Finite Group Databasebeta

← C2×#20243616 #26217134

Subgroups — C2×#20243616 #26217134

Order 12032.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic
4C2×#15688993C211yesmaximal, characteristic, complemented
5#20243616C218yesmaximal, direct product factor
6C2×#15688891C211yesmaximal, characteristic, complemented
7C2×#15688883C211yesmaximal, characteristic, complemented
8C22×#104×C47C211yesmaximal, characteristic
9C22×#10625329C2211yescharacteristic
10#15689606—22no—
11#15689606—22no—
12#15689606—22no—
13#15689606C2214yes—
14C2×Q8×Dic47—21no—
15C2×D4×Dic47—22no—
16C2×D4×Dic47—21no—
17C2×#13093983—21no—
18C2×#13093983C2211yescharacteristic
19C22×C4⋊Dic47—21no—
20C22×C4⋊Dic47—22no—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689165—22no—
24#15689165C2214yescomplemented
25C2×#13093958—21no—
26C2×#13093956—21no—
27#15689124—22no—
28#15689124C2214yes—
29C2×#13093948—21no—
30C2×#13093945—21no—
31#15688993—22no—
32#15688993C2214yescomplemented
33#15688891C2214yescomplemented
34#15688883C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C2×#104×C47C418yescomplemented
39C2×#104×C47C2214yes—
40C23×Dic47—22no—
41C23×Dic47—21no—
42C23×Dic47D412yes—
43C2×#10625329—21no—
44C2×#10625329—22no—
45C2×#10625329—22no—
46C2×#10625329C2312yes—
47C22×C47⋊C8—21no—
48Q8×Dic47—24no—
49Q8×Dic47—24no—
50D4×Dic47—28no—
51D4×Dic47—24no—
52D4×Dic47—24no—
53D4×Dic47—28no—
54#13093983—24no—
55#13093983—21no—
56#13093983—21no—
57#13093983—24no—
58#13093983D412yes—
59#13093983C2312yes—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—28no—
73C2×C4⋊Dic47—48no—
74C2×C4⋊Dic47D412yes—
75C2×C4⋊Dic47D411yescharacteristic
76C2×C4⋊Dic47D411yescharacteristic
77C2×C4⋊Dic47C2312yes—
78C2×C4⋊Dic47C2312yes—
79C2×C22⋊Dic47—22no—
80C2×C22⋊Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47—21no—
83#13093958—24no—
84#13093956—24no—
85#13093948—24no—
86#13093945—24no—
87C2×#41×C47—21no—
88C2×#41×C47—21no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47C2×C414yes—
92C2×#41×C47D412yes—
93C2×#41×C47Q814yes—
94C2×#41×C47C2312yes—
95C22×Q8×C47D411yescharacteristic
96C22×D4×C47—21no—
97C22×D4×C47D411yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—21no—
102C2×M16×C47—22no—
103C2×M16×C47C2×C414yescomplemented
104C2×M16×C47C2312yes—
105C22×C8×C47—21no—
106C2×C42×C47—21no—
107C2×C42×C47—22no—
108C2×C42×C47C2×C414yescomplemented
109C2×C42×C47C2312yes—
110#104×C47C2×C4116yescomplemented
111C24×C47—21no—
112C24×C47C2×D411yescharacteristic
113#41×C47—24no—
114#41×C47—24no—
115#41×C47—24no—
116#41×C47—24no—
117#41×C47—24no—
118#41×C47—22no—
119#41×C47—22no—
120#41×C47C4⋊C418yescomplemented
121#41×C47C22×C414yes—
122#41×C47C2×Q814yes—
123C2×Q8×C47—21no—
124C2×Q8×C47—22no—
125C2×Q8×C47C4⋊C414yescomplemented
126C2×Q8×C47C2×D412yes—
127C2×D4×C47—24no—
128C2×D4×C47—22no—
129C2×D4×C47—24no—
130C2×D4×C47—22no—
131C2×D4×C47—21no—
132C2×D4×C47—22no—
133C2×D4×C47C4⋊C414yescomplemented
134C2×D4×C47C2×D412yes—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—21no—
139C22×C4×C47—22no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×Q812yes—
151C22×C4×C47#4112yes—
152C22×C4×C47#4112yes—
153C22×C4×C47C2411yescharacteristic
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—28no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—44no—
161C22×Dic47—44no—
162C22×Dic47—41no—
163C22×Dic47—41no—
164C22×Dic47—84no—
165C22×Dic47C2×D414yes—
166M16×C47—24no—
167M16×C47—22no—
168M16×C47C22×C414yescomplemented
169C2×C8×C47—24no—
170C2×C8×C47—22no—
171C42×C47—24no—
172C42×C47—22no—
173C42×C47C22×C414yescomplemented
174C22⋊Dic47—28no—
175C22⋊Dic47—22no—
176C22⋊Dic47—22no—
177C4⋊Dic47—28no—
178C4⋊Dic47—22no—
179C4⋊Dic47—21no—
180C4⋊Dic47—22no—
181C4⋊Dic47—21no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—416no—
187C4⋊Dic47—41no—
188C4⋊Dic47—41no—
189C4⋊Dic47C2×D412yes—
190C4⋊Dic47C2×D412yes—
191C4×Dic47—28no—
192C4×Dic47—22no—
193C4×Dic47—22no—
194#10625329—22no—
195#10625329—22no—
196#10625329—42no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—42no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—24no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47C4×D412yes—
206C23×C47#12011yescharacteristic
207C23×C47#12011yescharacteristic
208C23×C47C22⋊Q812yes—
209C23×C47C22×D411yescharacteristic
210Q8×C47—22no—
211Q8×C47—24no—
212Q8×C47C2×C4⋊C414yescomplemented
213D4×C47—24no—
214D4×C47—24no—
215D4×C47—28no—
216D4×C47—22no—
217D4×C47—24no—
218D4×C47C2×C4⋊C414yescomplemented
219C2×C4×C47—24no—
220C2×C4×C47—216no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—28no—
228C2×C4×C47C2×C4⋊C412yes—
229C2×C4×C47C2×C4⋊C412yes—
230C2×C4×C47C4×D412yes—
231C2×C4×C47C4×D412yes—
232C2×C4×C47C4×D412yes—
233C2×C4×C47#12011yescharacteristic
234C2×C4×C47#12011yescharacteristic
235C2×C4×C47C22⋊Q812yes—
236C2×C4×C47C22⋊Q812yes—
237C2×C4×C47C22⋊Q812yes—
238C2×C4×C47#12312yes—
239C2×C4×C47#12312yes—
240C2×C4×C47C23×C411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×Q811yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×Dic47—22no—
247C2×Dic47—24no—
248C2×Dic47—22no—
249C2×Dic47—44no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—44no—
253C2×Dic47—44no—
254C2×Dic47—42no—
255C2×Dic47—82no—
256C2×Dic47—88no—
257C2×Dic47—82no—
258C2×Dic47—82no—
259C8×C47—24no—
260C47⋊C8—42no—
261C2×#1859—471nomaximal, Sylow
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—21no—
265C22×C47—21no—
266C22×C47—21no—
267C22×C47—21no—
268C22×C47—24no—
269C22×C47—48no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47#38512yes—
273C22×C47#38512yes—
274C22×C47#45312yes—
275C22×C47#45612yes—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×#12011yescharacteristic
278C22×C47C2×C22⋊Q811yescharacteristic
279C4×C47—28no—
280C4×C47—24no—
281C4×C47#38512yes—
282C4×C47#38512yes—
283C4×C47C22×C4⋊C411yescharacteristic
284C4×C47C2×C4×D411yescharacteristic
285C4×C47C2×C22⋊Q811yescharacteristic
286C4×C47C2×#12311yescharacteristic
287Dic47—44no—
288Dic47—42no—
289Dic47—42no—
290Dic47—82no—
291C2×#520—471no—
292C2×#428—471no—
293C22×#104D4711yescharacteristic
294C2×#417—471no—
295C2×#384—471no—
296#1859—478no—
297C2×#343—471no—
298C2×#339—471no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47—22no—
302C2×C47—44no—
303C2×C47#185914yescomplemented
304C2×C47C2×#38511yescharacteristic
305C2×C47C2×#45311yescharacteristic
306C2×C47C2×#45611yescharacteristic
307C22×#41C2×D4711yescharacteristic
308C22×M16—471no—
309C22×M16C2×D4711yescharacteristic
310#520—474no—
311#520—942no—
312#520—942no—
313#520—942no—
314C2×C4×Q8—941no—
315C2×C4×D4—942no—
316C2×C4×D4—941no—
317C2×#117—471no—
318C2×#117—941no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—471no—
321C22×C4⋊C4—941no—
322C22×C4⋊C4—942no—
323C22×C42C2×D4711yescharacteristic
324#428—474no—
325#428—942no—
326C2×#107—941no—
327C2×#106—941no—
328C2×#104Dic4718yescomplemented
329C2×#104C2×D4714yes—
330#417—474no—
331#417—942no—
332C2×#103—941no—
333C2×#102—941no—
334#384—474no—
335#384—942no—
336#343—474no—
337#339—474no—
338C47C2×#185911yescharacteristic, complemented, Sylow
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41Dic9414yes—
344C2×#41D18812yes—
345C2×#41C2×Dic4714yes—
346C2×#41C22×D4712yes—
347C22×Q8D18811yescharacteristic
348C22×D4—21no—
349C22×D4D18811yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—472no—
353C23×C4—942no—
354C23×C4—941no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—472no—
359C2×M16—941no—
360C2×M16—942no—
361C2×M16—942no—
362C2×M16C2×Dic4714yescomplemented
363C2×M16C22×D4712yes—
364C22×C8—21no—
365C22×C8—941no—
366C4×Q8—944no—
367C4×Q8—944no—
368C4×D4—948no—
369C4×D4—948no—
370C4×D4—944no—
371C4×D4—944no—
372#117—472no—
373#117—472no—
374#117—944no—
375#117—944no—
376#117—941no—
377#117—941no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4—472no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—944no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—948no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—1888no—
397C2×C22⋊C4—942no—
398C2×C22⋊C4—941no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—942no—
402C2×C42—941no—
403C2×C42C2×Dic4714yescomplemented
404C2×C42C22×D4712yes—
405#107—944no—
406#106—944no—
407#104C2×Dic47116yescomplemented
408#103—944no—
409#102—944no—
410C24—21no—
411C24D4×D4711yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41C4⋊Dic4718yescomplemented
420#41C2×Dic9414yes—
421#41C22×Dic4714yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8C4⋊Dic4714yescomplemented
425C2×Q8C2×D18812yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4C4⋊Dic4714yescomplemented
433C2×D4C2×D18812yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—474no—
444C22×C4—948no—
445C22×C4—942no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—3764no—
451C22×C4—1884no—
452C22×C4—1881no—
453C22×C4—1881no—
454C22×C4—1884no—
455C22×C4C2×Dic9412yes—
456C22×C4C2×D18811yescharacteristic
457C22×C4C2×D18811yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4D4×D4711yescharacteristic
460C22×C4D4×D4711yescharacteristic
461C22×C4#1062535212yes—
462C22×C4#1062535212yes—
463C22×C4C22×Dic4712yes—
464C22×C4C23×D4711yescharacteristic
465M16—24no—
466M16—22no—
467M16—942no—
468M16—942no—
469M16—1882no—
470M16C22×Dic4714yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—942no—
474C2×C8—1882no—
475C4⋊C4—472no—
476C4⋊C4—472no—
477C4⋊C4—948no—
478C4⋊C4—941no—
479C4⋊C4—942no—
480C4⋊C4—942no—
481C4⋊C4—941no—
482C4⋊C4—942no—
483C4⋊C4—942no—
484C4⋊C4—942no—
485C4⋊C4—942no—
486C4⋊C4—18816no—
487C4⋊C4—1881no—
488C4⋊C4—1881no—
489C22⋊C4—948no—
490C22⋊C4—942no—
491C22⋊C4—942no—
492C42—24no—
493C42—22no—
494C42—948no—
495C42—942no—
496C42—942no—
497C42C22×Dic4714yescomplemented
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23D4×Dic4712yes—
505C23#1309399411yescharacteristic
506C23#1309399411yescharacteristic
507C23#1309400112yes—
508C23C2×D4×D4711yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×C4⋊Dic4714yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×C4⋊Dic4714yescomplemented
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—944no—
530C2×C4—3762no—
531C2×C4—3762no—
532C2×C4—3762no—
533C2×C4—3768no—
534C2×C4—1884no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1884no—
538C2×C4—1882no—
539C2×C4—1884no—
540C2×C4C2×C4⋊Dic4712yes—
541C2×C4C2×C4⋊Dic4712yes—
542C2×C4D4×Dic4712yes—
543C2×C4D4×Dic4712yes—
544C2×C4D4×Dic4712yes—
545C2×C4#1309399411yescharacteristic
546C2×C4#1309399411yescharacteristic
547C2×C4#1309400112yes—
548C2×C4#1309400112yes—
549C2×C4#1309400112yes—
550C2×C4#1309400812yes—
551C2×C4#1309400812yes—
552C2×C4C23×Dic4711yescharacteristic
553C2×C4C2×D4×D4711yescharacteristic
554C2×C4C22×D18811yescharacteristic
555C2×C4C22×Dic9411yescharacteristic
556C2×C4C2×#1062535211yescharacteristic
557C2×C4C2×#1062535211yescharacteristic
558C8—24no—
559C8—1882no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1568899812yes—
571C22#1568899812yes—
572C22#1568929212yes—
573C22#1568930712yes—
574C22C2×D4×Dic4711yescharacteristic
575C22C2×#1309399411yescharacteristic
576C22C2×#1309400111yescharacteristic
577C4—28no—
578C4—24no—
579C4—3762no—
580C4—1882no—
581C4—1884no—
582C4—1882no—
583C4#1568899812yes—
584C4#1568899812yes—
585C4C22×C4⋊Dic4711yescharacteristic
586C4C2×D4×Dic4711yescharacteristic
587C4C2×#1309400111yescharacteristic
588C4C2×#1309400811yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2#2024361614yesdirect product factor
594C2C2×#1568899811yescharacteristic
595C2C2×#1568929211yescharacteristic
596C2C2×#1568930711yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#104×C47C2
socleC23×C47#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC22×#104D47
hypocenterC47C2×#1859
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
4. C2×#15688993C2semidirect
5. #20243616C2direct (N × H)
6. C2×#15688891C2semidirect
7. C2×#15688883C2semidirect
24. #15689165C22semidirect
32. #15688993C22semidirect
33. #15688891C22semidirect
34. #15688883C22semidirect
38. C2×#104×C47C4semidirect
103. C2×M16×C47C2×C4semidirect
108. C2×C42×C47C2×C4semidirect
110. #104×C47C2×C4semidirect
120. #41×C47C4⋊C4semidirect
125. C2×Q8×C47C4⋊C4semidirect
133. C2×D4×C47C4⋊C4semidirect
168. M16×C47C22×C4semidirect
173. C42×C47C22×C4semidirect
212. Q8×C47C2×C4⋊C4semidirect
218. D4×C47C2×C4⋊C4semidirect
303. C2×C47#1859semidirect
328. C2×#104Dic47semidirect
338. C47C2×#1859semidirect
362. C2×M16C2×Dic47semidirect
403. C2×C42C2×Dic47semidirect
407. #104C2×Dic47semidirect
419. #41C4⋊Dic47semidirect
424. C2×Q8C4⋊Dic47semidirect
432. C2×D4C4⋊Dic47semidirect
470. M16C22×Dic47semidirect
497. C42C22×Dic47semidirect
511. Q8C2×C4⋊Dic47semidirect
517. D4C2×C4⋊Dic47semidirect
593. C2#20243616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#156891652
C2×#156891242
C2×#156889932
#202436162
C2×#156888912
C2×#156888832
C22×#104×C472
C2×#185947