Finite Group Databasebeta

← C2×#20241512 #26131244

Subgroups — C2×#20241512 #26131244

Order 12032.

1154 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093939C211yesmaximal, characteristic
2C2×#15689026C211yesmaximal, characteristic
3#20241512C214yesmaximal, direct product factor
4C22×#99×C47C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C22×C22⋊Dic47—21no—
7C22×C22⋊Dic47—21no—
8C22×C22⋊Dic47—21no—
9C22×C4×Dic47C411yescharacteristic, complemented
10C2×#13093939—21no—
11C2×#13093939—22no—
12C2×#13093939C414yescomplemented
13C2×#13093939C2212yes—
14#15689026—21no—
15#15689026—21no—
16#15689026C412yescomplemented
17#15689026C2212yes—
18C23×D4×C47C2211yescharacteristic
19C22×C22⋊C4×C47—21no—
20C2×#99×C47—22no—
21C2×#99×C47—21no—
22C2×#99×C47C414yescomplemented
23C2×#99×C47C2212yes—
24C23×Dic47—21no—
25C23×Dic47—21no—
26C23×Dic47—41no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—41no—
29C2×C4⋊Dic47—41no—
30C2×C4⋊Dic47C2×C412yescomplemented
31C2×C22⋊Dic47—24no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—22no—
37C2×C22⋊Dic47—44no—
38C2×C22⋊Dic47—44no—
39C2×C22⋊Dic47—41no—
40C2×C22⋊Dic47—42no—
41C2×C4×Dic47—21no—
42C2×C4×Dic47—42no—
43C2×C4×Dic47C2×C412yescomplemented
44#13093939—24no—
45#13093939—22no—
46#13093939C2×C414yescomplemented
47C25×C47D411yescharacteristic
48C25×C47D411yescharacteristic
49C22×D4×C47—42no—
50C22×D4×C47—41no—
51C22×D4×C47—41no—
52C22×D4×C47C2×C412yes—
53C22×D4×C47C2×C412yes—
54C22×D4×C47C2×C411yescharacteristic
55C22×D4×C47D411yescharacteristic
56C22×D4×C47D411yescharacteristic
57C22×D4×C47D412yes—
58C22×D4×C47Q812yes—
59C22×D4×C47C2311yescharacteristic
60C23×C4×C47—41no—
61C23×C4×C47C2×C411yescharacteristic
62C2×C22⋊C4×C47—24no—
63C2×C22⋊C4×C47—22no—
64C2×C22⋊C4×C47—41no—
65C2×C22⋊C4×C47—42no—
66#99×C47—24no—
67#99×C47—22no—
68#99×C47C2×C414yescomplemented
69C24×C47—21no—
70C24×C47—21no—
71C24×C47—22no—
72C24×C47—21no—
73C24×C47—22no—
74C24×C47—21no—
75C24×C47—42no—
76C24×C47—42no—
77C24×C47—42no—
78C24×C47—42no—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C22⋊C412yes—
81C24×C47C22⋊C411yescharacteristic
82C24×C47C22⋊C411yescharacteristic
83C24×C47C22⋊C412yes—
84C24×C47C4⋊C412yes—
85C24×C47C4⋊C412yes—
86C24×C47C2×D411yescharacteristic
87C24×C47C2×D411yescharacteristic
88C2×D4×C47—42no—
89C2×D4×C47—42no—
90C2×D4×C47—42no—
91C2×D4×C47—42no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—42no—
95C2×D4×C47—42no—
96C2×D4×C47—41no—
97C2×D4×C47—41no—
98C2×D4×C47—41no—
99C2×D4×C47—41no—
100C2×D4×C47—42no—
101C2×D4×C47—42no—
102C2×D4×C47—41no—
103C2×D4×C47—41no—
104C2×D4×C47C4212yes—
105C2×D4×C47C22⋊C412yes—
106C2×D4×C47C22⋊C412yes—
107C2×D4×C47C4⋊C412yes—
108C2×D4×C47C4⋊C412yes—
109C2×D4×C47C4⋊C412yes—
110C2×D4×C47C22×C411yescharacteristic
111C2×D4×C47C22×C411yescharacteristic
112C2×D4×C47C2×D411yescharacteristic
113C2×D4×C47C2×Q811yescharacteristic
114C22×C4×C47—22no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—44no—
118C22×C4×C47—42no—
119C22×C4×C47—82no—
120C22×C4×C47—82no—
121C22×C4×C47C4212yes—
122C22×C4×C47C22⋊C411yescharacteristic
123C22×C4×C47C4⋊C412yes—
124C22×C4×C47C22×C411yescharacteristic
125C22×Dic47—24no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—42no—
129C22×Dic47—42no—
130C22×Dic47—44no—
131C22×Dic47—42no—
132C22×Dic47—42no—
133C22×Dic47—42no—
134C22×Dic47—41no—
135C22×Dic47—41no—
136C22×Dic47—82no—
137C22×Dic47—82no—
138C22⋊C4×C47—24no—
139C22⋊C4×C47—44no—
140C22⋊C4×C47—42no—
141C22⋊Dic47—24no—
142C22⋊Dic47—44no—
143C22⋊Dic47—44no—
144C22⋊Dic47—44no—
145C22⋊Dic47—44no—
146C22⋊Dic47—44no—
147C22⋊Dic47—42no—
148C4⋊Dic47—42no—
149C4⋊Dic47—42no—
150C4×Dic47—42no—
151C4×Dic47—42no—
152C23×C47—22no—
153C23×C47—22no—
154C23×C47—22no—
155C23×C47—22no—
156C23×C47—22no—
157C23×C47—22no—
158C23×C47—21no—
159C23×C47—21no—
160C23×C47—21no—
161C23×C47—21no—
162C23×C47—22no—
163C23×C47—22no—
164C23×C47—22no—
165C23×C47—22no—
166C23×C47—21no—
167C23×C47—21no—
168C23×C47—21no—
169C23×C47—21no—
170C23×C47—41no—
171C23×C47—41no—
172C23×C47—41no—
173C23×C47—41no—
174C23×C47—41no—
175C23×C47—41no—
176C23×C47—44no—
177C23×C47—44no—
178C23×C47—44no—
179C23×C47—44no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—81no—
187C23×C47—81no—
188C23×C47—81no—
189C23×C47—81no—
190C23×C47—84no—
191C23×C47—84no—
192C23×C47#9512yes—
193C23×C47#9512yes—
194C23×C47#9512yes—
195C23×C47#9911yescharacteristic
196C23×C47#9912yes—
197C23×C47#9911yescharacteristic
198C23×C47C2×C22⋊C411yescharacteristic
199C23×C47C2×C22⋊C411yescharacteristic
200C23×C47C2×C22⋊C411yescharacteristic
201C23×C47C2×C4⋊C411yescharacteristic
202C23×C47C2×C4⋊C411yescharacteristic
203D4×C47—42no—
204D4×C47—42no—
205D4×C47—42no—
206D4×C47—42no—
207D4×C47—42no—
208D4×C47—42no—
209D4×C47—41no—
210D4×C47—41no—
211D4×C47—41no—
212D4×C47—41no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—44no—
222C2×C4×C47—82no—
223C2×C4×C47—82no—
224C2×C4×C47—88no—
225C2×C4×C47#9512yes—
226C2×C4×C47C2×C4211yescharacteristic
227C2×C4×C47C2×C4⋊C411yescharacteristic
228C2×Dic47—24no—
229C2×Dic47—42no—
230C2×Dic47—48no—
231C2×Dic47—42no—
232C2×Dic47—44no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—82no—
237C2×Dic47—82no—
238C2×Dic47—82no—
239C2×Dic47—88no—
240C2×Dic47—82no—
241C2×#1393—471nomaximal, Sylow
242C22×C47—21no—
243C22×C47—21no—
244C22×C47—21no—
245C22×C47—21no—
246C22×C47—24no—
247C22×C47—24no—
248C22×C47—21no—
249C22×C47—21no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—21no—
253C22×C47—21no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—44no—
269C22×C47—44no—
270C22×C47—44no—
271C22×C47—44no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—82no—
277C22×C47—82no—
278C22×C47—82no—
279C22×C47—82no—
280C22×C47#34212yes—
281C22×C47#34212yes—
282C22×C47#35312yes—
283C22×C47#35412yes—
284C22×C47C2×#9511yescharacteristic
285C22×C47C2×#9911yescharacteristic
286C22×C47C2×#9911yescharacteristic
287C4×C47—22no—
288C4×C47—21no—
289C4×C47—21no—
290C4×C47—84no—
291Dic47—44no—
292Dic47—84no—
293Dic47—82no—
294C22×#99—471no—
295C22×#99D4711yescharacteristic
296C2×#390—471no—
297#1393—474no—
298C2×C47—21no—
299C2×C47—21no—
300C2×C47—22no—
301C2×C47—41no—
302C2×C47—41no—
303C2×C47—42no—
304C2×C47—42no—
305C2×C47—41no—
306C2×C47—41no—
307C2×C47#139314yescomplemented
308C2×C47C2×#34211yescharacteristic
309C2×C47C2×#35311yescharacteristic
310C2×C47C2×#35411yescharacteristic
311C23×D4C2×D4711yescharacteristic
312C22×C4⋊C4—471no—
313C22×C22⋊C4—21no—
314C22×C22⋊C4—941no—
315C22×C22⋊C4—941no—
316C22×C22⋊C4—941no—
317C22×C42—471no—
318C2×#99—21no—
319C2×#99—22no—
320C2×#99—474no—
321C2×#99—472no—
322C2×#99—941no—
323C2×#99—942no—
324C2×#99Dic4714yescomplemented
325C2×#99C2×D4712yes—
326#390—472no—
327#390—472no—
328#390—941no—
329#390—941no—
330C47C2×#139311yescharacteristic, complemented, Sylow
331C25D18811yescharacteristic
332C25C22⋊D4711yescharacteristic
333C22×D4—42no—
334C22×D4—41no—
335C22×D4—41no—
336C22×D4Dic9412yes—
337C22×D4C4×D4712yes—
338C22×D4C4×D4711yescharacteristic
339C22×D4D18811yescharacteristic
340C22×D4C2×Dic4712yes—
341C22×D4C22⋊D4712yes—
342C22×D4C22⋊D4711yescharacteristic
343C22×D4C22×D4711yescharacteristic
344C23×C4—41no—
345C23×C4—941no—
346C23×C4—941no—
347C23×C4—1881no—
348C23×C4C4×D4711yescharacteristic
349C2×C4⋊C4—472no—
350C2×C4⋊C4—941no—
351C2×C4⋊C4—1881no—
352C2×C4⋊C4—1881no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—944no—
358C2×C22⋊C4—942no—
359C2×C22⋊C4—942no—
360C2×C22⋊C4—942no—
361C2×C22⋊C4—942no—
362C2×C22⋊C4—942no—
363C2×C22⋊C4—1884no—
364C2×C22⋊C4—1884no—
365C2×C22⋊C4—1882no—
366C2×C22⋊C4—1881no—
367C2×C42—472no—
368C2×C42—941no—
369C2×C42—1882no—
370#99—24no—
371#99—22no—
372#99—474no—
373#99—944no—
374#99—942no—
375#99C2×Dic4714yescomplemented
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24Dic47⋊C412yes—
387C24C4⋊Dic4712yes—
388C24(C2×D47)⋊C411yescharacteristic
389C24(C2×D47)⋊C411yescharacteristic
390C24(C2×D47)⋊C412yes—
391C24(C2×D47)⋊C411yescharacteristic
392C24C22⋊Dic4712yes—
393C24C2×D18811yescharacteristic
394C24C2×C22⋊D4711yescharacteristic
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C4×Dic4712yes—
412C2×D4Dic47⋊C412yes—
413C2×D4Dic47⋊C412yes—
414C2×D4C4⋊Dic4712yes—
415C2×D4(C2×D47)⋊C412yes—
416C2×D4C22⋊Dic4712yes—
417C2×D4C2×Dic9411yescharacteristic
418C2×D4C2×C4×D4711yescharacteristic
419C2×D4C22×Dic4711yescharacteristic
420C2×D4C2×C22⋊D4711yescharacteristic
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—944no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—3762no—
432C22×C4—3762no—
433C22×C4—1882no—
434C22×C4—1884no—
435C22×C4—1882no—
436C22×C4—1882no—
437C22×C4—1881no—
438C22×C4—1881no—
439C22×C4—1882no—
440C22×C4—1882no—
441C22×C4C4×Dic4712yes—
442C22×C4Dic47⋊C412yes—
443C22×C4(C2×D47)⋊C411yescharacteristic
444C22×C4C2×C4×D4711yescharacteristic
445C4⋊C4—1882no—
446C4⋊C4—1882no—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—944no—
451C22⋊C4—1884no—
452C22⋊C4—1884no—
453C22⋊C4—1884no—
454C22⋊C4—1884no—
455C22⋊C4—1882no—
456C22⋊C4—1884no—
457C42—1882no—
458C42—1882no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—44no—
483C23—41no—
484C23—41no—
485C23—44no—
486C23—44no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—84no—
498C23—84no—
499C23C47⋊#9512yes—
500C23C47⋊#9512yes—
501C23C47⋊#9512yes—
502C23#1309393811yescharacteristic
503C23#1309393812yes—
504C23#1309393811yescharacteristic
505C23C2×(C2×D47)⋊C411yescharacteristic
506C23C2×(C2×D47)⋊C411yescharacteristic
507C23C2×C22⋊Dic4711yescharacteristic
508C23C2×Dic47⋊C411yescharacteristic
509C23C2×C4⋊Dic4711yescharacteristic
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—88no—
532C2×C4—944no—
533C2×C4—3762no—
534C2×C4—3762no—
535C2×C4—3762no—
536C2×C4—3768no—
537C2×C4—3762no—
538C2×C4—1884no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1888no—
544C2×C4—1882no—
545C2×C4C47⋊#9512yes—
546C2×C4C2×C4×Dic4711yescharacteristic
547C2×C4C2×Dic47⋊C411yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—24no—
553C22—24no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#1568888912yes—
587C22#1568888912yes—
588C22#1568891512yes—
589C22#1568891712yes—
590C22C2×C47⋊#9511yescharacteristic
591C22C2×#1309393811yescharacteristic
592C22C2×#1309393811yescharacteristic
593C4—22no—
594C4—21no—
595C4—21no—
596C4—84no—
597C4—3764no—
598C4—3762no—
599C4—1884no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—41no—
605C2—42no—
606C2—42no—
607C2—41no—
608C2—41no—
609C2#2024151214yesdirect product factor
610C2C2×#1568888911yescharacteristic
611C2C2×#1568891511yescharacteristic
612C2C2×#1568891711yescharacteristic
subgroupG/H
centerC23#13093938
commutatorC2×C4×C47C2×C42
FrattiniC22×D4C22×D47
FittingC22×#99×C47C2
socleC23×C47#99
prime radicalC23×D4×C47C22
cosocleC22×D4×C47C23
hypercenterC22×#99D47
hypocenterC47C2×#1393
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
3. #20241512C2direct (N × H)
5. C22×C4⋊Dic47C4semidirect
9. C22×C4×Dic47C4semidirect
12. C2×#13093939C4semidirect
16. #15689026C4semidirect
22. C2×#99×C47C4semidirect
30. C2×C4⋊Dic47C2×C4semidirect
43. C2×C4×Dic47C2×C4semidirect
46. #13093939C2×C4semidirect
68. #99×C47C2×C4semidirect
307. C2×C47#1393semidirect
324. C2×#99Dic47semidirect
330. C47C2×#1393semidirect
375. #99C2×Dic47semidirect
609. C2#20241512direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939392
C2×#156890262
#202415122
C22×#99×C472
C2×#139347