Finite Group Databasebeta

← C2×#20246293 #26375569

Subgroups — C2×#20246293 #26375569

Order 12032.

4198 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic47C211yesmaximal, characteristic
2C22×D4×Dic47C211yesmaximal, characteristic
3#20246293C2116yesmaximal, direct product factor
4C2×#15689021C211yesmaximal, characteristic, complemented
5C2×#15689019C211yesmaximal, characteristic, complemented
6C2×#15688996C212yesmaximal, complemented
7C2×#15688995C212yesmaximal, complemented
8C2×#15688988C211yesmaximal, characteristic, complemented
9C2×#15688981C212yesmaximal, complemented
10C2×C4×Dic47⋊C4C211yesmaximal, characteristic, complemented
11C22×C47⋊#95C212yesmaximal, complemented
12C22×C4×D4×C47C211yesmaximal, characteristic
13C24×Dic47C2212yes—
14C24×Dic47C2212yes—
15C2×D4×Dic47—28no—
16C2×D4×Dic47C4116yescomplemented
17C2×D4×Dic47C2218yes—
18C2×D4×Dic47C2218yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2211yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×C22⋊Dic47C2212yes—
26C22×C22⋊Dic47C2212yes—
27C22×C4×Dic47C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29#15689021C2218yescomplemented
30#15689019C2218yescomplemented
31#15688996C22116yescomplemented
32#15688995C22116yescomplemented
33#15688988C2218yescomplemented
34#15688981C22116yescomplemented
35C4×Dic47⋊C4C2218yescomplemented
36C2×C47⋊#95C2212yes—
37C2×C47⋊#95C2212yes—
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2211yescharacteristic
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C22116yescomplemented
43C23×D4×C47C2211yescharacteristic
44C24×C4×C47C2212yes—
45C2×C4×D4×C47—28no—
46C2×C4×D4×C47C2218yes—
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C22⋊C4×C47C2212yes—
49C22×C42×C47C2211yescharacteristic
50C23×Dic47—28no—
51C23×Dic47—28no—
52C23×Dic47—44no—
53C23×Dic47—44no—
54C23×Dic47—44no—
55C23×Dic47C2×C4116yescomplemented
56C23×Dic47C2312yes—
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2318yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2311yescharacteristic
61C23×Dic47C2312yes—
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2318yes—
64D4×Dic47—232no—
65D4×Dic47C2×C4164yescomplemented
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47C2×C418yescomplemented
68C2×C4⋊Dic47C2314yes—
69C2×C4⋊Dic47C2314yes—
70C2×Dic47⋊C4—24no—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4—44no—
73C2×Dic47⋊C4—44no—
74C2×Dic47⋊C4C2314yes—
75C2×Dic47⋊C4C2314yes—
76C2×Dic47⋊C4C2318yes—
77C2×Dic47⋊C4C2318yes—
78C2×C22⋊Dic47—28no—
79C2×C22⋊Dic47C2×C4116yescomplemented
80C2×C22⋊Dic47C2318yes—
81C2×C22⋊Dic47C2318yes—
82C2×C4×Dic47—24no—
83C2×C4×Dic47C2×C418yescomplemented
84C2×C4×Dic47C2314yes—
85C2×C4×Dic47C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2318yes—
88C47⋊#95C2318yes—
89C47⋊#95C2318yes—
90C47⋊#95C2314yes—
91C47⋊#95C2318yes—
92C25×C47C2312yes—
93C22×D4×C47C2×C418yes—
94C22×D4×C47D418yes—
95C22×D4×C47Q818yes—
96C22×D4×C47C2314yes—
97C23×C4×C47—28no—
98C23×C4×C47—28no—
99C23×C4×C47—44no—
100C23×C4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2318yes—
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C4×D4×C47—232no—
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47C2314yes—
108C2×C22⋊C4×C47—28no—
109C2×C22⋊C4×C47C2318yes—
110C2×C42×C47—24no—
111C2×C42×C47C2314yes—
112C24×C47—24no—
113C24×C47—24no—
114C24×C47—24no—
115C24×C47—24no—
116C24×C47C22×C418yes—
117C24×C47C2×D418yes—
118C24×C47C2×Q818yes—
119C24×C47C2414yes—
120C24×C47C2411yescharacteristic
121C2×D4×C47C4⋊C4164yescomplemented
122C2×D4×C47C22×C4116yes—
123C2×D4×C47C2×D4116yes—
124C2×D4×C47C2×Q8116yes—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—24no—
129C22×C4×C47—216no—
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—24no—
133C22×C4×C47—22no—
134C22×C4×C47—216no—
135C22×C4×C47—48no—
136C22×C4×C47—432no—
137C22×C4×C47C22×C414yes—
138C22×C4×C47C2×D414yes—
139C22×C4×C47C2×Q814yes—
140C22×C4×C47#4114yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2414yes—
143C22×C4×C47C2412yes—
144C22×C4×C47C2412yes—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—24no—
155C22×Dic47—216no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—216no—
161C22×Dic47—432no—
162C22×Dic47—48no—
163C22×Dic47—48no—
164C22×Dic47—48no—
165C22×Dic47—432no—
166C22×Dic47C22×C414yes—
167C22×Dic47C22×C414yes—
168C22×Dic47C22×C418yes—
169C22×Dic47C22×C4132yescomplemented
170C22×Dic47#4114yes—
171C22×Dic47C2414yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C22×Dic47C2412yes—
175C22×Dic47C2414yes—
176C22×Dic47C2412yes—
177C4⋊C4×C47—28no—
178C22⋊C4×C47—216no—
179C42×C47—28no—
180C22⋊Dic47—216no—
181C22⋊Dic47C22×C4132yescomplemented
182C4⋊Dic47—28no—
183C4⋊Dic47C22×C4116yescomplemented
184Dic47⋊C4—28no—
185Dic47⋊C4—28no—
186Dic47⋊C4—48no—
187Dic47⋊C4—48no—
188C4×Dic47—28no—
189C4×Dic47C22×C4116yescomplemented
190C23×C47—216no—
191C23×C47—216no—
192C23×C47—216no—
193C23×C47—216no—
194C23×C47—216no—
195C23×C47—216no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47C2×C4⋊C4132yes—
201C23×C47C23×C418yes—
202C23×C47C23×C412yes—
203C23×C47C22×D418yes—
204C23×C47C22×D412yes—
205C23×C47C22×Q818yes—
206C23×C47C22×Q812yes—
207C23×C47C2×#4112yes—
208C23×C47C2×#4112yes—
209C23×C47H(5,𝔽2)12yes—
210C23×C47H(5,𝔽2)12yes—
211C23×C47C2511yescharacteristic
212D4×C47C2×C4⋊C4164yescomplemented
213C2×C4×C47—28no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—432no—
223C2×C4×C47—44no—
224C2×C4×C47—48no—
225C2×C4×C47C2×C4⋊C4116yes—
226C2×C4×C47C23×C414yes—
227C2×C4×C47C22×D414yes—
228C2×C4×C47C22×Q814yes—
229C2×C4×C47C2×#4114yes—
230C2×Dic47—28no—
231C2×Dic47—28no—
232C2×Dic47—216no—
233C2×Dic47—28no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—24no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—22no—
247C2×Dic47—24no—
248C2×Dic47—432no—
249C2×Dic47—432no—
250C2×Dic47—44no—
251C2×Dic47—48no—
252C2×Dic47C23×C414yes—
253C2×Dic47C23×C414yes—
254C2×Dic47C23×C418yes—
255C2×Dic47C2×#4114yes—
256C2×#2349—471nomaximal, Sylow
257C22×C47—264no—
258C22×C47—28no—
259C22×C47—28no—
260C22×C47—28no—
261C22×C47—28no—
262C22×C47—28no—
263C22×C47—28no—
264C22×C47C22×C4⋊C4116yes—
265C22×C47C22×C4⋊C414yes—
266C22×C47C4×#4114yes—
267C22×C47#51814yes—
268C22×C47#54614yes—
269C22×C47#54614yes—
270C22×C47#55414yes—
271C22×C47#55414yes—
272C22×C47C24×C411yescharacteristic
273C22×C47C23×D411yescharacteristic
274C22×C47C23×Q811yescharacteristic
275C22×C47C22×#4111yescharacteristic
276C22×C47C22×#4111yescharacteristic
277C22×C47C2×H(5,𝔽2)11yescharacteristic
278C22×C47C2×H(5,𝔽2)11yescharacteristic
279C4×C47—24no—
280C4×C47—42no—
281C4×C47—44no—
282C4×C47C22×C4⋊C418yes—
283Dic47—24no—
284Dic47—24no—
285Dic47—28no—
286Dic47—24no—
287Dic47—42no—
288Dic47—44no—
289C22×C4×D4—471no—
290C22×C4×D4—471no—
291C22×C4×D4D4711yescharacteristic
292#2349—4716no—
293C2×#389—471no—
294C2×#389—471no—
295C2×#385—472no—
296C2×#385—472no—
297C2×#382—471no—
298C2×#380—472no—
299C2×C4×C4⋊C4—471no—
300C22×#95—472no—
301C2×C47—216no—
302C2×C47#234918yescomplemented
303C2×C47C23×C4⋊C411yescharacteristic
304C2×C47C2×C4×#4111yescharacteristic
305C2×C47C2×#51811yescharacteristic
306C2×C47C2×#54611yescharacteristic
307C2×C47C2×#54611yescharacteristic
308C2×C47C2×#55411yescharacteristic
309C2×C47C2×#55411yescharacteristic
310C23×D4C2×D4711yescharacteristic
311C24×C4—472no—
312C24×C4—472no—
313C24×C4C2×D4712yes—
314C2×C4×D4—28no—
315C2×C4×D4—4716no—
316C2×C4×D4—478no—
317C2×C4×D4—478no—
318C2×C4×D4—948no—
319C2×C4×D4C2×D4718yes—
320C22×C4⋊C4—471no—
321C22×C4⋊C4—471no—
322C22×C4⋊C4—471no—
323C22×C4⋊C4—471no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—472no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4C2×D4712yes—
330C22×C42—471no—
331C22×C42—471no—
332C22×C42C2×D4711yescharacteristic
333#389—478no—
334#389—478no—
335#385—4716no—
336#385—4716no—
337#382—478no—
338#380—4716no—
339C4×C4⋊C4—478no—
340C2×#95—471no—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—472no—
344C2×#95—471no—
345C2×#95—472no—
346C2×#95—4716no—
347C47C2×#234911yescharacteristic, complemented, Sylow
348C25C22×D4712yes—
349C22×D4Dic9418yes—
350C22×D4C4×D4718yes—
351C22×D4C22⋊D4718yes—
352C22×D4C22×D4714yes—
353C23×C4—28no—
354C23×C4—28no—
355C23×C4—44no—
356C23×C4—471no—
357C23×C4—472no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—4716no—
361C23×C4—478no—
362C23×C4—471no—
363C23×C4—472no—
364C23×C4—478no—
365C23×C4—948no—
366C23×C4—948no—
367C23×C4—1884no—
368C23×C4—1884no—
369C23×C4—1884no—
370C23×C4C22×D4711yescharacteristic
371C23×C4C22×D4712yes—
372C23×C4C22×D4718yes—
373C23×C4C22×D4711yescharacteristic
374C23×C4C22×D4711yescharacteristic
375C4×D4—232no—
376C4×D4—4764no—
377C4×D4—9432no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—478no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—1884no—
390C2×C4⋊C4—1884no—
391C2×C4⋊C4C22×D4714yes—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—4716no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—478no—
396C2×C22⋊C4—948no—
397C2×C22⋊C4C22×D4718yes—
398C2×C42—24no—
399C2×C42—478no—
400C2×C42—474no—
401C2×C42—474no—
402C2×C42—944no—
403C2×C42C22×D4714yes—
404#95—474no—
405#95—478no—
406#95—478no—
407#95—478no—
408#95—474no—
409#95—478no—
410C24—24no—
411C24—24no—
412C24—24no—
413C24—24no—
414C24C2×Dic9418yes—
415C24C2×C4×D4718yes—
416C24C2×C22⋊D4718yes—
417C24C23×D4714yes—
418C24C23×D4711yescharacteristic
419C2×D4Dic47⋊C4164yescomplemented
420C2×D4C2×Dic94116yes—
421C2×D4C2×C4×D47116yes—
422C2×D4C2×C22⋊D47116yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—216no—
433C22×C4—48no—
434C22×C4—432no—
435C22×C4—474no—
436C22×C4—474no—
437C22×C4—478no—
438C22×C4—474no—
439C22×C4—472no—
440C22×C4—472no—
441C22×C4—472no—
442C22×C4—474no—
443C22×C4—4732no—
444C22×C4—474no—
445C22×C4—472no—
446C22×C4—9416no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—944no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—944no—
457C22×C4—9416no—
458C22×C4—944no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—944no—
462C22×C4—18832no—
463C22×C4—1888no—
464C22×C4—1888no—
465C22×C4—1888no—
466C22×C4—18832no—
467C22×C4C2×Dic9414yes—
468C22×C4C2×C4×D4714yes—
469C22×C4#1062535214yes—
470C22×C4C2×C22⋊D4714yes—
471C22×C4C23×D4714yes—
472C22×C4C23×D4712yes—
473C22×C4C23×D4712yes—
474C22×C4C23×D4712yes—
475C4⋊C4—28no—
476C4⋊C4—4716no—
477C4⋊C4—948no—
478C4⋊C4—948no—
479C4⋊C4—948no—
480C4⋊C4—1888no—
481C4⋊C4—1888no—
482C22⋊C4—216no—
483C22⋊C4—4732no—
484C22⋊C4—9416no—
485C42—28no—
486C42—4716no—
487C42—948no—
488C23—216no—
489C23—216no—
490C23—216no—
491C23—216no—
492C23—216no—
493C23—216no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23C2×Dic47⋊C4132yes—
499C23C22×C4×D4718yes—
500C23C22×C4×D4712yes—
501C23C22×C22⋊D4718yes—
502C23C22×C22⋊D4712yes—
503C23C22×Dic9418yes—
504C23C22×Dic9412yes—
505C23#41×D4712yes—
506C23C2×#1062535212yes—
507C23#1309406912yes—
508C23#1309407012yes—
509C23C24×D4711yescharacteristic
510D4C2×Dic47⋊C4164yescomplemented
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—432no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—474no—
524C2×C4—474no—
525C2×C4—478no—
526C2×C4—474no—
527C2×C4—948no—
528C2×C4—948no—
529C2×C4—9416no—
530C2×C4—948no—
531C2×C4—942no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—944no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—944no—
545C2×C4—18832no—
546C2×C4—18832no—
547C2×C4—1888no—
548C2×C4—1884no—
549C2×C4C2×Dic47⋊C4116yes—
550C2×C4C22×C4×D4714yes—
551C2×C4C22×C22⋊D4714yes—
552C2×C4C22×Dic9414yes—
553C2×C4C2×#1062535214yes—
554C22—264no—
555C22—28no—
556C22—28no—
557C22—28no—
558C22—28no—
559C22—28no—
560C22—28no—
561C22C22×Dic47⋊C4116yes—
562C22C22×Dic47⋊C414yes—
563C22C4×#1062535214yes—
564C22#1568959514yes—
565C22#1568976114yes—
566C22#1568976714yes—
567C22#1568983014yes—
568C22#1568983214yes—
569C22C23×C4×D4711yescharacteristic
570C22C23×C22⋊D4711yescharacteristic
571C22C23×Dic9411yescharacteristic
572C22C2×#41×D4711yescharacteristic
573C22C22×#1062535211yescharacteristic
574C22C2×#1309406911yescharacteristic
575C22C2×#1309407011yescharacteristic
576C4—24no—
577C4—42no—
578C4—44no—
579C4—944no—
580C4—944no—
581C4—948no—
582C4—944no—
583C4—1884no—
584C4—1882no—
585C4C22×Dic47⋊C418yes—
586C2—216no—
587C2#2024629318yesdirect product factor
588C2C23×Dic47⋊C411yescharacteristic
589C2C2×C4×#1062535211yescharacteristic
590C2C2×#1568959511yescharacteristic
591C2C2×#1568976111yescharacteristic
592C2C2×#1568976711yescharacteristic
593C2C2×#1568983011yescharacteristic
594C2C2×#1568983211yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC22×C4×D4×C47C2
socleC24×C47C24
prime radicalC23×D4×C47C22
cosocleC23×C47C25
hypercenterC22×C4×D4D47
hypocenterC47C2×#2349
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
3. #20246293C2direct (N × H)
4. C2×#15689021C2semidirect
5. C2×#15689019C2semidirect
6. C2×#15688996C2semidirect
7. C2×#15688995C2semidirect
8. C2×#15688988C2semidirect
9. C2×#15688981C2semidirect
10. C2×C4×Dic47⋊C4C2semidirect
11. C22×C47⋊#95C2semidirect
16. C2×D4×Dic47C4semidirect
29. #15689021C22semidirect
30. #15689019C22semidirect
31. #15688996C22semidirect
32. #15688995C22semidirect
33. #15688988C22semidirect
34. #15688981C22semidirect
35. C4×Dic47⋊C4C22semidirect
42. C2×C47⋊#95C22semidirect
55. C23×Dic47C2×C4semidirect
65. D4×Dic47C2×C4semidirect
67. C2×C4⋊Dic47C2×C4semidirect
79. C2×C22⋊Dic47C2×C4semidirect
83. C2×C4×Dic47C2×C4semidirect
121. C2×D4×C47C4⋊C4semidirect
169. C22×Dic47C22×C4semidirect
181. C22⋊Dic47C22×C4semidirect
183. C4⋊Dic47C22×C4semidirect
189. C4×Dic47C22×C4semidirect
212. D4×C47C2×C4⋊C4semidirect
302. C2×C47#2349semidirect
347. C47C2×#2349semidirect
419. C2×D4Dic47⋊C4semidirect
510. D4C2×Dic47⋊C4semidirect
587. C2#20246293direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic472
#202462932
C2×#156890212
C2×#156890192
C2×#156889962
C2×#156889952
C2×#156889882
C2×#156889812
C2×C4×Dic47⋊C42
C22×C47⋊#952
C22×C4×D4×C472
C2×#234947