Finite Group Databasebeta

← C2×#20249134 #26382640

Subgroups — C2×#20249134 #26382640

Order 12032.

3718 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×#13094007C211yesmaximal, characteristic, complemented
2C22×#13093996C211yesmaximal, characteristic
3C23×C4⋊Dic47C211yesmaximal, characteristic, complemented
4#20249134C2116yesmaximal, direct product factor
5C2×#15689056C212yesmaximal, complemented
6C2×#15689046C212yesmaximal, complemented
7C2×#15689045C212yesmaximal, complemented
8C2×#15689039C212yesmaximal, complemented
9C2×#15689003C211yesmaximal, characteristic, complemented
10C2×#15689000C211yesmaximal, characteristic, complemented
11C2×#15688981C211yesmaximal, characteristic, complemented
12C22×#123×C47C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2211yescharacteristic
14C23×C22⋊D47C2211yescharacteristic
15C24×Dic47C2211yescharacteristic
16C24×Dic47C2211yescharacteristic
17C2×#13094007—28no—
18C2×#13094007C2218yescomplemented
19C2×#13093996—28no—
20C2×#13093996C2218yes—
21C22×C4⋊Dic47—28no—
22C22×C4⋊Dic47C2212yes—
23C22×C4⋊Dic47C2212yes—
24C22×C4⋊Dic47C2218yescomplemented
25C22×Dic47⋊C4C2212yes—
26C22×Dic47⋊C4C2211yescharacteristic
27C22×Dic47⋊C4C2212yes—
28C22×C22⋊Dic47C2211yescharacteristic
29C22×C22⋊Dic47C2211yescharacteristic
30C22×C22⋊Dic47C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×(C2×D47)⋊C4C2211yescharacteristic
33C22×(C2×D47)⋊C4C2212yes—
34#15689056C22116yescomplemented
35#15689046C22116yescomplemented
36#15689045C22116yescomplemented
37#15689039C22116yescomplemented
38#15689003C2218yescomplemented
39#15689000C2218yescomplemented
40#15688981C2218yescomplemented
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2212yes—
44C23×D4×C47C2211yescharacteristic
45C24×C4×C47C2211yescharacteristic
46C2×#123×C47—28no—
47C2×#123×C47C2218yescomplemented
48C22×C4⋊C4×C47C2212yes—
49C22×C22⋊C4×C47C2211yescharacteristic
50C22×C22⋊C4×C47C2212yes—
51C24×D47C2311yescharacteristic
52C22×C22⋊D47—24no—
53C22×C22⋊D47—24no—
54C22×C22⋊D47—24no—
55C22×C22⋊D47—24no—
56C22×C22⋊D47—44no—
57C22×C22⋊D47C2314yes—
58C22×C22⋊D47C2314yes—
59C23×Dic47—24no—
60C23×Dic47—44no—
61C23×Dic47—44no—
62C23×Dic47D418yes—
63C23×Dic47C2312yes—
64C23×Dic47C2312yes—
65C23×Dic47C2311yescharacteristic
66C23×Dic47C2311yescharacteristic
67C23×Dic47C2311yescharacteristic
68C23×Dic47C2314yes—
69C23×Dic47C2314yes—
70#13094007—232no—
71#13093996—232no—
72C2×C4⋊Dic47—28no—
73C2×C4⋊Dic47—28no—
74C2×C4⋊Dic47—232no—
75C2×C4⋊Dic47C2318yes—
76C2×C4⋊Dic47C2318yes—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4—24no—
79C2×Dic47⋊C4—44no—
80C2×Dic47⋊C4C2318yes—
81C2×Dic47⋊C4C2314yes—
82C2×Dic47⋊C4C2318yes—
83C2×C22⋊Dic47—24no—
84C2×C22⋊Dic47—24no—
85C2×C22⋊Dic47—44no—
86C2×C22⋊Dic47C2314yes—
87C2×C22⋊Dic47C2314yes—
88C2×C22⋊Dic47C2318yes—
89C2×(C2×D47)⋊C4—28no—
90C2×(C2×D47)⋊C4—24no—
91C2×(C2×D47)⋊C4—44no—
92C2×(C2×D47)⋊C4C2318yes—
93C2×(C2×D47)⋊C4C2314yes—
94C2×(C2×D47)⋊C4C2318yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C47⋊#95C2318yes—
98C25×C47C2311yescharacteristic
99C25×C47C2311yescharacteristic
100C22×D4×C47—24no—
101C22×D4×C47—24no—
102C22×D4×C47—24no—
103C22×D4×C47C2314yes—
104C23×C4×C47—24no—
105C23×C4×C47—24no—
106C23×C4×C47D418yescomplemented
107C23×C4×C47C2312yes—
108C23×C4×C47C2311yescharacteristic
109C23×C4×C47C2312yes—
110C23×C4×C47C2314yes—
111#123×C47—232no—
112C2×C4⋊C4×C47—28no—
113C2×C4⋊C4×C47C2318yes—
114C2×C22⋊C4×C47—24no—
115C2×C22⋊C4×C47—28no—
116C2×C22⋊C4×C47C2314yes—
117C2×C22⋊C4×C47C2318yes—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47—22no—
121C24×C47—22no—
122C24×C47—22no—
123C24×C47—24no—
124C24×C47—24no—
125C24×C47C2×D414yes—
126C24×C47C2×D414yes—
127C24×C47#4114yes—
128C24×C47C2412yes—
129C24×C47C2412yes—
130C24×C47C2411yescharacteristic
131C23×D47—22no—
132C23×D47—22no—
133C23×D47—22no—
134C23×D47—24no—
135C23×D47—24no—
136C23×D47C2412yes—
137C2×D4×C47—28no—
138C2×D4×C47—28no—
139C2×D4×C47—28no—
140C2×D4×C47—416no—
141C22×C4×C47—232no—
142C22×C4×C47—24no—
143C22×C4×C47—28no—
144C22×C4×C47—24no—
145C22×C4×C47—28no—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—24no—
151C22×C4×C47—24no—
152C22×C4×C47C2×D418yes—
153C22×C4×C47C2×D4116yescomplemented
154C22×C4×C47C2414yes—
155C22×C4×C47C2412yes—
156C22×C4×C47C2414yes—
157C2×C22⋊D47—28no—
158C2×C22⋊D47—28no—
159C2×C22⋊D47—28no—
160C2×C22⋊D47—28no—
161C2×C22⋊D47—48no—
162C2×C22⋊D47—416no—
163C2×C22⋊D47—416no—
164C22×Dic47—24no—
165C22×Dic47—24no—
166C22×Dic47—24no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—28no—
170C22×Dic47—24no—
171C22×Dic47—22no—
172C22×Dic47—24no—
173C22×Dic47—24no—
174C22×Dic47—24no—
175C22×Dic47—24no—
176C22×Dic47—416no—
177C22×Dic47—48no—
178C22×Dic47—48no—
179C22×Dic47—416no—
180C22×Dic47C2×D414yes—
181C22×Dic47C2×D414yes—
182C22×Dic47C2×D4116yes—
183C22×Dic47C2414yes—
184C22×Dic47C2414yes—
185C22×Dic47C2412yes—
186C22×Dic47C2412yes—
187C22×Dic47C2412yes—
188C4⋊C4×C47—216no—
189C22⋊C4×C47—28no—
190C22⋊C4×C47—216no—
191C22⋊Dic47—28no—
192C22⋊Dic47—28no—
193C22⋊Dic47—48no—
194(C2×D47)⋊C4—216no—
195(C2×D47)⋊C4—28no—
196(C2×D47)⋊C4—48no—
197C4⋊Dic47—216no—
198C4⋊Dic47—216no—
199Dic47⋊C4—216no—
200Dic47⋊C4—28no—
201Dic47⋊C4—48no—
202C23×C47—28no—
203C23×C47—28no—
204C23×C47—216no—
205C23×C47—216no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—24no—
212C23×C47—24no—
213C23×C47—44no—
214C23×C47—44no—
215C23×C47—44no—
216C23×C47—48no—
217C23×C47—48no—
218C23×C47#121116yes—
219C23×C47C22×D414yes—
220C23×C47C22×D412yes—
221C23×C47C22×D414yes—
222C23×C47C22×D412yes—
223C23×C47C2×#4114yes—
224C23×C47C2×#4112yes—
225C23×C47H(5,𝔽2)14yes—
226C23×C472−1+414yes—
227C23×C47C2511yescharacteristic
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—24no—
232C22×D47—24no—
233C22×D47—44no—
234C22×D47—44no—
235C22×D47—44no—
236C22×D47—48no—
237C22×D47—48no—
238D4×C47—416no—
239C2×C4×C47—216no—
240C2×C4×C47—232no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—22no—
244C2×C4×C47—24no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—44no—
250C2×C4×C47—48no—
251C2×C4×C47C22×D418yes—
252C22⋊D47—416no—
253C22⋊D47—416no—
254C2×Dic47—28no—
255C2×Dic47—28no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—416no—
268C2×Dic47—416no—
269C2×Dic47—44no—
270C2×Dic47—48no—
271C2×Dic47—48no—
272C2×Dic47C22×D414yes—
273C2×Dic47C22×D414yes—
274C2×#2618—471nomaximal, Sylow
275C22×C47—232no—
276C22×C47—24no—
277C22×C47—24no—
278C22×C47—28no—
279C22×C47—28no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—44no—
284C22×C47—44no—
285C22×C47—88no—
286C22×C47C2×#12118yes—
287C22×C47C2×#12114yes—
288C22×C47#53414yes—
289C22×C47#53614yes—
290C22×C47#53718yes—
291C22×C47#53918yes—
292C22×C47C23×D411yescharacteristic
293C22×C47C23×D411yescharacteristic
294C22×C47C22×#4111yescharacteristic
295C22×C47C2×H(5,𝔽2)12yes—
296C22×C47C2×2−1+412yes—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—44no—
301C2×D47—44no—
302C2×D47—88no—
303C4×C47—28no—
304C4×C47—42no—
305C4×C47—44no—
306Dic47—24no—
307Dic47—24no—
308Dic47—42no—
309Dic47—44no—
310Dic47—44no—
311C22×#123—471no—
312C22×#123D4711yescharacteristic, complemented
313C22×#121—471no—
314C23×C4⋊C4—471no—
315#2618—4716no—
316C2×#398—472no—
317C2×#396—472no—
318C2×#396—472no—
319C2×#394—472no—
320C2×#386—471no—
321C2×#386—471no—
322C2×#380—471no—
323C2×C47—28no—
324C2×C47—82no—
325C2×C47#261818yescomplemented
326C2×C47C22×#12111yescharacteristic
327C2×C47C2×#53411yescharacteristic
328C2×C47C2×#53611yescharacteristic
329C2×C47C2×#53712yes—
330C2×C47C2×#53912yes—
331D47—82no—
332C23×D4—471no—
333C23×D4—471no—
334C23×D4C2×D4711yescharacteristic
335C24×C4—471no—
336C24×C4—471no—
337C24×C4C2×D4711yescharacteristic
338C2×#123—28no—
339C2×#123—478no—
340C2×#123—948no—
341C2×#123C2×D4718yescomplemented
342C2×#121—478no—
343C2×#121—948no—
344C22×C4⋊C4—478no—
345C22×C4⋊C4—472no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4—472no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4—472no—
350C22×C4⋊C4—948no—
351C22×C4⋊C4C2×D4712yes—
352C22×C22⋊C4—471no—
353C22×C22⋊C4—471no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4—471no—
356C22×C22⋊C4—472no—
357C22×C22⋊C4—472no—
358C22×C22⋊C4C2×D4712yes—
359C22×C22⋊C4C2×D4711yescharacteristic
360#398—4716no—
361#396—4716no—
362#396—4716no—
363#394—4716no—
364#386—478no—
365#386—478no—
366#380—478no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C47C2×#261811yescharacteristic, complemented, Sylow
371C25—471no—
372C25C22×D4711yescharacteristic
373C25C22×D4711yescharacteristic
374C22×D4—24no—
375C22×D4—24no—
376C22×D4—24no—
377C22×D4—474no—
378C22×D4—474no—
379C22×D4—944no—
380C22×D4—944no—
381C22×D4—944no—
382C22×D4—944no—
383C22×D4—1884no—
384C22×D4C22×D4714yes—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—478no—
390C23×C4—474no—
391C23×C4—472no—
392C23×C4—472no—
393C23×C4—471no—
394C23×C4—474no—
395C23×C4—944no—
396C23×C4—1884no—
397C23×C4—1884no—
398C23×C4C22⋊D4718yescomplemented
399C23×C4C22×D4712yes—
400C23×C4C22×D4711yescharacteristic
401C23×C4C22×D4712yes—
402C23×C4C22×D4714yes—
403#123—232no—
404#123—9432no—
405#121—9432no—
406C2×C4⋊C4—28no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—478no—
409C2×C4⋊C4—474no—
410C2×C4⋊C4—478no—
411C2×C4⋊C4—478no—
412C2×C4⋊C4—948no—
413C2×C4⋊C4—948no—
414C2×C4⋊C4—9432no—
415C2×C4⋊C4—944no—
416C2×C4⋊C4—948no—
417C2×C4⋊C4—1884no—
418C2×C4⋊C4C22×D4718yes—
419C2×C22⋊C4—24no—
420C2×C22⋊C4—28no—
421C2×C22⋊C4—474no—
422C2×C22⋊C4—474no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—478no—
425C2×C22⋊C4—478no—
426C2×C22⋊C4—478no—
427C2×C22⋊C4—944no—
428C2×C22⋊C4—944no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—948no—
431C2×C22⋊C4—1884no—
432C2×C22⋊C4—1884no—
433C2×C22⋊C4C22×D4714yes—
434C2×C22⋊C4C22×D4718yes—
435#95—478no—
436#95—478no—
437#95—478no—
438C24—24no—
439C24—24no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—24no—
444C24—24no—
445C24—472no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24—944no—
450C24—944no—
451C24D4×D4714yes—
452C24#1062535214yes—
453C24C2×C22⋊D4714yes—
454C24C23×D4712yes—
455C24C23×D4712yes—
456C24C23×D4711yescharacteristic
457C2×D4—28no—
458C2×D4—28no—
459C2×D4—28no—
460C2×D4—416no—
461C2×D4—948no—
462C2×D4—948no—
463C2×D4—948no—
464C2×D4—948no—
465C2×D4—1888no—
466C2×D4—18816no—
467C2×D4—18816no—
468C22×C4—232no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—474no—
480C22×C4—474no—
481C22×C4—472no—
482C22×C4—472no—
483C22×C4—4716no—
484C22×C4—474no—
485C22×C4—474no—
486C22×C4—472no—
487C22×C4—944no—
488C22×C4—944no—
489C22×C4—942no—
490C22×C4—948no—
491C22×C4—944no—
492C22×C4—944no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—944no—
496C22×C4—944no—
497C22×C4—944no—
498C22×C4—944no—
499C22×C4—18816no—
500C22×C4—1888no—
501C22×C4—1888no—
502C22×C4—18816no—
503C22×C4C2×C22⋊D4718yes—
504C22×C4C2×C22⋊D47116yescomplemented
505C22×C4C23×D4714yes—
506C22×C4C23×D4712yes—
507C22×C4C23×D4714yes—
508C4⋊C4—216no—
509C4⋊C4—948no—
510C4⋊C4—9416no—
511C4⋊C4—9416no—
512C4⋊C4—9416no—
513C4⋊C4—1888no—
514C22⋊C4—28no—
515C22⋊C4—216no—
516C22⋊C4—948no—
517C22⋊C4—948no—
518C22⋊C4—948no—
519C22⋊C4—9416no—
520C22⋊C4—1888no—
521C22⋊C4—1888no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—216no—
527C23—216no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—48no—
538C23—942no—
539C23—942no—
540C23—942no—
541C23—944no—
542C23—944no—
543C23—1884no—
544C23—1884no—
545C23—1884no—
546C23—1888no—
547C23—1888no—
548C23#13093996116yes—
549C23C2×D4×D4714yes—
550C23C2×D4×D4712yes—
551C23C22×C22⋊D4714yes—
552C23C22×C22⋊D4712yes—
553C23C2×#1062535214yes—
554C23C2×#1062535212yes—
555C23#1309407014yes—
556C23#1309407114yes—
557C23C24×D4711yescharacteristic
558D4—416no—
559D4—18816no—
560D4—18816no—
561C2×C4—216no—
562C2×C4—232no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—474no—
574C2×C4—474no—
575C2×C4—948no—
576C2×C4—948no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—942no—
580C2×C4—944no—
581C2×C4—944no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—18816no—
589C2×C4—18816no—
590C2×C4—1884no—
591C2×C4—1888no—
592C2×C4—1888no—
593C2×C4C22×C22⋊D4718yes—
594C22—232no—
595C22—24no—
596C22—24no—
597C22—28no—
598C22—28no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—88no—
605C22—3768no—
606C22—1882no—
607C22—1882no—
608C22—1882no—
609C22—1884no—
610C22—1884no—
611C22C2×#1309399618yes—
612C22C2×#1309399614yes—
613C22#1568968414yes—
614C22#1568969514yes—
615C22#1568970118yes—
616C22#1568971718yes—
617C22C22×D4×D4711yescharacteristic
618C22C23×C22⋊D4711yescharacteristic
619C22C22×#1062535211yescharacteristic
620C22C2×#1309407012yes—
621C22C2×#1309407112yes—
622C4—28no—
623C4—42no—
624C4—44no—
625C4—944no—
626C4—944no—
627C4—1882no—
628C4—1884no—
629C4—1884no—
630C2—28no—
631C2—82no—
632C2—3762no—
633C2#2024913418yesdirect product factor
634C2C22×#1309399611yescharacteristic
635C2C2×#1568968411yescharacteristic
636C2C2×#1568969511yescharacteristic
637C2C2×#1568970112yes—
638C2C2×#1568971712yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#123×C47C2
socleC24×C47C24
prime radicalC22×#13093996C2
hypercenterC22×#123D47
hypocenterC47C2×#2618
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C22×#13094007C2semidirect
3. C23×C4⋊Dic47C2semidirect
4. #20249134C2direct (N × H)
5. C2×#15689056C2semidirect
6. C2×#15689046C2semidirect
7. C2×#15689045C2semidirect
8. C2×#15689039C2semidirect
9. C2×#15689003C2semidirect
10. C2×#15689000C2semidirect
11. C2×#15688981C2semidirect
12. C22×#123×C47C2semidirect
18. C2×#13094007C22semidirect
24. C22×C4⋊Dic47C22semidirect
34. #15689056C22semidirect
35. #15689046C22semidirect
36. #15689045C22semidirect
37. #15689039C22semidirect
38. #15689003C22semidirect
39. #15689000C22semidirect
40. #15688981C22semidirect
47. C2×#123×C47C22semidirect
106. C23×C4×C47D4semidirect
153. C22×C4×C47C2×D4semidirect
312. C22×#123D47semidirect
325. C2×C47#2618semidirect
341. C2×#123C2×D47semidirect
370. C47C2×#2618semidirect
398. C23×C4C22⋊D47semidirect
504. C22×C4C2×C22⋊D47semidirect
633. C2#20249134direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940072
C22×#130939962
C23×C4⋊Dic472
#202491342
C2×#156890562
C2×#156890462
C2×#156890452
C2×#156890392
C2×#156890032
C2×#156890002
C2×#156889812
C22×#123×C472
C2×#261847