Finite Group Databasebeta

← C2×#20246789 #26384976

Subgroups — C2×#20246789 #26384976

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×#13094008C211yesmaximal, characteristic
2C22×#13094007C211yesmaximal, characteristic
3C23×Dic47⋊C4C211yesmaximal, characteristic, complemented
4#20246789C2116yesmaximal, direct product factor
5C2×#15689053C212yesmaximal, complemented
6C2×#15689045C212yesmaximal, complemented
7C2×#15689021C212yesmaximal, complemented
8C2×#15689014C212yesmaximal, complemented
9C2×#15689000C211yesmaximal, characteristic
10C2×#15688998C211yesmaximal, characteristic, complemented
11C2×#15688984C211yesmaximal, characteristic, complemented
12C22×#117×C47C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2211yescharacteristic
14C24×Dic47C2211yescharacteristic
15C24×Dic47C2211yescharacteristic
16C2×#13094008—28no—
17C2×#13094008C2218yes—
18C2×#13094007—28no—
19C2×#13094007C2218yes—
20C22×C4⋊Dic47C2212yes—
21C22×C4⋊Dic47C2212yes—
22C22×Dic47⋊C4—28no—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2211yescharacteristic
26C22×Dic47⋊C4C2211yescharacteristic
27C22×Dic47⋊C4C2218yescomplemented
28C22×C22⋊Dic47C2211yescharacteristic
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2211yescharacteristic
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2211yescharacteristic
33#15689053C22116yescomplemented
34#15689045C22116yescomplemented
35#15689021C22116yescomplemented
36#15689014C22116yescomplemented
37#15689000C2218yes—
38#15688998C2218yescomplemented
39#15688984C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C24×C4×C47C2211yescharacteristic
45C2×#117×C47—28no—
46C2×#117×C47C2218yescomplemented
47C22×C4⋊C4×C47C2212yes—
48C22×C22⋊C4×C47C2211yescharacteristic
49C22×C22⋊C4×C47C2211yescharacteristic
50C22×C42×C47C2212yes—
51C24×D47C2311yescharacteristic
52C22×C22⋊D47—24no—
53C22×C22⋊D47—24no—
54C22×C22⋊D47—24no—
55C22×C22⋊D47C2314yes—
56C23×Dic47—24no—
57C23×Dic47—28no—
58C23×Dic47—24no—
59C23×Dic47—24no—
60C23×Dic47—24no—
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2312yes—
64C23×Dic47C2311yescharacteristic
65C23×Dic47C2314yes—
66C23×Dic47C2314yes—
67C23×Dic47C2312yes—
68#13094008—232no—
69#13094007—232no—
70C2×C4⋊Dic47—28no—
71C2×C4⋊Dic47—28no—
72C2×C4⋊Dic47C2318yes—
73C2×C4⋊Dic47C2318yes—
74C2×Dic47⋊C4—28no—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—232no—
77C2×Dic47⋊C4—24no—
78C2×Dic47⋊C4—24no—
79C2×Dic47⋊C4C2318yes—
80C2×Dic47⋊C4C2314yes—
81C2×Dic47⋊C4C2318yes—
82C2×Dic47⋊C4C2314yes—
83C2×C22⋊Dic47—24no—
84C2×C22⋊Dic47C2314yes—
85C2×(C2×D47)⋊C4—24no—
86C2×(C2×D47)⋊C4—24no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2314yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2314yes—
93C47⋊#95C2318yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C25×C47C2311yescharacteristic
98C23×C4×C47—24no—
99C23×C4×C47—24no—
100C23×C4×C47D418yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2312yes—
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2314yes—
106#117×C47—232no—
107C2×C4⋊C4×C47—28no—
108C2×C4⋊C4×C47C2318yes—
109C2×C22⋊C4×C47—24no—
110C2×C22⋊C4×C47D418yes—
111C2×C22⋊C4×C47C2314yes—
112C2×C22⋊C4×C47C2314yes—
113C2×C42×C47—28no—
114C2×C42×C47C2318yes—
115C24×C47—24no—
116C24×C47—24no—
117C24×C47C2×D414yes—
118C24×C47C2×D414yes—
119C24×C47#4114yes—
120C24×C47C2412yes—
121C24×C47C2411yescharacteristic
122C23×D47—22no—
123C23×D47—22no—
124C23×D47—24no—
125C23×D47—22no—
126C23×D47—24no—
127C23×D47C2412yes—
128C22×C4×C47—232no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—28no—
132C22×C4×C47—28no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47C2×D414yes—
141C22×C4×C47C2×D414yes—
142C22×C4×C47C2×D4116yes—
143C22×C4×C47C2×D418yes—
144C22×C4×C47C2414yes—
145C22×C4×C47C2412yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2412yes—
148C2×C22⋊D47—28no—
149C2×C22⋊D47—28no—
150C2×C22⋊D47—28no—
151C2×C22⋊D47—416no—
152C22×Dic47—232no—
153C22×Dic47—28no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—216no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—28no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—28no—
165C22×Dic47—24no—
166C22×Dic47—28no—
167C22×Dic47—24no—
168C22×Dic47—22no—
169C22×Dic47—24no—
170C22×Dic47—416no—
171C22×Dic47#4118yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C22×Dic47C2414yes—
175C22×Dic47C2412yes—
176C22×Dic47C2414yes—
177C4⋊C4×C47—216no—
178C22⋊C4×C47—28no—
179C22⋊C4×C47C2×D4116yes—
180C42×C47—216no—
181C22⋊Dic47—28no—
182(C2×D47)⋊C4—28no—
183(C2×D47)⋊C4—28no—
184(C2×D47)⋊C4—216no—
185(C2×D47)⋊C4—216no—
186C4⋊Dic47—216no—
187C4⋊Dic47—216no—
188Dic47⋊C4—216no—
189Dic47⋊C4—216no—
190Dic47⋊C4—28no—
191Dic47⋊C4—28no—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—216no—
195C23×C47—216no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47#121116yes—
199C23×C47C22×D414yes—
200C23×C47C22×D412yes—
201C23×C47C22×D414yes—
202C23×C47C22×D412yes—
203C23×C47C2×#4114yes—
204C23×C47C2×#4114yes—
205C23×C47C2×#4112yes—
206C23×C472−1+414yes—
207C23×C47C2511yescharacteristic
208C22×D47—22no—
209C22×D47—22no—
210C22×D47—24no—
211C22×D47—22no—
212C22×D47—24no—
213C22×D47—44no—
214C22×D47—44no—
215C22×D47—44no—
216C22×D47—48no—
217C22×D47—48no—
218C2×C4×C47—232no—
219C2×C4×C47—28no—
220C2×C4×C47—28no—
221C2×C4×C47—216no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—48no—
232C2×C4×C47C22×D418yes—
233C2×C4×C47C22×D414yes—
234C2×C4×C47C22×D414yes—
235C22⋊D47—416no—
236C2×Dic47—232no—
237C2×Dic47—216no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—416no—
251C2×Dic47—48no—
252C2×Dic47—44no—
253C2×Dic47—44no—
254C2×Dic47—44no—
255C2×Dic47C2×#4118yes—
256C2×#2413—471nomaximal, Sylow
257C22×C47—24no—
258C22×C47—232no—
259C22×C47—24no—
260C22×C47—28no—
261C22×C47—28no—
262C22×C47C2×#12118yes—
263C22×C47C2×#12114yes—
264C22×C47#52514yes—
265C22×C47#53614yes—
266C22×C47#54718yes—
267C22×C47#55218yes—
268C22×C47C23×D411yescharacteristic
269C22×C47C23×D411yescharacteristic
270C22×C47C22×#4112yes—
271C22×C47C22×#4111yescharacteristic
272C22×C47C2×2−1+412yes—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—44no—
277C2×D47—44no—
278C2×D47—88no—
279C4×C47—24no—
280C4×C47—24no—
281C4×C47—28no—
282C4×C47—44no—
283Dic47—28no—
284Dic47—44no—
285Dic47—42no—
286Dic47—42no—
287Dic47—42no—
288C22×#123—471no—
289C22×#123—471no—
290C22×#117D4711yescharacteristic, complemented
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#241318yescomplemented
302C2×C47C22×#12111yescharacteristic
303C2×C47C2×#52511yescharacteristic
304C2×C47C2×#53611yescharacteristic
305C2×C47C2×#54712yes—
306C2×C47C2×#55212yes—
307D47—82no—
308C23×D4—471no—
309C24×C4—471no—
310C24×C4—471no—
311C24×C4C2×D4711yescharacteristic
312C2×#123—478no—
313C2×#123—478no—
314C2×#123—948no—
315C2×#123—948no—
316C2×#117—28no—
317C2×#117C2×D4718yescomplemented
318C22×C4⋊C4—472no—
319C22×C4⋊C4—472no—
320C22×C4⋊C4—471no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—478no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4—948no—
326C22×C4⋊C4C2×D4712yes—
327C22×C22⋊C4—471no—
328C22×C22⋊C4—471no—
329C22×C22⋊C4—472no—
330C22×C22⋊C4—472no—
331C22×C22⋊C4—471no—
332C22×C22⋊C4C2×D4711yescharacteristic
333C22×C22⋊C4C2×D4711yescharacteristic
334C22×C42C2×D4712yes—
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
347C25—471no—
348C25C22×D4711yescharacteristic
349C22×D4—474no—
350C22×D4—944no—
351C22×D4—944no—
352C22×D4—944no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—472no—
356C23×C4—474no—
357C23×C4—474no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—472no—
361C23×C4—471no—
362C23×C4—948no—
363C23×C4—944no—
364C23×C4—944no—
365C23×C4—944no—
366C23×C4—944no—
367C23×C4D18818yes—
368C23×C4C22×D4712yes—
369C23×C4C22×D4714yes—
370C23×C4C22×D4711yescharacteristic
371C23×C4C22×D4711yescharacteristic
372C23×C4C22×D4712yes—
373#123—9432no—
374#123—9432no—
375#117—232no—
376C2×C4⋊C4—28no—
377C2×C4⋊C4—478no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—474no—
380C2×C4⋊C4—474no—
381C2×C4⋊C4—478no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—948no—
384C2×C4⋊C4—948no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—9432no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—948no—
390C2×C4⋊C4C22×D4718yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—474no—
393C2×C22⋊C4—474no—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—948no—
398C2×C22⋊C4—944no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4C22⋊D4718yes—
403C2×C22⋊C4C22×D4714yes—
404C2×C22⋊C4C22×D4714yes—
405C2×C42—28no—
406C2×C42C22×D4718yes—
407#95—478no—
408#95—478no—
409#95—478no—
410#95—478no—
411C24—24no—
412C24—24no—
413C24—472no—
414C24—942no—
415C24—942no—
416C24—944no—
417C24—942no—
418C24—944no—
419C24C2×D18814yes—
420C24#1062535014yes—
421C24C2×C22⋊D4714yes—
422C24C23×D4712yes—
423C24C23×D4711yescharacteristic
424C2×D4—948no—
425C2×D4—948no—
426C2×D4—948no—
427C2×D4—18816no—
428C22×C4—232no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—478no—
441C22×C4—474no—
442C22×C4—472no—
443C22×C4—472no—
444C22×C4—474no—
445C22×C4—472no—
446C22×C4—9416no—
447C22×C4—9432no—
448C22×C4—948no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—942no—
453C22×C4—948no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—948no—
457C22×C4—944no—
458C22×C4—942no—
459C22×C4—944no—
460C22×C4—948no—
461C22×C4—944no—
462C22×C4—942no—
463C22×C4—944no—
464C22×C4—18816no—
465C22×C4C2×D18814yes—
466C22×C4C2×D18814yes—
467C22×C4C2×D188116yes—
468C22×C4C2×C22⋊D4718yes—
469C22×C4C23×D4714yes—
470C22×C4C23×D4714yes—
471C22×C4C23×D4712yes—
472C22×C4C23×D4712yes—
473C4⋊C4—216no—
474C4⋊C4—9416no—
475C4⋊C4—9416no—
476C4⋊C4—9416no—
477C4⋊C4—9416no—
478C4⋊C4—948no—
479C4⋊C4—948no—
480C22⋊C4—28no—
481C22⋊C4—9416no—
482C22⋊C4—948no—
483C22⋊C4—9416no—
484C22⋊C4—948no—
485C22⋊C4—948no—
486C22⋊C4C2×C22⋊D47116yes—
487C42—216no—
488C23—216no—
489C23—216no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—942no—
495C23—942no—
496C23—944no—
497C23—942no—
498C23—944no—
499C23—1888no—
500C23—1888no—
501C23—1884no—
502C23—1884no—
503C23—1884no—
504C23#13093998116yes—
505C23C22×C22⋊D4714yes—
506C23C22×C22⋊D4712yes—
507C23C22×D18814yes—
508C23C22×D18812yes—
509C23#41×D4714yes—
510C23C2×#1062535014yes—
511C23C2×#1062535012yes—
512C23#1309407114yes—
513C23C24×D4711yescharacteristic
514D4—18816no—
515C2×C4—216no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—478no—
530C2×C4—9432no—
531C2×C4—9416no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—944no—
539C2×C4—942no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—942no—
543C2×C4—944no—
544C2×C4—18816no—
545C2×C4—1888no—
546C2×C4—1884no—
547C2×C4—1884no—
548C2×C4—1884no—
549C2×C4C22×C22⋊D4718yes—
550C2×C4C22×D18814yes—
551C2×C4C22×D18814yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—3768no—
558C22—1884no—
559C22—1882no—
560C22—1882no—
561C22—1884no—
562C22—1882no—
563C22C2×#1309399818yes—
564C22C2×#1309399814yes—
565C22#1568963514yes—
566C22#1568969914yes—
567C22#1568977218yes—
568C22#1568980618yes—
569C22C23×C22⋊D4711yescharacteristic
570C22C23×D18811yescharacteristic
571C22C2×#41×D4712yes—
572C22C22×#1062535011yescharacteristic
573C22C2×#1309407112yes—
574C4—28no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—948no—
579C4—1884no—
580C4—1882no—
581C4—1882no—
582C4—1882no—
583C2—28no—
584C2—3762no—
585C2#2024678918yesdirect product factor
586C2C22×#1309399811yescharacteristic
587C2C2×#1568963511yescharacteristic
588C2C2×#1568969911yescharacteristic
589C2C2×#1568977212yes—
590C2C2×#1568980612yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#117×C47C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC22×#117D47
hypocenterC47C2×#2413
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C23×Dic47⋊C4C2semidirect
4. #20246789C2direct (N × H)
5. C2×#15689053C2semidirect
6. C2×#15689045C2semidirect
7. C2×#15689021C2semidirect
8. C2×#15689014C2semidirect
10. C2×#15688998C2semidirect
11. C2×#15688984C2semidirect
12. C22×#117×C47C2semidirect
27. C22×Dic47⋊C4C22semidirect
33. #15689053C22semidirect
34. #15689045C22semidirect
35. #15689021C22semidirect
36. #15689014C22semidirect
38. #15688998C22semidirect
39. #15688984C22semidirect
46. C2×#117×C47C22semidirect
290. C22×#117D47semidirect
301. C2×C47#2413semidirect
317. C2×#117C2×D47semidirect
346. C47C2×#2413semidirect
585. C2#20246789direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940082
C22×#130940072
C23×Dic47⋊C42
#202467892
C2×#156890532
C2×#156890452
C2×#156890212
C2×#156890142
C2×#156890002
C2×#156889982
C2×#156889842
C22×#117×C472
C2×#241347