Finite Group Databasebeta

← C2×#20244561 #26243703

Subgroups — C2×#20244561 #26243703

Order 12032.

1234 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689680C211yesmaximal, characteristic, complemented
2C2×#15689139C211yesmaximal, characteristic
3C22×#13093952C211yesmaximal, characteristic
4C2×#15689096C211yesmaximal, characteristic
5#20244561C218yesmaximal, direct product factor
6C2×#15688894C211yesmaximal, characteristic, complemented
7C2×#15688879C211yesmaximal, characteristic, complemented
8C2×C47×#339C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689680—22no—
12#15689680—22no—
13#15689680—22no—
14#15689680C2214yescomplemented
15C2×#13094024—21no—
16C2×#13094016—21no—
17C2×#13094003—22no—
18C2×C4×Dic94—22no—
19C2×#13093983C2211yescharacteristic
20C2×#13093983C2211yescharacteristic
21#15689139—22no—
22#15689139C2214yes—
23C2×#13093952—21no—
24C2×#13093952—21no—
25C2×#13093952—22no—
26C2×#13093952C2214yes—
27C2×#13093950—22no—
28#15689096—22no—
29#15689096C2214yes—
30C2×#13093938—22no—
31#15688894C2214yescomplemented
32#15688879C2214yescomplemented
33C22×M16×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C22×C42×C47C2211yescharacteristic
36C47×#339C2214yescomplemented
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—21no—
44C2×#10625350—21no—
45C2×#10625350—21no—
46C2×#10625350—21no—
47C2×#10625350Q812yescomplemented
48C2×#10625350C2312yes—
49C22×Dic94—22no—
50C22×Dic94Q811yescharacteristic, complemented
51C22×D188Q811yescharacteristic, complemented
52C22×C22⋊D47—21no—
53C22×C4×D47—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329C2312yes—
60C22×C47⋊C8—21no—
61#13094024—24no—
62#13094024—24no—
63#13094016—24no—
64#13094016—24no—
65#13094003—28no—
66#13094003—28no—
67C4×Dic94—28no—
68C4×Dic94—28no—
69#13093983—21no—
70#13093983—21no—
71#13093983—21no—
72#13093983—21no—
73#13093983D412yes—
74#13093983D412yes—
75#13093983C2312yes—
76#13093983C2312yes—
77C2×C4⋊Dic47—22no—
78C2×C4⋊Dic47—21no—
79C2×C4⋊Dic47—21no—
80C2×Dic47⋊C4—24no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47D412yes—
83C2×C4×Dic47—21no—
84C2×C4×Dic47D412yes—
85#13093952—24no—
86#13093952—24no—
87#13093950—28no—
88#13093938—28no—
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91C2×M16×C47—21no—
92C2×M16×C47—21no—
93C2×M16×C47—21no—
94C2×M16×C47—21no—
95C2×M16×C47—21no—
96C2×M16×C47C2312yes—
97C22×C8×C47—21no—
98#117×C47—21no—
99#117×C47—21no—
100#117×C47—21no—
101#117×C47C2312yes—
102C2×C4⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—21no—
105C2×C42×C47—21no—
106C2×C42×C47—21no—
107C2×C42×C47—21no—
108C2×C42×C47D411yescharacteristic
109C2×C42×C47D411yescharacteristic
110C2×C42×C47D412yes—
111C2×C42×C47C2312yes—
112C24×C47C2×D411yescharacteristic
113C23×D47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—44no—
119C22×C4×C47—41no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×Q811yescharacteristic
126C22×C4×C47C2×Q811yescharacteristic
127C22×C4×C47#4111yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×C4×C47#4111yescharacteristic
130C22×C4×C47#4111yescharacteristic
131C22×C4×C47#4111yescharacteristic
132C22×C4×C47#4111yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—22no—
135C2×C22⋊D47—22no—
136C2×C22⋊D47—41no—
137C2×C22⋊D47—41no—
138C2×C22⋊D47—41no—
139C2×C22⋊D47—41no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—21no—
143#10625350—22no—
144#10625350—22no—
145#10625350—22no—
146#10625350—22no—
147#10625350—22no—
148#10625350—22no—
149#10625350—22no—
150#10625350—22no—
151#10625350—41no—
152#10625350—41no—
153#10625350—41no—
154#10625350—41no—
155#10625350—41no—
156#10625350—41no—
157#10625350—41no—
158#10625350—41no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188—21no—
163C2×D188—21no—
164C2×D188C2×Q812yescomplemented
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×C4×D47—41no—
168C2×C4×D47—41no—
169C2×C4×D47—41no—
170C2×C4×D47—41no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94—28no—
176C2×Dic94—21no—
177C2×Dic94—48no—
178C2×Dic94C2×Q812yescomplemented
179M16×C47—22no—
180M16×C47—22no—
181M16×C47—41no—
182M16×C47—41no—
183C2×C8×C47—22no—
184C2×C8×C47—41no—
185C2×C8×C47—41no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47—41no—
188C22⋊C4×C47—22no—
189C22⋊C4×C47—41no—
190C42×C47—22no—
191C42×C47—21no—
192C42×C47—21no—
193C42×C47—22no—
194C42×C47—22no—
195C42×C47—41no—
196C42×C47C2×D412yes—
197C42×C47C2×D412yes—
198C22⋊Dic47—22no—
199C22⋊Dic47—22no—
200C22⋊Dic47—22no—
201C22⋊Dic47C2×D414yes—
202C4⋊Dic47—28no—
203C4⋊Dic47—22no—
204C4⋊Dic47—22no—
205C4⋊Dic47—22no—
206C4⋊Dic47—22no—
207Dic47⋊C4—216no—
208C4×Dic47—22no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211C4×Dic47C2×D414yes—
212#10625329—22no—
213#10625329—22no—
214#10625329—41no—
215#10625329—41no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—41no—
218C2×C47⋊C8—41no—
219C23×C47—42no—
220C23×C47#12011yescharacteristic
221C23×C47#12011yescharacteristic
222C23×C47C22⋊Q811yescharacteristic
223C23×C47C22⋊Q811yescharacteristic
224C23×C47#12411yescharacteristic
225C23×C47#12411yescharacteristic
226C23×C47C22×D411yescharacteristic
227C22×D47—22no—
228C22×D47—41no—
229C22×D47—41no—
230C22×D47—82no—
231C2×C4×C47—24no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—44no—
235C2×C4×C47—44no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—82no—
239C2×C4×C47#12011yescharacteristic
240C2×C4×C47#12011yescharacteristic
241C2×C4×C47C22⋊Q811yescharacteristic
242C2×C4×C47C22⋊Q811yescharacteristic
243C2×C4×C47C22⋊Q811yescharacteristic
244C2×C4×C47C22⋊Q811yescharacteristic
245C2×C4×C47C22⋊Q811yescharacteristic
246C2×C4×C47C22⋊Q811yescharacteristic
247C2×C4×C47C22⋊Q811yescharacteristic
248C2×C4×C47C22⋊Q811yescharacteristic
249C2×C4×C47C22⋊Q811yescharacteristic
250C2×C4×C47C22⋊Q811yescharacteristic
251C2×C4×C47#12411yescharacteristic
252C2×C4×C47#12411yescharacteristic
253C2×C4×C47#12411yescharacteristic
254C2×C4×C47#12411yescharacteristic
255C2×C4×C47#12411yescharacteristic
256C2×C4×C47#12411yescharacteristic
257C2×C4×C47#12411yescharacteristic
258C2×C4×C47#12411yescharacteristic
259C2×C4×C47#12411yescharacteristic
260C2×C4×C47#12411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×Q811yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C22⋊D47—42no—
268C22⋊D47—42no—
269C22⋊D47—42no—
270C22⋊D47—42no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—22no—
274C2×Dic47—44no—
275C2×Dic47—44no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278D188—22no—
279D188—22no—
280D188—41no—
281D188—41no—
282C4×D47—42no—
283C4×D47—42no—
284C4×D47—42no—
285C4×D47—42no—
286Dic94—22no—
287Dic94—22no—
288Dic94—416no—
289Dic94—41no—
290Dic94—41no—
291C8×C47—42no—
292C47⋊C8—42no—
293C2×#2031—471nomaximal, Sylow
294C22×C47—22no—
295C22×C47—22no—
296C22×C47—22no—
297C22×C47—41no—
298C22×C47—41no—
299C22×C47—41no—
300C22×C47—41no—
301C22×C47#39311yescharacteristic
302C22×C47#39311yescharacteristic
303C22×C47#39311yescharacteristic
304C22×C47#39311yescharacteristic
305C22×C47C2×#12011yescharacteristic
306C22×C47C2×C22⋊Q811yescharacteristic
307C22×C47C2×#12411yescharacteristic
308C2×D47—42no—
309C2×D47—82no—
310C2×D47—82no—
311C2×D47—82no—
312C4×C47—24no—
313C4×C47—42no—
314C4×C47—82no—
315C4×C47#39311yescharacteristic
316C4×C47#39311yescharacteristic
317C4×C47#39311yescharacteristic
318C4×C47#39311yescharacteristic
319C4×C47C2×C22⋊Q811yescharacteristic
320C4×C47C2×C22⋊Q811yescharacteristic
321C4×C47C2×#12411yescharacteristic
322C4×C47C2×#12411yescharacteristic
323Dic47—44no—
324Dic47—44no—
325Dic47—42no—
326C2×#533—471no—
327C2×#421—471no—
328C22×#104—471no—
329C2×#410—471no—
330#2031—478no—
331C2×#344—471no—
332C2×#339D4711yescharacteristic, complemented
333C2×H(3,ℤ/4)—471no—
334C2×C47—21no—
335C2×C47—21no—
336C2×C47—21no—
337C2×C47—21no—
338C2×C47—21no—
339C2×C47—21no—
340C2×C47#203112yescomplemented
341C2×C47C2×#39311yescharacteristic
342D47—82no—
343C22×#41—471no—
344C22×M16—471no—
345C22×M16C2×D4711yescharacteristic
346#533—474no—
347#533—942no—
348#533—942no—
349#533—942no—
350C2×C4⋊Q8—941no—
351C2×#125—941no—
352C2×C22⋊Q8—942no—
353C2×C4×Q8—942no—
354C2×#117—471no—
355C2×#117—471no—
356C2×#117C2×D4711yescharacteristic
357C22×C42C2×D4711yescharacteristic
358#421—474no—
359#421—942no—
360C2×#104—474no—
361C2×#104—942no—
362C2×#104—941no—
363C2×#104—941no—
364C2×#104—942no—
365#410—474no—
366#410—942no—
367C2×#99—942no—
368#344—474no—
369#339C2×D4714yescomplemented
370H(3,ℤ/4)—474no—
371C47C2×#203111yescharacteristic, complemented, Sylow
372C2×#41—472no—
373C2×#41—472no—
374C2×#41—941no—
375C2×#41—941no—
376C2×#41—941no—
377C2×#41—941no—
378C2×#41—941no—
379C2×#41—941no—
380C2×#41—941no—
381C2×#41—941no—
382C2×#41—941no—
383C2×#41—941no—
384C22×Q8—471no—
385C22×Q8—942no—
386C22×D4—471no—
387C22×D4—941no—
388C23×C4—21no—
389C23×C4—941no—
390C23×C4C22×D4711yescharacteristic
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—21no—
394C2×M16—21no—
395C2×M16—21no—
396C2×M16—472no—
397C2×M16—941no—
398C2×M16—941no—
399C2×M16—941no—
400C2×M16—941no—
401C2×M16—941no—
402C2×M16C22×D4712yes—
403C22×C8—21no—
404C22×C8—941no—
405C4⋊Q8—944no—
406C4⋊Q8—944no—
407#125—944no—
408#125—944no—
409C22⋊Q8—948no—
410C22⋊Q8—948no—
411C4×Q8—948no—
412C4×Q8—948no—
413#117—21no—
414#117—21no—
415#117—21no—
416#117—472no—
417#117—472no—
418#117—472no—
419#117—472no—
420#117—941no—
421#117—941no—
422#117—941no—
423#117—941no—
424#117C22×D4712yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—944no—
427C2×C4⋊C4—942no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—941no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—472no—
432C2×C22⋊C4—941no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—472no—
438C2×C42—941no—
439C2×C42D18811yescharacteristic
440C2×C42D18811yescharacteristic
441C2×C42D18812yes—
442C2×C42C22×D4712yes—
443#104—948no—
444#104—944no—
445#104—944no—
446#99—948no—
447C24—941no—
448C24D4×D4711yescharacteristic
449#41—942no—
450#41—942no—
451#41—942no—
452#41—942no—
453#41—942no—
454#41—942no—
455#41—942no—
456#41—942no—
457#41—1881no—
458#41—1881no—
459#41—1881no—
460#41—1881no—
461#41—1881no—
462#41—1881no—
463#41—1881no—
464#41—1881no—
465C2×Q8—472no—
466C2×Q8—948no—
467C2×Q8—941no—
468C2×Q8—941no—
469C2×Q8—941no—
470C2×Q8—941no—
471C2×Q8—941no—
472C2×Q8—1888no—
473C2×D4—472no—
474C2×D4—942no—
475C2×D4—942no—
476C2×D4—941no—
477C2×D4—941no—
478C2×D4—941no—
479C2×D4—941no—
480C2×D4—941no—
481C2×D4—1881no—
482C2×D4—1881no—
483C2×D4—1881no—
484C2×D4—1881no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—41no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—941no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—1881no—
497C22×C4—1881no—
498C22×C4—1881no—
499C22×C4—1881no—
500C22×C4C2×D18811yescharacteristic
501C22×C4C2×D18811yescharacteristic
502C22×C4#1062535011yescharacteristic
503C22×C4#1062535011yescharacteristic
504C22×C4#1062535011yescharacteristic
505C22×C4#1062535011yescharacteristic
506C22×C4D4×D4711yescharacteristic
507C22×C4D4×D4711yescharacteristic
508C22×C4D4×D4711yescharacteristic
509C22×C4#1062535211yescharacteristic
510C22×C4#1062535211yescharacteristic
511C22×C4Q8×D4711yescharacteristic
512C22×C4Q8×D4711yescharacteristic
513C22×C4C23×D4711yescharacteristic
514M16—22no—
515M16—22no—
516M16—41no—
517M16—41no—
518M16—942no—
519M16—942no—
520M16—1881no—
521M16—1881no—
522C2×C8—22no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—942no—
526C2×C8—1881no—
527C2×C8—1881no—
528C4⋊C4—22no—
529C4⋊C4—41no—
530C4⋊C4—9416no—
531C4⋊C4—948no—
532C4⋊C4—942no—
533C4⋊C4—942no—
534C4⋊C4—942no—
535C4⋊C4—942no—
536C22⋊C4—22no—
537C22⋊C4—41no—
538C22⋊C4—474no—
539C22⋊C4—942no—
540C22⋊C4—942no—
541C22⋊C4—942no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—474no—
549C42—942no—
550C42—942no—
551C42—942no—
552C42C2×D18812yes—
553C42C2×D18812yes—
554C23—42no—
555C23—942no—
556C23—3762no—
557C23—1881no—
558C23—1881no—
559C23#1309399411yescharacteristic
560C23#1309399411yescharacteristic
561C23#1309400211yescharacteristic
562C23#1309400211yescharacteristic
563C23#1309401011yescharacteristic
564C23#1309401011yescharacteristic
565C23C2×D4×D4711yescharacteristic
566Q8—942no—
567Q8—942no—
568Q8—18816no—
569Q8—1881no—
570Q8—1881no—
571D4—942no—
572D4—942no—
573D4—1882no—
574D4—1882no—
575D4—1882no—
576D4—1882no—
577D4—1881no—
578D4—1881no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—944no—
588C2×C4—944no—
589C2×C4—942no—
590C2×C4—1884no—
591C2×C4—1884no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4#1309399411yescharacteristic
599C2×C4#1309399411yescharacteristic
600C2×C4#1309400211yescharacteristic
601C2×C4#1309400211yescharacteristic
602C2×C4#1309400211yescharacteristic
603C2×C4#1309400211yescharacteristic
604C2×C4#1309400211yescharacteristic
605C2×C4#1309400211yescharacteristic
606C2×C4#1309400411yescharacteristic
607C2×C4#1309400411yescharacteristic
608C2×C4#1309400411yescharacteristic
609C2×C4#1309400411yescharacteristic
610C2×C4#1309401011yescharacteristic
611C2×C4#1309401011yescharacteristic
612C2×C4#1309401011yescharacteristic
613C2×C4#1309401011yescharacteristic
614C2×C4#1309401011yescharacteristic
615C2×C4#1309401011yescharacteristic
616C2×C4#1309401211yescharacteristic
617C2×C4#1309401211yescharacteristic
618C2×C4#1309401211yescharacteristic
619C2×C4#1309401211yescharacteristic
620C2×C4C2×D4×D4711yescharacteristic
621C2×C4C22×D18811yescharacteristic
622C2×C4C2×Q8×D4711yescharacteristic
623C2×C4C2×#1062535211yescharacteristic
624C2×C4C2×#1062535011yescharacteristic
625C2×C4C2×#1062535011yescharacteristic
626C8—42no—
627C8—1882no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—3762no—
636C22—3762no—
637C22—3762no—
638C22—1882no—
639C22#1568903611yescharacteristic
640C22#1568903611yescharacteristic
641C22#1568903611yescharacteristic
642C22#1568903611yescharacteristic
643C22C2×#1309399411yescharacteristic
644C22C2×#1309400211yescharacteristic
645C22C2×#1309401011yescharacteristic
646C4—24no—
647C4—42no—
648C4—82no—
649C4—1884no—
650C4—1884no—
651C4—1882no—
652C4#1568903611yescharacteristic
653C4#1568903611yescharacteristic
654C4#1568903611yescharacteristic
655C4#1568903611yescharacteristic
656C4C2×#1309400211yescharacteristic
657C4C2×#1309400411yescharacteristic
658C4C2×#1309401011yescharacteristic
659C4C2×#1309401211yescharacteristic
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—3762no—
667C2#2024456112yesdirect product factor
668C2C2×#1568903611yescharacteristic
subgroupG/H
centerC2×C4#13094010
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#339C2
socleC22×C47#393
prime radicalC22×#10625350C22
hypercenterC2×#339D47
hypocenterC47C2×#2031
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. C2×#15689680C2semidirect
5. #20244561C2direct (N × H)
6. C2×#15688894C2semidirect
7. C2×#15688879C2semidirect
8. C2×C47×#339C2semidirect
14. #15689680C22semidirect
31. #15688894C22semidirect
32. #15688879C22semidirect
36. C47×#339C22semidirect
47. C2×#10625350Q8semidirect
50. C22×Dic94Q8semidirect
51. C22×D188Q8semidirect
164. C2×D188C2×Q8semidirect
178. C2×Dic94C2×Q8semidirect
332. C2×#339D47semidirect
340. C2×C47#2031semidirect
369. #339C2×D47semidirect
371. C47C2×#2031semidirect
667. C2#20244561direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896802
C2×#156891392
C22×#130939522
C2×#156890962
#202445612
C2×#156888942
C2×#156888792
C2×C47×#3392
C2×#203147