Finite Group Databasebeta

← C2×#20246796 #26384982

Subgroups — C2×#20246796 #26384982

Order 12032.

3570 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094005C211yesmaximal, characteristic
2C22×#13093983C211yesmaximal, characteristic, complemented
3C23×Dic47⋊C4C211yesmaximal, characteristic, complemented
4#20246796C2116yesmaximal, direct product factor
5C2×#15689056C212yesmaximal, complemented
6C2×#15689046C212yesmaximal, complemented
7C2×#15689018C212yesmaximal, complemented
8C2×#15689017C212yesmaximal, complemented
9C2×#15689003C211yesmaximal, characteristic
10C2×#15688995C211yesmaximal, characteristic, complemented
11C2×#15688984C211yesmaximal, characteristic, complemented
12C22×#123×C47C211yesmaximal, characteristic
13C24×Dic47C2211yescharacteristic
14C24×Dic47C2211yescharacteristic
15C2×#13094005—28no—
16C2×#13094005C2218yes—
17C2×#13093983—28no—
18C2×#13093983C2218yescomplemented
19C22×C4⋊Dic47C2212yes—
20C22×Dic47⋊C4—28no—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2218yescomplemented
26C22×C22⋊Dic47C2211yescharacteristic
27C22×C22⋊Dic47C2212yes—
28C22×C22⋊Dic47C2212yes—
29C22×C22⋊Dic47C2211yescharacteristic
30C22×C4×Dic47C2212yes—
31#15689056C22116yescomplemented
32#15689046C22116yescomplemented
33#15689018C22116yescomplemented
34#15689017C22116yescomplemented
35#15689003C2218yes—
36#15688995C2218yescomplemented
37#15688984C2218yescomplemented
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C23×D4×C47C2211yescharacteristic
43C24×C4×C47C2211yescharacteristic
44C2×#123×C47—28no—
45C2×#123×C47C2218yes—
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C22⋊C4×C47C2211yescharacteristic
49C22×C22⋊C4×C47C2211yescharacteristic
50C22×C22⋊C4×C47C2211yescharacteristic
51C23×Dic47—24no—
52C23×Dic47—28no—
53C23×Dic47—24no—
54C23×Dic47—24no—
55C23×Dic47—24no—
56C23×Dic47C2312yes—
57C23×Dic47C2312yes—
58C23×Dic47C2314yes—
59C23×Dic47C2314yes—
60C23×Dic47C2312yes—
61C23×Dic47C2312yes—
62#13094005—232no—
63#13093983—232no—
64C2×C4⋊Dic47—28no—
65C2×C4⋊Dic47C2318yes—
66C2×Dic47⋊C4—28no—
67C2×Dic47⋊C4—28no—
68C2×Dic47⋊C4—232no—
69C2×Dic47⋊C4—28no—
70C2×Dic47⋊C4—28no—
71C2×Dic47⋊C4C2318yes—
72C2×Dic47⋊C4C2318yes—
73C2×Dic47⋊C4C2318yes—
74C2×Dic47⋊C4C2318yes—
75C2×C22⋊Dic47—24no—
76C2×C22⋊Dic47—28no—
77C2×C22⋊Dic47—28no—
78C2×C22⋊Dic47D418yes—
79C2×C22⋊Dic47C2314yes—
80C2×C22⋊Dic47C2318yes—
81C2×C22⋊Dic47C2318yes—
82C2×C22⋊Dic47C2314yes—
83C2×C4×Dic47—28no—
84C2×C4×Dic47C2318yes—
85C47⋊#95C2318yes—
86C47⋊#95C2318yes—
87C47⋊#95C2318yes—
88C47⋊#95C2318yes—
89C25×C47C2311yescharacteristic
90C25×C47C2311yescharacteristic
91C22×D4×C47—24no—
92C22×D4×C47—24no—
93C22×D4×C47—24no—
94C22×D4×C47C2314yes—
95C23×C4×C47—24no—
96C23×C4×C47—24no—
97C23×C4×C47D418yes—
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2311yescharacteristic
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2314yes—
104#123×C47—232no—
105C2×C4⋊C4×C47—24no—
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47C2314yes—
108C2×C4⋊C4×C47C2314yes—
109C2×C22⋊C4×C47—24no—
110C2×C22⋊C4×C47—24no—
111C2×C22⋊C4×C47—24no—
112C2×C22⋊C4×C47C2314yes—
113C2×C22⋊C4×C47C2314yes—
114C2×C22⋊C4×C47C2314yes—
115C24×C47—24no—
116C24×C47—24no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—22no—
120C24×C47—24no—
121C24×C47—24no—
122C24×C47C2×D414yes—
123C24×C47C2×D414yes—
124C24×C47#4114yes—
125C24×C47C2412yes—
126C24×C47C2412yes—
127C24×C47C2411yescharacteristic
128C2×D4×C47—28no—
129C2×D4×C47—28no—
130C2×D4×C47—28no—
131C2×D4×C47—416no—
132C22×C4×C47—232no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—28no—
136C22×C4×C47—28no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47C2×D414yes—
147C22×C4×C47C2×D414yes—
148C22×C4×C47C2×D4116yes—
149C22×C4×C47C2412yes—
150C22×C4×C47C2412yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2412yes—
153C22×C4×C47C2412yes—
154C22×Dic47—232no—
155C22×Dic47—28no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—216no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—28no—
164C22×Dic47—24no—
165C22×Dic47—28no—
166C22×Dic47—24no—
167C22×Dic47—28no—
168C22×Dic47—24no—
169C22×Dic47—24no—
170C22×Dic47—416no—
171C22×Dic47C2×D418yes—
172C22×Dic47#4118yes—
173C22×Dic47C2414yes—
174C22×Dic47C2414yes—
175C22×Dic47C2414yes—
176C22×Dic47C2414yes—
177C4⋊C4×C47—28no—
178C4⋊C4×C47—28no—
179C22⋊C4×C47—28no—
180C22⋊C4×C47—28no—
181C22⋊C4×C47—28no—
182C22⋊Dic47—28no—
183C22⋊Dic47—216no—
184C22⋊Dic47—216no—
185C22⋊Dic47C2×D4116yes—
186C4⋊Dic47—216no—
187Dic47⋊C4—216no—
188Dic47⋊C4—216no—
189Dic47⋊C4—216no—
190Dic47⋊C4—216no—
191C4×Dic47—216no—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—216no—
195C23×C47—216no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—24no—
201C23×C47—22no—
202C23×C47—24no—
203C23×C47—44no—
204C23×C47—44no—
205C23×C47—44no—
206C23×C47—48no—
207C23×C47—48no—
208C23×C47#121116yes—
209C23×C47C22×D414yes—
210C23×C47C22×D412yes—
211C23×C47C22×D414yes—
212C23×C47C22×D412yes—
213C23×C47C2×#4114yes—
214C23×C47C2×#4114yes—
215C23×C47C2×#4112yes—
216C23×C472−1+414yes—
217C23×C47C2511yescharacteristic
218D4×C47—416no—
219C2×C4×C47—232no—
220C2×C4×C47—28no—
221C2×C4×C47—28no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—44no—
234C2×C4×C47—44no—
235C2×C4×C47—44no—
236C2×C4×C47C22×D414yes—
237C2×C4×C47C22×D414yes—
238C2×Dic47—232no—
239C2×Dic47—216no—
240C2×Dic47—216no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—416no—
252C2×Dic47—48no—
253C2×Dic47—48no—
254C2×Dic47C22×D418yes—
255C2×Dic47C2×#4118yes—
256C2×#2413—471nomaximal, Sylow
257C22×C47—24no—
258C22×C47—232no—
259C22×C47—24no—
260C22×C47—28no—
261C22×C47—28no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—44no—
266C22×C47—44no—
267C22×C47—88no—
268C22×C47C2×#12118yes—
269C22×C47C2×#12114yes—
270C22×C47#52514yes—
271C22×C47#53614yes—
272C22×C47#54718yes—
273C22×C47#55218yes—
274C22×C47C23×D411yescharacteristic
275C22×C47C23×D411yescharacteristic
276C22×C47C22×#4112yes—
277C22×C47C22×#4111yescharacteristic
278C22×C47C2×2−1+412yes—
279C4×C47—24no—
280C4×C47—24no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—42no—
284Dic47—28no—
285Dic47—28no—
286Dic47—44no—
287Dic47—44no—
288C22×#123—471no—
289C22×#123D4711yescharacteristic
290C22×#117—471no—
291C23×C4⋊C4—471no—
292#2413—4716no—
293C2×#398—472no—
294C2×#396—472no—
295C2×#389—472no—
296C2×#388—472no—
297C2×#386—471no—
298C2×#385—471no—
299C2×#381—471no—
300C2×C47—28no—
301C2×C47—82no—
302C2×C47#241318yescomplemented
303C2×C47C22×#12111yescharacteristic
304C2×C47C2×#52511yescharacteristic
305C2×C47C2×#53611yescharacteristic
306C2×C47C2×#54712yes—
307C2×C47C2×#55212yes—
308C23×D4C2×D4711yescharacteristic
309C24×C4—471no—
310C24×C4—471no—
311C24×C4C2×D4711yescharacteristic
312C2×#123—28no—
313C2×#123—478no—
314C2×#123—948no—
315C2×#123C2×D4718yes—
316C2×#117—478no—
317C2×#117—948no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4—472no—
320C22×C4⋊C4—472no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—478no—
324C22×C4⋊C4—948no—
325C22×C4⋊C4C2×D4711yescharacteristic
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—471no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4—472no—
330C22×C22⋊C4—471no—
331C22×C22⋊C4C2×D4711yescharacteristic
332C22×C22⋊C4C2×D4711yescharacteristic
333C22×C22⋊C4C2×D4711yescharacteristic
334C22×C42—472no—
335#398—4716no—
336#396—4716no—
337#389—4716no—
338#388—4716no—
339#386—478no—
340#385—478no—
341#381—478no—
342C2×#95—472no—
343C2×#95—472no—
344C2×#95—472no—
345C2×#95—472no—
346C47C2×#241311yescharacteristic, complemented, Sylow
347C25C22×D4711yescharacteristic
348C25C22×D4711yescharacteristic
349C22×D4—24no—
350C22×D4—24no—
351C22×D4—24no—
352C22×D4C22×D4714yes—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—472no—
356C23×C4—472no—
357C23×C4—474no—
358C23×C4—474no—
359C23×C4—472no—
360C23×C4—472no—
361C23×C4—948no—
362C23×C4—944no—
363C23×C4—944no—
364C23×C4—944no—
365C23×C4—944no—
366C23×C4C22⋊D4718yes—
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C23×C4C22×D4714yes—
371C23×C4C22×D4711yescharacteristic
372C23×C4C22×D4711yescharacteristic
373#123—232no—
374#123—9432no—
375#117—9432no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—478no—
380C2×C4⋊C4—478no—
381C2×C4⋊C4—478no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—948no—
384C2×C4⋊C4—948no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—948no—
388C2×C4⋊C4—9432no—
389C2×C4⋊C4C22×D4714yes—
390C2×C4⋊C4C22×D4714yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—474no—
397C2×C22⋊C4—478no—
398C2×C22⋊C4—478no—
399C2×C22⋊C4—948no—
400C2×C22⋊C4—948no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4C22×D4714yes—
403C2×C22⋊C4C22×D4714yes—
404C2×C22⋊C4C22×D4714yes—
405C2×C42—478no—
406C2×C42—948no—
407#95—478no—
408#95—478no—
409#95—478no—
410#95—478no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—22no—
417C24—24no—
418C24D4×D4714yes—
419C24#1062535214yes—
420C24C2×C22⋊D4714yes—
421C24C23×D4712yes—
422C24C23×D4712yes—
423C24C23×D4711yescharacteristic
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—416no—
428C22×C4—232no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—478no—
443C22×C4—478no—
444C22×C4—474no—
445C22×C4—474no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—9416no—
449C22×C4—9432no—
450C22×C4—948no—
451C22×C4—944no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—948no—
458C22×C4—948no—
459C22×C4—944no—
460C22×C4—944no—
461C22×C4—948no—
462C22×C4—944no—
463C22×C4—944no—
464C22×C4—18816no—
465C22×C4C2×C22⋊D4714yes—
466C22×C4C2×C22⋊D4714yes—
467C22×C4C2×C22⋊D47116yes—
468C22×C4C23×D4712yes—
469C22×C4C23×D4712yes—
470C22×C4C23×D4712yes—
471C22×C4C23×D4712yes—
472C22×C4C23×D4712yes—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—9416no—
476C4⋊C4—9416no—
477C4⋊C4—9416no—
478C4⋊C4—9416no—
479C4⋊C4—9416no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—4716no—
484C22⋊C4—9416no—
485C22⋊C4—9416no—
486C22⋊C4—948no—
487C42—9416no—
488C23—216no—
489C23—216no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—48no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23#13093996116yes—
505C23C2×D4×D4714yes—
506C23C2×D4×D4712yes—
507C23C22×C22⋊D4714yes—
508C23C22×C22⋊D4712yes—
509C23#41×D4714yes—
510C23C2×#1062535214yes—
511C23C2×#1062535212yes—
512C23#1309407114yes—
513C23C24×D4711yescharacteristic
514D4—416no—
515C2×C4—232no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—478no—
533C2×C4—478no—
534C2×C4—9432no—
535C2×C4—9416no—
536C2×C4—9416no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—944no—
543C2×C4—944no—
544C2×C4—944no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—1888no—
548C2×C4—18816no—
549C2×C4—1888no—
550C2×C4C22×C22⋊D4714yes—
551C2×C4C22×C22⋊D4714yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—88no—
563C22C2×#1309399618yes—
564C22C2×#1309399614yes—
565C22#1568963014yes—
566C22#1568969414yes—
567C22#1568977118yes—
568C22#1568981018yes—
569C22C22×D4×D4711yescharacteristic
570C22C23×C22⋊D4711yescharacteristic
571C22C2×#41×D4712yes—
572C22C22×#1062535211yescharacteristic
573C22C2×#1309407112yes—
574C4—24no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—948no—
580C4—948no—
581C4—1884no—
582C4—1884no—
583C2—28no—
584C2—82no—
585C2#2024679618yesdirect product factor
586C2C22×#1309399611yescharacteristic
587C2C2×#1568963011yescharacteristic
588C2C2×#1568969411yescharacteristic
589C2C2×#1568977112yes—
590C2C2×#1568981012yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#123×C47C2
socleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC22×#123D47
hypocenterC47C2×#2413
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C22×#13093983C2semidirect
3. C23×Dic47⋊C4C2semidirect
4. #20246796C2direct (N × H)
5. C2×#15689056C2semidirect
6. C2×#15689046C2semidirect
7. C2×#15689018C2semidirect
8. C2×#15689017C2semidirect
10. C2×#15688995C2semidirect
11. C2×#15688984C2semidirect
18. C2×#13093983C22semidirect
25. C22×Dic47⋊C4C22semidirect
31. #15689056C22semidirect
32. #15689046C22semidirect
33. #15689018C22semidirect
34. #15689017C22semidirect
36. #15688995C22semidirect
37. #15688984C22semidirect
302. C2×C47#2413semidirect
346. C47C2×#2413semidirect
585. C2#20246796direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940052
C22×#130939832
C23×Dic47⋊C42
#202467962
C2×#156890562
C2×#156890462
C2×#156890182
C2×#156890172
C2×#156890032
C2×#156889952
C2×#156889842
C22×#123×C472
C2×#241347