Finite Group Databasebeta

← C2×#20241919 #26183512

Subgroups — C2×#20241919 #26183512

Order 12032.

1446 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D47C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic
3C2×#15688985C211yesmaximal, characteristic
4#20241919C218yesmaximal, direct product factor
5C2×#15688861C211yesmaximal, characteristic, complemented
6C2×#15688860C211yesmaximal, characteristic, complemented
7C2×#15688859C211yesmaximal, characteristic, complemented
8C2×C47×#328C211yesmaximal, characteristic, complemented
9C22×Q8×D47C411yescharacteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625324—21no—
12C2×#13094004C411yescharacteristic, complemented
13C2×#13094003C2211yescharacteristic
14C2×#13094002—21no—
15C2×#13094001—21no—
16C2×#13094000C411yescharacteristic, complemented
17C22⋊Q8×D47—22no—
18C22⋊Q8×D47—22no—
19C22⋊Q8×D47C414yescomplemented
20C22⋊Q8×D47C2214yes—
21C2×C4⋊C4×D47—21no—
22C2×C4⋊C4×D47C411yescharacteristic, complemented
23C22×(C2×D47)⋊C4—21no—
24C2×C22⋊C4×D47—21no—
25#15689085—22no—
26#15689085C2214yes—
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×(C2×D47)⋊C8—21no—
29#15688985—22no—
30#15688985C2214yes—
31C2×C47⋊#95—21no—
32C2×C47⋊#95C2211yescharacteristic
33#15688861C2214yescomplemented
34#15688860C2214yescomplemented
35#15688859C2214yescomplemented
36C2×C22⋊Q8×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C47×#328C2214yescomplemented
40C24×D47D411yescharacteristic
41C22×Dic94—21no—
42C22×Dic94C2×C411yescharacteristic
43C2×Q8×D47—22no—
44C2×Q8×D47—44no—
45C2×Q8×D47C2×C414yescomplemented
46C23×Dic47—21no—
47C23×Dic47D411yescharacteristic
48C22×C4×D47—21no—
49C22×C4×D47—21no—
50C22×C4×D47—44no—
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C412yes—
53C22×C4×D47C2×C411yescharacteristic
54C22×C4×D47D411yescharacteristic
55C22×C4×D47D412yes—
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D412yes—
58C22×C4×D47C2312yes—
59C2×#10625324—24no—
60C2×#10625324—44no—
61C22×C47⋊C8—21no—
62#13094004—22no—
63#13094004C2×C414yescomplemented
64#13094003—21no—
65#13094003—21no—
66#13094003C2×C412yes—
67#13094003C2312yes—
68#13094002—24no—
69#13094002—24no—
70#13094001—24no—
71#13094001—24no—
72#13094000—22no—
73#13094000C2×C414yescomplemented
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2×C411yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C4⋊C4×D47—24no—
80C4⋊C4×D47—22no—
81C4⋊C4×D47—24no—
82C4⋊C4×D47C2×C414yescomplemented
83C2×C22⋊Dic47—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—24no—
87C2×(C2×D47)⋊C4—44no—
88C22⋊C4×D47—24no—
89C22⋊C4×D47—24no—
90C22⋊(C47⋊C8)—21no—
91C22⋊(C47⋊C8)C2312yes—
92(C2×D47)⋊C8—24no—
93C47⋊#95—21no—
94C47⋊#95—24no—
95C47⋊#95C2312yes—
96C22×Q8×C47C2×C411yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47—21no—
100C22⋊Q8×C47—21no—
101C22⋊Q8×C47—21no—
102C22⋊Q8×C47C2×C412yes—
103C22⋊Q8×C47C2312yes—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47C2×C411yescharacteristic
106C2×C22⋊C4×C47—21no—
107C22⋊C8×C47—21no—
108C22⋊C8×C47C2312yes—
109#95×C47—21no—
110#95×C47C2312yes—
111C24×C47C2×D411yescharacteristic
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—24no—
115C23×D47—42no—
116C23×D47C22⋊C411yescharacteristic
117C23×D47C22⋊C412yes—
118C23×D47C22⋊C411yescharacteristic
119C23×D47C2×D412yes—
120C2×Q8×C47—21no—
121C2×Q8×C47—42no—
122C2×Q8×C47C22×C412yes—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C22×C411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—22no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C22×Dic47—42no—
142C22×Dic47—42no—
143C22×Dic47C22⋊C411yescharacteristic
144C22×Dic47C22⋊C411yescharacteristic
145C22×Dic47C22⋊C412yes—
146C22×Dic47C2×D412yes—
147Q8×D47—48no—
148Q8×D47—48no—
149C2×C4×D47—24no—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—24no—
153C2×C4×D47—44no—
154C2×C4×D47—44no—
155C2×C4×D47—44no—
156C2×C4×D47—44no—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×C4×D47—44no—
160C2×C4×D47—44no—
161C2×C4×D47C22⋊C412yes—
162C2×C4×D47C22⋊C412yes—
163C2×C4×D47C22⋊C412yes—
164C2×C4×D47C22⋊C412yes—
165C2×C4×D47C22×C412yes—
166C2×C4×D47C22×C412yes—
167C2×C4×D47C2×D412yes—
168C2×C4×D47C2×D412yes—
169C2×Dic94—21no—
170C2×Dic94—24no—
171C2×Dic94—44no—
172C2×Dic94—42no—
173C2×Dic94C22×C412yes—
174C2×C8×C47—22no—
175C2×C8×C47—42no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47C22×C412yes—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C22⋊Dic47—22no—
183C22⋊Dic47—22no—
184(C2×D47)⋊C4—24no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—48no—
187C4⋊Dic47—21no—
188C4⋊Dic47—24no—
189C4⋊Dic47C22×C412yes—
190Dic47⋊C4—22no—
191Dic47⋊C4—22no—
192Dic47⋊C4—24no—
193Dic47⋊C4—24no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—42no—
196#10625324—48no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47C2×C22⋊C411yescharacteristic
200C23×C47C2×C22⋊C411yescharacteristic
201C23×C47#13712yes—
202C23×C47#13712yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—24no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47—48no—
217C22×D47—48no—
218C22×D47#9914yes—
219C22×D47C2×C22⋊C412yes—
220C22×D47C2×C22⋊C412yes—
221Q8×C47—42no—
222Q8×C47—42no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—84no—
237C2×C4×C47C2×C22⋊C411yescharacteristic
238C2×C4×C47C2×C22⋊C411yescharacteristic
239C2×C4×C47C23×C411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—84no—
253C2×Dic47C2×C22⋊C412yes—
254C2×Dic47C2×C22⋊C412yes—
255C4×D47—24no—
256C4×D47—24no—
257C4×D47—24no—
258C4×D47—24no—
259C4×D47—42no—
260C4×D47—42no—
261C4×D47—42no—
262C4×D47—42no—
263C4×D47—48no—
264C4×D47—48no—
265Dic94—48no—
266Dic94—42no—
267Dic94—42no—
268C8×C47—42no—
269C47⋊C8—42no—
270C2×#1511—471nomaximal, Sylow
271C22×C47—21no—
272C22×C47—21no—
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—24no—
276C22×C47—24no—
277C22×C47C2×#9912yes—
278C22×C47#41912yes—
279C22×C47#41912yes—
280C22×C47#42112yes—
281C22×C47C22×C22⋊C411yescharacteristic
282C22×C47C2×#13711yescharacteristic
283C22×C47C2×#13711yescharacteristic
284C2×D47—22no—
285C2×D47—22no—
286C2×D47—28no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—44no—
291C2×D47—44no—
292C2×D47—48no—
293C2×D47—44no—
294C2×D47C2×#9914yes—
295C4×C47—21no—
296C4×C47—21no—
297C4×C47—21no—
298C4×C47—21no—
299C4×C47—42no—
300C4×C47—42no—
301C4×C47—82no—
302Dic47—22no—
303Dic47—22no—
304Dic47—42no—
305Dic47—42no—
306Dic47—82no—
307C22×C22⋊Q8—471no—
308C2×#407—471no—
309C2×#381—471no—
310#1511—478no—
311C2×#328—471no—
312C2×#328—471no—
313C2×#328—471no—
314C2×#328D4711yescharacteristic, complemented
315C2×C47—21no—
316C2×C47—21no—
317C2×C47—22no—
318C2×C47#151114yescomplemented
319C2×C47C22×#9911yescharacteristic
320C2×C47C2×#41911yescharacteristic
321C2×C47C2×#42111yescharacteristic
322D47—24no—
323D47—42no—
324C23×Q8—471no—
325C24×C4—471no—
326C22×M16—941no—
327C2×C22⋊Q8—471no—
328C2×C22⋊Q8—471no—
329C2×C22⋊Q8—474no—
330C2×C22⋊Q8—474no—
331C2×C22⋊Q8—471no—
332C2×C22⋊Q8—941no—
333C2×C22⋊Q8—941no—
334C2×C22⋊Q8—942no—
335C2×C22⋊Q8—942no—
336C2×C22⋊Q8C2×D4711yescharacteristic
337C22×C4⋊C4—471no—
338C22×C4⋊C4—941no—
339C22×C22⋊C4—941no—
340C22×C22⋊C4—941no—
341#407—474no—
342#407—942no—
343C2×C22⋊C8—471no—
344C2×C22⋊C8—941no—
345C2×C22⋊C8C2×D4711yescharacteristic
346#381—474no—
347#381—942no—
348C2×#95—471no—
349C2×#95—941no—
350C2×#95C2×D4711yescharacteristic
351#328—474no—
352#328—474no—
353#328—474no—
354#328C2×D4714yescomplemented
355C47C2×#151111yescharacteristic, complemented, Sylow
356C25—471no—
357C22×Q8—474no—
358C22×Q8—471no—
359C22×Q8—941no—
360C22×Q8—942no—
361C22×Q8—1884no—
362C22×Q8C4×D4711yescharacteristic
363C23×C4—21no—
364C23×C4—471no—
365C23×C4—471no—
366C23×C4—471no—
367C23×C4—471no—
368C23×C4—472no—
369C23×C4—472no—
370C23×C4—471no—
371C23×C4—472no—
372C23×C4—472no—
373C23×C4—941no—
374C23×C4—941no—
375C23×C4—941no—
376C23×C4—1884no—
377C23×C4C22×D4711yescharacteristic
378C2×M16—944no—
379C2×M16—1884no—
380C22×C8—21no—
381C22×C8—941no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—474no—
385C22⋊Q8—472no—
386C22⋊Q8—472no—
387C22⋊Q8—474no—
388C22⋊Q8—944no—
389C22⋊Q8—944no—
390C22⋊Q8—944no—
391C22⋊Q8—944no—
392C22⋊Q8—942no—
393C22⋊Q8—942no—
394C22⋊Q8—941no—
395C22⋊Q8—941no—
396C22⋊Q8C4×D4712yes—
397C22⋊Q8C22×D4712yes—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—474no—
401C2×C4⋊C4—942no—
402C2×C4⋊C4—944no—
403C2×C4⋊C4—944no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4—941no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4C4×D4711yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—944no—
411C2×C22⋊C4—944no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—941no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—1884no—
417C22⋊C8—21no—
418C22⋊C8—472no—
419C22⋊C8—944no—
420C22⋊C8—941no—
421C22⋊C8C22×D4712yes—
422#95—21no—
423#95—472no—
424#95—944no—
425#95—941no—
426#95C22×D4712yes—
427C24—471no—
428C24—471no—
429C24—472no—
430C24—472no—
431C24—944no—
432C24—942no—
433C24—942no—
434C24—1882no—
435C24D4×D4711yescharacteristic
436C2×Q8—21no—
437C2×Q8—42no—
438C2×Q8—472no—
439C2×Q8—944no—
440C2×Q8—941no—
441C2×Q8—1882no—
442C2×Q8—1884no—
443C2×Q8—1888no—
444C2×Q8—1888no—
445C2×Q8C2×C4×D4712yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—472no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—472no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—944no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—942no—
472C22×C4—1882no—
473C22×C4—1882no—
474C22×C4—1884no—
475C22×C4—1884no—
476C22×C4—1884no—
477C22×C4—1884no—
478C22×C4—1884no—
479C22×C4—1884no—
480C22×C4—1882no—
481C22×C4—1882no—
482C22×C4—1882no—
483C22×C4—1882no—
484C22×C4C2×C4×D4711yescharacteristic
485C22×C4C2×C4×D4711yescharacteristic
486C22×C4D4×D4711yescharacteristic
487C22×C4D4×D4711yescharacteristic
488C22×C4D4×D4711yescharacteristic
489C22×C4C23×D4711yescharacteristic
490M16—1888no—
491C2×C8—22no—
492C2×C8—42no—
493C2×C8—942no—
494C2×C8—1882no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—472no—
499C4⋊C4—944no—
500C4⋊C4—944no—
501C4⋊C4—944no—
502C4⋊C4—942no—
503C4⋊C4—941no—
504C4⋊C4—942no—
505C4⋊C4C2×C4×D4712yes—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—944no—
509C22⋊C4—944no—
510C22⋊C4—942no—
511C22⋊C4—942no—
512C22⋊C4—1888no—
513C23—22no—
514C23—22no—
515C23—472no—
516C23—474no—
517C23—472no—
518C23—944no—
519C23—944no—
520C23—942no—
521C23—942no—
522C23—942no—
523C23—944no—
524C23—942no—
525C23—942no—
526C23—942no—
527C23—942no—
528C23—1882no—
529C23—1882no—
530C23—1888no—
531C23—1888no—
532C23C22⋊C4×D4711yescharacteristic
533C23C22⋊C4×D4711yescharacteristic
534C23#1309405312yes—
535C23#1309405612yes—
536C23C2×D4×D4711yescharacteristic
537Q8—42no—
538Q8—42no—
539Q8—1882no—
540Q8—1882no—
541Q8—1888no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—472no—
557C2×C4—472no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—3764no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1888no—
572C2×C4—1888no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4C22⋊C4×D4711yescharacteristic
581C2×C4C22⋊C4×D4711yescharacteristic
582C2×C4C22×C4×D4711yescharacteristic
583C2×C4C2×D4×D4711yescharacteristic
584C8—42no—
585C8—1882no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—24no—
591C22—24no—
592C22—474no—
593C22—942no—
594C22—942no—
595C22—948no—
596C22—942no—
597C22—942no—
598C22—942no—
599C22—1884no—
600C22—1884no—
601C22—1888no—
602C22—1884no—
603C22#99×D4712yes—
604C22#1568913012yes—
605C22#1568913012yes—
606C22#1568913812yes—
607C22C2×C22⋊C4×D4711yescharacteristic
608C22C2×#1309405311yescharacteristic
609C22C2×#1309405611yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4—942no—
618C4—942no—
619C4—3762no—
620C4—1882no—
621C4—1882no—
622C2—21no—
623C2—21no—
624C2—22no—
625C2—944no—
626C2—1882no—
627C2#2024191914yesdirect product factor
628C2C2×#99×D4711yescharacteristic
629C2C2×#1568913011yescharacteristic
630C2C2×#1568913811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#328C2
socleC23×C47C2×C22⋊C4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC2×#328D47
hypocenterC47C2×#1511
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
4. #20241919C2direct (N × H)
5. C2×#15688861C2semidirect
6. C2×#15688860C2semidirect
7. C2×#15688859C2semidirect
8. C2×C47×#328C2semidirect
9. C22×Q8×D47C4semidirect
12. C2×#13094004C4semidirect
16. C2×#13094000C4semidirect
19. C22⋊Q8×D47C4semidirect
22. C2×C4⋊C4×D47C4semidirect
33. #15688861C22semidirect
34. #15688860C22semidirect
35. #15688859C22semidirect
39. C47×#328C22semidirect
45. C2×Q8×D47C2×C4semidirect
63. #13094004C2×C4semidirect
73. #13094000C2×C4semidirect
82. C4⋊C4×D47C2×C4semidirect
314. C2×#328D47semidirect
318. C2×C47#1511semidirect
354. #328C2×D47semidirect
355. C47C2×#1511semidirect
627. C2#20241919direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D472
C2×#156890852
C2×#156889852
#202419192
C2×#156888612
C2×#156888602
C2×#156888592
C2×C47×#3282
C2×#151147