Finite Group Databasebeta

← C2×#21446744 #29684063

Subgroups — C2×#21446744 #29684063

Order 15616.

2374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519773C211yesmaximal, characteristic
2C22×#14519706C211yesmaximal, characteristic, complemented
3C22×#14519699C211yesmaximal, characteristic, complemented
4C22×#14519698C211yesmaximal, characteristic, complemented
5C22×#14519697C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
7#21446744C218yesmaximal, direct product factor
8C22×#102×C61C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C23×C22⋊D61—21no—
11C23×C22⋊D61—21no—
12C23×#92418C2211yescharacteristic
13C2×#14519773—24no—
14C2×#14519773—22no—
15C2×#14519773C418yescomplemented
16C2×#14519773C2214yes—
17C22×C4⋊Dic61C2211yescharacteristic
18C22×C4×Dic61—21no—
19C22×C4×Dic61C2211yescharacteristic
20C2×#14519706—24no—
21C2×#14519706C2214yescomplemented
22C2×#14519699—24no—
23C2×#14519699C2214yescomplemented
24C2×#14519698—24no—
25C2×#14519698C2214yescomplemented
26C2×#14519697—24no—
27C2×#14519697C2214yescomplemented
28C4⋊C4×Dic61—22no—
29C4⋊C4×Dic61C2214yescomplemented
30C2×#14519684—21no—
31C23×D4×C61C2211yescharacteristic
32C23×C8×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C2×#102×C61—24no—
35C2×#102×C61C2214yescomplemented
36C24×D61—21no—
37C22×D244—22no—
38C22×D244—22no—
39C22×D244—22no—
40C22×D244—22no—
41C22×D244—21no—
42C22×D244—22no—
43C22×D244C2×C414yes—
44C22×D244C2312yes—
45C22×C22⋊D61—28no—
46C22×C22⋊D61—28no—
47C22×C22⋊D61—24no—
48C22×C22⋊D61—24no—
49C22×C22⋊D61—44no—
50C22×C22⋊D61—44no—
51C22×C22⋊D61—44no—
52C22×C22⋊D61—44no—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C22×#92418—22no—
57C22×#92418—22no—
58C22×#92418—22no—
59C22×#92418C2312yes—
60#14519773—216no—
61#14519773—28no—
62#14519773C2×C4116yescomplemented
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D414yescomplemented
66C2×C4⋊Dic61C2312yes—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—44no—
69C2×C4×Dic61C2×C414yes—
70C2×C4×Dic61D412yes—
71C2×C4×Dic61D414yes—
72C2×C4×Dic61C2312yes—
73#14519706—28no—
74#14519699—28no—
75#14519698—28no—
76#14519697—28no—
77#14519684—24no—
78C25×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61—22no—
84C22×D4×C61—22no—
85C22×D4×C61C2×C414yes—
86C22×D4×C61C2312yes—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61—22no—
90C22×C8×C61—22no—
91C22×C8×C61—22no—
92C22×C8×C61C2312yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61D414yescomplemented
96C2×C4⋊C4×C61C2312yes—
97#102×C61—28no—
98C24×C61—24no—
99C24×C61—22no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61—42no—
104C24×C61—42no—
105C24×C61—42no—
106C24×C61C2×D411yescharacteristic
107C23×D61—24no—
108C23×D61—22no—
109C23×D61—42no—
110C23×D61—42no—
111C23×D61—42no—
112C23×D61—42no—
113C23×D61—42no—
114C23×D61—42no—
115C2×D4×C61—24no—
116C2×D4×C61—28no—
117C2×D4×C61—28no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—44no—
126C2×D4×C61—44no—
127C2×D4×C61C22×C414yes—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—216no—
141C2×C22⋊D61—216no—
142C2×C22⋊D61—48no—
143C2×C22⋊D61—48no—
144C2×C22⋊D61—432no—
145C2×C22⋊D61—432no—
146C2×C22⋊D61—48no—
147C2×C22⋊D61—48no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—24no—
151C22×Dic61—22no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61C22⋊C414yes—
155C22×Dic61C22⋊C414yes—
156C22×Dic61D814yes—
157C22×Dic61SD1614yes—
158C22×Dic61C2×D412yes—
159C22×Dic61C2×D412yes—
160C2×D244—28no—
161C2×D244—28no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—24no—
167C2×D244—22no—
168C2×D244—42no—
169C2×D244—42no—
170C2×D244—44no—
171C2×D244—44no—
172C2×D244C22×C414yes—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—44no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61C2×D414yescomplemented
180C4⋊Dic61—24no—
181C4⋊Dic61—22no—
182C4⋊Dic61C2×D414yescomplemented
183C4×Dic61—48no—
184C4×Dic61C22⋊C418yes—
185C4×Dic61C22×C414yes—
186C4×Dic61C2×D414yes—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—44no—
191C23×C61—24no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—48no—
199C23×C61—48no—
200C23×C61—48no—
201C23×C61—84no—
202C23×C61—84no—
203C23×C61—84no—
204C23×C61—84no—
205C23×C61C2×C22⋊C412yes—
206C23×C61#12112yes—
207C23×C61#12112yes—
208C23×C61C2×D812yes—
209C23×C61C2×SD1612yes—
210C23×C61AGL(1,ℤ/8)12yes—
211C23×C61AGL(1,ℤ/8)12yes—
212C23×C61C22×D411yescharacteristic
213C22×D61—24no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—48no—
223C22×D61—84no—
224C22×D61—84no—
225C22×D61—84no—
226C22×D61—84no—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—48no—
232C2×C4×C61—24no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—48no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C4×D414yes—
238C2×C4×C61C4×D414yes—
239C2×C4×C61#12112yes—
240C2×C4×C61#12112yes—
241C2×C4×C61#12414yes—
242C2×C4×C61#12714yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C22⋊D61—432no—
250C22⋊D61—432no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—28no—
254C2×Dic61—24no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—48no—
258C2×Dic61#102116yescomplemented
259C2×Dic61C2×C22⋊C414yes—
260C2×Dic61C2×C22⋊C414yes—
261C2×Dic61C2×D814yes—
262C2×Dic61C2×SD1614yes—
263D244—24no—
264D244—24no—
265D244—42no—
266D244—42no—
267D244—48no—
268C8×C61—42no—
269#92418—42no—
270C2×#1982—611nomaximal, Sylow
271C22×C61—44no—
272C22×C61—44no—
273C22×C61—44no—
274C22×C61—816no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61#39014yes—
280C22×C61C2×#10214yes—
281C22×C61#41814yes—
282C22×C61#45914yes—
283C22×C61#45914yes—
284C22×C61#46014yes—
285C22×C61#46014yes—
286C22×C61C22×C22⋊C411yescharacteristic
287C22×C61C2×#12111yescharacteristic
288C22×C61C2×#12111yescharacteristic
289C22×C61C22×D811yescharacteristic
290C22×C61C22×SD1611yescharacteristic
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C2×D61—44no—
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—816no—
297C2×D61—82no—
298C2×D61—82no—
299C2×D61—82no—
300C2×D61—82no—
301C4×C61—44no—
302C4×C61#39014yes—
303C4×C61C2×C4×D411yescharacteristic
304C4×C61C2×C4×D411yescharacteristic
305C4×C61C2×#12411yescharacteristic
306C4×C61C2×#12711yescharacteristic
307Dic61—24no—
308Dic61—44no—
309Dic61C2×#10218yescomplemented
310C22×#127—611no—
311C22×C4⋊C8—611no—
312C22×#102—611no—
313C22×#102—611no—
314C22×#102—611no—
315C22×#102D6111yescharacteristic, complemented
316C2×C4×C4⋊C4—611no—
317#1982—618no—
318C2×C61—84no—
319C2×C61#198218yescomplemented
320C2×C61C2×#39011yescharacteristic
321C2×C61C22×#10211yescharacteristic
322C2×C61C2×#41811yescharacteristic
323C2×C61C2×#45911yescharacteristic
324C2×C61C2×#45911yescharacteristic
325C2×C61C2×#46011yescharacteristic
326C2×C61C2×#46011yescharacteristic
327D61—84no—
328C23×D4—611no—
329C23×D4—1221no—
330C23×D4—1221no—
331C23×D4C2×D6111yescharacteristic
332C23×C8—611no—
333C23×C8C2×D6111yescharacteristic
334C2×#127—618no—
335C2×#127—614no—
336C2×#127—1222no—
337C2×#127—1224no—
338C22×C4⋊C4—611no—
339C22×C4⋊C4C2×D6111yescharacteristic
340C22×C42—611no—
341C22×C42—1221no—
342C2×C4⋊C8—614no—
343C2×C4⋊C8—1224no—
344C2×#102—24no—
345C2×#102—614no—
346C2×#102—614no—
347C2×#102—614no—
348C2×#102—1224no—
349C2×#102—1224no—
350C2×#102—1224no—
351C2×#102C2×D6114yescomplemented
352C4×C4⋊C4—614no—
353C4×C4⋊C4—1222no—
354C2×#95—1221no—
355C61C2×#198211yescharacteristic, complemented, Sylow
356C25—21no—
357C25—1221no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—21no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—614no—
365C22×D4—612no—
366C22×D4—1224no—
367C22×D4—1228no—
368C22×D4—1224no—
369C22×D4—1228no—
370C22×D4—1222no—
371C22×D4—1222no—
372C22×D4—1222no—
373C22×D4—1221no—
374C22×D4—1222no—
375C22×D4—1222no—
376C22×D4—2444no—
377C22×D4—2444no—
378C22×D4—2444no—
379C22×D4—2444no—
380C22×D4C4×D6114yes—
381C22×D4C22×D6112yes—
382C23×C4—21no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—1221no—
386C23×C4C22×D6111yescharacteristic
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—612no—
391C22×C8—1222no—
392C22×C8—1222no—
393C22×C8—1222no—
394C22×C8C22×D6112yes—
395#127—6116no—
396#127—1228no—
397#127—12216no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—614no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4C22⋊D6114yescomplemented
405C2×C4⋊C4C22×D6112yes—
406C2×C42—614no—
407C2×C42—612no—
408C2×C42—614no—
409C2×C42—612no—
410C2×C42—1224no—
411C2×C42—2444no—
412C4⋊C8—1228no—
413#102—28no—
414#102—1228no—
415#102—1228no—
416#102—1228no—
417#95—1224no—
418C24—22no—
419C24—24no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—1222no—
427C24—1224no—
428C24—2442no—
429C24—2442no—
430C24—2442no—
431C24—2442no—
432C24—2442no—
433C24—2442no—
434C24D4×D6111yescharacteristic
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—614no—
448C2×D4—12216no—
449C2×D4—12216no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1224no—
456C2×D4—1228no—
457C2×D4—1228no—
458C2×D4—24432no—
459C2×D4—2448no—
460C2×D4—2448no—
461C2×D4—24432no—
462C2×D4—2448no—
463C2×D4—2448no—
464C2×D4—2442no—
465C2×D4—2442no—
466C2×D4—2444no—
467C2×D4—2444no—
468C2×D4C2×C4×D6114yes—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—612no—
474C22×C4—614no—
475C22×C4—614no—
476C22×C4—614no—
477C22×C4—612no—
478C22×C4—614no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×C4×D6112yes—
486C22×C4#1175822212yes—
487C22×C4D4×D6112yes—
488C22×C4D4×D6112yes—
489C22×C4D4×D6111yescharacteristic
490C22×C4#1175822412yes—
491C22×C4C2×C22⋊D6112yes—
492C22×C4C23×D6111yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—2444no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—614no—
504C4⋊C4—1222no—
505C4⋊C4—1224no—
506C4⋊C4C2×C22⋊D6114yescomplemented
507C42—614no—
508C42—614no—
509C42—618no—
510C42—2448no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—48no—
515C23—42no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—48no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—1224no—
526C23—4884no—
527C23—4884no—
528C23—4884no—
529C23—4884no—
530C23—2442no—
531C23—2442no—
532C23—2448no—
533C23—2442no—
534C23—2448no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2448no—
539C23C22⋊C4×D6112yes—
540C23#1451974912yes—
541C23#1451975012yes—
542C23D8×D6112yes—
543C23SD16×D6112yes—
544C23#1451980112yes—
545C23#1451980412yes—
546C23C2×D4×D6111yescharacteristic
547D4—24no—
548D4—24no—
549D4—42no—
550D4—42no—
551D4—48no—
552D4—1224no—
553D4—1224no—
554D4—24432no—
555D4—24432no—
556D4—2442no—
557D4—2442no—
558D4—2448no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—614no—
564C2×C4—6116no—
565C2×C4—614no—
566C2×C4—614no—
567C2×C4—614no—
568C2×C4—1228no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1224no—
572C2×C4—2448no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4C22⋊C4×D6112yes—
576C2×C4#1451973714yes—
577C2×C4C4×C22⋊D6114yes—
578C2×C4#1451974912yes—
579C2×C4#1451975012yes—
580C2×C4#1451976314yes—
581C2×C4#1451977314yes—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C22×C22⋊D6111yescharacteristic
586C2×C4C2×#1175822411yescharacteristic
587C2×C4C2×#1175822211yescharacteristic
588C8—42no—
589C8—2442no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—816no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—48816no—
599C22—4882no—
600C22—4882no—
601C22—4882no—
602C22—4882no—
603C22—2444no—
604C22—2444no—
605C22—2444no—
606C22#1683966214yes—
607C22#102×D6114yes—
608C22#1683976514yes—
609C22#1683995514yes—
610C22#1683995614yes—
611C22#1683996014yes—
612C22#1683996114yes—
613C22C2×C22⋊C4×D6111yescharacteristic
614C22C2×#1451974911yescharacteristic
615C22C2×#1451975011yescharacteristic
616C22C2×D8×D6111yescharacteristic
617C22C2×SD16×D6111yescharacteristic
618C22C2×#1451980111yescharacteristic
619C22C2×#1451980411yescharacteristic
620C4—44no—
621C4—618no—
622C4—1224no—
623C4—2444no—
624C4#1683966214yes—
625C4C2×#1451973711yescharacteristic
626C4C2×C4×C22⋊D6111yescharacteristic
627C4C2×#1451976311yescharacteristic
628C4C2×#1451977311yescharacteristic
629C2—84no—
630C2—4884no—
631C2#2144674418yesdirect product factor
632C2C2×#1683966211yescharacteristic
633C2C2×#102×D6111yescharacteristic
634C2C2×#1683976511yescharacteristic
635C2C2×#1683995511yescharacteristic
636C2C2×#1683995611yescharacteristic
637C2C2×#1683996011yescharacteristic
638C2C2×#1683996111yescharacteristic
subgroupG/H
centerC24D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#102×C61C2
socleC24×C61C2×D4
prime radicalC22×#14519773C2
cosocleC22×C4×C61C24
hypercenterC22×#102D61
hypocenterC61C2×#1982
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×#14519706C2semidirect
3. C22×#14519699C2semidirect
4. C22×#14519698C2semidirect
5. C22×#14519697C2semidirect
6. C2×C4⋊C4×Dic61C2semidirect
7. #21446744C2direct (N × H)
8. C22×#102×C61C2semidirect
15. C2×#14519773C4semidirect
21. C2×#14519706C22semidirect
23. C2×#14519699C22semidirect
25. C2×#14519698C22semidirect
27. C2×#14519697C22semidirect
29. C4⋊C4×Dic61C22semidirect
35. C2×#102×C61C22semidirect
62. #14519773C2×C4semidirect
65. C2×C4⋊Dic61D4semidirect
95. C2×C4⋊C4×C61D4semidirect
179. C4⋊C4×C61C2×D4semidirect
182. C4⋊Dic61C2×D4semidirect
258. C2×Dic61#102semidirect
309. Dic61C2×#102semidirect
315. C22×#102D61semidirect
319. C2×C61#1982semidirect
351. C2×#102C2×D61semidirect
355. C61C2×#1982semidirect
404. C2×C4⋊C4C22⋊D61semidirect
506. C4⋊C4C2×C22⋊D61semidirect
631. C2#21446744direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197732
C22×#145197062
C22×#145196992
C22×#145196982
C22×#145196972
C2×C4⋊C4×Dic612
#214467442
C22×#102×C612
C2×#198261