Finite Group Databasebeta

← C2×#20241737 #26182223

Subgroups — C2×#20241737 #26182223

Order 12032.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094007C211yesmaximal, characteristic
2C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3#20241737C218yesmaximal, direct product factor
4C2×#15688848C212yesmaximal, complemented
5C2×#15688846C212yesmaximal, complemented
6C2×C47×#407C211yesmaximal, characteristic, complemented
7C23×C22⋊D47C411yescharacteristic
8C23×C47⋊C8—21no—
9C2×#13094007—21no—
10C2×#13094007—21no—
11C2×#13094007—22no—
12C2×#13094007—21no—
13C2×#13094007—21no—
14C2×#13094007—22no—
15C2×#13094007C414yes—
16C2×#13094007C2214yes—
17C22×Dic47⋊C4C411yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×C22⋊Dic47C411yescharacteristic
20C22×(C2×D47)⋊C4C411yescharacteristic
21C22×(C2×D47)⋊C4C2211yescharacteristic
22C2×C22⋊(C47⋊C8)—22no—
23C2×C22⋊(C47⋊C8)—21no—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2214yescomplemented
26#15688848C2218yescomplemented
27#15688846C2218yescomplemented
28C24×C4×C47C2211yescharacteristic
29C22×M16×C47—21no—
30C47×#407—22no—
31C47×#407C2214yescomplemented
32C2×C22⋊C8×C47—21no—
33C2×C22⋊C8×C47C2212yes—
34C24×D47C2×C411yescharacteristic
35C22×C22⋊D47—21no—
36C22×C22⋊D47—22no—
37C22×C22⋊D47—21no—
38C22×C22⋊D47—44no—
39C22×C22⋊D47C2×C414yes—
40C23×Dic47C2×C411yescharacteristic
41C23×Dic47C2×C411yescharacteristic
42C23×Dic47C2×C411yescharacteristic
43C22×C47⋊C8—22no—
44C22×C47⋊C8—24no—
45C22×C47⋊C8—44no—
46#13094007—24no—
47#13094007—24no—
48#13094007—24no—
49#13094007—24no—
50#13094007—24no—
51#13094007—24no—
52#13094007—24no—
53#13094007—24no—
54C2×Dic47⋊C4—21no—
55C2×Dic47⋊C4—21no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—21no—
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—21no—
60C2×Dic47⋊C4—21no—
61C2×Dic47⋊C4C2×C414yes—
62C2×Dic47⋊C4C2×C412yes—
63C2×Dic47⋊C4C2312yes—
64C2×C22⋊Dic47—21no—
65C2×C22⋊Dic47—22no—
66C2×C22⋊Dic47—21no—
67C2×C22⋊Dic47C2×C414yes—
68C2×(C2×D47)⋊C4—21no—
69C2×(C2×D47)⋊C4—21no—
70C2×(C2×D47)⋊C4—22no—
71C2×(C2×D47)⋊C4—21no—
72C2×(C2×D47)⋊C4—21no—
73C2×(C2×D47)⋊C4—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4C2×C414yes—
76C2×(C2×D47)⋊C4C2×C412yes—
77C2×(C2×D47)⋊C4C2312yes—
78C22⋊(C47⋊C8)—22no—
79C22⋊(C47⋊C8)—24no—
80C22⋊(C47⋊C8)C2314yes—
81C25×C47C2×C411yescharacteristic
82C23×C4×C47—44no—
83C23×C4×C47C2×C411yescharacteristic
84C23×C4×C47C2×C412yes—
85C23×C4×C47D411yescharacteristic
86C23×C4×C47D411yescharacteristic
87C23×C4×C47D411yescharacteristic
88C23×C4×C47D412yes—
89C23×C4×C47D411yescharacteristic
90C23×C4×C47D412yes—
91C23×C4×C47C2311yescharacteristic
92C23×C4×C47C2312yes—
93C2×M16×C47—24no—
94C2×M16×C47—44no—
95C22×C8×C47—22no—
96C22⋊C8×C47—24no—
97C22⋊C8×C47—22no—
98C22⋊C8×C47C2314yes—
99C24×C47—24no—
100C24×C47—24no—
101C24×C47—42no—
102C24×C47C22⋊C411yescharacteristic
103C24×C47C22⋊C412yes—
104C24×C47C22⋊C411yescharacteristic
105C24×C47C22×C412yes—
106C24×C47C22×C411yescharacteristic
107C23×D47—21no—
108C23×D47—21no—
109C23×D47—24no—
110C23×D47—42no—
111C23×D47—42no—
112C23×D47C22×C412yes—
113C22×C4×C47—24no—
114C22×C4×C47—24no—
115C22×C4×C47—42no—
116C22×C4×C47—44no—
117C22×C4×C47—44no—
118C22×C4×C47—42no—
119C22×C4×C47—42no—
120C22×C4×C47—44no—
121C22×C4×C47—42no—
122C22×C4×C47—44no—
123C22×C4×C47C22⋊C411yescharacteristic
124C22×C4×C47C22⋊C411yescharacteristic
125C22×C4×C47C22⋊C412yes—
126C22×C4×C47C22⋊C412yes—
127C22×C4×C47C22⋊C412yes—
128C22×C4×C47C22⋊C412yes—
129C22×C4×C47C22⋊C412yes—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C22×C411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2411yescharacteristic
141C2×C22⋊D47—24no—
142C2×C22⋊D47—24no—
143C2×C22⋊D47—48no—
144C2×C22⋊D47—44no—
145C2×C22⋊D47—44no—
146C2×C22⋊D47—48no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—24no—
154C22×Dic47—42no—
155C22×Dic47—42no—
156C22×Dic47C22×C412yes—
157C22×Dic47C22×C412yes—
158C22×Dic47C22×C412yes—
159M16×C47—48no—
160C2×C8×C47—24no—
161C2×C8×C47—44no—
162C22⋊Dic47—24no—
163C22⋊Dic47—24no—
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—24no—
166(C2×D47)⋊C4—22no—
167(C2×D47)⋊C4—22no—
168(C2×D47)⋊C4—22no—
169(C2×D47)⋊C4—22no—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172Dic47⋊C4—22no—
173Dic47⋊C4—22no—
174Dic47⋊C4—22no—
175Dic47⋊C4—22no—
176C2×C47⋊C8—24no—
177C2×C47⋊C8—44no—
178C2×C47⋊C8—48no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—22no—
186C23×C47—24no—
187C23×C47—22no—
188C23×C47—24no—
189C23×C47—22no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—416no—
193C23×C47#9914yes—
194C23×C47C2×C22⋊C412yes—
195C23×C47C2×C22⋊C411yescharacteristic
196C23×C47C2×C22⋊C412yes—
197C23×C47C2×C22⋊C411yescharacteristic
198C23×C47#13114yes—
199C23×C47C23×C411yescharacteristic
200C22×D47—22no—
201C22×D47—24no—
202C22×D47—42no—
203C22×D47—42no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—88no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—41no—
223C2×C4×C47—42no—
224C2×C4×C47—41no—
225C2×C4×C47—42no—
226C2×C4×C47—41no—
227C2×C4×C47—42no—
228C2×C4×C47—41no—
229C2×C4×C47C2×C22⋊C412yes—
230C2×C4×C47C2×C22⋊C412yes—
231C2×C4×C47C2×C22⋊C411yescharacteristic
232C2×C4×C47C2×C22⋊C411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C22⋊D47—48no—
236C22⋊D47—48no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C8×C47—44no—
250C47⋊C8—44no—
251C2×#1468—471nomaximal, Sylow
252C22×C47—22no—
253C22×C47—21no—
254C22×C47—21no—
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—22no—
258C22×C47—28no—
259C22×C47—28no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—48no—
264C22×C47—48no—
265C22×C47—44no—
266C22×C47#40812yes—
267C22×C47#40812yes—
268C22×C47C2×#9914yes—
269C22×C47C2×#9912yes—
270C22×C47#41312yes—
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47C2×#13112yes—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—84no—
277C2×D47—84no—
278C2×D47—84no—
279C4×C47—21no—
280C4×C47—21no—
281C4×C47—22no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—22no—
285Dic47—42no—
286Dic47—42no—
287Dic47—42no—
288C22×#123—471no—
289C2×#407D4711yescharacteristic, complemented
290C22×C22⋊C8—471no—
291#1468—478no—
292C2×#324—472no—
293C2×#323—472no—
294C2×C47—22no—
295C2×C47—22no—
296C2×C47—24no—
297C2×C47—42no—
298C2×C47#146814yescomplemented
299C2×C47C2×#40811yescharacteristic
300C2×C47C22×#9911yescharacteristic
301C2×C47C2×#41311yescharacteristic
302D47—82no—
303C23×D4—471no—
304C24×C4C2×D4711yescharacteristic
305C22×M16—21no—
306C23×C8—941no—
307C2×#123—474no—
308C2×#123—474no—
309C2×#123—941no—
310C2×#123—942no—
311C2×#123—941no—
312C2×#123—941no—
313C2×#123—941no—
314C2×#123—942no—
315C22×C4⋊C4—471no—
316C22×C4⋊C4—471no—
317C22×C22⋊C4—471no—
318C22×C22⋊C4—471no—
319C22×C22⋊C4—471no—
320#407—22no—
321#407C2×D4714yescomplemented
322C2×C22⋊C8—21no—
323C2×C22⋊C8—472no—
324C2×C22⋊C8—474no—
325C2×C22⋊C8—942no—
326C2×C22⋊C8—941no—
327C2×C22⋊C8C2×D4712yes—
328#324—478no—
329#323—478no—
330C47C2×#146811yescharacteristic, complemented, Sylow
331C25—471no—
332C25C4×D4711yescharacteristic
333C22×D4—474no—
334C22×D4—942no—
335C22×D4—941no—
336C22×D4—941no—
337C22×D4—1884no—
338C23×C4—44no—
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4C4×D4711yescharacteristic
343C23×C4C4×D4712yes—
344C23×C4D18811yescharacteristic
345C23×C4D18812yes—
346C23×C4D18811yescharacteristic
347C23×C4C22⋊D4711yescharacteristic
348C23×C4C22⋊D4711yescharacteristic
349C23×C4C22⋊D4712yes—
350C23×C4C22×D4711yescharacteristic
351C23×C4C22×D4712yes—
352C2×M16—24no—
353C2×M16—44no—
354C22×C8—22no—
355C22×C8—944no—
356C22×C8—942no—
357C22×C8—1884no—
358#123—944no—
359#123—944no—
360#123—944no—
361#123—944no—
362#123—944no—
363#123—944no—
364#123—944no—
365#123—944no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C22⋊C4—474no—
377C2×C22⋊C4—474no—
378C2×C22⋊C4—472no—
379C2×C22⋊C4—472no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—942no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C22⋊C8—24no—
391C22⋊C8—22no—
392C22⋊C8—474no—
393C22⋊C8—942no—
394C22⋊C8—944no—
395C22⋊C8C22×D4714yes—
396C24—24no—
397C24—24no—
398C24—42no—
399C24—472no—
400C24—944no—
401C24—941no—
402C24—941no—
403C24—1882no—
404C24—1882no—
405C24(C2×D47)⋊C411yescharacteristic
406C24(C2×D47)⋊C412yes—
407C24(C2×D47)⋊C411yescharacteristic
408C24C2×C4×D4712yes—
409C24C2×C4×D4711yescharacteristic
410C2×D4—944no—
411C2×D4—944no—
412C2×D4—1888no—
413C2×D4—1884no—
414C2×D4—1884no—
415C2×D4—1888no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—472no—
427C22×C4—472no—
428C22×C4—472no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—944no—
436C22×C4—1882no—
437C22×C4—1882no—
438C22×C4(C2×D47)⋊C412yes—
439C22×C4(C2×D47)⋊C412yes—
440C22×C4(C2×D47)⋊C411yescharacteristic
441C22×C4(C2×D47)⋊C411yescharacteristic
442C22×C4(C2×D47)⋊C412yes—
443C22×C4(C2×D47)⋊C412yes—
444C22×C4(C2×D47)⋊C412yes—
445C22×C4C2×C4×D4712yes—
446C22×C4C2×C4×D4711yescharacteristic
447C22×C4C2×D18812yes—
448C22×C4C2×D18811yescharacteristic
449C22×C4C2×D18811yescharacteristic
450C22×C4C2×D18812yes—
451C22×C4C2×C22⋊D4712yes—
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4712yes—
455C22×C4C23×D4711yescharacteristic
456M16—48no—
457C2×C8—24no—
458C2×C8—44no—
459C2×C8—944no—
460C2×C8—1888no—
461C2×C8—1884no—
462C4⋊C4—944no—
463C4⋊C4—944no—
464C4⋊C4—942no—
465C4⋊C4—942no—
466C4⋊C4—942no—
467C4⋊C4—942no—
468C22⋊C4—944no—
469C22⋊C4—944no—
470C22⋊C4—944no—
471C22⋊C4—944no—
472C22⋊C4—942no—
473C22⋊C4—942no—
474C22⋊C4—942no—
475C22⋊C4—942no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—416no—
490C23—944no—
491C23—942no—
492C23—3768no—
493C23—1882no—
494C23—1882no—
495C23—1882no—
496C23—1882no—
497C23—1882no—
498C23—1882no—
499C23—1882no—
500C23—1882no—
501C23#1309393814yes—
502C23C2×(C2×D47)⋊C412yes—
503C23C2×(C2×D47)⋊C411yescharacteristic
504C23C2×(C2×D47)⋊C412yes—
505C23C2×(C2×D47)⋊C411yescharacteristic
506C23#1309403214yes—
507C23C22×C4×D4711yescharacteristic
508D4—1888no—
509D4—1888no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—942no—
531C2×C4—944no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4C2×(C2×D47)⋊C411yescharacteristic
541C2×C4C2×(C2×D47)⋊C411yescharacteristic
542C2×C4C2×(C2×D47)⋊C412yes—
543C2×C4C2×(C2×D47)⋊C412yes—
544C2×C4C22×C22⋊D4711yescharacteristic
545C2×C4C22×D18811yescharacteristic
546C8—44no—
547C8—1884no—
548C22—28no—
549C22—28no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—3764no—
563C22—3764no—
564C22—3764no—
565C22—1882no—
566C22—1882no—
567C22—1882no—
568C22#1568909112yes—
569C22#1568909112yes—
570C22C2×#1309393814yes—
571C22C2×#1309393812yes—
572C22#1568910512yes—
573C22C22×(C2×D47)⋊C411yescharacteristic
574C22C2×#1309403212yes—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—21no—
580C4—22no—
581C4—1882no—
582C4—1882no—
583C4—1882no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—3762no—
589C2#2024173714yesdirect product factor
590C2C2×#1568909111yescharacteristic
591C2C22×#1309393811yescharacteristic
592C2C2×#1568910511yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#407C2
socleC23×C47C2×C22⋊C4
prime radicalC23×C22⋊D47C4
cosocleC22×C4×C47C24
hypercenterC2×#407D47
hypocenterC47C2×#1468
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊(C47⋊C8)C2semidirect
3. #20241737C2direct (N × H)
4. C2×#15688848C2semidirect
5. C2×#15688846C2semidirect
6. C2×C47×#407C2semidirect
25. C2×C22⋊(C47⋊C8)C22semidirect
26. #15688848C22semidirect
27. #15688846C22semidirect
31. C47×#407C22semidirect
289. C2×#407D47semidirect
298. C2×C47#1468semidirect
321. #407C2×D47semidirect
330. C47C2×#1468semidirect
589. C2#20241737direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940072
C22×C22⋊(C47⋊C8)2
#202417372
C2×#156888482
C2×#156888462
C2×C47×#4072
C2×#146847