Finite Group Databasebeta

← C2×#20247701 #26385447

Subgroups — C2×#20247701 #26385447

Order 12032.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D47C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
3#20247701C2116yesmaximal, direct product factor
4C2×#15689064C212yesmaximal, complemented
5C2×#15689059C212yesmaximal, complemented
6C2×#15689022C211yesmaximal, characteristic, complemented
7C2×#15689019C211yesmaximal, characteristic, complemented
8C2×#15688999C212yesmaximal
9C2×#15688997C212yesmaximal, complemented
10C2×#15688983C211yesmaximal, characteristic, complemented
11C2×#15688980C211yesmaximal, characteristic, complemented
12C2×C47×#380C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2212yes—
16C2×C4×C22⋊D47—28no—
17C2×C4×C22⋊D47C2218yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2211yescharacteristic
23C22×Dic47⋊C4C2212yes—
24C2×C4⋊C4×D47—28no—
25C2×C4⋊C4×D47C2218yescomplemented
26C22×C22⋊Dic47C2211yescharacteristic
27C22×C22⋊Dic47C2212yes—
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×(C2×D47)⋊C4C2212yes—
30C22×C4×Dic47C2211yescharacteristic
31#15689064C22116yescomplemented
32#15689059C22116yescomplemented
33#15689022C2218yescomplemented
34#15689019C2218yescomplemented
35#15688999C22116yes—
36#15688997C22116yescomplemented
37#15688983C2218yescomplemented
38#15688980C2218yescomplemented
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C24×C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C22⋊C4×C47C2212yes—
47C47×#380C2218yescomplemented
48C2×#95×C47C2212yes—
49C24×D47C2311yescharacteristic
50C22×C22⋊D47—24no—
51C22×C22⋊D47—24no—
52C22×C22⋊D47Q818yes—
53C22×C22⋊D47C2314yes—
54C23×Dic47C2312yes—
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C22×C4×D47—24no—
60C22×C4×D47—24no—
61C22×C4×D47—28no—
62C22×C4×D47—44no—
63C22×C4×D47—44no—
64C22×C4×D47D418yes—
65C22×C4×D47C2314yes—
66C22×C4×D47C2318yes—
67C4×C22⋊D47—232no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47—24no—
70C2×C4⋊Dic47—44no—
71C2×C4⋊Dic47C2314yes—
72C2×C4⋊Dic47C2314yes—
73C2×C4⋊Dic47C2318yes—
74C2×Dic47⋊C4—28no—
75C2×Dic47⋊C4—24no—
76C2×Dic47⋊C4—44no—
77C2×Dic47⋊C4C2318yes—
78C2×Dic47⋊C4C2314yes—
79C2×Dic47⋊C4C2318yes—
80C4⋊C4×D47—232no—
81C2×C22⋊Dic47—24no—
82C2×C22⋊Dic47—44no—
83C2×C22⋊Dic47C2314yes—
84C2×C22⋊Dic47C2318yes—
85C2×(C2×D47)⋊C4—24no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2314yes—
89C2×C4×Dic47—24no—
90C2×C4×Dic47C2314yes—
91C47⋊#95C2318yes—
92C47⋊#95C2318yes—
93C47⋊#95C2318yes—
94C47⋊#95C2318yes—
95C25×C47C2311yescharacteristic
96C23×C4×C47—24no—
97C23×C4×C47—24no—
98C23×C4×C47—24no—
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2312yes—
102C23×C4×C47C2314yes—
103C23×C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47—24no—
105C2×C4⋊C4×C47—24no—
106C2×C4⋊C4×C47C2314yes—
107C2×C4⋊C4×C47C2314yes—
108C2×C22⋊C4×C47—44no—
109C2×C22⋊C4×C47C2318yes—
110#95×C47C2318yes—
111C24×C47—22no—
112C24×C47—22no—
113C24×C47—22no—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47C2×Q814yes—
118C24×C47C2412yes—
119C24×C47C2411yescharacteristic
120C23×D47—22no—
121C23×D47—22no—
122C23×D47—22no—
123C23×D47—22no—
124C23×D47C2×D414yes—
125C23×D47C2×Q814yes—
126C23×D47#4114yes—
127C23×D47C2412yes—
128C22×C4×C47—24no—
129C22×C4×C47—22no—
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—28no—
138C22×C4×C47—28no—
139C22×C4×C47—28no—
140C22×C4×C47—416no—
141C22×C4×C47C2×D414yes—
142C22×C4×C47#4114yes—
143C22×C4×C47#4114yes—
144C22×C4×C47C2414yes—
145C22×C4×C47C2414yes—
146C22×C4×C47C2412yes—
147C22×C4×C47C2412yes—
148C2×C22⋊D47—232no—
149C2×C22⋊D47—28no—
150C2×C22⋊D47—28no—
151C2×C22⋊D47C2×Q8116yes—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—24no—
165C22×Dic47C2×D414yes—
166C22×Dic47C2×Q814yes—
167C22×Dic47C2414yes—
168C22×Dic47C2414yes—
169C22×Dic47C2412yes—
170C22×Dic47C2412yes—
171C22×Dic47C2412yes—
172C2×C4×D47—232no—
173C2×C4×D47—28no—
174C2×C4×D47—28no—
175C2×C4×D47—216no—
176C2×C4×D47—48no—
177C2×C4×D47—48no—
178C2×C4×D47—432no—
179C2×C4×D47C2×D4116yes—
180C4⋊C4×C47—28no—
181C4⋊C4×C47—28no—
182C22⋊C4×C47—48no—
183C22⋊Dic47—28no—
184C22⋊Dic47—48no—
185(C2×D47)⋊C4—28no—
186(C2×D47)⋊C4—216no—
187C4⋊Dic47—28no—
188C4⋊Dic47—28no—
189C4⋊Dic47—48no—
190Dic47⋊C4—216no—
191Dic47⋊C4—28no—
192Dic47⋊C4—48no—
193C4×Dic47—28no—
194C23×C47—28no—
195C23×C47—28no—
196C23×C47—28no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—44no—
204C23×C47—44no—
205C23×C47—44no—
206C23×C47—44no—
207C23×C47C22×D412yes—
208C23×C47C22×Q814yes—
209C23×C47C22×Q812yes—
210C23×C47C2×#4112yes—
211C23×C47C2×#4112yes—
212C23×C47C2×#4112yes—
213C23×C47H(5,𝔽2)12yes—
214C23×C47H(5,𝔽2)12yes—
215C23×C47C2511yescharacteristic
216C22×D47—28no—
217C22×D47—28no—
218C22×D47—28no—
219C22×D47—28no—
220C22×D47—28no—
221C22×D47—28no—
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47C22⋊Q8116yes—
227C22×D47C22×D414yes—
228C22×D47C22×Q814yes—
229C22×D47C2×#4114yes—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—44no—
240C2×C4×C47—48no—
241C2×C4×C47—48no—
242C2×C4×C47—416no—
243C2×C4×C47#123116yes—
244C2×C4×C47C22×D414yes—
245C2×C4×C47C2×#4114yes—
246C2×C4×C47C2×#4114yes—
247C22⋊D47—232no—
248C2×Dic47—28no—
249C2×Dic47—28no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—48no—
264C2×Dic47—44no—
265C2×Dic47—48no—
266C2×Dic47C22×D414yes—
267C2×Dic47C22×Q814yes—
268C4×D47—232no—
269C4×D47—432no—
270C2×#2487—471nomaximal, Sylow
271C22×C47—24no—
272C22×C47—24no—
273C22×C47—24no—
274C22×C47—42no—
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—42no—
278C22×C47—416no—
279C22×C47C2×C22⋊Q814yes—
280C22×C47C2×#12314yes—
281C22×C47#53514yes—
282C22×C47#55314yes—
283C22×C47#55314yes—
284C22×C47#55414yes—
285C22×C47#55414yes—
286C22×C47C23×D411yescharacteristic
287C22×C47C23×Q811yescharacteristic
288C22×C47C22×#4111yescharacteristic
289C22×C47C22×#4111yescharacteristic
290C22×C47C22×#4111yescharacteristic
291C22×C47C2×H(5,𝔽2)11yescharacteristic
292C22×C47C2×H(5,𝔽2)11yescharacteristic
293C2×D47—24no—
294C2×D47—24no—
295C2×D47—24no—
296C2×D47—24no—
297C2×D47—24no—
298C2×D47—24no—
299C2×D47—232no—
300C2×D47C2×C22⋊Q818yes—
301C4×C47—42no—
302C4×C47—44no—
303C4×C47—44no—
304C4×C47C2×#12318yes—
305Dic47—24no—
306Dic47—24no—
307Dic47—44no—
308Dic47—42no—
309Dic47—44no—
310C22×C4×D4—471no—
311C23×C4⋊C4—471no—
312#2487—4716no—
313C2×#400—472no—
314C2×#399—472no—
315C2×#389—471no—
316C2×#389—471no—
317C2×#386—472no—
318C2×#385—472no—
319C2×#380—471no—
320C2×#380—471no—
321C2×#380D4711yescharacteristic, complemented
322C2×C47—44no—
323C2×C47#248718yescomplemented
324C2×C47C22×C22⋊Q811yescharacteristic
325C2×C47C22×#12311yescharacteristic
326C2×C47C2×#53511yescharacteristic
327C2×C47C2×#55311yescharacteristic
328C2×C47C2×#55311yescharacteristic
329C2×C47C2×#55411yescharacteristic
330C2×C47C2×#55411yescharacteristic
331D47—28no—
332C23×D4—471no—
333C24×C4—471no—
334C24×C4—472no—
335C24×C4C2×D4711yescharacteristic
336C2×C4×D4—478no—
337C2×C4×D4—948no—
338C22×C4⋊C4—472no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—471no—
341C22×C4⋊C4—471no—
342C22×C4⋊C4—472no—
343C22×C4⋊C4—472no—
344C22×C4⋊C4—478no—
345C22×C4⋊C4—948no—
346C22×C4⋊C4C2×D4711yescharacteristic
347C22×C4⋊C4C2×D4711yescharacteristic
348C22×C22⋊C4—471no—
349C22×C22⋊C4—471no—
350C22×C22⋊C4—472no—
351C22×C22⋊C4—472no—
352C22×C22⋊C4C2×D4712yes—
353C22×C42—471no—
354#400—4716no—
355#399—4716no—
356#389—478no—
357#389—478no—
358#386—4716no—
359#385—4716no—
360#380—478no—
361#380—478no—
362#380C2×D4718yescomplemented
363C2×#95—472no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95—472no—
367C2×#95C2×D4712yes—
368C47C2×#248711yescharacteristic, complemented, Sylow
369C25—471no—
370C25C22×D4711yescharacteristic
371C22×D4—478no—
372C22×D4—474no—
373C22×D4—944no—
374C22×D4—944no—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—478no—
381C23×C4—474no—
382C23×C4—471no—
383C23×C4—472no—
384C23×C4—472no—
385C23×C4—478no—
386C23×C4—948no—
387C23×C4—944no—
388C23×C4—944no—
389C23×C4—1884no—
390C23×C4—1884no—
391C23×C4C22×D4714yes—
392C23×C4C22×D4712yes—
393C23×C4C22×D4711yescharacteristic
394C23×C4C22×D4712yes—
395C23×C4C22×D4711yescharacteristic
396C4×D4—9432no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—478no—
400C2×C4⋊C4—474no—
401C2×C4⋊C4—474no—
402C2×C4⋊C4—474no—
403C2×C4⋊C4—478no—
404C2×C4⋊C4—478no—
405C2×C4⋊C4—9432no—
406C2×C4⋊C4—948no—
407C2×C4⋊C4—944no—
408C2×C4⋊C4—944no—
409C2×C4⋊C4—944no—
410C2×C4⋊C4—1884no—
411C2×C4⋊C4—1884no—
412C2×C4⋊C4C22×D4714yes—
413C2×C4⋊C4C22×D4714yes—
414C2×C22⋊C4—44no—
415C2×C22⋊C4—474no—
416C2×C22⋊C4—474no—
417C2×C22⋊C4—478no—
418C2×C22⋊C4—478no—
419C2×C22⋊C4—944no—
420C2×C22⋊C4—948no—
421C2×C22⋊C4—944no—
422C2×C22⋊C4—1884no—
423C2×C22⋊C4C22×D4718yes—
424C2×C42—474no—
425C2×C42—944no—
426#95—478no—
427#95—478no—
428#95—478no—
429#95—478no—
430#95C22×D4718yes—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—474no—
438C24—474no—
439C24—474no—
440C24—472no—
441C24—942no—
442C24—942no—
443C24—942no—
444C24—942no—
445C24Q8×D4714yes—
446C24C23×D4712yes—
447C24C23×D4711yescharacteristic
448C2×D4—4716no—
449C2×D4—948no—
450C2×D4—9432no—
451C2×D4—948no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—416no—
465C22×C4—474no—
466C22×C4—474no—
467C22×C4—472no—
468C22×C4—472no—
469C22×C4—4716no—
470C22×C4—474no—
471C22×C4—474no—
472C22×C4—472no—
473C22×C4—9416no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—9432no—
479C22×C4—948no—
480C22×C4—948no—
481C22×C4—942no—
482C22×C4—944no—
483C22×C4—944no—
484C22×C4—944no—
485C22×C4—942no—
486C22×C4—944no—
487C22×C4—944no—
488C22×C4—944no—
489C22×C4—942no—
490C22×C4—1888no—
491C22×C4—1888no—
492C22×C4—18832no—
493C22×C4#1062535014yes—
494C22×C4D4×D4714yes—
495C22×C4#1062535214yes—
496C22×C4C23×D4712yes—
497C22×C4C23×D4714yes—
498C22×C4C23×D4714yes—
499C22×C4C23×D4712yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—9416no—
503C4⋊C4—948no—
504C4⋊C4—948no—
505C4⋊C4—948no—
506C4⋊C4—1888no—
507C4⋊C4—1888no—
508C22⋊C4—48no—
509C22⋊C4—9416no—
510C22⋊C4—948no—
511C22⋊C4—948no—
512C22⋊C4—1888no—
513C42—948no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—4716no—
528C23—474no—
529C23—474no—
530C23—474no—
531C23—948no—
532C23—948no—
533C23—948no—
534C23—948no—
535C23—948no—
536C23—948no—
537C23—942no—
538C23—942no—
539C23—942no—
540C23—942no—
541C23C2×D4×D4712yes—
542C23C2×Q8×D4714yes—
543C23C2×Q8×D4712yes—
544C23#41×D4712yes—
545C23C2×#1062535212yes—
546C23C2×#1062535012yes—
547C23#1309406912yes—
548C23#1309407012yes—
549C23C24×D4711yescharacteristic
550D4—9432no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—416no—
564C2×C4—474no—
565C2×C4—474no—
566C2×C4—948no—
567C2×C4—948no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—9432no—
573C2×C4—942no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—942no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—942no—
582C2×C4—18832no—
583C2×C4—1888no—
584C2×C4—1884no—
585C2×C4—1888no—
586C2×C4#13094006116yes—
587C2×C4C2×D4×D4714yes—
588C2×C4C2×#1062535214yes—
589C2×C4C2×#1062535014yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—416no—
597C22—42no—
598C22—478no—
599C22—944no—
600C22—944no—
601C22—944no—
602C22—944no—
603C22—9432no—
604C22—944no—
605C22—944no—
606C22C22⋊Q8×D4714yes—
607C22C2×#1309400614yes—
608C22#1568968914yes—
609C22#1568981714yes—
610C22#1568982014yes—
611C22#1568982814yes—
612C22#1568983114yes—
613C22C22×D4×D4711yescharacteristic
614C22C22×Q8×D4711yescharacteristic
615C22C2×#41×D4711yescharacteristic
616C22C22×#1062535211yescharacteristic
617C22C22×#1062535011yescharacteristic
618C22C2×#1309406911yescharacteristic
619C22C2×#1309407011yescharacteristic
620C4—42no—
621C4—44no—
622C4—44no—
623C4—944no—
624C4—944no—
625C4—1884no—
626C4—1884no—
627C4—1882no—
628C4C2×#1309400618yes—
629C2—44no—
630C2—948no—
631C2#2024770118yesdirect product factor
632C2C2×C22⋊Q8×D4711yescharacteristic
633C2C22×#1309400611yescharacteristic
634C2C2×#1568968911yescharacteristic
635C2C2×#1568981711yescharacteristic
636C2C2×#1568982011yescharacteristic
637C2C2×#1568982811yescharacteristic
638C2C2×#1568983111yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#380C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC2×#380D47
hypocenterC47C2×#2487
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D47C2semidirect
3. #20247701C2direct (N × H)
4. C2×#15689064C2semidirect
5. C2×#15689059C2semidirect
6. C2×#15689022C2semidirect
7. C2×#15689019C2semidirect
9. C2×#15688997C2semidirect
10. C2×#15688983C2semidirect
11. C2×#15688980C2semidirect
12. C2×C47×#380C2semidirect
25. C2×C4⋊C4×D47C22semidirect
31. #15689064C22semidirect
32. #15689059C22semidirect
33. #15689022C22semidirect
34. #15689019C22semidirect
36. #15688997C22semidirect
37. #15688983C22semidirect
38. #15688980C22semidirect
47. C47×#380C22semidirect
321. C2×#380D47semidirect
323. C2×C47#2487semidirect
362. #380C2×D47semidirect
368. C47C2×#2487semidirect
631. C2#20247701direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D472
C22×C4⋊C4×D472
#202477012
C2×#156890642
C2×#156890592
C2×#156890222
C2×#156890192
C2×#156889992
C2×#156889972
C2×#156889832
C2×#156889802
C2×C47×#3802
C2×#248747