Finite Group Databasebeta

← C2×#20241931 #26183533

Subgroups — C2×#20241931 #26183533

Order 12032.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689087C211yesmaximal, characteristic, complemented
2C2×#15688983C211yesmaximal, characteristic, complemented
3#20241931C218yesmaximal, direct product factor
4C2×#15688860C212yesmaximal, complemented
5C2×#15688857C212yesmaximal, complemented
6C2×C47×#525C211yesmaximal, characteristic
7C22×#10625329—21no—
8C22×C4⋊Dic47—21no—
9C22×C22⋊Dic47—21no—
10#15689087—22no—
11#15689087C2214yescomplemented
12C2×C22⋊(C47⋊C8)—21no—
13C2×C22⋊(C47⋊C8)C2212yes—
14#15688983—22no—
15#15688983C2214yescomplemented
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—
35C2×#10625329—24no—
36C2×#10625329—44no—
37C22×C47⋊C8—22no—
38C2×C4⋊Dic47—24no—
39C2×C4⋊Dic47—44no—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47—44no—
42C22⋊(C47⋊C8)—24no—
43C22⋊(C47⋊C8)—22no—
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—22no—
101C24×C47—22no—
102C24×C47—21no—
103C24×C47—42no—
104C24×C47—42no—
105C24×C47C22⋊C411yescharacteristic
106C24×C47C22⋊C411yescharacteristic
107C24×C47C22⋊C412yes—
108C24×C47C2×D411yescharacteristic
109C24×C47C2×D412yes—
110#41×C47—48no—
111#41×C47—48no—
112C2×Q8×C47—21no—
113C2×Q8×C47—42no—
114C2×Q8×C47C22×C412yes—
115C2×D4×C47—22no—
116C2×D4×C47—24no—
117C2×D4×C47—24no—
118C2×D4×C47—22no—
119C2×D4×C47—21no—
120C2×D4×C47—42no—
121C2×D4×C47—44no—
122C2×D4×C47—44no—
123C2×D4×C47—42no—
124C2×D4×C47—42no—
125C2×D4×C47C22×C412yes—
126C22×C4×C47—24no—
127C22×C4×C47—24no—
128C22×C4×C47—24no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—24no—
133C22×C4×C47—44no—
134C22×C4×C47—44no—
135C22×C4×C47—44no—
136C22×C4×C47—44no—
137C22×C4×C47—42no—
138C22×C4×C47—42no—
139C22×C4×C47—42no—
140C22×C4×C47—44no—
141C22×C4×C47C22⋊C411yescharacteristic
142C22×C4×C47C22⋊C412yes—
143C22×C4×C47C22⋊C412yes—
144C22×C4×C47C22⋊C412yes—
145C22×C4×C47C22⋊C412yes—
146C22×C4×C47C22⋊C412yes—
147C22×C4×C47C22⋊C411yescharacteristic
148C22×C4×C47C22×C411yescharacteristic
149C22×C4×C47C22×C412yes—
150C22×C4×C47C22×C411yescharacteristic
151C22×C4×C47C22×C412yes—
152C22×C4×C47C2×D412yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D412yes—
158C22×C4×C47C2411yescharacteristic
159C22×Dic47—24no—
160C22×Dic47—44no—
161C22×Dic47—44no—
162C22×Dic47—44no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—24no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47C22×C412yes—
169C4⋊C4×C47C22×C412yes—
170C22⋊C4×C47—24no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—24no—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C22⋊C4×C47—24no—
176C22⋊C4×C47—22no—
177C42×C47—24no—
178C22⋊Dic47—48no—
179C4⋊Dic47—48no—
180#10625329—48no—
181C2×C47⋊C8—24no—
182C2×C47⋊C8—44no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—24no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—48no—
200C23×C47—48no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47C2×C22⋊C412yes—
205C23×C47C2×C22⋊C412yes—
206C23×C47C2×C22⋊C411yescharacteristic
207C23×C47C2×C22⋊C411yescharacteristic
208C23×C47AGL(1,ℤ/8)12yes—
209C23×C47#13712yes—
210C23×C47C22×D411yescharacteristic
211Q8×C47—42no—
212Q8×C47—42no—
213D4×C47—42no—
214D4×C47—42no—
215D4×C47—48no—
216D4×C47—48no—
217D4×C47—42no—
218D4×C47—42no—
219D4×C47—44no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—41no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—48no—
244C2×C4×C47#9914yes—
245C2×C4×C47C2×C22⋊C411yescharacteristic
246C2×C4×C47C2×C22⋊C412yes—
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C2×C22⋊C412yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×Dic47—44no—
252C2×Dic47—44no—
253C2×Dic47—44no—
254C2×Dic47—88no—
255C47⋊C8—44no—
256C2×#1513—471nomaximal, Sylow
257C22×C47—21no—
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—21no—
261C22×C47—24no—
262C22×C47—24no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—44no—
270C22×C47—44no—
271C22×C47—48no—
272C22×C47—44no—
273C22×C47—44no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—84no—
278C22×C47C2×#9912yes—
279C22×C47#41712yes—
280C22×C47#41712yes—
281C22×C47#42112yes—
282C22×C47C22×C22⋊C411yescharacteristic
283C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C47C2×#13711yescharacteristic
285C4×C47—21no—
286C4×C47—21no—
287C4×C47—22no—
288C4×C47—21no—
289C4×C47—21no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47C2×#9914yes—
294Dic47—84no—
295C2×#525D4711yescharacteristic
296C2×#407—471no—
297C2×#380—471no—
298#1513—478no—
299C2×#328—472no—
300C2×#327—472no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—22no—
304C2×C47—42no—
305C2×C47—42no—
306C2×C47—82no—
307C2×C47#151314yescomplemented
308C2×C47C22×#9911yescharacteristic
309C2×C47C2×#41711yescharacteristic
310C2×C47C2×#42111yescharacteristic
311C22×#41Dic4711yescharacteristic, complemented
312C24×C4C2×D4711yescharacteristic
313C22×M16—941no—
314#525—22no—
315#525—22no—
316#525Dic4714yescomplemented
317#525C2×D4714yes—
318C2×#123—21no—
319C2×C22⋊Q8Dic4711yescharacteristic, complemented
320C2×C22⋊Q8C2×D4711yescharacteristic
321C2×#121Dic4711yescharacteristic, complemented
322C2×#121C2×D4711yescharacteristic
323C2×#120—21no—
324C2×C4×D4—21no—
325C2×C4×D4—21no—
326C2×#117Dic4711yescharacteristic, complemented
327C22×C4⋊C4—941no—
328C22×C22⋊C4—941no—
329#407—474no—
330#407—942no—
331C2×C22⋊C8—472no—
332C2×C22⋊C8—941no—
333#380—474no—
334#380—942no—
335C2×#95—472no—
336#328—478no—
337#327—478no—
338C47C2×#151311yescharacteristic, complemented, Sylow
339C25C22⋊D4711yescharacteristic
340C2×#41—22no—
341C2×#41—44no—
342C2×#41C2×Dic4714yescomplemented
343C22×Q8C2×Dic4711yescharacteristic
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4C2×Dic4711yescharacteristic
348C23×C4—21no—
349C23×C4—44no—
350C23×C4—942no—
351C23×C4C2×Dic4711yescharacteristic
352C23×C4C2×Dic4711yescharacteristic
353C23×C4C2×Dic4712yes—
354C23×C4C22⋊D4711yescharacteristic
355C23×C4C22⋊D4711yescharacteristic
356C23×C4C22⋊D4712yes—
357C23×C4C22⋊D4711yescharacteristic
358C23×C4C22⋊D4712yes—
359C23×C4C22×D4711yescharacteristic
360C23×C4C22×D4712yes—
361C2×M16—944no—
362C2×M16—1884no—
363C22×C8—942no—
364#123—24no—
365#123—24no—
366C22⋊Q8—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8C2×Dic4712yes—
370C22⋊Q8C2×Dic4714yescomplemented
371C22⋊Q8C22×D4712yes—
372#121—22no—
373#121—21no—
374#121—21no—
375#121C2×Dic4712yes—
376#121C2×Dic4714yescomplemented
377#121C22×D4712yes—
378#120—24no—
379#120—24no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—24no—
383C4×D4—24no—
384#117—22no—
385#117C2×Dic4714yescomplemented
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—1884no—
390C2×C4⋊C4C2×Dic4711yescharacteristic
391C2×C4⋊C4C2×Dic4711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—1884no—
399C2×C42—21no—
400C22⋊C8—474no—
401C22⋊C8—942no—
402C22⋊C8—944no—
403#95—474no—
404#95—942no—
405C24—22no—
406C24—22no—
407C24—21no—
408C24—42no—
409C24—42no—
410C24C22⋊Dic4711yescharacteristic
411C24C22⋊Dic4711yescharacteristic
412C24C22⋊Dic4712yes—
413C24C2×C22⋊D4711yescharacteristic
414C24C2×C22⋊D4712yes—
415#41—48no—
416#41—48no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8C22×Dic4712yes—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4C22×Dic4712yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—944no—
447C22×C4—1884no—
448C22×C4—1884no—
449C22×C4—1884no—
450C22×C4C22⋊Dic4712yes—
451C22×C4C22⋊Dic4712yes—
452C22×C4C22⋊Dic4712yes—
453C22×C4C22⋊Dic4711yescharacteristic
454C22×C4C22⋊Dic4712yes—
455C22×C4C22⋊Dic4712yes—
456C22×C4C22⋊Dic4711yescharacteristic
457C22×C4C22×Dic4712yes—
458C22×C4C22×Dic4711yescharacteristic
459C22×C4C22×Dic4711yescharacteristic
460C22×C4C22×Dic4712yes—
461C22×C4C2×C22⋊D4712yes—
462C22×C4C2×C22⋊D4712yes—
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4712yes—
467C22×C4C23×D4711yescharacteristic
468M16—1888no—
469C2×C8—944no—
470C2×C8—1884no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—1888no—
477C4⋊C4C22×Dic4712yes—
478C4⋊C4C22×Dic4712yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—1888no—
487C42—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—48no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23C2×C22⋊Dic4712yes—
510C23C2×C22⋊Dic4712yes—
511C23C2×C22⋊Dic4711yescharacteristic
512C23C2×C22⋊Dic4711yescharacteristic
513C23#1309404912yes—
514C23#1309405412yes—
515C23C22×C22⋊D4711yescharacteristic
516Q8—42no—
517Q8—42no—
518D4—42no—
519D4—42no—
520D4—48no—
521D4—48no—
522D4—44no—
523D4—42no—
524D4—42no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—48no—
549C2×C4—3768no—
550C2×C4—1884no—
551C2×C4—1884no—
552C2×C4—1884no—
553C2×C4#1309393914yes—
554C2×C4C2×C22⋊Dic4711yescharacteristic
555C2×C4C2×C22⋊Dic4712yes—
556C2×C4C2×C22⋊Dic4711yescharacteristic
557C2×C4C2×C22⋊Dic4712yes—
558C2×C4C23×Dic4711yescharacteristic
559C2×C4C22×C22⋊D4711yescharacteristic
560C8—1884no—
561C22—24no—
562C22—24no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—84no—
582C22C2×#1309393912yes—
583C22#1568912412yes—
584C22#1568912412yes—
585C22#1568914012yes—
586C22C22×C22⋊Dic4711yescharacteristic
587C22C2×#1309404911yescharacteristic
588C22C2×#1309405411yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—3764no—
598C4C2×#1309393914yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—42no—
603C2—42no—
604C2—82no—
605C2#2024193114yesdirect product factor
606C2C22×#1309393911yescharacteristic
607C2C2×#1568912411yescharacteristic
608C2C2×#1568914011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#525C2
socleC23×C47C2×C22⋊C4
cosocleC22×C4×C47C24
hypercenterC2×#525D47
hypocenterC47C2×#1513
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C2×#15689087C2semidirect
2. C2×#15688983C2semidirect
3. #20241931C2direct (N × H)
4. C2×#15688860C2semidirect
5. C2×#15688857C2semidirect
11. #15689087C22semidirect
15. #15688983C22semidirect
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
307. C2×C47#1513semidirect
311. C22×#41Dic47semidirect
316. #525Dic47semidirect
319. C2×C22⋊Q8Dic47semidirect
321. C2×#121Dic47semidirect
326. C2×#117Dic47semidirect
338. C47C2×#1513semidirect
342. C2×#41C2×Dic47semidirect
370. C22⋊Q8C2×Dic47semidirect
376. #121C2×Dic47semidirect
385. #117C2×Dic47semidirect
605. C2#20241931direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156890872
C2×#156889832
#202419312
C2×#156888602
C2×#156888572
C2×C47×#5252
C2×#151347