Finite Group Databasebeta

← C2×#20243758 #26217769

Subgroups — C2×#20243758 #26217769

Order 12032.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094057C211yesmaximal, characteristic
2C2×#15689603C211yesmaximal, characteristic
3C22×#13093952C211yesmaximal, characteristic
4C2×#15689131C211yesmaximal, characteristic, complemented
5C2×#15689029C211yesmaximal, characteristic, complemented
6#20243758C218yesmaximal, direct product factor
7C2×#15688891C211yesmaximal, characteristic, complemented
8C2×C47×#339C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C23×Dic94C2211yescharacteristic
11C2×#13094057—24no—
12C2×#13094057—22no—
13C2×#13094057C418yescomplemented
14C2×#13094057C2214yes—
15C22×Dic188—21no—
16C22×#10625325—21no—
17C22×#10625329C2211yescharacteristic
18#15689603—22no—
19#15689603—22no—
20#15689603—22no—
21#15689603C2214yes—
22C2×C4×Dic94—21no—
23C2×C4×D188—21no—
24C2×(C22⋊D47)⋊C4—22no—
25C2×#13093982—21no—
26C22×C4⋊Dic47C2211yescharacteristic
27C2×C4⋊C4×D47—21no—
28C2×#13093952—24no—
29C2×#13093952C2214yes—
30#15689131—22no—
31#15689131C2214yescomplemented
32C2×#13093947—21no—
33C2×#13093947—21no—
34#15689029—22no—
35#15689029C2214yescomplemented
36C2×C47⋊#95—22no—
37#15688891C2214yescomplemented
38C22×M16×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C42×C47C2211yescharacteristic
41C47×#339C2214yescomplemented
42C2×#10625350—21no—
43C2×#10625350—21no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350C2×C414yes—
47C2×#10625350D414yes—
48C2×#10625350D412yes—
49C2×#10625350C2312yes—
50C22×Dic94—21no—
51C22×Dic94—21no—
52C22×Dic94—21no—
53C22×Dic94—22no—
54C22×Dic94—21no—
55C22×Dic94—22no—
56C22×Dic94—21no—
57C22×Dic94—22no—
58C22×Dic94C2×C414yescomplemented
59C22×Dic94D411yescharacteristic
60C22×Dic94C2312yes—
61C22×D188D411yescharacteristic
62C22×C22⋊D47—21no—
63C23×Dic47—21no—
64C23×Dic47D412yes—
65C22×C4×D47—21no—
66C22×C4×D47—22no—
67#13094057—216no—
68#13094057—28no—
69#13094057C2×C4116yescomplemented
70C2×Dic188—28no—
71C2×Dic188—24no—
72C2×#10625325—28no—
73C2×#10625325—24no—
74C2×#10625329—21no—
75C2×#10625329—22no—
76C2×#10625329—22no—
77C2×#10625329C2312yes—
78C22×C47⋊C8—21no—
79C4×Dic94—24no—
80C4×Dic94—24no—
81C4×D188—24no—
82C4×D188—24no—
83(C22⋊D47)⋊C4—28no—
84(C22⋊D47)⋊C4—28no—
85#13093982—24no—
86#13093982—24no—
87C2×C4⋊Dic47—21no—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—21no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47D412yes—
92C2×C4⋊Dic47C2312yes—
93C2×Dic47⋊C4—22no—
94C4⋊C4×D47—24no—
95C4⋊C4×D47—24no—
96C2×(C2×D47)⋊C4—22no—
97C2×C4×Dic47—22no—
98#13093952—28no—
99#13093947—24no—
100#13093947—24no—
101C47⋊#95—28no—
102C23×C4×C47—21no—
103C23×C4×C47C2311yescharacteristic
104C2×M16×C47—22no—
105C2×M16×C47—21no—
106C2×M16×C47C2×C414yescomplemented
107C2×M16×C47C2312yes—
108C22×C8×C47—21no—
109#117×C47—21no—
110#117×C47—21no—
111#117×C47D412yes—
112#117×C47C2312yes—
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C22⋊C4×C47—21no—
116C2×C42×C47—21no—
117C2×C42×C47—22no—
118C2×C42×C47—22no—
119C2×C42×C47—21no—
120C2×C42×C47C2312yes—
121C24×C47C2×D411yescharacteristic
122C23×D47—21no—
123C22×C4×C47—24no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47#4112yes—
137C22×C4×C47#4112yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—24no—
140C2×C22⋊D47—22no—
141C2×C22⋊D47—24no—
142C2×C22⋊D47—22no—
143C22×Dic47—21no—
144C22×Dic47—24no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—41no—
149C22×Dic47—41no—
150C22×Dic47—42no—
151C22×Dic47—44no—
152C22×Dic47C2×D414yes—
153#10625350—24no—
154#10625350—24no—
155#10625350—24no—
156#10625350—24no—
157#10625350—24no—
158#10625350—22no—
159#10625350—22no—
160#10625350C22⋊C418yes—
161#10625350C22×C414yes—
162#10625350C2×D414yes—
163C2×D188—21no—
164C2×D188—22no—
165C2×D188C22⋊C414yes—
166C2×D188C2×D412yes—
167C2×C4×D47—24no—
168C2×C4×D47—22no—
169C2×C4×D47—24no—
170C2×C4×D47—22no—
171C2×C4×D47—28no—
172C2×C4×D47—84no—
173C2×Dic94—21no—
174C2×Dic94—22no—
175C2×Dic94—24no—
176C2×Dic94—24no—
177C2×Dic94—24no—
178C2×Dic94—24no—
179C2×Dic94—22no—
180C2×Dic94—22no—
181C2×Dic94—22no—
182C2×Dic94—22no—
183C2×Dic94—24no—
184C2×Dic94—22no—
185C2×Dic94—41no—
186C2×Dic94—41no—
187C2×Dic94—42no—
188C2×Dic94—42no—
189C2×Dic94—42no—
190C2×Dic94—42no—
191C2×Dic94—42no—
192C2×Dic94C22⋊C414yes—
193C2×Dic94C22×C414yescomplemented
194C2×Dic94C2×D412yes—
195M16×C47—24no—
196M16×C47—22no—
197M16×C47C22×C414yescomplemented
198C2×C8×C47—24no—
199C2×C8×C47—22no—
200C4⋊C4×C47—21no—
201C4⋊C4×C47—21no—
202C4⋊C4×C47C2×D412yes—
203C4⋊C4×C47C2×D412yes—
204C22⋊C4×C47—22no—
205C22⋊C4×C47—22no—
206C42×C47—22no—
207C42×C47—22no—
208C42×C47—22no—
209C42×C47—22no—
210C42×C47—42no—
211(C2×D47)⋊C4—28no—
212C4⋊Dic47—22no—
213C4⋊Dic47—22no—
214C4⋊Dic47—22no—
215C4⋊Dic47—22no—
216Dic47⋊C4—28no—
217C4×Dic47—28no—
218#10625329—22no—
219#10625329—22no—
220#10625329—42no—
221C2×C47⋊C8—22no—
222C2×C47⋊C8—42no—
223Dic188—216no—
224#10625325—216no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47C4×D412yes—
228C23×C47#12011yescharacteristic
229C23×C47#12011yescharacteristic
230C23×C47#12112yes—
231C23×C47C22×D411yescharacteristic
232C22×D47—22no—
233C22×D47—24no—
234C22×D47—42no—
235C22×D47—42no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—48no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C4×D412yes—
247C2×C4×C47C4×D412yes—
248C2×C4×C47C4×D412yes—
249C2×C4×C47#12011yescharacteristic
250C2×C4×C47#12011yescharacteristic
251C2×C4×C47#12112yes—
252C2×C4×C47#12112yes—
253C2×C4×C47#12112yes—
254C2×C4×C47#12312yes—
255C2×C4×C47#12312yes—
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C22×D411yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C22⋊D47—24no—
263C22⋊D47—24no—
264C22⋊D47—28no—
265C2×Dic47—22no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—44no—
272C2×Dic47—44no—
273C2×Dic47—44no—
274C2×Dic47—44no—
275C2×Dic47—82no—
276D188—22no—
277D188—24no—
278D188C2×C22⋊C414yes—
279C4×D47—24no—
280C4×D47—24no—
281C4×D47—28no—
282C4×D47—88no—
283Dic94—22no—
284Dic94—24no—
285Dic94—24no—
286Dic94—24no—
287Dic94—42no—
288Dic94—42no—
289Dic94—44no—
290Dic94—44no—
291Dic94C2×C22⋊C414yes—
292C8×C47—24no—
293C47⋊C8—42no—
294C2×#1882—471nomaximal, Sylow
295C22×C47—24no—
296C22×C47—24no—
297C22×C47—21no—
298C22×C47—21no—
299C22×C47—21no—
300C22×C47—21no—
301C22×C47#38612yes—
302C22×C47#38612yes—
303C22×C47#45512yes—
304C22×C47#45612yes—
305C22×C47C2×C4×D411yescharacteristic
306C22×C47C2×#12011yescharacteristic
307C22×C47C2×#12111yescharacteristic
308C2×D47—24no—
309C2×D47—48no—
310C2×D47—42no—
311C2×D47—42no—
312C4×C47—44no—
313C4×C47—42no—
314C4×C47—42no—
315C4×C47#38612yes—
316C4×C47#38612yes—
317C4×C47C22×C22⋊C411yescharacteristic
318C4×C47C2×C4×D411yescharacteristic
319C4×C47C2×#12111yescharacteristic
320C4×C47C2×#12311yescharacteristic
321Dic47—24no—
322Dic47—44no—
323Dic47—44no—
324C22×#137—471no—
325C2×#520—471no—
326C22×#104—471no—
327C2×#419—471no—
328C2×#391—471no—
329#1882—478no—
330C2×#343—471no—
331C2×#339D4711yescharacteristic, complemented
332C2×C47—21no—
333C2×C47—21no—
334C2×C47—22no—
335C2×C47#188214yescomplemented
336C2×C47C2×#38611yescharacteristic
337C2×C47C2×#45511yescharacteristic
338C2×C47C2×#45611yescharacteristic
339D47—44no—
340C22×#41—471no—
341C23×Q8—471no—
342C2×#137—478no—
343C2×#137—474no—
344C2×#137—942no—
345C2×#137—944no—
346C22×Q16—941no—
347C22×SD16—941no—
348C22×M16—471no—
349C22×M16C2×D4711yescharacteristic
350#520—474no—
351#520—942no—
352#520—942no—
353#520—942no—
354C2×C4×Q8—941no—
355C2×C4×D4—941no—
356C2×C4×D4—942no—
357C2×#117—941no—
358C2×#117C2×D4711yescharacteristic
359C22×C4⋊C4—471no—
360C22×C4⋊C4—941no—
361C22×C42C2×D4711yescharacteristic
362C2×#104—474no—
363C2×#104—944no—
364#419—474no—
365#419—942no—
366C2×#103—941no—
367C2×#103—941no—
368#391—474no—
369#391—942no—
370C2×#95—942no—
371#343—474no—
372#339C2×D4714yescomplemented
373C47C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—474no—
375C2×#41—472no—
376C2×#41—474no—
377C2×#41—472no—
378C2×#41—941no—
379C2×#41—942no—
380C2×#41—941no—
381C2×#41—942no—
382C22×Q8—474no—
383C22×Q8—471no—
384C22×Q8—472no—
385C22×Q8—941no—
386C22×Q8—942no—
387C22×Q8—941no—
388C22×Q8—941no—
389C22×Q8—942no—
390C22×Q8—941no—
391C22×Q8—941no—
392C22×Q8—942no—
393C22×D4—471no—
394C22×D4—941no—
395C23×C4—21no—
396C23×C4—472no—
397C23×C4—942no—
398C23×C4—941no—
399C23×C4—941no—
400C23×C4C22×D4711yescharacteristic
401#137—4716no—
402#137—9416no—
403#137—948no—
404C2×Q16—948no—
405C2×Q16—944no—
406C2×SD16—948no—
407C2×SD16—944no—
408C2×M16—22no—
409C2×M16—21no—
410C2×M16—472no—
411C2×M16—941no—
412C2×M16—942no—
413C2×M16—942no—
414C2×M16C4×D4714yescomplemented
415C2×M16C22×D4712yes—
416C22×C8—21no—
417C22×C8—941no—
418C4×Q8—944no—
419C4×Q8—944no—
420C4×D4—944no—
421C4×D4—944no—
422C4×D4—948no—
423C4×D4—948no—
424#117—21no—
425#117—21no—
426#117—944no—
427#117—944no—
428#117D18812yes—
429#117C22×D4712yes—
430C2×C4⋊C4—472no—
431C2×C4⋊C4—472no—
432C2×C4⋊C4—944no—
433C2×C4⋊C4—942no—
434C2×C4⋊C4—941no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—944no—
439C2×C4⋊C4D18811yescharacteristic
440C2×C4⋊C4D18811yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—942no—
443C2×C42—21no—
444C2×C42—22no—
445C2×C42—22no—
446C2×C42—21no—
447C2×C42—942no—
448C2×C42C22×D4712yes—
449#104—948no—
450#103—944no—
451#103—944no—
452#95—948no—
453C24—941no—
454C24D4×D4711yescharacteristic
455#41—478no—
456#41—474no—
457#41—474no—
458#41—944no—
459#41—944no—
460#41—944no—
461#41—944no—
462#41—944no—
463#41—942no—
464#41—942no—
465C2×Q8—474no—
466C2×Q8—474no—
467C2×Q8—472no—
468C2×Q8—944no—
469C2×Q8—941no—
470C2×Q8—942no—
471C2×Q8—942no—
472C2×Q8—944no—
473C2×Q8—944no—
474C2×Q8—944no—
475C2×Q8—944no—
476C2×Q8—942no—
477C2×Q8—942no—
478C2×Q8—942no—
479C2×Q8—942no—
480C2×Q8—1881no—
481C2×Q8—1881no—
482C2×Q8—1882no—
483C2×Q8—1882no—
484C2×Q8—1882no—
485C2×Q8—1882no—
486C2×Q8—1882no—
487C2×D4—474no—
488C2×D4—472no—
489C2×D4—944no—
490C2×D4—942no—
491C2×D4—944no—
492C2×D4—942no—
493C2×D4—941no—
494C2×D4—942no—
495C22×C4—24no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—474no—
502C22×C4—948no—
503C22×C4—941no—
504C22×C4—944no—
505C22×C4—942no—
506C22×C4—944no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—944no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—3764no—
513C22×C4—1884no—
514C22×C4—1881no—
515C22×C4—1881no—
516C22×C4—1882no—
517C22×C4C2×C4×D4712yes—
518C22×C4C2×D18811yescharacteristic
519C22×C4C2×D18811yescharacteristic
520C22×C4#1062535012yes—
521C22×C4D4×D4711yescharacteristic
522C22×C4D4×D4711yescharacteristic
523C22×C4D4×D4712yes—
524C22×C4D4×D4711yescharacteristic
525C22×C4#1062535412yes—
526C22×C4C23×D4711yescharacteristic
527Q16—9416no—
528SD16—9416no—
529M16—24no—
530M16—22no—
531M16—942no—
532M16—942no—
533M16—1882no—
534M16C2×C4×D4714yescomplemented
535C2×C8—24no—
536C2×C8—22no—
537C2×C8—942no—
538C2×C8—1882no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—948no—
542C4⋊C4—942no—
543C4⋊C4—942no—
544C4⋊C4—942no—
545C4⋊C4—942no—
546C4⋊C4C2×D18812yes—
547C4⋊C4C2×D18812yes—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—948no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—948no—
557C23—22no—
558C23—22no—
559C23—944no—
560C23—942no—
561C23—1882no—
562C23—1882no—
563C23#1309398612yes—
564C23#1309399411yescharacteristic
565C23#1309399411yescharacteristic
566C23#1309399712yes—
567C23C2×D4×D4711yescharacteristic
568Q8—474no—
569Q8—944no—
570Q8—942no—
571Q8—944no—
572Q8—944no—
573Q8—1884no—
574Q8—1884no—
575Q8—1882no—
576Q8—1882no—
577D4—474no—
578D4—948no—
579D4—944no—
580D4—944no—
581D4—944no—
582D4—942no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—944no—
592C2×C4—942no—
593C2×C4—948no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—3768no—
599C2×C4—3762no—
600C2×C4—1884no—
601C2×C4—1884no—
602C2×C4—1884no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1884no—
606C2×C4C22⋊C4×D4712yes—
607C2×C4C22⋊C4×D4712yes—
608C2×C4#1309398612yes—
609C2×C4C4×D18812yes—
610C2×C4C4×D18812yes—
611C2×C4#1309399411yescharacteristic
612C2×C4#1309399411yescharacteristic
613C2×C4#1309399712yes—
614C2×C4#1309399912yes—
615C2×C4#1309399912yes—
616C2×C4#1309400912yes—
617C2×C4#1309400912yes—
618C2×C4C22×C4×D4711yescharacteristic
619C2×C4C2×D4×D4711yescharacteristic
620C2×C4C2×D4×D4711yescharacteristic
621C2×C4C22×D18811yescharacteristic
622C2×C4C2×#1062535011yescharacteristic
623C2×C4C2×#1062535411yescharacteristic
624C8—24no—
625C8—1882no—
626C22—24no—
627C22—24no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—944no—
633C22—1888no—
634C22—1882no—
635C22—1882no—
636C22#1568900112yes—
637C22#1568900112yes—
638C22#1568930212yes—
639C22#1568930712yes—
640C22C2×#1309398611yescharacteristic
641C22C2×#1309399411yescharacteristic
642C22C2×#1309399711yescharacteristic
643C4—44no—
644C4—42no—
645C4—42no—
646C4—944no—
647C4—1884no—
648C4—1884no—
649C4#1568900112yes—
650C4#1568900112yes—
651C4C2×C22⋊C4×D4711yescharacteristic
652C4C2×C4×D18811yescharacteristic
653C4C2×#1309399911yescharacteristic
654C4C2×#1309400911yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—1884no—
659C2#2024375814yesdirect product factor
660C2C2×#1568900111yescharacteristic
661C2C2×#1568930211yescharacteristic
662C2C2×#1568930711yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#339C2
socleC23×C47#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#339D47
hypocenterC47C2×#1882
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. C2×#15689131C2semidirect
5. C2×#15689029C2semidirect
6. #20243758C2direct (N × H)
7. C2×#15688891C2semidirect
8. C2×C47×#339C2semidirect
13. C2×#13094057C4semidirect
31. #15689131C22semidirect
35. #15689029C22semidirect
37. #15688891C22semidirect
41. C47×#339C22semidirect
58. C22×Dic94C2×C4semidirect
69. #13094057C2×C4semidirect
106. C2×M16×C47C2×C4semidirect
193. C2×Dic94C22×C4semidirect
197. M16×C47C22×C4semidirect
331. C2×#339D47semidirect
335. C2×C47#1882semidirect
372. #339C2×D47semidirect
373. C47C2×#1882semidirect
414. C2×M16C4×D47semidirect
534. M16C2×C4×D47semidirect
659. C2#20243758direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940572
C2×#156896032
C22×#130939522
C2×#156891312
C2×#156890292
#202437582
C2×#156888912
C2×C47×#3392
C2×#188247