Finite Group Databasebeta

← C2×#20246718 #26384964

Subgroups — C2×#20246718 #26384964

Order 12032.

5718 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
1C23×#10625352C211yesmaximal, characteristic
2C22×#13094006C212yesmaximal, complemented
3C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20246718C2116yesmaximal, direct product factor
5C2×#15689044C211yesmaximal, characteristic, complemented
6C2×#15689043C211yesmaximal, characteristic, complemented
7C2×#15689037C211yesmaximal, characteristic, complemented
8C2×#15688999C212yesmaximal, complemented
9C2×#15688998C212yesmaximal, complemented
10C2×#15688995C212yesmaximal, complemented
11C2×#15688985C211yesmaximal, characteristic, complemented
12C2×C47×#394C211yesmaximal, characteristic, complemented
13C22×#10625352—216no—
14C22×#10625352—28no—
15C22×#10625352C2218yes—
16C23×Dic94C2211yescharacteristic
17C23×C22⋊D47C2212yes—
18C24×Dic47C2212yes—
19C23×C4×D47C2211yescharacteristic
20C23×C4×D47C2211yescharacteristic
21C23×C4×D47C2211yescharacteristic
22C2×#13094006—216no—
23C2×#13094006C22116yescomplemented
24C22×C4⋊Dic47C2211yescharacteristic
25C22×C4⋊Dic47C2212yes—
26C22×Dic47⋊C4C2211yescharacteristic
27C22×Dic47⋊C4C2211yescharacteristic
28C22×Dic47⋊C4C2212yes—
29C2×C4⋊C4×D47—28no—
30C2×C4⋊C4×D47C2218yescomplemented
31C22×C22⋊Dic47C2212yes—
32C22×C22⋊Dic47C2212yes—
33C22×(C2×D47)⋊C4C2212yes—
34#15689044C2218yescomplemented
35#15689043C2218yescomplemented
36#15689037C2218yescomplemented
37#15688999C22116yescomplemented
38#15688998C22116yescomplemented
39#15688995C22116yescomplemented
40#15688985C2218yescomplemented
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C2×C47⋊#95C2212yes—
44C2×C47⋊#95C2212yes—
45C23×D4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C22×C22⋊C4×C47C2212yes—
49C47×#394C2218yescomplemented
50C2×#95×C47C2211yescharacteristic
51C24×D47C2311yescharacteristic
52C2×#10625352—2128no—
53C2×#10625352—264no—
54C2×#10625352—232no—
55C22×Dic94—24no—
56C22×Dic94—28no—
57C22×Dic94C2314yes—
58C22×C22⋊D47—28no—
59C22×C22⋊D47—28no—
60C22×C22⋊D47D4116yes—
61C22×C22⋊D47C2318yes—
62C23×Dic47—28no—
63C23×Dic47—28no—
64C23×Dic47—28no—
65C23×Dic47C2312yes—
66C23×Dic47C2318yes—
67C23×Dic47C2311yescharacteristic
68C23×Dic47C2311yescharacteristic
69C23×Dic47C2312yes—
70C23×Dic47C2311yescharacteristic
71C22×C4×D47—24no—
72C22×C4×D47—28no—
73C22×C4×D47—24no—
74C22×C4×D47—24no—
75C22×C4×D47—24no—
76C22×C4×D47—28no—
77C22×C4×D47D418yes—
78C22×C4×D47C2314yes—
79C22×C4×D47C2314yes—
80C22×C4×D47C2314yes—
81#13094006—264no—
82C2×C4⋊Dic47—24no—
83C2×C4⋊Dic47—28no—
84C2×C4⋊Dic47C2314yes—
85C2×C4⋊Dic47C2318yes—
86C2×Dic47⋊C4—24no—
87C2×Dic47⋊C4—24no—
88C2×Dic47⋊C4—28no—
89C2×Dic47⋊C4C2314yes—
90C2×Dic47⋊C4C2318yes—
91C2×Dic47⋊C4C2314yes—
92C4⋊C4×D47—232no—
93C2×C22⋊Dic47—28no—
94C2×C22⋊Dic47—28no—
95C2×C22⋊Dic47C2318yes—
96C2×C22⋊Dic47C2318yes—
97C2×(C2×D47)⋊C4—28no—
98C2×(C2×D47)⋊C4C2318yes—
99C47⋊#95C2318yes—
100C47⋊#95C2314yes—
101C47⋊#95C2318yes—
102C47⋊#95C2318yes—
103C25×C47C2312yes—
104C22×D4×C47—28no—
105C22×D4×C47—24no—
106C22×D4×C47C2314yes—
107C23×C4×C47C2312yes—
108C23×C4×C47C2311yescharacteristic
109C23×C4×C47C2311yescharacteristic
110C23×C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47—24no—
112C2×C4⋊C4×C47C2314yes—
113C2×C22⋊C4×C47—28no—
114C2×C22⋊C4×C47—28no—
115C2×C22⋊C4×C47C2318yes—
116C2×C22⋊C4×C47C2318yes—
117#95×C47C2314yes—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47—24no—
121C24×C47—24no—
122C24×C47—24no—
123C24×C47—24no—
124C24×C47C2×D418yes—
125C24×C47C2414yes—
126C24×C47C2411yescharacteristic
127C23×D47—22no—
128C23×D47—22no—
129C23×D47—24no—
130C23×D47C2×D418yes—
131C23×D47C2×D414yes—
132C23×D47C2412yes—
133C2×D4×C47—232no—
134C2×D4×C47—216no—
135C2×D4×C47—28no—
136C22×C4×C47—22no—
137C22×C4×C47—24no—
138C22×C4×C47—22no—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—24no—
143C22×C4×C47—24no—
144C22×C4×C47C2×D414yes—
145C22×C4×C47#4114yescomplemented
146C22×C4×C47C2412yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C2×C22⋊D47—264no—
151C2×C22⋊D47—216no—
152C2×C22⋊D47—216no—
153C2×C22⋊D47C2×D4132yes—
154C22×Dic47—264no—
155C22×Dic47—22no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—216no—
160C22×Dic47—22no—
161C22×Dic47—216no—
162C22×Dic47—216no—
163C22×Dic47—24no—
164C22×Dic47—24no—
165C22×Dic47—22no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47C2×D414yes—
169C22×Dic47C2×D418yes—
170C22×Dic47#4114yescomplemented
171C22×Dic47C2414yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C22×Dic47C2414yes—
175C22×Dic47C2412yes—
176#10625352—2256no—
177C2×C4×D47—232no—
178C2×C4×D47—232no—
179C2×C4×D47—28no—
180C2×C4×D47—216no—
181C2×C4×D47—28no—
182C2×C4×D47—28no—
183C2×C4×D47—28no—
184C2×C4×D47—216no—
185C2×C4×D47—416no—
186C2×C4×D47C2×D4116yes—
187C2×Dic94—232no—
188C2×Dic94—28no—
189C2×Dic94—216no—
190C4⋊C4×C47—28no—
191C22⋊C4×C47—216no—
192C22⋊C4×C47—216no—
193C22⋊Dic47—216no—
194C22⋊Dic47—216no—
195(C2×D47)⋊C4—216no—
196C4⋊Dic47—28no—
197C4⋊Dic47—216no—
198Dic47⋊C4—28no—
199Dic47⋊C4—28no—
200Dic47⋊C4—216no—
201C23×C47—216no—
202C23×C47—216no—
203C23×C47—216no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—24no—
210C23×C47—48no—
211C23×C47—48no—
212C23×C47—48no—
213C23×C47—48no—
214C23×C47C22×D418yes—
215C23×C47C22×D412yes—
216C23×C47C22×D414yes—
217C23×C47C2×#4112yes—
218C23×C47C2×#4112yes—
219C23×C472−1+414yes—
220C23×C47C2511yescharacteristic
221C22×D47—28no—
222C22×D47—28no—
223C22×D47—216no—
224C22×D47—216no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—24no—
228C22×D47#120116yes—
229C22×D47C22×D418yes—
230C22×D47C22×D414yes—
231D4×C47—232no—
232C2×C4×C47—28no—
233C2×C4×C47—28no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47—44no—
243C2×C4×C47—48no—
244C2×C4×C47C22×D414yes—
245C2×C4×C47C2×#4114yescomplemented
246C22⋊D47—264no—
247C2×Dic47—264no—
248C2×Dic47—216no—
249C2×Dic47—28no—
250C2×Dic47—28no—
251C2×Dic47—22no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—44no—
262C2×Dic47—48no—
263C2×Dic47C22×D418yes—
264C2×Dic47C22×D414yes—
265C2×Dic47C2×#4114yescomplemented
266C4×D47—232no—
267C4×D47—232no—
268C4×D47—416no—
269Dic94—232no—
270C2×#2406—471nomaximal, Sylow
271C22×C47—28no—
272C22×C47—28no—
273C22×C47—28no—
274C22×C47—432no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—44no—
278C22×C47—44no—
279C22×C47C2×#12014yes—
280C22×C47#52514yes—
281C22×C47#53614yes—
282C22×C47#54718yes—
283C22×C47#54718yes—
284C22×C47C23×D411yescharacteristic
285C22×C47C23×D412yes—
286C22×C47C22×#4111yescharacteristic
287C22×C47C22×#4111yescharacteristic
288C22×C47C2×2−1+412yes—
289C2×D47—232no—
290C2×D47—24no—
291C2×D47—24no—
292C2×D47—28no—
293C2×D47—28no—
294C2×D47C2×#12018yes—
295C4×C47—24no—
296C4×C47—24no—
297C4×C47—42no—
298C4×C47—44no—
299Dic47—28no—
300Dic47—24no—
301Dic47—24no—
302Dic47—42no—
303Dic47—44no—
304C23×#41—471no—
305C22×#123—472no—
306C23×C4⋊C4—471no—
307#2406—4716no—
308C2×#395—471no—
309C2×#395—471no—
310C2×#394—471no—
311C2×#394D4711yescharacteristic, complemented
312C2×#386—472no—
313C2×#385—472no—
314C2×#385—472no—
315C2×#381—471no—
316C2×C47—48no—
317C2×C47#240618yescomplemented
318C2×C47C22×#12011yescharacteristic
319C2×C47C2×#52511yescharacteristic
320C2×C47C2×#53611yescharacteristic
321C2×C47C2×#54712yes—
322C2×C47C2×#54712yes—
323D47—28no—
324C22×#41—478no—
325C22×#41—948no—
326C22×#41—9416no—
327C23×Q8—471no—
328C23×D4—472no—
329C23×D4C2×D4711yescharacteristic
330C24×C4—471no—
331C24×C4—472no—
332C24×C4—471no—
333C24×C4—471no—
334C2×#123—4716no—
335C2×#123—9416no—
336C22×C4⋊C4—471no—
337C22×C4⋊C4—472no—
338C22×C4⋊C4—472no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—471no—
341C22×C4⋊C4—478no—
342C22×C4⋊C4—948no—
343C22×C4⋊C4C2×D4711yescharacteristic
344C22×C22⋊C4—472no—
345C22×C22⋊C4—472no—
346C22×C22⋊C4—472no—
347C22×C22⋊C4C2×D4712yes—
348C22×C22⋊C4C2×D4712yes—
349#395—478no—
350#395—478no—
351#394—478no—
352#394C2×D4718yescomplemented
353#386—4716no—
354#385—4716no—
355#385—4716no—
356#381—478no—
357C2×#95—471no—
358C2×#95—472no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95C2×D4711yescharacteristic
362C47C2×#240611yescharacteristic, complemented, Sylow
363C25—471no—
364C25C22×D4712yes—
365C2×#41—9432no—
366C2×#41—9464no—
367C2×#41—94128no—
368C22×Q8—474no—
369C22×Q8—948no—
370C22×Q8—944no—
371C22×D4—28no—
372C22×D4—24no—
373C22×D4—4716no—
374C22×D4—478no—
375C22×D4—948no—
376C22×D4—948no—
377C22×D4C22×D4714yes—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—472no—
381C23×C4—478no—
382C23×C4—474no—
383C23×C4—478no—
384C23×C4—472no—
385C23×C4—471no—
386C23×C4—474no—
387C23×C4—474no—
388C23×C4—944no—
389C23×C4—944no—
390C23×C4—948no—
391C23×C4—944no—
392C23×C4—948no—
393C23×C4—948no—
394C23×C4—948no—
395C23×C4—948no—
396C23×C4—944no—
397C23×C4C22×D4712yes—
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401#123—9464no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—474no—
404C2×C4⋊C4—474no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—478no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—9432no—
409C2×C4⋊C4—944no—
410C2×C4⋊C4—944no—
411C2×C4⋊C4—948no—
412C2×C4⋊C4—948no—
413C2×C4⋊C4—944no—
414C2×C4⋊C4C22×D4714yes—
415C2×C22⋊C4—28no—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—478no—
418C2×C22⋊C4—478no—
419C2×C22⋊C4—478no—
420C2×C22⋊C4—948no—
421C2×C22⋊C4—948no—
422C2×C22⋊C4—948no—
423C2×C22⋊C4C22×D4718yes—
424C2×C22⋊C4C22×D4718yes—
425#95—478no—
426#95—474no—
427#95—478no—
428#95—478no—
429#95C22×D4714yes—
430C24—24no—
431C24—24no—
432C24—24no—
433C24—24no—
434C24—24no—
435C24—24no—
436C24—478no—
437C24—474no—
438C24—472no—
439C24—942no—
440C24—942no—
441C24—944no—
442C24D4×D4718yes—
443C24C23×D4714yes—
444C24C23×D4711yescharacteristic
445#41—94256no—
446C2×Q8—9416no—
447C2×Q8—9432no—
448C2×Q8—948no—
449C2×D4—232no—
450C2×D4—216no—
451C2×D4—28no—
452C2×D4—4732no—
453C2×D4—9464no—
454C2×D4—9416no—
455C2×D4—9416no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—474no—
465C22×C4—478no—
466C22×C4—474no—
467C22×C4—472no—
468C22×C4—472no—
469C22×C4—474no—
470C22×C4—4716no—
471C22×C4—474no—
472C22×C4—472no—
473C22×C4—948no—
474C22×C4—948no—
475C22×C4—9432no—
476C22×C4—9416no—
477C22×C4—9432no—
478C22×C4—948no—
479C22×C4—9416no—
480C22×C4—9416no—
481C22×C4—9416no—
482C22×C4—9464no—
483C22×C4—942no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—9416no—
488C22×C4—942no—
489C22×C4—948no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—942no—
493C22×C4—944no—
494C22×C4—944no—
495C22×C4—18816no—
496C22×C4D4×D4714yes—
497C22×C4#1062535214yescomplemented
498C22×C4C23×D4714yes—
499C22×C4C23×D4712yes—
500C22×C4C23×D4712yes—
501C22×C4C23×D4712yes—
502C4⋊C4—28no—
503C4⋊C4—948no—
504C4⋊C4—9416no—
505C4⋊C4—948no—
506C4⋊C4—9416no—
507C4⋊C4—948no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—9416no—
511C22⋊C4—9416no—
512C22⋊C4—9416no—
513C23—216no—
514C23—216no—
515C23—216no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—4716no—
527C23—478no—
528C23—474no—
529C23—9416no—
530C23—9416no—
531C23—948no—
532C23—942no—
533C23—948no—
534C23—942no—
535C23—944no—
536C23C2×D4×D4718yes—
537C23C2×D4×D4714yes—
538C23C2×D4×D4712yes—
539C23#41×D4712yes—
540C23C2×#1062535212yes—
541C23#1309407114yes—
542C23C24×D4711yescharacteristic
543Q8—9432no—
544D4—232no—
545D4—9464no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—474no—
559C2×C4—478no—
560C2×C4—474no—
561C2×C4—9432no—
562C2×C4—9432no—
563C2×C4—9464no—
564C2×C4—948no—
565C2×C4—9416no—
566C2×C4—948no—
567C2×C4—942no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—942no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—942no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—18816no—
578C2×C4—1888no—
579C2×C4—1884no—
580C2×C4C2×D4×D4714yes—
581C2×C4C2×#1062535214yescomplemented
582C22—28no—
583C22—28no—
584C22—28no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—432no—
590C22—478no—
591C22—948no—
592C22—948no—
593C22—944no—
594C22—944no—
595C22—9432no—
596C22#120×D4714yes—
597C22#1568963314yes—
598C22#1568969314yes—
599C22#1568977118yes—
600C22#1568977318yes—
601C22C22×D4×D4712yes—
602C22C22×D4×D4711yescharacteristic
603C22C2×#41×D4711yescharacteristic
604C22C22×#1062535211yescharacteristic
605C22C2×#1309407112yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—42no—
610C4—944no—
611C4—948no—
612C4—944no—
613C4—1884no—
614C4—1882no—
615C2—48no—
616C2—948no—
617C2#2024671818yesdirect product factor
618C2C2×#120×D4711yescharacteristic
619C2C2×#1568963311yescharacteristic
620C2C2×#1568969311yescharacteristic
621C2C2×#1568977112yes—
622C2C2×#1568977312yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#394C2
socleC24×C47C24
prime radicalC23×#10625352C2
hypercenterC2×#394D47
hypocenterC47C2×#2406
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#13094006C2semidirect
3. C22×C4⋊C4×D47C2semidirect
4. #20246718C2direct (N × H)
5. C2×#15689044C2semidirect
6. C2×#15689043C2semidirect
7. C2×#15689037C2semidirect
8. C2×#15688999C2semidirect
9. C2×#15688998C2semidirect
10. C2×#15688995C2semidirect
11. C2×#15688985C2semidirect
12. C2×C47×#394C2semidirect
23. C2×#13094006C22semidirect
30. C2×C4⋊C4×D47C22semidirect
34. #15689044C22semidirect
35. #15689043C22semidirect
36. #15689037C22semidirect
37. #15688999C22semidirect
38. #15688998C22semidirect
39. #15688995C22semidirect
40. #15688985C22semidirect
49. C47×#394C22semidirect
145. C22×C4×C47#41semidirect
170. C22×Dic47#41semidirect
245. C2×C4×C47C2×#41semidirect
265. C2×Dic47C2×#41semidirect
311. C2×#394D47semidirect
317. C2×C47#2406semidirect
352. #394C2×D47semidirect
362. C47C2×#2406semidirect
497. C22×C4#10625352semidirect
581. C2×C4C2×#10625352semidirect
617. C2#20246718direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253522
C22×#130940062
C22×C4⋊C4×D472
#202467182
C2×#156890442
C2×#156890432
C2×#156890372
C2×#156889992
C2×#156889982
C2×#156889952
C2×#156889852
C2×C47×#3942
C2×#240647