571C2×C4—4884no—
572C2×C4—4882no—
573C2×C4—4882no—
574C2×C4—2444no—
575C2×C4#1451974512yes—
576C2×C4#1451974511yescharacteristic
577C2×C4#1451974511yescharacteristic
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4C2×D4×D6112yes—
580C2×C4#41×D6112yes—
581C2×C4C2×#1175822211yescharacteristic
582C8—1224no—
583C8—2442no—
584C22—21no—
585C22—21no—
586C22—24no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—1224no—
597C22—4888no—
598C22—2442no—
599C22—2442no—
600C22C2×C4×C22⋊D6111yescharacteristic
601C22C2×#1451974511yescharacteristic
602C22#1684026911yescharacteristic
603C4—24no—
604C4—21no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—1224no—
610C4—2442no—
611C4—2442no—
612C4#1683992912yesdirect product factor
613C4C4×D4×D6112yes—
614C4#1684027112yes—
615C2—21no—
616C2—21no—
617C2—44no—
618C2—82no—
619C2—2444no—
620C2C4×#1451974511yescharacteristic
621C2C2×#1683992911yescharacteristic
622C2#2145516711yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#127×C61C2
socleC22×C61C2×#120
prime radicalC2×#16839929C2
cosocleC22×C4×C61C24
hypercenterC4×#127D61
hypocenterC61C4×#453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C4×#14519822C2semidirect
3. C4×#14519803C2semidirect
4. #21446218C2semidirect
5. #21446198C2semidirect
6. #21446044C2semidirect
7. #21446028C2semidirect
8. #21445772C2semidirect
9. C4×#14519705C2semidirect
10. C4×#14519694C2semidirect
11. C4×#127×C61C2semidirect
23. #16839929C4direct (N × H)
28. #16839802C22semidirect
31. #16839767C22semidirect
35. C4×#11758194C22semidirect
36. #16839531C22semidirect
37. #16839522C22semidirect
38. #16839522C22semidirect
42. C2×C4×D4×C61C22semidirect
45. C61×#390C22semidirect
46. C61×#384C22semidirect
47. C43×C61C22semidirect
50. #14519822C2×C4semidirect
57. #14519803C2×C4semidirect
62. C4×Dic122D4semidirect
63. C4×Dic122D4semidirect
65. C4×D244D4semidirect
66. C4×D244D4semidirect
91. #14519705C2×C4semidirect
97. #14519694C2×C4semidirect
106. #127×C61C2×C4semidirect
136. C2×D4×C61C22×C4semidirect
184. C42×C61C22×C4semidirect
206. #11758194C22×C4semidirect
262. D244C4×D4semidirect
269. Dic122C4×D4semidirect
298. C4×C61#453semidirect
307. C4×#127D61semidirect
337. C2×C4×D4C2×D61semidirect
353. #390C2×D61semidirect
354. #384C2×D61semidirect
355. C43C2×D61semidirect
359. C61C4×#453semidirect
380. #127C4×D61semidirect
462. C2×D4C2×C4×D61semidirect
518. C42C2×C4×D61semidirect
612. C4#16839929direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#145198222
C2×#168399292
C4×#145198032
#214462182
#214461982
#214460442
#214460282
#214457722
C4×#145197052
C4×#145196942
C4×#127×C612
C4×#45361