Finite Group Databasebeta

← C2×#20241750 #26182247

Subgroups — C2×#20241750 #26182247

Order 12032.

1426 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094002C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic
3C2×C22⋊C8×D47C211yesmaximal, characteristic
4#20241750C218yesmaximal, direct product factor
5C2×#15688849C211yesmaximal, characteristic, complemented
6C2×#15688848C211yesmaximal, characteristic, complemented
7C2×#15688848C211yesmaximal, characteristic, complemented
8C2×C47×#324C211yesmaximal, characteristic, complemented
9C23×Dic94C411yescharacteristic
10C23×C4×D47C2211yescharacteristic
11C22×#10625324—21no—
12C22×C8×D47—21no—
13C2×#13094002—21no—
14C2×#13094002—21no—
15C2×#13094002—22no—
16C2×#13094002—21no—
17C2×#13094002—21no—
18C2×#13094002—22no—
19C2×#13094002C414yes—
20C2×#13094002C2214yes—
21C22×Dic47⋊C4C411yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C22×(C2×D47)⋊C4C411yescharacteristic
24C22×(C2×D47)⋊C4C411yescharacteristic
25#15689085—22no—
26#15689085C2214yes—
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×(C2×D47)⋊C8—21no—
30C2×(C2×D47)⋊C8—21no—
31C22⋊C8×D47—22no—
32C22⋊C8×D47C2214yes—
33#15688849C2214yescomplemented
34#15688848C2214yescomplemented
35#15688848C2214yescomplemented
36C22×C4⋊C4×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2211yescharacteristic
38C2×C22⋊C8×C47C2211yescharacteristic
39C47×#324C2214yescomplemented
40C24×D47C2×C411yescharacteristic
41C22×Dic94—21no—
42C22×Dic94—22no—
43C22×Dic94—21no—
44C22×Dic94—44no—
45C22×Dic94C2×C414yes—
46C23×Dic47C2×C411yescharacteristic
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47—44no—
50C22×C4×D47C2×C412yes—
51C22×C4×D47D411yescharacteristic
52C22×C4×D47D411yescharacteristic
53C22×C4×D47D412yes—
54C22×C4×D47D411yescharacteristic
55C22×C4×D47D411yescharacteristic
56C22×C4×D47D412yes—
57C22×C4×D47C2312yes—
58C2×#10625324—24no—
59C2×#10625324—44no—
60C22×C47⋊C8—21no—
61C22×C47⋊C8—21no—
62C2×C8×D47—24no—
63C2×C8×D47—44no—
64#13094002—24no—
65#13094002—24no—
66#13094002—24no—
67#13094002—24no—
68#13094002—24no—
69#13094002—24no—
70#13094002—24no—
71#13094002—24no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—22no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4C2×C414yes—
80C2×Dic47⋊C4C2×C412yes—
81C2×Dic47⋊C4C2312yes—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—22no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—22no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4C2×C414yes—
89C2×(C2×D47)⋊C4C2×C414yes—
90C22⋊(C47⋊C8)—21no—
91C22⋊(C47⋊C8)—21no—
92C22⋊(C47⋊C8)C2312yes—
93C22⋊(C47⋊C8)C2312yes—
94(C2×D47)⋊C8—24no—
95(C2×D47)⋊C8—24no—
96C23×C4×C47C2×C411yescharacteristic
97C23×C4×C47C2×C411yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47—21no—
100C22×C8×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47C2×C412yes—
106C2×C4⋊C4×C47C2312yes—
107C22⋊C8×C47—21no—
108C22⋊C8×C47—21no—
109C22⋊C8×C47C2312yes—
110C22⋊C8×C47C2312yes—
111C24×C47C22×C411yescharacteristic
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—24no—
115C23×D47—42no—
116C23×D47C22⋊C411yescharacteristic
117C23×D47C22⋊C412yes—
118C23×D47C22⋊C411yescharacteristic
119C23×D47C22×C412yes—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—24no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—42no—
144C22×Dic47C22⋊C411yescharacteristic
145C22×Dic47C22⋊C412yes—
146C22×Dic47C22⋊C411yescharacteristic
147C22×Dic47C22×C412yes—
148C22×Dic47C22×C412yes—
149C22×Dic47C22×C412yes—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—44no—
153C2×C4×D47—44no—
154C2×C4×D47—44no—
155C2×C4×D47—44no—
156C2×C4×D47—42no—
157C2×C4×D47—42no—
158C2×C4×D47C22⋊C412yes—
159C2×C4×D47C22⋊C412yes—
160C2×C4×D47C22⋊C412yes—
161C2×C4×D47C22⋊C412yes—
162C2×C4×D47C2×D412yes—
163C2×C4×D47C2×D412yes—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×D412yes—
166C2×Dic94—24no—
167C2×Dic94—24no—
168C2×Dic94—44no—
169C2×Dic94—48no—
170C2×Dic94—44no—
171C2×Dic94—48no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—42no—
175C2×C8×C47—42no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180(C2×D47)⋊C4—24no—
181(C2×D47)⋊C4—24no—
182(C2×D47)⋊C4—24no—
183(C2×D47)⋊C4—24no—
184Dic47⋊C4—22no—
185Dic47⋊C4—22no—
186Dic47⋊C4—22no—
187Dic47⋊C4—22no—
188Dic47⋊C4—24no—
189Dic47⋊C4—24no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—42no—
193C2×C47⋊C8—42no—
194#10625324—48no—
195C8×D47—48no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47C2×C22⋊C411yescharacteristic
199C23×C47C2×C22⋊C411yescharacteristic
200C23×C47#13112yes—
201C23×C47#13112yes—
202C23×C47C23×C411yescharacteristic
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—24no—
208C22×D47—22no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47—48no—
216C22×D47—48no—
217C22×D47#10114yes—
218C22×D47C2×C22⋊C412yes—
219C22×D47C2×C22⋊C412yes—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47C2×C22⋊C411yescharacteristic
236C2×C4×C47C2×C22⋊C411yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C22×D411yescharacteristic
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47#9914yes—
252C2×Dic47C2×C22⋊C412yes—
253C2×Dic47C2×C22⋊C412yes—
254C4×D47—24no—
255C4×D47—24no—
256C4×D47—24no—
257C4×D47—24no—
258C4×D47—42no—
259C4×D47—42no—
260C4×D47—42no—
261C4×D47—42no—
262Dic94—48no—
263Dic94—48no—
264C8×C47—42no—
265C8×C47—42no—
266C47⋊C8—42no—
267C47⋊C8—42no—
268C2×#1472—471nomaximal, Sylow
269C22×C47—21no—
270C22×C47—21no—
271C22×C47—21no—
272C22×C47—21no—
273C22×C47—24no—
274C22×C47—24no—
275C22×C47#40812yes—
276C22×C47#40812yes—
277C22×C47C2×#9912yes—
278C22×C47C2×#10112yes—
279C22×C47C22×C22⋊C411yescharacteristic
280C22×C47C2×#13111yescharacteristic
281C22×C47C2×#13111yescharacteristic
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—28no—
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—44no—
289C2×D47—44no—
290C2×D47—48no—
291C2×D47—44no—
292C2×D47C2×#10114yes—
293C4×C47—21no—
294C4×C47—21no—
295C4×C47—21no—
296C4×C47—21no—
297C4×C47—42no—
298C4×C47—42no—
299Dic47—22no—
300Dic47—42no—
301Dic47—42no—
302Dic47C2×#9914yes—
303C22×C22⋊Q8—471no—
304C2×#407—471no—
305C22×C22⋊C8—471no—
306#1472—478no—
307C2×#324—471no—
308C2×#324—471no—
309C2×#324—471no—
310C2×#324D4711yescharacteristic, complemented
311C2×C47—21no—
312C2×C47—21no—
313C2×C47—22no—
314C2×C47#147214yescomplemented
315C2×C47C2×#40811yescharacteristic
316C2×C47C22×#9911yescharacteristic
317C2×C47C22×#10111yescharacteristic
318D47—24no—
319D47—42no—
320C23×Q8—471no—
321C24×C4—471no—
322C22×M16—941no—
323C23×C8—941no—
324C2×C22⋊Q8—474no—
325C2×C22⋊Q8—474no—
326C2×C22⋊Q8—941no—
327C2×C22⋊Q8—941no—
328C2×C22⋊Q8—942no—
329C2×C22⋊Q8—941no—
330C2×C22⋊Q8—941no—
331C2×C22⋊Q8—942no—
332C22×C4⋊C4—471no—
333C22×C4⋊C4—471no—
334C22×C4⋊C4C2×D4711yescharacteristic
335C22×C22⋊C4—471no—
336C22×C22⋊C4—471no—
337#407—474no—
338#407—942no—
339C2×C22⋊C8—471no—
340C2×C22⋊C8—471no—
341C2×C22⋊C8—474no—
342C2×C22⋊C8—942no—
343C2×C22⋊C8—941no—
344C2×C22⋊C8—941no—
345C2×C22⋊C8C2×D4711yescharacteristic
346C2×C22⋊C8C2×D4711yescharacteristic
347#324—474no—
348#324—474no—
349#324—474no—
350#324C2×D4714yescomplemented
351C47C2×#147211yescharacteristic, complemented, Sylow
352C25—471no—
353C22×Q8—474no—
354C22×Q8—942no—
355C22×Q8—941no—
356C22×Q8—941no—
357C22×Q8—1884no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—472no—
362C23×C4—472no—
363C23×C4—471no—
364C23×C4—471no—
365C23×C4—472no—
366C23×C4—472no—
367C23×C4—471no—
368C23×C4—471no—
369C23×C4—1884no—
370C23×C4C4×D4711yescharacteristic
371C23×C4C4×D4711yescharacteristic
372C23×C4C22×D4711yescharacteristic
373C2×M16—944no—
374C2×M16—1884no—
375C22×C8—21no—
376C22×C8—21no—
377C22×C8—944no—
378C22×C8—941no—
379C22×C8—941no—
380C22×C8—1884no—
381C22⋊Q8—944no—
382C22⋊Q8—944no—
383C22⋊Q8—944no—
384C22⋊Q8—944no—
385C22⋊Q8—944no—
386C22⋊Q8—944no—
387C22⋊Q8—944no—
388C22⋊Q8—944no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—474no—
394C2×C4⋊C4—472no—
395C2×C4⋊C4—472no—
396C2×C4⋊C4—942no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4C4×D4712yes—
404C2×C4⋊C4C22×D4712yes—
405C2×C22⋊C4—474no—
406C2×C22⋊C4—474no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—942no—
409C2×C22⋊C4—941no—
410C2×C22⋊C4—941no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4—941no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—472no—
416C22⋊C8—472no—
417C22⋊C8—944no—
418C22⋊C8—944no—
419C22⋊C8—941no—
420C22⋊C8—941no—
421C22⋊C8C22×D4712yes—
422C22⋊C8C22×D4712yes—
423C24—471no—
424C24—471no—
425C24—472no—
426C24—472no—
427C24—944no—
428C24—942no—
429C24—942no—
430C24—1882no—
431C24C2×C4×D4711yescharacteristic
432C2×Q8—944no—
433C2×Q8—944no—
434C2×Q8—1884no—
435C2×Q8—1888no—
436C2×Q8—1888no—
437C2×Q8—1884no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—471no—
447C22×C4—472no—
448C22×C4—471no—
449C22×C4—472no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—944no—
461C22×C4—944no—
462C22×C4—944no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—1882no—
470C22×C4—1882no—
471C22×C4—1884no—
472C22×C4—1884no—
473C22×C4—1884no—
474C22×C4—1884no—
475C22×C4—1882no—
476C22×C4C2×C4×D4712yes—
477C22×C4C2×C4×D4712yes—
478C22×C4C2×C4×D4711yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4D4×D4711yescharacteristic
483C22×C4C23×D4711yescharacteristic
484M16—1888no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—942no—
490C2×C8—942no—
491C2×C8—1888no—
492C2×C8—1882no—
493C2×C8—1882no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—944no—
499C4⋊C4—944no—
500C4⋊C4—942no—
501C4⋊C4—942no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C22⋊C4—944no—
505C22⋊C4—944no—
506C22⋊C4—944no—
507C22⋊C4—944no—
508C23—22no—
509C23—22no—
510C23—472no—
511C23—474no—
512C23—472no—
513C23—944no—
514C23—944no—
515C23—942no—
516C23—942no—
517C23—944no—
518C23—942no—
519C23—942no—
520C23—942no—
521C23—942no—
522C23—942no—
523C23—1882no—
524C23—1882no—
525C23—1888no—
526C23—1888no—
527C23C22⋊C4×D4711yescharacteristic
528C23C22⋊C4×D4711yescharacteristic
529C23#1309403012yes—
530C23#1309403212yes—
531C23C22×C4×D4711yescharacteristic
532Q8—1888no—
533Q8—1888no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—472no—
550C2×C4—474no—
551C2×C4—472no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—944no—
555C2×C4—944no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—1882no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4C22⋊C4×D4711yescharacteristic
573C2×C4C22⋊C4×D4711yescharacteristic
574C2×C4C2×D4×D4711yescharacteristic
575C2×C4C2×D4×D4711yescharacteristic
576C8—42no—
577C8—42no—
578C8—1882no—
579C8—1882no—
580C22—24no—
581C22—24no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—474no—
587C22—948no—
588C22—942no—
589C22—942no—
590C22—942no—
591C22—942no—
592C22—942no—
593C22—1888no—
594C22—1884no—
595C22—1884no—
596C22—1884no—
597C22#1568908812yes—
598C22#1568908812yes—
599C22#99×D4712yes—
600C22#101×D4712yes—
601C22C2×C22⋊C4×D4711yescharacteristic
602C22C2×#1309403011yescharacteristic
603C22C2×#1309403211yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—474no—
611C4—942no—
612C4—1882no—
613C4—1882no—
614C2—22no—
615C2—21no—
616C2—21no—
617C2—944no—
618C2—1882no—
619C2#2024175014yesdirect product factor
620C2C2×#1568908811yescharacteristic
621C2C2×#99×D4711yescharacteristic
622C2C2×#101×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#324C2
socleC23×C47C2×C22⋊C4
prime radicalC24×D47C2×C4
cosocleC22×C4×C47C24
hypercenterC2×#324D47
hypocenterC47C2×#1472
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
4. #20241750C2direct (N × H)
5. C2×#15688849C2semidirect
6. C2×#15688848C2semidirect
7. C2×#15688848C2semidirect
8. C2×C47×#324C2semidirect
33. #15688849C22semidirect
34. #15688848C22semidirect
35. #15688848C22semidirect
39. C47×#324C22semidirect
310. C2×#324D47semidirect
314. C2×C47#1472semidirect
350. #324C2×D47semidirect
351. C47C2×#1472semidirect
619. C2#20241750direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940022
C2×#156890852
C2×C22⋊C8×D472
#202417502
C2×#156888492
C2×#156888482
C2×C47×#3242
C2×#147247