Finite Group Databasebeta

← C2×#21447004 #29707464

Subgroups — C2×#21447004 #29707464

Order 15616.

2006 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×#14519752C211yesmaximal, characteristic
2C22×#14519710C211yesmaximal, characteristic, complemented
3C22×#14519699C211yesmaximal, characteristic, complemented
4C22×#14519697C211yesmaximal, characteristic, complemented
5C22×#14519689C211yesmaximal, characteristic, complemented
6C2×#16839619C211yesmaximal, characteristic, complemented
7#21447004C218yesmaximal, direct product factor
8C2×C61×#340C211yesmaximal, characteristic, complemented
9C23×D244—21no—
10C23×D244C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13C2×#14519752—24no—
14C2×#14519752—24no—
15C2×#14519752—22no—
16C2×#14519752C2214yes—
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758198—21no—
19C22×#11758198—21no—
20C2×#14519710—24no—
21C2×#14519710C2214yescomplemented
22C2×#14519699—24no—
23C2×#14519699C2214yescomplemented
24C2×#14519697—24no—
25C2×#14519697C2214yescomplemented
26C2×#14519689—24no—
27C2×#14519689C2214yescomplemented
28#16839619—22no—
29#16839619C2214yescomplemented
30C2×#14519684—21no—
31C23×C8×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C61×#340C2214yescomplemented
35C24×D61—21no—
36C24×D61D411yescharacteristic
37C22×D244—22no—
38C22×D244—22no—
39C22×D244—22no—
40C22×D244—22no—
41C22×D244—28no—
42C22×D244—24no—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—44no—
46C22×D244—44no—
47C22×D244D414yes—
48C22×D244C2312yes—
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61—22no—
57C22×C4×D61D412yes—
58C22×C4×D61C2312yes—
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418D414yescomplemented
62C22×#92418C2312yes—
63#14519752—216no—
64#14519752—28no—
65#14519752—28no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758198—28no—
71C2×#11758198—24no—
72C2×#11758198—24no—
73C2×#11758198—44no—
74#14519710—28no—
75#14519699—28no—
76#14519697—28no—
77#14519689—28no—
78#14519684—24no—
79C23×C4×C61—21no—
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C22×C8×C61—22no—
83C22×C8×C61—22no—
84C22×C8×C61—22no—
85C22×C8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61D414yes—
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2312yes—
94C24×C61C2×D411yescharacteristic
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—24no—
102C23×D61—22no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61D814yes—
110C23×D61C2×D412yes—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C22×Dic61C2×D412yes—
135C2×D244—28no—
136C2×D244—28no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—216no—
142C2×D244—24no—
143C2×D244—22no—
144C2×D244—48no—
145C2×D244—432no—
146C2×D244—48no—
147C2×D244—44no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—44no—
151C2×D244C2×D414yes—
152C2×C4×D61—24no—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—44no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61C2×D414yes—
175#11758198—216no—
176#11758198—48no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C2×#92418—28no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418C2×D414yescomplemented
184C23×C61#12012yes—
185C23×C61#12112yes—
186C23×C61#12312yes—
187C23×C61C2×D812yes—
188C23×C61#13512yes—
189C23×C61AGL(1,ℤ/8)12yes—
190C23×C61#13712yes—
191C23×C61C22×D411yescharacteristic
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—28no—
199C22×D61—28no—
200C22×D61—28no—
201C22×D61—24no—
202C22×D61—44no—
203C22×D61—44no—
204C22×D61—44no—
205C22×D61—44no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—48no—
213C22×D61—48no—
214C22×D61—48no—
215C22×D61—84no—
216C22×D61—84no—
217C22×D61—84no—
218C22×D61—84no—
219C22×D61C2×D814yes—
220C2×C4×C61—24no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—48no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—84no—
228C2×C4×C61#12012yes—
229C2×C4×C61#12114yes—
230C2×C4×C61#12114yes—
231C2×C4×C61#12112yes—
232C2×C4×C61#12314yes—
233C2×C4×C61#12312yes—
234C2×C4×C61#12414yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—44no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—84no—
249D244—24no—
250D244—24no—
251D244—432no—
252D244—42no—
253D244—42no—
254D244—48no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—48no—
260C8×C61—42no—
261#92418—24no—
262C2×#2033—611nomaximal, Sylow
263C22×C61#39414yes—
264C22×C61#44714yes—
265C22×C61#45214yes—
266C22×C61#46614yes—
267C22×C61#46814yes—
268C22×C61#48014yes—
269C22×C61#48214yes—
270C22×C61C2×#12011yescharacteristic
271C22×C61C2×#12111yescharacteristic
272C22×C61C2×#12311yescharacteristic
273C22×C61C22×D811yescharacteristic
274C22×C61C2×#13511yescharacteristic
275C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C2×D61—24no—
278C2×D61—24no—
279C2×D61—24no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—416no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—44no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61—82no—
291C2×D61—82no—
292C2×D61—816no—
293C4×C61—44no—
294C4×C61—82no—
295C4×C61#39414yes—
296C4×C61C2×#12111yescharacteristic
297C4×C61C2×#12111yescharacteristic
298C4×C61C2×#12311yescharacteristic
299C4×C61C2×#12411yescharacteristic
300Dic61—42no—
301Dic61—82no—
302C22×#121—611no—
303C22×#107—611no—
304C22×#102—611no—
305C22×#102—611no—
306C22×C22⋊C8—611no—
307C2×#380—611no—
308#2033—618no—
309C2×#340D6111yescharacteristic, complemented
310C2×C61#203318yescomplemented
311C2×C61C2×#39411yescharacteristic
312C2×C61C2×#44711yescharacteristic
313C2×C61C2×#45211yescharacteristic
314C2×C61C2×#46611yescharacteristic
315C2×C61C2×#46811yescharacteristic
316C2×C61C2×#48011yescharacteristic
317C2×C61C2×#48211yescharacteristic
318D61—44no—
319D61—84no—
320C23×D4—611no—
321C23×D4—1221no—
322C24×C4—611no—
323C23×C8—611no—
324C23×C8C2×D6111yescharacteristic
325C2×#121—614no—
326C2×#121—1222no—
327C2×#121—1224no—
328C2×#121—1224no—
329C22×C4⋊C4—611no—
330C22×C4⋊C4C2×D6111yescharacteristic
331C22×C4⋊C4C2×D6111yescharacteristic
332C22×C22⋊C4—1221no—
333C22×C22⋊C4—1221no—
334C2×#107—614no—
335C2×#107—1224no—
336C2×#102—614no—
337C2×#102—614no—
338C2×#102—1224no—
339C2×#102—1224no—
340C2×C22⋊C8—614no—
341C2×C22⋊C8—1224no—
342#380—614no—
343#380—1222no—
344C2×#95—1221no—
345#340C2×D6114yescomplemented
346C61C2×#203311yescharacteristic, complemented, Sylow
347C25—611no—
348C25—1221no—
349C22×D4—614no—
350C22×D4—612no—
351C22×D4—1224no—
352C22×D4—1228no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4—1222no—
356C22×D4—1222no—
357C22×D4—1221no—
358C22×D4—1222no—
359C22×D4—2444no—
360C22×D4—2444no—
361C23×C4—21no—
362C23×C4—21no—
363C23×C4—611no—
364C23×C4—612no—
365C23×C4—612no—
366C23×C4—1222no—
367C23×C4—1222no—
368C23×C4—1222no—
369C23×C4—1222no—
370C23×C4—1222no—
371C23×C4—1222no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—614no—
378C22×C8—612no—
379C22×C8—1222no—
380C22×C8—1222no—
381C22×C8C22×D6112yes—
382#121—1228no—
383#121—12216no—
384#121—1228no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4D24414yes—
395C2×C4⋊C4C22×D6112yes—
396C2×C4⋊C4C22×D6112yes—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1228no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4—2444no—
401#107—1228no—
402#102—1228no—
403#102—1228no—
404C22⋊C8—1228no—
405#95—1224no—
406C24—612no—
407C24—614no—
408C24—1222no—
409C24—1222no—
410C24—1222no—
411C24—1222no—
412C24—1222no—
413C24—1222no—
414C24—1222no—
415C24—1224no—
416C24—2442no—
417C24—2442no—
418C24—2442no—
419C24—2442no—
420C24—2442no—
421C24—2442no—
422C24D4×D6111yescharacteristic
423C2×D4—614no—
424C2×D4—12216no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1224no—
431C2×D4—1228no—
432C2×D4—1228no—
433C2×D4—2448no—
434C2×D4—2448no—
435C2×D4—24432no—
436C2×D4—2442no—
437C2×D4—2444no—
438C2×D4—2442no—
439C2×D4—2444no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—612no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2444no—
466C22×C4—2444no—
467C22×C4—2444no—
468C22×C4—2444no—
469C22×C4C2×D24412yes—
470C22×C4#1175822212yes—
471C22×C4#1175822212yes—
472C22×C4D4×D6112yes—
473C22×C4D4×D6112yes—
474C22×C4D4×D6111yescharacteristic
475C22×C4#1175822612yes—
476C22×C4C23×D6111yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8—614no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1228no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—2442no—
493C4⋊C4C2×D24414yes—
494C22⋊C4—12216no—
495C22⋊C4—2448no—
496C23—614no—
497C23—1222no—
498C23—1228no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—1228no—
505C23—1228no—
506C23—1224no—
507C23—4884no—
508C23—4884no—
509C23—4884no—
510C23—4884no—
511C23—2448no—
512C23—2442no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2448no—
519C23—2448no—
520C23—2444no—
521C23—2444no—
522C23—2444no—
523C23—2444no—
524C23#1451974712yes—
525C23#1451975012yes—
526C23#1451976212yes—
527C23D8×D6112yes—
528C23#1451979812yes—
529C23#1451980412yes—
530C23#1451980912yes—
531C23C2×D4×D6111yescharacteristic
532D4—1224no—
533D4—1224no—
534D4—24432no—
535D4—2448no—
536D4—2442no—
537D4—2442no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—4884no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2444no—
554C2×C4—2448no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4#1451974712yes—
560C2×C4#1451975012yes—
561C2×C4#1451975014yes—
562C2×C4#1451975214yes—
563C2×C4#1451976212yes—
564C2×C4#1451976214yes—
565C2×C4#1451976514yes—
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C22×D24411yescharacteristic
569C2×C4C2×#1175822211yescharacteristic
570C2×C4C2×#1175822211yescharacteristic
571C2×C4C2×#1175822611yescharacteristic
572C8—42no—
573C8—1224no—
574C22—1224no—
575C22—1224no—
576C22—1224no—
577C22—4882no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22—48816no—
582C22—2444no—
583C22—2444no—
584C22—2444no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22—24416no—
589C22—2442no—
590C22#1683968014yes—
591C22#1683989714yes—
592C22#1683992714yes—
593C22#1683999714yes—
594C22#1684001114yes—
595C22#1684007714yes—
596C22#1684008814yes—
597C22C2×#1451974711yescharacteristic
598C22C2×#1451975011yescharacteristic
599C22C2×#1451976211yescharacteristic
600C22C2×D8×D6111yescharacteristic
601C22C2×#1451979811yescharacteristic
602C22C2×#1451980411yescharacteristic
603C22C2×#1451980911yescharacteristic
604C4—44no—
605C4—82no—
606C4—4882no—
607C4—2442no—
608C4#1683968014yes—
609C4C2×#1451975011yescharacteristic
610C4C2×#1451975211yescharacteristic
611C4C2×#1451976211yescharacteristic
612C4C2×#1451976511yescharacteristic
613C2—4884no—
614C2—2444no—
615C2#2144700418yesdirect product factor
616C2C2×#1683968011yescharacteristic
617C2C2×#1683989711yescharacteristic
618C2C2×#1683992711yescharacteristic
619C2C2×#1683999711yescharacteristic
620C2C2×#1684001111yescharacteristic
621C2C2×#1684007711yescharacteristic
622C2C2×#1684008811yescharacteristic
subgroupG/H
centerC24D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC24×C61C2×D4
prime radicalC22×#14519752C2
hypercenterC2×#340D61
hypocenterC61C2×#2033
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×#14519710C2semidirect
3. C22×#14519699C2semidirect
4. C22×#14519697C2semidirect
5. C22×#14519689C2semidirect
6. C2×#16839619C2semidirect
7. #21447004C2direct (N × H)
8. C2×C61×#340C2semidirect
21. C2×#14519710C22semidirect
23. C2×#14519699C22semidirect
25. C2×#14519697C22semidirect
27. C2×#14519689C22semidirect
29. #16839619C22semidirect
34. C61×#340C22semidirect
61. C22×#92418D4semidirect
183. C2×#92418C2×D4semidirect
309. C2×#340D61semidirect
310. C2×C61#2033semidirect
345. #340C2×D61semidirect
346. C61C2×#2033semidirect
615. C2#21447004direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197522
C22×#145197102
C22×#145196992
C22×#145196972
C22×#145196892
C2×#168396192
#214470042
C2×C61×#3402
C2×#203361