Finite Group Databasebeta

← C2×#20245869 #26374658

Subgroups — C2×#20245869 #26374658

Order 12032.

3854 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×#13093982C211yesmaximal, characteristic
2C22×#13093982C211yesmaximal, characteristic
3C22×C42×D47C211yesmaximal, characteristic
4#20245869C2116yesmaximal, direct product factor
5C2×#15689022C211yesmaximal, characteristic, complemented
6C2×#15689021C211yesmaximal, characteristic, complemented
7C2×#15689019C211yesmaximal, characteristic, complemented
8C2×#15688997C212yesmaximal
9C2×#15688994C212yesmaximal
10C2×C4⋊C4×Dic47C212yesmaximal, complemented
11C2×#15688974C212yesmaximal, complemented
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×C4×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C2×#13093982—28no—
17C2×#13093982C4116yescomplemented
18C2×#13093982C2218yes—
19C2×#13093982C2218yes—
20C22×C4⋊Dic47C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×C4×Dic47C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29C22×C4×Dic47C2211yescharacteristic
30C22×C4×Dic47C2212yes—
31C22×C4×Dic47C2212yes—
32C2×C42×D47—28no—
33C2×C42×D47C2218yes—
34#15689022C2218yescomplemented
35#15689021C2218yescomplemented
36#15689019C2218yescomplemented
37#15688997C22116yes—
38#15688994C22116yes—
39C4⋊C4×Dic47C22116yescomplemented
40#15688974C22116yescomplemented
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C22×C4⋊C4×C47C2212yes—
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C42×C47C2211yescharacteristic
48C47×#389C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2311yescharacteristic
51C23×Dic47C2312yes—
52C23×Dic47C2312yes—
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2311yescharacteristic
56C22×C4×D47—24no—
57C22×C4×D47—24no—
58C22×C4×D47—24no—
59C22×C4×D47—24no—
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47C2×C418yes—
63C22×C4×D47D418yes—
64C22×C4×D47Q818yes—
65C22×C4×D47C2314yes—
66C22×C4×D47C2314yes—
67C22×C4×D47C2314yes—
68#13093982—232no—
69#13093982C2×C4164yescomplemented
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47—28no—
72C2×C4⋊Dic47C2×C418yescomplemented
73C2×C4⋊Dic47C2318yes—
74C2×C4⋊Dic47C2314yes—
75C2×C4⋊Dic47C2314yes—
76C2×Dic47⋊C4—28no—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4C2318yes—
79C2×Dic47⋊C4C2318yes—
80C2×(C2×D47)⋊C4—28no—
81C2×(C2×D47)⋊C4C2×C4116yes—
82C2×(C2×D47)⋊C4C2318yes—
83C2×(C2×D47)⋊C4C2318yes—
84C2×C4×Dic47—24no—
85C2×C4×Dic47—24no—
86C2×C4×Dic47—24no—
87C2×C4×Dic47—28no—
88C2×C4×Dic47C2×C4116yescomplemented
89C2×C4×Dic47C2314yes—
90C2×C4×Dic47C2318yes—
91C2×C4×Dic47C2314yes—
92C2×C4×Dic47C2314yes—
93C2×C4×Dic47C2318yes—
94C42×D47—232no—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C47⋊#95C2318yes—
98C23×C4×C47C2312yes—
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47—24no—
104C2×C4⋊C4×C47—28no—
105C2×C4⋊C4×C47C2×C418yescomplemented
106C2×C4⋊C4×C47C2318yes—
107C2×C4⋊C4×C47C2314yes—
108C2×C4⋊C4×C47C2314yes—
109C2×C42×C47—24no—
110C2×C42×C47C2314yes—
111#95×C47C2318yes—
112C24×C47C2411yescharacteristic
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—22no—
117C23×D47C22×C414yes—
118C23×D47C2×D414yes—
119C23×D47C2×Q814yes—
120C23×D47C2412yes—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—24no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47C22×C418yes—
131C22×C4×C47C22×C414yes—
132C22×C4×C47C2×D414yes—
133C22×C4×C47C2×Q814yes—
134C22×C4×C47#4114yes—
135C22×C4×C47#4114yes—
136C22×C4×C47C2414yes—
137C22×C4×C47C2414yes—
138C22×C4×C47C2412yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—24no—
144C22×Dic47—24no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47C22×C418yes—
151C22×Dic47C22×C414yes—
152C22×Dic47C2×D414yes—
153C22×Dic47C2×Q814yes—
154C22×Dic47#4114yes—
155C22×Dic47#4114yes—
156C22×Dic47C2414yes—
157C22×Dic47C2414yes—
158C22×Dic47C2412yes—
159C22×Dic47C2412yes—
160C22×Dic47C2412yes—
161C2×C4×D47—232no—
162C2×C4×D47—232no—
163C2×C4×D47—28no—
164C2×C4×D47—28no—
165C2×C4×D47—28no—
166C2×C4×D47—28no—
167C2×C4×D47—28no—
168C2×C4×D47—28no—
169C2×C4×D47C4⋊C4164yescomplemented
170C2×C4×D47C22×C4116yes—
171C2×C4×D47C2×D4116yes—
172C2×C4×D47C2×Q8116yes—
173C4⋊C4×C47—28no—
174C4⋊C4×C47—216no—
175C4⋊C4×C47C22×C4116yescomplemented
176C42×C47—28no—
177(C2×D47)⋊C4—216no—
178(C2×D47)⋊C4C22×C4132yes—
179C4⋊Dic47—28no—
180C4⋊Dic47—216no—
181C4⋊Dic47C22×C4116yescomplemented
182Dic47⋊C4—216no—
183Dic47⋊C4—216no—
184C4×Dic47—28no—
185C4×Dic47—28no—
186C4×Dic47—28no—
187C4×Dic47—216no—
188C4×Dic47C22×C4132yescomplemented
189C23×C47C23×C412yes—
190C23×C47C22×D412yes—
191C23×C47C22×Q812yes—
192C23×C47C2×#4112yes—
193C23×C47C2×#4112yes—
194C23×C47C2×#4112yes—
195C23×C47C2×#4112yes—
196C23×C47C2511yescharacteristic
197C22×D47—28no—
198C22×D47—28no—
199C22×D47—28no—
200C22×D47—28no—
201C22×D47—28no—
202C22×D47—28no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47C2×C4⋊C4116yes—
208C22×D47C23×C414yes—
209C22×D47C22×D414yes—
210C22×D47C22×Q814yes—
211C2×C4×C47—216no—
212C2×C4×C47—28no—
213C2×C4×C47—28no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—48no—
224C2×C4×C47C2×C4⋊C4116yes—
225C2×C4×C47C23×C418yes—
226C2×C4×C47C23×C414yes—
227C2×C4×C47C22×D414yes—
228C2×C4×C47C22×Q814yes—
229C2×C4×C47C2×#4114yes—
230C2×C4×C47C2×#4114yes—
231C2×Dic47—216no—
232C2×Dic47—28no—
233C2×Dic47—28no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—48no—
244C2×Dic47C2×C4⋊C4116yes—
245C2×Dic47C23×C418yes—
246C2×Dic47C23×C414yes—
247C2×Dic47C22×D414yes—
248C2×Dic47C22×Q814yes—
249C2×Dic47C2×#4114yes—
250C2×Dic47C2×#4114yes—
251C4×D47—232no—
252C4×D47—232no—
253C4×D47C2×C4⋊C4164yescomplemented
254C2×#2296—471nomaximal, Sylow
255C22×C47C22×C4⋊C414yes—
256C22×C47C4×#4114yes—
257C22×C47C4×#4114yes—
258C22×C47#52514yes—
259C22×C47#52514yes—
260C22×C47#53314yes—
261C22×C47#53314yes—
262C22×C47C24×C411yescharacteristic
263C22×C47C23×D411yescharacteristic
264C22×C47C23×Q811yescharacteristic
265C22×C47C22×#4111yescharacteristic
266C22×C47C22×#4111yescharacteristic
267C22×C47C22×#4111yescharacteristic
268C22×C47C22×#4111yescharacteristic
269C2×D47—24no—
270C2×D47—24no—
271C2×D47—24no—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47—24no—
275C2×D47—232no—
276C2×D47C22×C4⋊C418yes—
277C4×C47—28no—
278C4×C47—24no—
279C4×C47—24no—
280C4×C47—44no—
281C4×C47C22×C4⋊C418yes—
282Dic47—28no—
283Dic47—24no—
284Dic47—24no—
285Dic47—44no—
286Dic47C22×C4⋊C418yes—
287C22×#117—471no—
288C22×#117—471no—
289C23×C42—471no—
290#2296—4716no—
291C2×#389—471no—
292C2×#389—471no—
293C2×#389—471no—
294C2×#389D4711yescharacteristic, complemented
295C2×#385—472no—
296C2×#385—472no—
297C2×C4×C4⋊C4—472no—
298C2×C4×C4⋊C4—472no—
299C2×C47#229618yescomplemented
300C2×C47C23×C4⋊C411yescharacteristic
301C2×C47C2×C4×#4111yescharacteristic
302C2×C47C2×C4×#4111yescharacteristic
303C2×C47C2×#52511yescharacteristic
304C2×C47C2×#52511yescharacteristic
305C2×C47C2×#53311yescharacteristic
306C2×C47C2×#53311yescharacteristic
307D47—28no—
308C24×C4—471no—
309C24×C4—471no—
310C24×C4—471no—
311C2×#117—4716no—
312C2×#117—478no—
313C2×#117—478no—
314C2×#117—948no—
315C22×C4⋊C4—471no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4—472no—
320C22×C4⋊C4C2×D4711yescharacteristic
321C22×C4⋊C4C2×D4712yes—
322C22×C4⋊C4C2×D4711yescharacteristic
323C22×C22⋊C4—472no—
324C22×C22⋊C4—472no—
325C22×C42—472no—
326C22×C42—471no—
327C22×C42—471no—
328C22×C42—471no—
329C22×C42—472no—
330C22×C42—478no—
331C22×C42—948no—
332C22×C42C2×D4711yescharacteristic
333#389—478no—
334#389—478no—
335#389—478no—
336#389C2×D4718yescomplemented
337#385—4716no—
338#385—4716no—
339C4×C4⋊C4—4716no—
340C4×C4⋊C4—4716no—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—472no—
344C2×#95C2×D4712yes—
345C47C2×#229611yescharacteristic, complemented, Sylow
346C25—471no—
347C23×C4—471no—
348C23×C4—478no—
349C23×C4—478no—
350C23×C4—478no—
351C23×C4—474no—
352C23×C4—472no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4—472no—
356C23×C4—474no—
357C23×C4—474no—
358C23×C4—944no—
359C23×C4—944no—
360C23×C4—944no—
361C23×C4—944no—
362C23×C4—944no—
363C23×C4—944no—
364C23×C4C22×D4712yes—
365C23×C4C22×D4712yes—
366C23×C4C22×D4711yescharacteristic
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4711yescharacteristic
369#117—4764no—
370#117—9432no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—478no—
374C2×C4⋊C4—474no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—478no—
377C2×C4⋊C4—478no—
378C2×C4⋊C4—478no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—948no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—948no—
383C2×C4⋊C4C4×D4718yescomplemented
384C2×C4⋊C4C22×D4714yes—
385C2×C4⋊C4C22×D4718yes—
386C2×C4⋊C4C22×D4714yes—
387C2×C22⋊C4—4716no—
388C2×C22⋊C4—478no—
389C2×C22⋊C4—478no—
390C2×C22⋊C4—948no—
391C2×C42—24no—
392C2×C42—4716no—
393C2×C42—478no—
394C2×C42—478no—
395C2×C42—474no—
396C2×C42—474no—
397C2×C42—474no—
398C2×C42—9432no—
399C2×C42—948no—
400C2×C42—944no—
401C2×C42—944no—
402C2×C42—944no—
403C2×C42C22×D4714yes—
404#95—478no—
405#95—478no—
406#95—478no—
407#95C22×D4718yes—
408C24—474no—
409C24—474no—
410C24—474no—
411C24—472no—
412C24—942no—
413C24—942no—
414C24—942no—
415C24—942no—
416C24C23×D4711yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—474no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—472no—
430C22×C4—4764no—
431C22×C4—4716no—
432C22×C4—4716no—
433C22×C4—4716no—
434C22×C4—478no—
435C22×C4—474no—
436C22×C4—474no—
437C22×C4—472no—
438C22×C4—472no—
439C22×C4—474no—
440C22×C4—474no—
441C22×C4—948no—
442C22×C4—948no—
443C22×C4—948no—
444C22×C4—948no—
445C22×C4—9432no—
446C22×C4—9432no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—944no—
450C22×C4—942no—
451C22×C4—944no—
452C22×C4—942no—
453C22×C4—948no—
454C22×C4—948no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—944no—
458C22×C4C2×C4×D4718yes—
459C22×C4C2×C4×D4714yes—
460C22×C4D4×D4714yes—
461C22×C4#1062535214yes—
462C22×C4Q8×D4714yes—
463C22×C4#1062535414yes—
464C22×C4C23×D4714yes—
465C22×C4C23×D4714yes—
466C22×C4C23×D4712yes—
467C22×C4C23×D4712yes—
468C22×C4C23×D4712yes—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—4716no—
472C4⋊C4—9416no—
473C4⋊C4—9416no—
474C4⋊C4—948no—
475C4⋊C4—9416no—
476C4⋊C4C2×C4×D47116yescomplemented
477C22⋊C4—4732no—
478C22⋊C4—9416no—
479C42—28no—
480C42—4732no—
481C42—948no—
482C42—948no—
483C42—9416no—
484C42—948no—
485C23—4716no—
486C23—474no—
487C23—474no—
488C23—474no—
489C23—948no—
490C23—948no—
491C23—948no—
492C23—948no—
493C23—948no—
494C23—948no—
495C23—942no—
496C23—942no—
497C23—942no—
498C23—942no—
499C23C22×C4×D4712yes—
500C23C2×D4×D4712yes—
501C23C2×Q8×D4712yes—
502C23#41×D4712yes—
503C23#41×D4712yes—
504C23C2×#1062535212yes—
505C23C2×#1062535412yes—
506C23C24×D4711yescharacteristic
507C2×C4—216no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—48no—
520C2×C4—4716no—
521C2×C4—474no—
522C2×C4—474no—
523C2×C4—474no—
524C2×C4—4764no—
525C2×C4—478no—
526C2×C4—474no—
527C2×C4—474no—
528C2×C4—9432no—
529C2×C4—9432no—
530C2×C4—9416no—
531C2×C4—948no—
532C2×C4—948no—
533C2×C4—942no—
534C2×C4—942no—
535C2×C4—944no—
536C2×C4—942no—
537C2×C4—944no—
538C2×C4—942no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—1888no—
543C2×C4C4⋊C4×D47116yes—
544C2×C4C22×C4×D4718yes—
545C2×C4C22×C4×D4714yes—
546C2×C4C2×D4×D4714yes—
547C2×C4C2×Q8×D4714yes—
548C2×C4C2×#1062535214yes—
549C2×C4C2×#1062535414yes—
550C22—478no—
551C22—944no—
552C22—944no—
553C22—9432no—
554C22—944no—
555C22—944no—
556C22—944no—
557C22—944no—
558C22C2×C4⋊C4×D4714yes—
559C22C4×#1062535214yes—
560C22C4×#1062535414yes—
561C22#1568963314yes—
562C22#1568963614yes—
563C22#1568967814yes—
564C22#1568967914yes—
565C22C23×C4×D4711yescharacteristic
566C22C22×D4×D4711yescharacteristic
567C22C22×Q8×D4711yescharacteristic
568C22C2×#41×D4711yescharacteristic
569C22C2×#41×D4711yescharacteristic
570C22C22×#1062535211yescharacteristic
571C22C22×#1062535411yescharacteristic
572C4—28no—
573C4—24no—
574C4—24no—
575C4—44no—
576C4—478no—
577C4—948no—
578C4—944no—
579C4—944no—
580C4—1884no—
581C4C2×C4⋊C4×D4718yes—
582C2—948no—
583C2#2024586918yesdirect product factor
584C2C22×C4⋊C4×D4711yescharacteristic
585C2C2×C4×#1062535211yescharacteristic
586C2C2×C4×#1062535411yescharacteristic
587C2C2×#1568963311yescharacteristic
588C2C2×#1568963611yescharacteristic
589C2C2×#1568967811yescharacteristic
590C2C2×#1568967911yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC24×D47C23
cosocleC23×C47C25
hypercenterC2×#389D47
hypocenterC47C2×#2296
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. #20245869C2direct (N × H)
5. C2×#15689022C2semidirect
6. C2×#15689021C2semidirect
7. C2×#15689019C2semidirect
10. C2×C4⋊C4×Dic47C2semidirect
11. C2×#15688974C2semidirect
12. C2×C47×#389C2semidirect
17. C2×#13093982C4semidirect
34. #15689022C22semidirect
35. #15689021C22semidirect
36. #15689019C22semidirect
39. C4⋊C4×Dic47C22semidirect
40. #15688974C22semidirect
48. C47×#389C22semidirect
69. #13093982C2×C4semidirect
72. C2×C4⋊Dic47C2×C4semidirect
88. C2×C4×Dic47C2×C4semidirect
105. C2×C4⋊C4×C47C2×C4semidirect
169. C2×C4×D47C4⋊C4semidirect
175. C4⋊C4×C47C22×C4semidirect
181. C4⋊Dic47C22×C4semidirect
188. C4×Dic47C22×C4semidirect
253. C4×D47C2×C4⋊C4semidirect
294. C2×#389D47semidirect
299. C2×C47#2296semidirect
336. #389C2×D47semidirect
345. C47C2×#2296semidirect
383. C2×C4⋊C4C4×D47semidirect
476. C4⋊C4C2×C4×D47semidirect
583. C2#20245869direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939822
C22×C42×D472
#202458692
C2×#156890222
C2×#156890212
C2×#156890192
C2×#156889972
C2×#156889942
C2×C4⋊C4×Dic472
C2×#156889742
C2×C47×#3892
C2×#229647