Finite Group Databasebeta

← C2×#20241879 #26183372

Subgroups — C2×#20241879 #26183372

Order 12032.

1606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic47C211yesmaximal, characteristic, complemented
3#20241879C218yesmaximal, direct product factor
4C2×#15688860C212yesmaximal, complemented
5C2×#15688857C212yesmaximal, complemented
6C2×C47×#525C211yesmaximal, characteristic
7C23×C47⋊C8—21no—
8C22×C22⋊Dic47—21no—
9C22×C4×Dic47—21no—
10C2×C22⋊(C47⋊C8)—21no—
11C2×C22⋊(C47⋊C8)—22no—
12C2×C22⋊(C47⋊C8)C2212yes—
13C2×C22⋊(C47⋊C8)C2214yescomplemented
14C4×C22⋊Dic47—22no—
15C4×C22⋊Dic47C2214yescomplemented
16C2×C47⋊#95C2212yes—
17#15688860C2218yescomplemented
18#15688857C2218yescomplemented
19C22×#41×C47C411yescharacteristic, complemented
20C24×C4×C47C2211yescharacteristic
21C47×#525—22no—
22C47×#525—22no—
23C47×#525C414yescomplemented
24C47×#525C2214yes—
25C2×#123×C47—21no—
26C2×C22⋊Q8×C47C411yescharacteristic, complemented
27C2×C22⋊Q8×C47C2211yescharacteristic
28C2×#121×C47C411yescharacteristic, complemented
29C2×#121×C47C2211yescharacteristic
30C2×#120×C47—21no—
31C2×C4×D4×C47—21no—
32C2×C4×D4×C47—21no—
33C2×#117×C47C411yescharacteristic, complemented
34C23×Dic47—22no—
35C22×C47⋊C8—22no—
36C22×C47⋊C8—24no—
37C22×C47⋊C8—44no—
38C2×C22⋊Dic47—28no—
39C2×C22⋊Dic47—24no—
40C2×C4×Dic47—24no—
41C2×C4×Dic47—44no—
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)—24no—
44C22⋊(C47⋊C8)C2314yes—
45C47⋊#95—22no—
46C47⋊#95C2314yes—
47C25×C47D411yescharacteristic
48C2×#41×C47—22no—
49C2×#41×C47—44no—
50C2×#41×C47C2×C414yescomplemented
51C22×Q8×C47C2×C411yescharacteristic
52C22×D4×C47—21no—
53C22×D4×C47—21no—
54C22×D4×C47—21no—
55C22×D4×C47C2×C411yescharacteristic
56C23×C4×C47—21no—
57C23×C4×C47—44no—
58C23×C4×C47C2×C411yescharacteristic
59C23×C4×C47C2×C411yescharacteristic
60C23×C4×C47C2×C412yes—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47D412yes—
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D412yes—
66C23×C4×C47C2311yescharacteristic
67C23×C4×C47C2312yes—
68#123×C47—24no—
69#123×C47—24no—
70C22⋊Q8×C47—22no—
71C22⋊Q8×C47—21no—
72C22⋊Q8×C47—21no—
73C22⋊Q8×C47C2×C412yes—
74C22⋊Q8×C47C2×C414yescomplemented
75C22⋊Q8×C47C2312yes—
76#121×C47—22no—
77#121×C47—21no—
78#121×C47—21no—
79#121×C47C2×C412yes—
80#121×C47C2×C414yescomplemented
81#121×C47C2312yes—
82#120×C47—24no—
83#120×C47—24no—
84C4×D4×C47—24no—
85C4×D4×C47—24no—
86C4×D4×C47—24no—
87C4×D4×C47—24no—
88#117×C47—22no—
89#117×C47C2×C414yescomplemented
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47C2×C411yescharacteristic
93C2×C4⋊C4×C47C2×C411yescharacteristic
94C2×C22⋊C4×C47—21no—
95C2×C22⋊C4×C47—21no—
96C2×C22⋊C4×C47—21no—
97C2×C22⋊C4×C47—21no—
98C2×C22⋊C4×C47—21no—
99C2×C42×C47—21no—
100C24×C47—24no—
101C24×C47—24no—
102C24×C47—21no—
103C24×C47—42no—
104C24×C47C22⋊C411yescharacteristic
105C24×C47C22⋊C412yes—
106C24×C47C22⋊C411yescharacteristic
107C24×C47C2×D412yes—
108C24×C47C2×D411yescharacteristic
109#41×C47—48no—
110#41×C47—48no—
111C2×Q8×C47—21no—
112C2×Q8×C47—42no—
113C2×Q8×C47C22×C412yes—
114C2×D4×C47—22no—
115C2×D4×C47—24no—
116C2×D4×C47—24no—
117C2×D4×C47—22no—
118C2×D4×C47—21no—
119C2×D4×C47—42no—
120C2×D4×C47—44no—
121C2×D4×C47—44no—
122C2×D4×C47—44no—
123C2×D4×C47C22×C412yes—
124C22×C4×C47—24no—
125C22×C4×C47—24no—
126C22×C4×C47—24no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—24no—
131C22×C4×C47—44no—
132C22×C4×C47—44no—
133C22×C4×C47—44no—
134C22×C4×C47—44no—
135C22×C4×C47—42no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47—44no—
139C22×C4×C47C22⋊C411yescharacteristic
140C22×C4×C47C22⋊C412yes—
141C22×C4×C47C22⋊C412yes—
142C22×C4×C47C22⋊C412yes—
143C22×C4×C47C22⋊C412yes—
144C22×C4×C47C22⋊C412yes—
145C22×C4×C47C22⋊C411yescharacteristic
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C22×C412yes—
148C22×C4×C47C22×C411yescharacteristic
149C22×C4×C47C22×C412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D412yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—28no—
158C22×Dic47—24no—
159C22×Dic47—44no—
160C22×Dic47—44no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—24no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C22×C412yes—
168C22⋊C4×C47—24no—
169C22⋊C4×C47—22no—
170C22⋊C4×C47—24no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—24no—
174C22⋊C4×C47—22no—
175C42×C47—24no—
176C22⋊Dic47—216no—
177C4×Dic47—48no—
178C2×C47⋊C8—24no—
179C2×C47⋊C8—44no—
180C2×C47⋊C8—48no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—24no—
186C23×C47—22no—
187C23×C47—24no—
188C23×C47—22no—
189C23×C47—24no—
190C23×C47—24no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—416no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47#10414yes—
200C23×C47C2×C22⋊C412yes—
201C23×C47C2×C22⋊C411yescharacteristic
202C23×C47C2×C22⋊C412yes—
203C23×C47C2×C22⋊C411yescharacteristic
204C23×C47#13514yes—
205C23×C47C22×D411yescharacteristic
206Q8×C47—42no—
207Q8×C47—42no—
208D4×C47—42no—
209D4×C47—42no—
210D4×C47—48no—
211D4×C47—44no—
212D4×C47—44no—
213D4×C47—48no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—41no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—48no—
238C2×C4×C47#9914yes—
239C2×C4×C47C2×C22⋊C411yescharacteristic
240C2×C4×C47C2×C22⋊C412yes—
241C2×C4×C47C2×C22⋊C411yescharacteristic
242C2×C4×C47C2×C22⋊C412yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×Dic47—28no—
246C2×Dic47—44no—
247C2×Dic47—416no—
248C2×Dic47—44no—
249C47⋊C8—44no—
250C2×#1501—471nomaximal, Sylow
251C22×C47—21no—
252C22×C47—21no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—28no—
258C22×C47—28no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—48no—
263C22×C47—48no—
264C22×C47—44no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—84no—
269C22×C47C2×#9912yes—
270C22×C47#41612yes—
271C22×C47#41612yes—
272C22×C47C2×#10414yes—
273C22×C47C2×#10412yes—
274C22×C47C22×C22⋊C411yescharacteristic
275C22×C47C2×#13512yes—
276C4×C47—21no—
277C4×C47—21no—
278C4×C47—22no—
279C4×C47—21no—
280C4×C47—21no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—42no—
284C4×C47C2×#9914yes—
285Dic47—48no—
286C2×#525D4711yescharacteristic
287C22×C22⋊C8—471no—
288C2×C4×C22⋊C4—471no—
289#1501—478no—
290C2×#328—472no—
291C2×#327—472no—
292C2×C47—22no—
293C2×C47—22no—
294C2×C47—24no—
295C2×C47—42no—
296C2×C47—82no—
297C2×C47#150114yescomplemented
298C2×C47C22×#9911yescharacteristic
299C2×C47C2×#41611yescharacteristic
300C2×C47C22×#10411yescharacteristic
301C22×#41Dic4711yescharacteristic, complemented
302C24×C4C2×D4711yescharacteristic
303C23×C8—941no—
304#525—22no—
305#525—22no—
306#525Dic4714yescomplemented
307#525C2×D4714yes—
308C2×#123—21no—
309C2×C22⋊Q8Dic4711yescharacteristic, complemented
310C2×C22⋊Q8C2×D4711yescharacteristic
311C2×#121Dic4711yescharacteristic, complemented
312C2×#121C2×D4711yescharacteristic
313C2×#120—21no—
314C2×C4×D4—21no—
315C2×C4×D4—21no—
316C2×#117Dic4711yescharacteristic, complemented
317C22×C22⋊C4—941no—
318C22×C42—941no—
319C2×C22⋊C8—472no—
320C2×C22⋊C8—474no—
321C2×C22⋊C8—941no—
322C2×C22⋊C8—942no—
323C4×C22⋊C4—474no—
324C4×C22⋊C4—942no—
325C2×#95—472no—
326#328—478no—
327#327—478no—
328C47C2×#150111yescharacteristic, complemented, Sylow
329C25C22⋊D4711yescharacteristic
330C2×#41—22no—
331C2×#41—44no—
332C2×#41C2×Dic4714yescomplemented
333C22×Q8C2×Dic4711yescharacteristic
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—21no—
337C22×D4C2×Dic4711yescharacteristic
338C23×C4—21no—
339C23×C4—44no—
340C23×C4—942no—
341C23×C4C2×Dic4711yescharacteristic
342C23×C4C2×Dic4711yescharacteristic
343C23×C4C2×Dic4712yes—
344C23×C4C22⋊D4711yescharacteristic
345C23×C4C22⋊D4711yescharacteristic
346C23×C4C22⋊D4712yes—
347C23×C4C22⋊D4711yescharacteristic
348C23×C4C22⋊D4712yes—
349C23×C4C22×D4711yescharacteristic
350C23×C4C22×D4712yes—
351C22×C8—942no—
352C22×C8—944no—
353C22×C8—1884no—
354#123—24no—
355#123—24no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8C2×Dic4712yes—
360C22⋊Q8C2×Dic4714yescomplemented
361C22⋊Q8C22×D4712yes—
362#121—22no—
363#121—21no—
364#121—21no—
365#121C2×Dic4712yes—
366#121C2×Dic4714yescomplemented
367#121C22×D4712yes—
368#120—24no—
369#120—24no—
370C4×D4—24no—
371C4×D4—24no—
372C4×D4—24no—
373C4×D4—24no—
374#117—22no—
375#117C2×Dic4714yescomplemented
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C2×Dic4711yescharacteristic
379C2×C4⋊C4C2×Dic4711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—948no—
386C2×C22⋊C4—944no—
387C2×C42—21no—
388C2×C42—944no—
389C2×C42—1884no—
390C22⋊C8—474no—
391C22⋊C8—942no—
392C22⋊C8—944no—
393#95—474no—
394#95—942no—
395C24—24no—
396C24—24no—
397C24—21no—
398C24—42no—
399C24C22⋊Dic4711yescharacteristic
400C24C22⋊Dic4712yes—
401C24C22⋊Dic4711yescharacteristic
402C24C2×C22⋊D4712yes—
403C24C2×C22⋊D4711yescharacteristic
404#41—48no—
405#41—48no—
406C2×Q8—21no—
407C2×Q8—42no—
408C2×Q8C22×Dic4712yes—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4C22×Dic4712yes—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—948no—
435C22×C4—944no—
436C22×C4—1884no—
437C22×C4—1884no—
438C22×C4C22⋊Dic4711yescharacteristic
439C22×C4C22⋊Dic4712yes—
440C22×C4C22⋊Dic4712yes—
441C22×C4C22⋊Dic4712yes—
442C22×C4C22⋊Dic4712yes—
443C22×C4C22⋊Dic4712yes—
444C22×C4C22⋊Dic4711yescharacteristic
445C22×C4C22×Dic4711yescharacteristic
446C22×C4C22×Dic4712yes—
447C22×C4C22×Dic4711yescharacteristic
448C22×C4C22×Dic4712yes—
449C22×C4C2×C22⋊D4712yes—
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4712yes—
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4712yes—
455C22×C4C23×D4711yescharacteristic
456C2×C8—944no—
457C2×C8—1884no—
458C2×C8—1888no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4C22×Dic4712yes—
465C4⋊C4C22×Dic4712yes—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—9416no—
474C42—24no—
475C42—1888no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—416no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23#1309395114yes—
495C23C2×C22⋊Dic4712yes—
496C23C2×C22⋊Dic4711yescharacteristic
497C23C2×C22⋊Dic4712yes—
498C23C2×C22⋊Dic4711yescharacteristic
499C23#1309404414yes—
500C23C22×C22⋊D4711yescharacteristic
501Q8—42no—
502Q8—42no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—44no—
507D4—44no—
508D4—48no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—948no—
534C2×C4—1884no—
535C2×C4—18816no—
536C2×C4—1884no—
537C2×C4#1309393914yes—
538C2×C4C2×C22⋊Dic4711yescharacteristic
539C2×C4C2×C22⋊Dic4712yes—
540C2×C4C2×C22⋊Dic4711yescharacteristic
541C2×C4C2×C22⋊Dic4712yes—
542C2×C4C23×Dic4711yescharacteristic
543C2×C4C22×C22⋊D4711yescharacteristic
544C8—1884no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—48no—
557C22—48no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—84no—
563C22C2×#1309393912yes—
564C22#1568911912yes—
565C22#1568911912yes—
566C22C2×#1309395114yes—
567C22C2×#1309395112yes—
568C22C22×C22⋊Dic4711yescharacteristic
569C22C2×#1309404412yes—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—1888no—
579C4C2×#1309393914yes—
580C2—22no—
581C2—22no—
582C2—24no—
583C2—42no—
584C2—82no—
585C2#2024187914yesdirect product factor
586C2C22×#1309393911yescharacteristic
587C2C2×#1568911911yescharacteristic
588C2C22×#1309395111yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#525C2
socleC23×C47C2×C22⋊C4
cosocleC22×C4×C47C24
hypercenterC2×#525D47
hypocenterC47C2×#1501
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C47⋊C8)C2semidirect
2. C2×C4×C22⋊Dic47C2semidirect
3. #20241879C2direct (N × H)
4. C2×#15688860C2semidirect
5. C2×#15688857C2semidirect
13. C2×C22⋊(C47⋊C8)C22semidirect
15. C4×C22⋊Dic47C22semidirect
17. #15688860C22semidirect
18. #15688857C22semidirect
19. C22×#41×C47C4semidirect
23. C47×#525C4semidirect
26. C2×C22⋊Q8×C47C4semidirect
28. C2×#121×C47C4semidirect
33. C2×#117×C47C4semidirect
50. C2×#41×C47C2×C4semidirect
74. C22⋊Q8×C47C2×C4semidirect
80. #121×C47C2×C4semidirect
89. #117×C47C2×C4semidirect
297. C2×C47#1501semidirect
301. C22×#41Dic47semidirect
306. #525Dic47semidirect
309. C2×C22⋊Q8Dic47semidirect
311. C2×#121Dic47semidirect
316. C2×#117Dic47semidirect
328. C47C2×#1501semidirect
332. C2×#41C2×Dic47semidirect
360. C22⋊Q8C2×Dic47semidirect
366. #121C2×Dic47semidirect
375. #117C2×Dic47semidirect
585. C2#20241879direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊(C47⋊C8)2
C2×C4×C22⋊Dic472
#202418792
C2×#156888602
C2×#156888572
C2×C47×#5252
C2×#150147