Finite Group Databasebeta

← C2×#20245916 #26374721

Subgroups — C2×#20245916 #26374721

Order 12032.

3854 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C22×#13093989C211yesmaximal, characteristic, complemented
3C23×C4×Dic47C211yesmaximal, characteristic
4#20245916C2116yesmaximal, direct product factor
5C2×#15689028C211yesmaximal, characteristic, complemented
6C2×#15689019C211yesmaximal, characteristic, complemented
7C2×#15689005C212yesmaximal, complemented
8C2×#15688995C212yesmaximal
9C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
10C2×#15688974C212yesmaximal, complemented
11C2×C22⋊C4×Dic47C212yesmaximal
12C2×C47×#380C211yesmaximal, characteristic
13C23×Dic94C2211yescharacteristic
14C24×Dic47C2211yescharacteristic
15C24×Dic47C2211yescharacteristic
16C2×#13094003C4116yescomplemented
17C2×#13094003C2218yes—
18C2×#13093989—28no—
19C2×#13093989C2218yescomplemented
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×C22⋊Dic47C2212yes—
25C22×C4×Dic47—28no—
26C22×C4×Dic47C2211yescharacteristic
27C22×C4×Dic47C2211yescharacteristic
28C22×C4×Dic47C2218yes—
29C22×C4×Dic47C2211yescharacteristic
30C22×C4×Dic47C2212yes—
31C22×C4×Dic47C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33#15689028C2218yescomplemented
34#15689019C2218yescomplemented
35#15689005C22116yescomplemented
36#15688995C22116yes—
37C4⋊C4×Dic47C2218yescomplemented
38#15688974C22116yescomplemented
39C22⋊C4×Dic47C22116yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C24×C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C47×#380C2218yes—
49C2×#95×C47C2212yes—
50C22×Dic94—24no—
51C22×Dic94—24no—
52C22×Dic94C2×C418yescomplemented
53C22×Dic94C2314yes—
54C23×Dic47—24no—
55C23×Dic47—24no—
56C23×Dic47—24no—
57C23×Dic47—24no—
58C23×Dic47—24no—
59C23×Dic47Q818yescomplemented
60C23×Dic47C2314yes—
61C23×Dic47C2312yes—
62C23×Dic47C2312yes—
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2311yescharacteristic
65C23×Dic47C2314yes—
66C23×Dic47C2311yescharacteristic
67C23×Dic47C2311yescharacteristic
68#13094003C2×C4164yescomplemented
69#13093989—232no—
70C2×C4⋊Dic47C2×C418yescomplemented
71C2×C4⋊Dic47C2314yes—
72C2×Dic47⋊C4—28no—
73C2×Dic47⋊C4—28no—
74C2×Dic47⋊C4C2×C4116yescomplemented
75C2×Dic47⋊C4C2318yes—
76C2×Dic47⋊C4C2318yes—
77C2×Dic47⋊C4C2318yes—
78C2×C22⋊Dic47C2×C4116yes—
79C2×C22⋊Dic47C2318yes—
80C2×C4×Dic47—24no—
81C2×C4×Dic47—24no—
82C2×C4×Dic47—24no—
83C2×C4×Dic47—28no—
84C2×C4×Dic47—28no—
85C2×C4×Dic47—232no—
86C2×C4×Dic47D418yes—
87C2×C4×Dic47C2318yes—
88C2×C4×Dic47C2318yes—
89C2×C4×Dic47C2314yes—
90C2×C4×Dic47C2314yes—
91C2×C4×Dic47C2314yes—
92C2×C4×Dic47C2314yes—
93C47⋊#95C2318yes—
94C47⋊#95C2318yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C25×C47C2311yescharacteristic
98C23×C4×C47—24no—
99C23×C4×C47—24no—
100C23×C4×C47C2×C418yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2312yes—
103C23×C4×C47C2311yescharacteristic
104C23×C4×C47C2314yes—
105C23×C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47C2314yes—
109C2×C4⋊C4×C47C2314yes—
110C2×C22⋊C4×C47—28no—
111C2×C22⋊C4×C47C2318yes—
112#95×C47C2318yes—
113C24×C47—22no—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47C22×C414yes—
118C24×C47C2×Q814yes—
119C24×C47#4114yes—
120C24×C47C2412yes—
121C24×C47C2411yescharacteristic
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—232no—
125C22×C4×C47—28no—
126C22×C4×C47—28no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C4116yes—
135C22×C4×C47C22×C414yes—
136C22×C4×C47C2×D414yes—
137C22×C4×C47#4114yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×Dic47—232no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—28no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—24no—
154C22×Dic47—232no—
155C22×Dic47—28no—
156C22×Dic47—28no—
157C22×Dic47—28no—
158C22×Dic47—28no—
159C22×Dic47C22×C418yes—
160C22×Dic47C22×C418yes—
161C22×Dic47C2×D414yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×Q814yes—
164C22×Dic47C2×Q814yes—
165C22×Dic47C2×Q8116yescomplemented
166C22×Dic47#4114yes—
167C22×Dic47#4114yes—
168C22×Dic47C2414yes—
169C22×Dic47C2414yes—
170C22×Dic47C2412yes—
171C22×Dic47C2412yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C2×Dic94—232no—
175C2×Dic94—28no—
176C2×Dic94—28no—
177C2×Dic94C22×C4116yescomplemented
178C4⋊C4×C47—28no—
179C4⋊C4×C47—28no—
180C22⋊C4×C47—216no—
181C22⋊Dic47C22×C4132yes—
182C4⋊Dic47C22×C4116yescomplemented
183Dic47⋊C4—216no—
184Dic47⋊C4—216no—
185Dic47⋊C4C22×C4132yescomplemented
186C4×Dic47—28no—
187C4×Dic47—28no—
188C4×Dic47—28no—
189C4×Dic47—216no—
190C4×Dic47—216no—
191C4×Dic47C2×D4116yes—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—28no—
195C23×C47—28no—
196C23×C47—28no—
197C23×C47—28no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47C4×Q8116yescomplemented
203C23×C47C23×C414yes—
204C23×C47C23×C412yes—
205C23×C47C22×D412yes—
206C23×C47C22×Q814yes—
207C23×C47C22×Q812yes—
208C23×C47C2×#4112yes—
209C23×C47C2×#4112yes—
210C23×C47C2×#4114yes—
211C23×C47C2×#4112yes—
212C23×C47C2×#4112yes—
213C23×C47C2511yescharacteristic
214C2×C4×C47—28no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—232no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—48no—
225C2×C4×C47—48no—
226C2×C4×C47C4×D4116yes—
227C2×C4×C47C23×C414yes—
228C2×C4×C47C23×C414yes—
229C2×C4×C47C22×D414yes—
230C2×C4×C47C2×#4114yes—
231C2×Dic47—232no—
232C2×Dic47—216no—
233C2×Dic47—216no—
234C2×Dic47—28no—
235C2×Dic47—28no—
236C2×Dic47—28no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—232no—
248C2×Dic47C22⋊Q8116yescomplemented
249C2×Dic47C23×C418yes—
250C2×Dic47C23×C418yes—
251C2×Dic47C22×D414yes—
252C2×Dic47C22×D414yes—
253C2×Dic47C22×Q814yes—
254C2×Dic47C22×Q814yes—
255C2×Dic47C2×#4114yes—
256C2×Dic47C2×#4114yes—
257Dic94—232no—
258C2×C4×C22⋊Q8—471nomaximal, Sylow
259C22×C47—24no—
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—232no—
266C22×C47C2×C4×D414yes—
267C22×C47C2×C4×Q814yes—
268C22×C47C2×C4×Q818yescomplemented
269C22×C47C4×#4114yes—
270C22×C47C2×C22⋊Q814yes—
271C22×C47#52514yes—
272C22×C47#52914yes—
273C22×C47#53314yes—
274C22×C47C24×C411yescharacteristic
275C22×C47C23×D411yescharacteristic
276C22×C47C23×Q811yescharacteristic
277C22×C47C22×#4111yescharacteristic
278C22×C47C22×#4111yescharacteristic
279C22×C47C22×#4111yescharacteristic
280C22×C47C22×#4111yescharacteristic
281C4×C47—24no—
282C4×C47—44no—
283C4×C47—44no—
284C4×C47C2×C4×D418yes—
285Dic47—28no—
286Dic47—28no—
287Dic47—24no—
288Dic47—24no—
289Dic47—24no—
290Dic47C2×C22⋊Q818yescomplemented
291C22×C22⋊Q8—471no—
292C22×C4×Q8—471no—
293C23×C42—471no—
294C4×C22⋊Q8—4716no—
295C2×#391—471no—
296C2×#389—471no—
297C2×#387—472no—
298C2×#385—472no—
299C2×#380D4711yescharacteristic
300C2×C4×C4⋊C4—472no—
301C2×C4×C4⋊C4—471no—
302C2×C4×C22⋊C4—472no—
303C2×C47—28no—
304C2×C47C4×C22⋊Q818yescomplemented
305C2×C47C22×C4×D411yescharacteristic
306C2×C47C22×C4×Q811yescharacteristic
307C2×C47C2×C4×#4111yescharacteristic
308C2×C47C22×C22⋊Q811yescharacteristic
309C2×C47C2×#52511yescharacteristic
310C2×C47C2×#52911yescharacteristic
311C2×C47C2×#53311yescharacteristic
312C23×Q8—471no—
313C24×C4—471no—
314C24×C4—471no—
315C24×C4C2×D4711yescharacteristic
316C2×C22⋊Q8—4716no—
317C2×C22⋊Q8—478no—
318C2×C4×Q8—478no—
319C2×C4×Q8—948no—
320C22×C4⋊C4—472no—
321C22×C4⋊C4—471no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4C2×D4711yescharacteristic
325C22×C4⋊C4C2×D4711yescharacteristic
326C22×C22⋊C4—472no—
327C22×C22⋊C4C2×D4712yes—
328C22×C42—471no—
329C22×C42—471no—
330C22×C42—471no—
331C22×C42—471no—
332C22×C42—478no—
333C22×C42—472no—
334C22×C42—472no—
335C22×C42—948no—
336#391—478no—
337#389—478no—
338#387—4716no—
339#385—4716no—
340#380C2×D4718yes—
341C4×C4⋊C4—478no—
342C4×C4⋊C4—4716no—
343C4×C22⋊C4—4716no—
344C2×#95—472no—
345C2×#95—472no—
346C2×#95—472no—
347C2×#95—472no—
348C2×#95C2×D4712yes—
349C47C2×C4×C22⋊Q811yescharacteristic, complemented, Sylow
350C25C22×D4711yescharacteristic
351C22×Q8—478no—
352C22×Q8—474no—
353C22×Q8—944no—
354C22×Q8—944no—
355C23×C4—24no—
356C23×C4—24no—
357C23×C4—471no—
358C23×C4—478no—
359C23×C4—474no—
360C23×C4—472no—
361C23×C4—472no—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—471no—
365C23×C4—474no—
366C23×C4—944no—
367C23×C4—944no—
368C23×C4—944no—
369C23×C4—944no—
370C23×C4—944no—
371C23×C4C4×D4718yes—
372C23×C4C22×D4712yes—
373C23×C4C22×D4712yes—
374C23×C4C22×D4714yes—
375C23×C4C22×D4711yescharacteristic
376C23×C4C22×D4711yescharacteristic
377C22⋊Q8—4764no—
378C4×Q8—9432no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—4716no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—478no—
384C2×C4⋊C4—474no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—478no—
387C2×C4⋊C4—948no—
388C2×C4⋊C4—948no—
389C2×C4⋊C4C22×D4714yes—
390C2×C4⋊C4C22×D4714yes—
391C2×C22⋊C4—28no—
392C2×C22⋊C4—4716no—
393C2×C22⋊C4—478no—
394C2×C22⋊C4C22×D4718yes—
395C2×C42—478no—
396C2×C42—474no—
397C2×C42—478no—
398C2×C42—478no—
399C2×C42—474no—
400C2×C42—474no—
401C2×C42—474no—
402C2×C42—9432no—
403C2×C42—948no—
404C2×C42—948no—
405C2×C42—944no—
406C2×C42—944no—
407C2×C42—944no—
408#95—478no—
409#95—478no—
410#95—478no—
411#95—478no—
412#95C22×D4718yes—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24C2×C4×D4714yes—
418C24#1062535214yes—
419C24Q8×D4714yes—
420C24C23×D4712yes—
421C24C23×D4711yescharacteristic
422C2×Q8—4716no—
423C2×Q8—9432no—
424C2×Q8—948no—
425C2×Q8—948no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—232no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—474no—
438C22×C4—474no—
439C22×C4—474no—
440C22×C4—472no—
441C22×C4—478no—
442C22×C4—478no—
443C22×C4—474no—
444C22×C4—474no—
445C22×C4—4716no—
446C22×C4—474no—
447C22×C4—472no—
448C22×C4—472no—
449C22×C4—472no—
450C22×C4—474no—
451C22×C4—474no—
452C22×C4—948no—
453C22×C4—948no—
454C22×C4—9432no—
455C22×C4—948no—
456C22×C4—948no—
457C22×C4—9432no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—944no—
461C22×C4—944no—
462C22×C4—944no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—948no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—944no—
469C22×C4C2×C4×D4714yes—
470C22×C4C2×C4×D47116yes—
471C22×C4C2×C4×D4714yes—
472C22×C4D4×D4714yes—
473C22×C4#1062535214yes—
474C22×C4C23×D4714yes—
475C22×C4C23×D4714yes—
476C22×C4C23×D4712yes—
477C22×C4C23×D4712yes—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—4732no—
481C4⋊C4—4716no—
482C4⋊C4—9416no—
483C4⋊C4—9416no—
484C22⋊C4—216no—
485C22⋊C4—4732no—
486C42—4716no—
487C42—9416no—
488C42—948no—
489C42—948no—
490C42—9416no—
491C42—948no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23#13093989116yescomplemented
503C23C22×C4×D4714yes—
504C23C22×C4×D4712yes—
505C23C2×D4×D4712yes—
506C23C2×Q8×D4714yes—
507C23C2×Q8×D4712yes—
508C23#41×D4712yes—
509C23#41×D4712yes—
510C23C2×#1062535214yes—
511C23C2×#1062535212yes—
512C23C2×#1062535212yes—
513C23C24×D4711yescharacteristic
514Q8—9432no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—4716no—
528C2×C4—474no—
529C2×C4—474no—
530C2×C4—474no—
531C2×C4—478no—
532C2×C4—478no—
533C2×C4—474no—
534C2×C4—474no—
535C2×C4—474no—
536C2×C4—9432no—
537C2×C4—9432no—
538C2×C4—9416no—
539C2×C4—9416no—
540C2×C4—948no—
541C2×C4—948no—
542C2×C4—948no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—944no—
548C2×C4—942no—
549C2×C4—942no—
550C2×C4—942no—
551C2×C4—942no—
552C2×C4—944no—
553C2×C4(C22⋊D47)⋊C4116yes—
554C2×C4C22×C4×D4714yes—
555C2×C4C22×C4×D4714yes—
556C2×C4C2×D4×D4714yes—
557C2×C4C2×#1062535214yes—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—232no—
564C22—24no—
565C22C2×(C22⋊D47)⋊C414yes—
566C22C2×#1309398914yes—
567C22C2×#1309398918yescomplemented
568C22#1568959114yes—
569C22C22⋊Q8×D4714yes—
570C22#1568963314yes—
571C22#1568965414yes—
572C22#1568967514yes—
573C22C23×C4×D4711yescharacteristic
574C22C22×D4×D4711yescharacteristic
575C22C22×Q8×D4711yescharacteristic
576C22C2×#41×D4711yescharacteristic
577C22C2×#41×D4711yescharacteristic
578C22C22×#1062535211yescharacteristic
579C22C22×#1062535211yescharacteristic
580C4—24no—
581C4—44no—
582C4—44no—
583C4—478no—
584C4—948no—
585C4—948no—
586C4—944no—
587C4—944no—
588C4—944no—
589C4C2×(C22⋊D47)⋊C418yes—
590C2—28no—
591C2#2024591618yesdirect product factor
592C2C22×(C22⋊D47)⋊C411yescharacteristic
593C2C22×#1309398911yescharacteristic
594C2C2×#1568959111yescharacteristic
595C2C2×C22⋊Q8×D4711yescharacteristic
596C2C2×#1568963311yescharacteristic
597C2C2×#1568965411yescharacteristic
598C2C2×#1568967511yescharacteristic
subgroupG/H
centerC24C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C47×#380C2
socleC24×C47C24
prime radicalC25×C47C23
cosocleC23×C47C25
hypercenterC2×#380D47
hypocenterC47C2×C4×C22⋊Q8
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#13093989C2semidirect
4. #20245916C2direct (N × H)
5. C2×#15689028C2semidirect
6. C2×#15689019C2semidirect
7. C2×#15689005C2semidirect
9. C2×C4⋊C4×Dic47C2semidirect
10. C2×#15688974C2semidirect
16. C2×#13094003C4semidirect
19. C2×#13093989C22semidirect
33. #15689028C22semidirect
34. #15689019C22semidirect
35. #15689005C22semidirect
37. C4⋊C4×Dic47C22semidirect
38. #15688974C22semidirect
52. C22×Dic94C2×C4semidirect
59. C23×Dic47Q8semidirect
68. #13094003C2×C4semidirect
70. C2×C4⋊Dic47C2×C4semidirect
74. C2×Dic47⋊C4C2×C4semidirect
165. C22×Dic47C2×Q8semidirect
177. C2×Dic94C22×C4semidirect
182. C4⋊Dic47C22×C4semidirect
185. Dic47⋊C4C22×C4semidirect
202. C23×C47C4×Q8semidirect
248. C2×Dic47C22⋊Q8semidirect
268. C22×C47C2×C4×Q8semidirect
290. Dic47C2×C22⋊Q8semidirect
304. C2×C47C4×C22⋊Q8semidirect
349. C47C2×C4×C22⋊Q8semidirect
502. C23#13093989semidirect
567. C22C2×#13093989semidirect
591. C2#20245916direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940032
C22×#130939892
C23×C4×Dic472
#202459162
C2×#156890282
C2×#156890192
C2×#156890052
C2×#156889952
C2×C4⋊C4×Dic472
C2×#156889742
C2×C22⋊C4×Dic472
C2×C47×#3802
C2×C4×C22⋊Q847