Finite Group Databasebeta

← C2×#20244559 #26243701

Subgroups — C2×#20244559 #26243701

Order 12032.

1234 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689676C211yesmaximal, characteristic, complemented
2C2×#15689141C211yesmaximal, characteristic
3C22×#13093950C211yesmaximal, characteristic
4C2×#15689096C211yesmaximal, characteristic
5#20244559C218yesmaximal, direct product factor
6C2×#15688895C211yesmaximal, characteristic, complemented
7C2×#15688882C211yesmaximal, characteristic, complemented
8C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689676—22no—
12#15689676—22no—
13#15689676—22no—
14#15689676C2214yescomplemented
15C2×#13094023—21no—
16C2×#13094015—21no—
17C2×#13094001—22no—
18C2×C4×Dic94—22no—
19C2×#13093983C2211yescharacteristic
20C22×C4×Dic47C2211yescharacteristic
21#15689141—22no—
22#15689141C2214yes—
23C2×#13093952—22no—
24C2×#13093950—21no—
25C2×#13093950—21no—
26C2×#13093950—22no—
27C2×#13093950C2214yes—
28#15689096—22no—
29#15689096C2214yes—
30C2×#13093938—22no—
31#15688895C2214yescomplemented
32#15688882C2214yescomplemented
33C22×M16×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×H(3,ℤ/4)C2214yescomplemented
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—
53C23×Dic47—21no—
54C22×C4×D47—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329—21no—
60C2×#10625329C2312yes—
61C22×C47⋊C8—21no—
62#13094023—24no—
63#13094023—24no—
64#13094015—24no—
65#13094015—24no—
66#13094001—28no—
67#13094001—28no—
68C4×Dic94—28no—
69C4×Dic94—28no—
70#13093983—21no—
71#13093983—21no—
72#13093983D412yes—
73#13093983C2312yes—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47—21no—
76C2×Dic47⋊C4—22no—
77C2×Dic47⋊C4—22no—
78C2×C22⋊Dic47D412yes—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D412yes—
86C2×C4×Dic47C2312yes—
87#13093952—28no—
88#13093950—24no—
89#13093950—24no—
90#13093938—28no—
91C23×C4×C47C2311yescharacteristic
92C2×M16×C47—21no—
93C2×M16×C47—21no—
94C2×M16×C47—21no—
95C2×M16×C47—21no—
96C2×M16×C47—21no—
97C2×M16×C47C2312yes—
98C22×C8×C47—21no—
99#117×C47—21no—
100#117×C47—21no—
101#117×C47—21no—
102#117×C47—21no—
103#117×C47—21no—
104#117×C47D412yes—
105#117×C47C2312yes—
106#117×C47C2312yes—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47D412yes—
113C24×C47C2×D411yescharacteristic
114C23×D47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—41no—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×Q811yescharacteristic
123C22×C4×C47C2×Q811yescharacteristic
124C22×C4×C47#4111yescharacteristic
125C22×C4×C47#4111yescharacteristic
126C22×C4×C47#4111yescharacteristic
127C22×C4×C47#4111yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×C4×C47#4111yescharacteristic
130C22×C4×C47C2411yescharacteristic
131C2×C22⋊D47—22no—
132C2×C22⋊D47—22no—
133C2×C22⋊D47—41no—
134C2×C22⋊D47—41no—
135C2×C22⋊D47—41no—
136C2×C22⋊D47—41no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—21no—
142C22×Dic47—22no—
143C22×Dic47—44no—
144#10625350—22no—
145#10625350—22no—
146#10625350—22no—
147#10625350—22no—
148#10625350—22no—
149#10625350—22no—
150#10625350—22no—
151#10625350—22no—
152#10625350—41no—
153#10625350—41no—
154#10625350—41no—
155#10625350—41no—
156#10625350—41no—
157#10625350—41no—
158#10625350—41no—
159#10625350—41no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188—21no—
163C2×D188—21no—
164C2×D188—21no—
165C2×D188C2×Q812yescomplemented
166C2×C4×D47—22no—
167C2×C4×D47—22no—
168C2×C4×D47—41no—
169C2×C4×D47—41no—
170C2×C4×D47—41no—
171C2×C4×D47—41no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94—21no—
176C2×Dic94—28no—
177C2×Dic94—21no—
178C2×Dic94—48no—
179C2×Dic94C2×Q812yescomplemented
180M16×C47—22no—
181M16×C47—22no—
182M16×C47—41no—
183M16×C47—41no—
184C2×C8×C47—22no—
185C2×C8×C47—41no—
186C2×C8×C47—41no—
187C4⋊C4×C47—22no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47—22no—
190C4⋊C4×C47—41no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—22no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—41no—
195C42×C47—22no—
196C42×C47—22no—
197C42×C47—41no—
198C42×C47C2×D414yes—
199C22⋊Dic47—22no—
200C22⋊Dic47C2×D414yes—
201C4⋊Dic47—28no—
202C4⋊Dic47—22no—
203C4⋊Dic47—22no—
204Dic47⋊C4—28no—
205Dic47⋊C4—28no—
206C4×Dic47—22no—
207C4×Dic47—21no—
208C4×Dic47—21no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211C4×Dic47—22no—
212C4×Dic47C2×D412yes—
213C4×Dic47C2×D412yes—
214#10625329—22no—
215#10625329—22no—
216#10625329—41no—
217#10625329—41no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—41no—
220C2×C47⋊C8—41no—
221C23×C47—42no—
222C23×C47#12011yescharacteristic
223C23×C47#12011yescharacteristic
224C23×C47C22⋊Q811yescharacteristic
225C23×C47C22⋊Q811yescharacteristic
226C23×C47#12411yescharacteristic
227C23×C47#12411yescharacteristic
228C23×C47C22×D411yescharacteristic
229C22×D47—22no—
230C22×D47—41no—
231C22×D47—41no—
232C22×D47—82no—
233C2×C4×C47—24no—
234C2×C4×C47—44no—
235C2×C4×C47—42no—
236C2×C4×C47—82no—
237C2×C4×C47#12011yescharacteristic
238C2×C4×C47#12011yescharacteristic
239C2×C4×C47C22⋊Q811yescharacteristic
240C2×C4×C47C22⋊Q811yescharacteristic
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×C47#12411yescharacteristic
250C2×C4×C47#12411yescharacteristic
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×C47C22×D411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×Q811yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C22⋊D47—42no—
266C22⋊D47—42no—
267C22⋊D47—42no—
268C22⋊D47—42no—
269C2×Dic47—24no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—24no—
273C2×Dic47—22no—
274C2×Dic47—44no—
275C2×Dic47—44no—
276C2×Dic47—44no—
277C2×Dic47—42no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280D188—22no—
281D188—22no—
282D188—41no—
283D188—41no—
284C4×D47—42no—
285C4×D47—42no—
286C4×D47—42no—
287C4×D47—42no—
288Dic94—22no—
289Dic94—22no—
290Dic94—416no—
291Dic94—41no—
292Dic94—41no—
293C8×C47—42no—
294C47⋊C8—42no—
295C2×#2031—471nomaximal, Sylow
296C22×C47—22no—
297C22×C47—22no—
298C22×C47—22no—
299C22×C47—41no—
300C22×C47—41no—
301C22×C47—41no—
302C22×C47—41no—
303C22×C47#39311yescharacteristic
304C22×C47#39311yescharacteristic
305C22×C47#39311yescharacteristic
306C22×C47#39311yescharacteristic
307C22×C47C2×#12011yescharacteristic
308C22×C47C2×C22⋊Q811yescharacteristic
309C22×C47C2×#12411yescharacteristic
310C2×D47—42no—
311C2×D47—82no—
312C2×D47—82no—
313C2×D47—82no—
314C4×C47—44no—
315C4×C47—82no—
316C4×C47#39311yescharacteristic
317C4×C47#39311yescharacteristic
318C4×C47#39311yescharacteristic
319C4×C47#39311yescharacteristic
320C4×C47C2×C22⋊Q811yescharacteristic
321C4×C47C2×C22⋊Q811yescharacteristic
322C4×C47C2×#12411yescharacteristic
323C4×C47C2×#12411yescharacteristic
324Dic47—24no—
325Dic47—44no—
326Dic47—42no—
327Dic47—42no—
328C2×#533—471no—
329C2×#421—471no—
330C22×#104—471no—
331C2×#410—471no—
332#2031—478no—
333C2×#344—471no—
334C2×#339—471no—
335C2×H(3,ℤ/4)D4711yescharacteristic, complemented
336C2×C47—21no—
337C2×C47—21no—
338C2×C47—21no—
339C2×C47—21no—
340C2×C47—21no—
341C2×C47—21no—
342C2×C47#203112yescomplemented
343C2×C47C2×#39311yescharacteristic
344D47—82no—
345C22×#41—471no—
346C22×M16—471no—
347C22×M16C2×D4711yescharacteristic
348#533—474no—
349#533—942no—
350#533—942no—
351#533—942no—
352C2×C4⋊Q8—941no—
353C2×#125—941no—
354C2×C22⋊Q8—942no—
355C2×C4×Q8—942no—
356C2×#117—471no—
357C2×#117C2×D4711yescharacteristic
358C2×#117C2×D4711yescharacteristic
359C22×C42—471no—
360#421—474no—
361#421—942no—
362C2×#104—474no—
363C2×#104—941no—
364C2×#104—941no—
365C2×#104—942no—
366C2×#104—942no—
367#410—474no—
368#410—942no—
369C2×#99—942no—
370#344—474no—
371#339—474no—
372H(3,ℤ/4)C2×D4714yescomplemented
373C47C2×#203111yescharacteristic, complemented, Sylow
374C2×#41—472no—
375C2×#41—472no—
376C2×#41—941no—
377C2×#41—941no—
378C2×#41—941no—
379C2×#41—941no—
380C2×#41—941no—
381C2×#41—941no—
382C2×#41—941no—
383C2×#41—941no—
384C2×#41—941no—
385C2×#41—941no—
386C22×Q8—471no—
387C22×Q8—942no—
388C22×D4—471no—
389C22×D4—941no—
390C23×C4—941no—
391C23×C4—941no—
392C23×C4C22×D4711yescharacteristic
393C2×M16—21no—
394C2×M16—21no—
395C2×M16—21no—
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—472no—
399C2×M16—941no—
400C2×M16—941no—
401C2×M16—941no—
402C2×M16—941no—
403C2×M16—941no—
404C2×M16C22×D4712yes—
405C22×C8—21no—
406C22×C8—941no—
407C4⋊Q8—944no—
408C4⋊Q8—944no—
409#125—944no—
410#125—944no—
411C22⋊Q8—948no—
412C22⋊Q8—948no—
413C4×Q8—948no—
414C4×Q8—948no—
415#117—21no—
416#117—21no—
417#117—21no—
418#117—21no—
419#117—21no—
420#117—472no—
421#117—472no—
422#117—941no—
423#117—941no—
424#117D18812yes—
425#117C22×D4712yes—
426#117C22×D4712yes—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—942no—
430C2×C4⋊C4—942no—
431C2×C4⋊C4—942no—
432C2×C4⋊C4—941no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—472no—
436C2×C42—21no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—472no—
440C2×C42—472no—
441C2×C42—941no—
442C2×C42—941no—
443C2×C42—941no—
444C2×C42—941no—
445C2×C42D18812yes—
446#104—944no—
447#104—944no—
448#104—948no—
449#99—948no—
450C24—941no—
451C24D4×D4711yescharacteristic
452#41—942no—
453#41—942no—
454#41—942no—
455#41—942no—
456#41—942no—
457#41—942no—
458#41—942no—
459#41—942no—
460#41—1881no—
461#41—1881no—
462#41—1881no—
463#41—1881no—
464#41—1881no—
465#41—1881no—
466#41—1881no—
467#41—1881no—
468C2×Q8—472no—
469C2×Q8—948no—
470C2×Q8—941no—
471C2×Q8—941no—
472C2×Q8—941no—
473C2×Q8—941no—
474C2×Q8—941no—
475C2×Q8—1888no—
476C2×D4—472no—
477C2×D4—942no—
478C2×D4—942no—
479C2×D4—941no—
480C2×D4—941no—
481C2×D4—941no—
482C2×D4—941no—
483C2×D4—941no—
484C2×D4—1881no—
485C2×D4—1881no—
486C2×D4—1881no—
487C2×D4—1881no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—941no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—1884no—
499C22×C4—1881no—
500C22×C4—1881no—
501C22×C4—1881no—
502C22×C4—1881no—
503C22×C4C2×D18811yescharacteristic
504C22×C4C2×D18811yescharacteristic
505C22×C4#1062535011yescharacteristic
506C22×C4#1062535011yescharacteristic
507C22×C4#1062535011yescharacteristic
508C22×C4#1062535011yescharacteristic
509C22×C4D4×D4711yescharacteristic
510C22×C4D4×D4711yescharacteristic
511C22×C4D4×D4711yescharacteristic
512C22×C4#1062535211yescharacteristic
513C22×C4#1062535211yescharacteristic
514C22×C4Q8×D4711yescharacteristic
515C22×C4Q8×D4711yescharacteristic
516C22×C4C23×D4711yescharacteristic
517M16—22no—
518M16—22no—
519M16—41no—
520M16—41no—
521M16—942no—
522M16—942no—
523M16—1881no—
524M16—1881no—
525C2×C8—22no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—942no—
529C2×C8—1881no—
530C2×C8—1881no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—41no—
535C4⋊C4—948no—
536C4⋊C4—948no—
537C4⋊C4—948no—
538C4⋊C4—942no—
539C4⋊C4—942no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—41no—
544C22⋊C4—474no—
545C22⋊C4—942no—
546C42—22no—
547C42—22no—
548C42—41no—
549C42—472no—
550C42—472no—
551C42—942no—
552C42—941no—
553C42—941no—
554C42—942no—
555C42—942no—
556C42—942no—
557C42C2×D18814yes—
558C23—42no—
559C23—942no—
560C23—3762no—
561C23—1881no—
562C23—1881no—
563C23#1309399411yescharacteristic
564C23#1309399411yescharacteristic
565C23#1309400211yescharacteristic
566C23#1309400211yescharacteristic
567C23#1309401011yescharacteristic
568C23#1309401011yescharacteristic
569C23C2×D4×D4711yescharacteristic
570Q8—942no—
571Q8—942no—
572Q8—18816no—
573Q8—1881no—
574Q8—1881no—
575D4—942no—
576D4—942no—
577D4—1882no—
578D4—1882no—
579D4—1882no—
580D4—1882no—
581D4—1881no—
582D4—1881no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—944no—
591C2×C4—942no—
592C2×C4—1884no—
593C2×C4—1884no—
594C2×C4—1882no—
595C2×C4—1884no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4#1309399411yescharacteristic
603C2×C4#1309399411yescharacteristic
604C2×C4#1309400211yescharacteristic
605C2×C4#1309400211yescharacteristic
606C2×C4#1309400211yescharacteristic
607C2×C4#1309400211yescharacteristic
608C2×C4#1309400211yescharacteristic
609C2×C4#1309400211yescharacteristic
610C2×C4#1309400411yescharacteristic
611C2×C4#1309400411yescharacteristic
612C2×C4#1309400411yescharacteristic
613C2×C4#1309400411yescharacteristic
614C2×C4#1309401011yescharacteristic
615C2×C4#1309401011yescharacteristic
616C2×C4#1309401011yescharacteristic
617C2×C4#1309401011yescharacteristic
618C2×C4#1309401011yescharacteristic
619C2×C4#1309401011yescharacteristic
620C2×C4#1309401211yescharacteristic
621C2×C4#1309401211yescharacteristic
622C2×C4#1309401211yescharacteristic
623C2×C4#1309401211yescharacteristic
624C2×C4C2×D4×D4711yescharacteristic
625C2×C4C22×D18811yescharacteristic
626C2×C4C2×Q8×D4711yescharacteristic
627C2×C4C2×#1062535211yescharacteristic
628C2×C4C2×#1062535011yescharacteristic
629C2×C4C2×#1062535011yescharacteristic
630C8—42no—
631C8—1882no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—3762no—
640C22—3762no—
641C22—3762no—
642C22—1882no—
643C22#1568903611yescharacteristic
644C22#1568903611yescharacteristic
645C22#1568903611yescharacteristic
646C22#1568903611yescharacteristic
647C22C2×#1309399411yescharacteristic
648C22C2×#1309400211yescharacteristic
649C22C2×#1309401011yescharacteristic
650C4—44no—
651C4—82no—
652C4—944no—
653C4—1882no—
654C4—1884no—
655C4—1882no—
656C4#1568903611yescharacteristic
657C4#1568903611yescharacteristic
658C4#1568903611yescharacteristic
659C4#1568903611yescharacteristic
660C4C2×#1309400211yescharacteristic
661C4C2×#1309400411yescharacteristic
662C4C2×#1309401011yescharacteristic
663C4C2×#1309401211yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—3762no—
671C2#2024455912yesdirect product factor
672C2C2×#1568903611yescharacteristic
subgroupG/H
centerC2×C4#13094012
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×H(3,ℤ/4)C2
socleC22×C47#393
prime radicalC22×#10625350C22
hypercenterC2×H(3,ℤ/4)D47
hypocenterC47C2×#2031
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#15689676C2semidirect
5. #20244559C2direct (N × H)
6. C2×#15688895C2semidirect
7. C2×#15688882C2semidirect
8. C2×C47×H(3,ℤ/4)C2semidirect
14. #15689676C22semidirect
31. #15688895C22semidirect
32. #15688882C22semidirect
36. C47×H(3,ℤ/4)C22semidirect
47. C2×#10625350Q8semidirect
50. C22×Dic94Q8semidirect
51. C22×D188Q8semidirect
165. C2×D188C2×Q8semidirect
179. C2×Dic94C2×Q8semidirect
335. C2×H(3,ℤ/4)D47semidirect
342. C2×C47#2031semidirect
372. H(3,ℤ/4)C2×D47semidirect
373. C47C2×#2031semidirect
671. C2#20244559direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896762
C2×#156891412
C22×#130939502
C2×#156890962
#202445592
C2×#156888952
C2×#156888822
C2×C47×H(3,ℤ/4)2
C2×#203147