Finite Group Databasebeta

← C22⋊C4×C22⋊Dic47 #26137680

Subgroups — C22⋊C4×C22⋊Dic47 #26137680

Order 12032.

4254 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic47C211yesmaximal, characteristic
2C2×#15688979C211yesmaximal, characteristic
3C2×C22⋊C4×Dic47C212yesmaximal, complemented
4C2×C4×C22⋊Dic47C212yesmaximal, complemented
5#20241625C214yesmaximal, complemented
6#20241611C212yesmaximal, complemented
7#20241610C212yesmaximal, complemented
8C23×C22⋊C4×C47C211yesmaximal, characteristic
9C24×Dic47C2212yes—
10C22×C22⋊Dic47—28no—
11C22×C22⋊Dic47C418yescomplemented
12C22×C22⋊Dic47C2211yescharacteristic
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C22⋊Dic47C2212yes—
16C22×C22⋊Dic47C2212yes—
17C22×C22⋊Dic47C2211yescharacteristic
18C22×C4×Dic47C2214yescomplemented
19#15688979C418yescomplemented
20C22⋊C4×Dic47—28no—
21C4×C22⋊Dic47—28no—
22C2×C47⋊#95C2214yescomplemented
23C2×C47⋊#95C2214yescomplemented
24C2×C47⋊#95C2214yescomplemented
25C26×C47C2211yescharacteristic
26C24×C4×C47C2212yes—
27C22×C22⋊C4×C47—28no—
28C22×C22⋊C4×C47C418yescomplemented
29C22×C22⋊C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2212yes—
31C22×C22⋊C4×C47C2211yescharacteristic
32C23×Dic47—24no—
33C23×Dic47—24no—
34C23×Dic47—28no—
35C23×Dic47—44no—
36C23×Dic47C2×C418yescomplemented
37C23×Dic47C2312yes—
38C23×Dic47C2314yes—
39C23×Dic47C2312yes—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47—24no—
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—24no—
45C2×C22⋊Dic47—28no—
46C2×C22⋊Dic47—232no—
47C2×C22⋊Dic47—44no—
48C2×C22⋊Dic47C2×C414yes—
49C2×C22⋊Dic47C2×C414yes—
50C2×C22⋊Dic47C2×C418yescomplemented
51C2×C22⋊Dic47C2×C418yescomplemented
52C2×C22⋊Dic47C2×C418yescomplemented
53C2×C22⋊Dic47C2×C418yescomplemented
54C2×C22⋊Dic47D418yes—
55C2×C4×Dic47—28no—
56C2×C4×Dic47—28no—
57C2×C4×Dic47—44no—
58C47⋊#95—28no—
59C47⋊#95—28no—
60C47⋊#95—44no—
61C25×C47—24no—
62C25×C47—24no—
63C25×C47—24no—
64C25×C47—24no—
65C25×C47—44no—
66C25×C47C2×C414yes—
67C25×C47C2×C414yes—
68C25×C47C2×C414yes—
69C25×C47C2311yescharacteristic
70C25×C47C2311yescharacteristic
71C25×C47C2311yescharacteristic
72C23×C4×C47—24no—
73C23×C4×C47—24no—
74C23×C4×C47—28no—
75C23×C4×C47—44no—
76C23×C4×C47C2×C418yescomplemented
77C23×C4×C47C2312yes—
78C23×C4×C47C2312yes—
79C2×C22⋊C4×C47—24no—
80C2×C22⋊C4×C47—28no—
81C2×C22⋊C4×C47—232no—
82C2×C22⋊C4×C47C2×C414yes—
83C2×C22⋊C4×C47C2×C414yes—
84C2×C22⋊C4×C47C2×C418yescomplemented
85C2×C22⋊C4×C47D418yes—
86C24×C47—22no—
87C24×C47—22no—
88C24×C47—22no—
89C24×C47—22no—
90C24×C47—24no—
91C24×C47—24no—
92C24×C47—28no—
93C24×C47—28no—
94C24×C47—28no—
95C24×C47—28no—
96C24×C47—216no—
97C24×C47—216no—
98C24×C47—216no—
99C24×C47—216no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—24no—
105C24×C47—48no—
106C24×C47—48no—
107C24×C47—48no—
108C24×C47—416no—
109C24×C47—416no—
110C24×C47—416no—
111C24×C47—416no—
112C24×C47C42116yescomplemented
113C24×C47C22×C412yes—
114C24×C47C22×C412yes—
115C24×C47C22×C412yes—
116C24×C47C22×C412yes—
117C24×C47C22×C412yes—
118C24×C47C22×C412yes—
119C24×C47C22×C412yes—
120C24×C47C22×C412yes—
121C24×C47C22×C412yes—
122C24×C47C2×D414yes—
123C24×C47C2×D414yes—
124C24×C47#4114yes—
125C24×C47#4114yes—
126C24×C47C2411yescharacteristic
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—216no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—24no—
135C22×C4×C47—216no—
136C22×C4×C47—48no—
137C22×C4×C47—416no—
138C22×C4×C47—416no—
139C22×C4×C47C22×C414yes—
140C22×C4×C47C22×C414yes—
141C22×C4×C47C2×D418yes—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—216no—
146C22×Dic47—22no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—22no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—216no—
157C22×Dic47—48no—
158C22×Dic47—416no—
159C22×Dic47—416no—
160C22×Dic47C22×C414yes—
161C22×Dic47C22×C418yes—
162C22×Dic47C22×C414yes—
163C22×Dic47C2×D418yes—
164C22⋊C4×C47—28no—
165C22⋊C4×C47—216no—
166C22⋊C4×C47C22⋊C4116yescomplemented
167C22⋊Dic47—28no—
168C22⋊Dic47—216no—
169C22⋊Dic47—216no—
170C22⋊Dic47—216no—
171C22⋊Dic47—48no—
172C22⋊Dic47C22⋊C4116yesdirect product factor
173C4×Dic47—416no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—24no—
179C23×C47—24no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—24no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—28no—
189C23×C47—28no—
190C23×C47—28no—
191C23×C47—28no—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—28no—
195C23×C47—28no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—24no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—232no—
208C23×C47—232no—
209C23×C47—416no—
210C23×C47—432no—
211C23×C47—432no—
212C23×C47—432no—
213C23×C47—432no—
214C23×C47—464no—
215C23×C47—44no—
216C23×C47—44no—
217C23×C47—44no—
218C23×C47—44no—
219C23×C47C2×C4214yes—
220C23×C47C2×C4214yes—
221C23×C47C2×C4214yes—
222C23×C47C2×C22⋊C414yes—
223C23×C47C2×C22⋊C414yes—
224C23×C47#11714yes—
225C23×C47#11714yes—
226C23×C47C4×D418yes—
227C23×C47C4×D418yes—
228C23×C47C4×D418yes—
229C23×C47C4×D418yes—
230C23×C47C23×C411yescharacteristic
231C23×C47C23×C411yescharacteristic
232C23×C47C23×C411yescharacteristic
233C23×C47C22×D412yes—
234C23×C47C22×D412yes—
235C23×C47C2×#4112yes—
236C23×C47C2×#4112yes—
237C23×C47H(5,𝔽2)14yes—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—28no—
241C2×C4×C47—28no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—432no—
246C2×C4×C47—44no—
247C2×C4×C47—44no—
248C2×C4×C47C2×C22⋊C418yescomplemented
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—28no—
252C2×Dic47—28no—
253C2×Dic47—24no—
254C2×Dic47—28no—
255C2×Dic47—28no—
256C2×Dic47—24no—
257C2×Dic47—28no—
258C2×Dic47—24no—
259C2×Dic47—432no—
260C2×Dic47—44no—
261C2×Dic47—44no—
262C2×Dic47—44no—
263C2×Dic47—44no—
264C2×Dic47—44no—
265C2×Dic47—44no—
266C2×Dic47C2×C22⋊C418yescomplemented
267C22⋊C42—471nomaximal, Sylow
268C22×C47—28no—
269C22×C47—28no—
270C22×C47—28no—
271C22×C47—28no—
272C22×C47—28no—
273C22×C47—28no—
274C22×C47—28no—
275C22×C47—28no—
276C22×C47—216no—
277C22×C47—216no—
278C22×C47—24no—
279C22×C47—24no—
280C22×C47—24no—
281C22×C47—24no—
282C22×C47—24no—
283C22×C47—24no—
284C22×C47—24no—
285C22×C47—464no—
286C22×C47—44no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47—44no—
290C22×C47—48no—
291C22×C47C4×C22⋊C418yescomplemented
292C22×C47C4×C22⋊C418yescomplemented
293C22×C47C22×C4211yescharacteristic
294C22×C47C22×C22⋊C411yescharacteristic
295C22×C47C22×C22⋊C411yescharacteristic
296C22×C47C2×#11711yescharacteristic
297C22×C47C2×#11711yescharacteristic
298C22×C47C2×C4×D412yes—
299C22×C47C2×C4×D412yes—
300C22×C47C2×C4×D412yes—
301C22×C47C2×C4×D412yes—
302C22×C47#51812yes—
303C22×C47#51812yes—
304C22×C47#51812yes—
305C22×C47D4214yes—
306C22×C47#54614yes—
307C22×C47#54614yes—
308C22×C47#55114yes—
309C4×C47—28no—
310C4×C47—44no—
311Dic47—28no—
312Dic47—44no—
313Dic47—48no—
314C23×C22⋊C4—471no—
315C23×C22⋊C4D4711yescharacteristic
316C2×#379—471no—
317C2×C4×C22⋊C4—472no—
318C2×C4×C22⋊C4—472no—
319#1437—474no—
320C22⋊#95—472no—
321C22⋊#95—472no—
322C2×C47—28no—
323C2×C47—28no—
324C2×C47—44no—
325C2×C47C2×C4×C22⋊C411yescharacteristic
326C2×C47C2×C4×C22⋊C411yescharacteristic
327C2×C47#227311yescharacteristic
328C2×C47D4×C22⋊C412yes—
329C2×C47D4×C22⋊C412yes—
330C2×C47#234212yes—
331C2×C47#234212yes—
332C2×C47#235814yes—
333C26C2×D4711yescharacteristic
334C24×C4—472no—
335C24×C4C2×D4712yes—
336C22×C22⋊C4—28no—
337C22×C22⋊C4—471no—
338C22×C22⋊C4—478no—
339C22×C22⋊C4—471no—
340C22×C22⋊C4—472no—
341C22×C22⋊C4—472no—
342C22×C22⋊C4—472no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—948no—
345C22×C22⋊C4Dic4718yescomplemented
346C22×C22⋊C4C2×D4711yescharacteristic
347C22×C22⋊C4C2×D4712yes—
348C22×C22⋊C4C2×D4711yescharacteristic
349C22×C42—474no—
350#379—478no—
351C4×C22⋊C4—948no—
352C4×C22⋊C4—948no—
353C2×#95—474no—
354C2×#95—474no—
355C2×#95—474no—
356C47C22⋊C4211yescharacteristic, complemented, Sylow
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25C4×D4714yes—
363C25C4×D4714yes—
364C25C2×Dic4714yes—
365C25C22×D4711yescharacteristic
366C25C22×D4711yescharacteristic
367C25C22×D4711yescharacteristic
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—28no—
371C23×C4—44no—
372C23×C4—472no—
373C23×C4—474no—
374C23×C4—472no—
375C23×C4—478no—
376C23×C4—948no—
377C23×C4—944no—
378C23×C4—944no—
379C23×C4—1884no—
380C23×C4C2×Dic4718yescomplemented
381C23×C4C22×D4712yes—
382C23×C4C22×D4712yes—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—28no—
385C2×C22⋊C4—232no—
386C2×C22⋊C4—474no—
387C2×C22⋊C4—478no—
388C2×C22⋊C4—474no—
389C2×C22⋊C4—478no—
390C2×C22⋊C4—478no—
391C2×C22⋊C4—478no—
392C2×C22⋊C4—478no—
393C2×C22⋊C4—9432no—
394C2×C22⋊C4—948no—
395C2×C22⋊C4—944no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—944no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—1884no—
401C2×C22⋊C4C2×Dic4714yes—
402C2×C22⋊C4C2×Dic4714yes—
403C2×C22⋊C4C2×Dic4718yescomplemented
404C2×C22⋊C4C22⋊D4718yes—
405C2×C42—948no—
406C2×C42—948no—
407C2×C42—1884no—
408#95—948no—
409#95—948no—
410#95—1884no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—28no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—216no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4×Dic47116yescomplemented
438C24C2×C4×D4712yes—
439C24C2×C4×D4712yes—
440C24C2×C4×D4712yes—
441C24C2×C4×D4712yes—
442C24C2×C4×D4712yes—
443C24C2×C4×D4712yes—
444C24#1062535014yes—
445C24D4×D4714yes—
446C24#1062535214yes—
447C24C22×Dic4712yes—
448C24C22×Dic4712yes—
449C24C22×Dic4712yes—
450C24C2×C22⋊D4714yes—
451C24C23×D4711yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—216no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—474no—
465C22×C4—478no—
466C22×C4—478no—
467C22×C4—474no—
468C22×C4—9416no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—944no—
472C22×C4—9416no—
473C22×C4—944no—
474C22×C4—944no—
475C22×C4—942no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—942no—
480C22×C4—944no—
481C22×C4—944no—
482C22×C4—944no—
483C22×C4—1888no—
484C22×C4—18816no—
485C22×C4—18816no—
486C22×C4C22×Dic4714yes—
487C22×C4C22×Dic4714yes—
488C22×C4C2×C22⋊D4718yes—
489C22⋊C4—28no—
490C22⋊C4—216no—
491C22⋊C4—4716no—
492C22⋊C4—9416no—
493C22⋊C4—948no—
494C22⋊C4—9416no—
495C22⋊C4—9416no—
496C22⋊C4—1888no—
497C22⋊C4C22⋊Dic47116yesdirect product factor
498C42—18816no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—232no—
522C23—232no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—416no—
535C23—464no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C4×Dic4714yes—
545C23C2×C4×Dic4714yes—
546C23C2×C4×Dic4714yes—
547C23C22⋊C4×D4714yes—
548C23C2×C22⋊Dic4714yes—
549C23#1309398014yes—
550C23#1309398314yes—
551C23(C22⋊D47)⋊C418yes—
552C23C4×C22⋊D4718yes—
553C23C4×C22⋊D4718yes—
554C23D4×Dic4718yes—
555C23C22×C4×D4711yescharacteristic
556C23C22×C4×D4711yescharacteristic
557C23C23×Dic4711yescharacteristic
558C23C2×D4×D4712yes—
559C23C22×C22⋊D4712yes—
560C23C2×#1062535212yes—
561C23C2×#1062535012yes—
562C23#1309406914yes—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—478no—
574C2×C4—948no—
575C2×C4—948no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—948no—
580C2×C4—948no—
581C2×C4—944no—
582C2×C4—948no—
583C2×C4—944no—
584C2×C4—18832no—
585C2×C4—1884no—
586C2×C4—1884no—
587C2×C4—1884no—
588C2×C4—1884no—
589C2×C4—1884no—
590C2×C4—1884no—
591C2×C4C2×C22⋊Dic4718yescomplemented
592C22—216no—
593C22—216no—
594C22—28no—
595C22—28no—
596C22—28no—
597C22—28no—
598C22—24no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22C4×C22⋊Dic4718yescomplemented
616C22C22⋊C4×Dic4718yescomplemented
617C22C22×C4×Dic4711yescharacteristic
618C22C2×C22⋊C4×D4711yescharacteristic
619C22C22×C22⋊Dic4711yescharacteristic
620C22C2×#1309398011yescharacteristic
621C22C2×#1309398311yescharacteristic
622C22C2×(C22⋊D47)⋊C412yes—
623C22C2×C4×C22⋊D4712yes—
624C22C2×C4×C22⋊D4712yes—
625C22C2×D4×Dic4712yes—
626C22#1568959612yes—
627C22#1568959612yes—
628C22#1568959812yes—
629C22D4×C22⋊D4714yes—
630C22#1568976314yes—
631C22#1568976714yes—
632C22#1568979914yes—
633C4—28no—
634C4—44no—
635C4—948no—
636C4—1888no—
637C4—1884no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×C4×C22⋊Dic4711yescharacteristic
642C2C2×C22⋊C4×Dic4711yescharacteristic
643C2#2024574111yescharacteristic
644C2C22⋊C4×C22⋊D4712yes—
645C2D4×C22⋊Dic4712yes—
646C2#2024623312yes—
647C2#2024623712yes—
648C2#2024636714yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC23×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC26×C47C22
hypercenterC23×C22⋊C4D47
hypocenterC47C22⋊C42
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C22⋊C4×Dic47C2semidirect
4. C2×C4×C22⋊Dic47C2semidirect
5. #20241625C2semidirect
6. #20241611C2semidirect
7. #20241610C2semidirect
11. C22×C22⋊Dic47C4semidirect
18. C22×C4×Dic47C22semidirect
19. #15688979C4semidirect
22. C2×C47⋊#95C22semidirect
23. C2×C47⋊#95C22semidirect
24. C2×C47⋊#95C22semidirect
28. C22×C22⋊C4×C47C4semidirect
36. C23×Dic47C2×C4semidirect
50. C2×C22⋊Dic47C2×C4semidirect
51. C2×C22⋊Dic47C2×C4semidirect
52. C2×C22⋊Dic47C2×C4semidirect
53. C2×C22⋊Dic47C2×C4semidirect
76. C23×C4×C47C2×C4semidirect
84. C2×C22⋊C4×C47C2×C4semidirect
112. C24×C47C42semidirect
166. C22⋊C4×C47C22⋊C4semidirect
172. C22⋊Dic47C22⋊C4direct (N × H)
248. C2×C4×C47C2×C22⋊C4semidirect
266. C2×Dic47C2×C22⋊C4semidirect
291. C22×C47C4×C22⋊C4semidirect
292. C22×C47C4×C22⋊C4semidirect
345. C22×C22⋊C4Dic47semidirect
356. C47C22⋊C42semidirect
380. C23×C4C2×Dic47semidirect
403. C2×C22⋊C4C2×Dic47semidirect
437. C24C4×Dic47semidirect
497. C22⋊C4C22⋊Dic47direct (N × H)
591. C2×C4C2×C22⋊Dic47semidirect
615. C22C4×C22⋊Dic47semidirect
616. C22C22⋊C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic472
C2×#156889792
C2×C22⋊C4×Dic472
C2×C4×C22⋊Dic472
#202416252
#202416112
#202416102
C23×C22⋊C4×C472
C22⋊C4247