Finite Group Databasebeta

← C2×#20243702 #26217475

Subgroups — C2×#20243702 #26217475

Order 12032.

2614 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093995C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
3C22×#13093945C211yesmaximal, characteristic, complemented
4C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
5#20243702C218yesmaximal, direct product factor
6C2×#15688886C211yesmaximal, characteristic, complemented
7C2×#15688885C211yesmaximal, characteristic, complemented
8C22×#102×C47C211yesmaximal, characteristic, complemented
9C23×C22⋊D47—21no—
10C23×C4×D47—21no—
11C23×C4×D47C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13C2×#13093995—24no—
14C2×#13093995—22no—
15C2×#13093995C418yescomplemented
16C2×#13093995C2214yes—
17C22×C4⋊Dic47C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C22×Dic47⋊C4—21no—
20C2×C4⋊C4×D47—22no—
21C2×C4⋊C4×D47—24no—
22C2×C4⋊C4×D47—24no—
23C2×C4⋊C4×D47C2214yescomplemented
24C22×C22⋊Dic47—21no—
25C2×#13093945—24no—
26C2×#13093945C2214yescomplemented
27C2×(C2×D47)⋊C8—24no—
28C2×(C2×D47)⋊C8C2214yescomplemented
29#15688886C2214yescomplemented
30#15688885C2214yescomplemented
31C23×D4×C47C2211yescharacteristic
32C23×C8×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C2×#102×C47—24no—
35C2×#102×C47C2214yescomplemented
36C24×D47D411yescharacteristic
37C22×C22⋊D47—28no—
38C22×C22⋊D47—24no—
39C22×C22⋊D47—44no—
40C22×C22⋊D47—44no—
41C23×Dic47—21no—
42C23×Dic47—21no—
43C23×Dic47D411yescharacteristic
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—28no—
49C22×C4×D47—24no—
50C22×C4×D47—44no—
51C22×C4×D47—44no—
52C22×C4×D47C2×C414yes—
53C22×C4×D47D412yes—
54C22×C4×D47D414yes—
55C22×C4×D47C2312yes—
56C22×C47⋊C8—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8—22no—
59C22×C47⋊C8C2312yes—
60#13093995—216no—
61#13093995—28no—
62#13093995C2×C4116yescomplemented
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47C2×C414yescomplemented
68C2×C4⋊Dic47D414yescomplemented
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—28no—
72C2×Dic47⋊C4—24no—
73C4⋊C4×D47—216no—
74C4⋊C4×D47—28no—
75C4⋊C4×D47—28no—
76C2×C22⋊Dic47—28no—
77C2×C22⋊Dic47—24no—
78#13093945—28no—
79(C2×D47)⋊C8—28no—
80C25×C47—21no—
81C22×D4×C47—22no—
82C22×D4×C47—22no—
83C22×D4×C47—21no—
84C22×D4×C47—22no—
85C22×D4×C47—22no—
86C22×D4×C47—22no—
87C22×D4×C47C2×C414yescomplemented
88C22×D4×C47C2312yes—
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91C22×C8×C47—22no—
92C22×C8×C47—22no—
93C22×C8×C47—22no—
94C22×C8×C47C2312yes—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47D414yescomplemented
98C2×C4⋊C4×C47C2312yes—
99#102×C47—28no—
100C24×C47—24no—
101C24×C47—22no—
102C24×C47—42no—
103C24×C47—42no—
104C24×C47—42no—
105C24×C47—42no—
106C24×C47—42no—
107C24×C47—42no—
108C24×C47C2×D411yescharacteristic
109C23×D47—22no—
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47C22⋊C414yes—
114C23×D47D814yes—
115C23×D47SD1614yes—
116C23×D47C2×D412yes—
117C2×D4×C47—24no—
118C2×D4×C47—28no—
119C2×D4×C47—28no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47—22no—
124C2×D4×C47—22no—
125C2×D4×C47—42no—
126C2×D4×C47—44no—
127C2×D4×C47—44no—
128C2×D4×C47—42no—
129C2×D4×C47C22×C414yescomplemented
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47#4112yes—
140C22×C4×C47#4112yes—
141C22×C4×C47C2411yescharacteristic
142C2×C22⋊D47—216no—
143C2×C22⋊D47—432no—
144C2×C22⋊D47—48no—
145C2×C22⋊D47—48no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—42no—
153C22×Dic47—42no—
154C22×Dic47—42no—
155C22×Dic47—42no—
156C22×Dic47C22⋊C414yes—
157C22×Dic47C2×D412yes—
158C2×C4×D47—28no—
159C2×C4×D47—28no—
160C2×C4×D47—24no—
161C2×C4×D47—28no—
162C2×C4×D47—28no—
163C2×C4×D47—24no—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×C4×D47—22no—
168C2×C4×D47—216no—
169C2×C4×D47—48no—
170C2×C4×D47—432no—
171C2×C4×D47—48no—
172C2×C4×D47C22⋊C418yes—
173C2×C4×D47C22×C414yes—
174C2×C4×D47C2×D414yes—
175C2×C8×C47—22no—
176C2×C8×C47—22no—
177C2×C8×C47—22no—
178C2×C8×C47—44no—
179C4⋊C4×C47—24no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47C2×D414yescomplemented
182C22⋊Dic47—216no—
183C4⋊Dic47—24no—
184C4⋊Dic47—24no—
185C4⋊Dic47—22no—
186C4⋊Dic47—22no—
187C4⋊Dic47C22×C414yescomplemented
188C4⋊Dic47C2×D414yescomplemented
189Dic47⋊C4—216no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—44no—
194C23×C47—24no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—48no—
202C23×C47—48no—
203C23×C47—48no—
204C23×C47—84no—
205C23×C47—84no—
206C23×C47—84no—
207C23×C47—84no—
208C23×C47C2×C22⋊C412yes—
209C23×C47#12012yes—
210C23×C47#12312yes—
211C23×C47C2×D812yes—
212C23×C47C2×SD1612yes—
213C23×C47AGL(1,ℤ/8)12yes—
214C23×C47#13712yes—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—28no—
221C22×D47—28no—
222C22×D47—28no—
223C22×D47—28no—
224C22×D47—28no—
225C22×D47—28no—
226C22×D47#102116yescomplemented
227C22×D47C2×C22⋊C414yes—
228C22×D47C2×D814yes—
229C22×D47C2×SD1614yes—
230D4×C47—24no—
231D4×C47—24no—
232D4×C47—42no—
233D4×C47—42no—
234D4×C47—48no—
235C2×C4×C47—24no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—48no—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47C4×D414yes—
241C2×C4×C47C4×D414yes—
242C2×C4×C47#12012yes—
243C2×C4×C47#12114yes—
244C2×C4×C47#12114yes—
245C2×C4×C47#12312yes—
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C22⋊D47—432no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—28no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—48no—
263C2×Dic47—48no—
264C2×Dic47C2×C22⋊C414yes—
265C4×D47—24no—
266C4×D47—24no—
267C4×D47—24no—
268C4×D47—24no—
269C4×D47—216no—
270C4×D47—432no—
271C8×C47—42no—
272C47⋊C8—42no—
273C2×#1874—471nomaximal, Sylow
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—82no—
278C22×C47—82no—
279C22×C47—816no—
280C22×C47—82no—
281C22×C47—82no—
282C22×C47#38614yes—
283C22×C47C2×#10214yes—
284C22×C47#41714yes—
285C22×C47#44714yes—
286C22×C47#44814yes—
287C22×C47#48014yes—
288C22×C47#48114yes—
289C22×C47C22×C22⋊C411yescharacteristic
290C22×C47C2×#12011yescharacteristic
291C22×C47C2×#12311yescharacteristic
292C22×C47C22×D811yescharacteristic
293C22×C47C22×SD1611yescharacteristic
294C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C47C2×#13711yescharacteristic
296C2×D47—24no—
297C2×D47—24no—
298C2×D47—24no—
299C2×D47—232no—
300C2×D47—24no—
301C2×D47—24no—
302C2×D47—24no—
303C2×D47C2×#10218yescomplemented
304C4×C47—44no—
305C4×C47#38614yes—
306C4×C47C2×C4×D411yescharacteristic
307C4×C47C2×C4×D411yescharacteristic
308C4×C47C2×#12111yescharacteristic
309C4×C47C2×#12111yescharacteristic
310Dic47—24no—
311Dic47—44no—
312Dic47—44no—
313C22×#121—471no—
314C23×C4⋊C4—471no—
315C22×#102—471no—
316C22×#102D4711yescharacteristic, complemented
317C22×C22⋊C8—471no—
318#1874—478no—
319C2×#340—471no—
320C2×#340—471no—
321C2×C47—84no—
322C2×C47#187418yescomplemented
323C2×C47C2×#38611yescharacteristic
324C2×C47C22×#10211yescharacteristic
325C2×C47C2×#41711yescharacteristic
326C2×C47C2×#44711yescharacteristic
327C2×C47C2×#44811yescharacteristic
328C2×C47C2×#48011yescharacteristic
329C2×C47C2×#48111yescharacteristic
330D47—28no—
331C23×D4—941no—
332C23×D4C2×D4711yescharacteristic
333C24×C4—471no—
334C24×C4—941no—
335C23×C8—471no—
336C23×C8C2×D4711yescharacteristic
337C2×#121—478no—
338C2×#121—474no—
339C2×#121—942no—
340C2×#121—944no—
341C22×C4⋊C4—471no—
342C22×C4⋊C4—471no—
343C22×C4⋊C4—474no—
344C22×C4⋊C4—944no—
345C22×C4⋊C4—942no—
346C22×C4⋊C4—941no—
347C22×C4⋊C4—944no—
348C22×C4⋊C4C2×D4711yescharacteristic
349C22×C22⋊C4—941no—
350C2×#102—24no—
351C2×#102—474no—
352C2×#102—944no—
353C2×#102C2×D4714yescomplemented
354C2×C22⋊C8—474no—
355C2×C22⋊C8—944no—
356#340—474no—
357#340—474no—
358C47C2×#187411yescharacteristic, complemented, Sylow
359C25—21no—
360C25—471no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—21no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—948no—
368C22×D4—944no—
369C22×D4—1884no—
370C22×D4—1884no—
371C22×D4C4×D4714yescomplemented
372C22×D4C22×D4712yes—
373C23×C4—21no—
374C23×C4—471no—
375C23×C4—474no—
376C23×C4—472no—
377C23×C4—474no—
378C23×C4—472no—
379C23×C4—944no—
380C23×C4—948no—
381C23×C4—942no—
382C23×C4—942no—
383C23×C4—942no—
384C23×C4—942no—
385C23×C4—941no—
386C23×C4—941no—
387C23×C4—1884no—
388C23×C4—1884no—
389C23×C4C22×D4711yescharacteristic
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—472no—
394C22×C8—942no—
395C22×C8—942no—
396C22×C8—942no—
397C22×C8C22×D4712yes—
398#121—4716no—
399#121—9416no—
400#121—948no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—474no—
404C2×C4⋊C4—474no—
405C2×C4⋊C4—472no—
406C2×C4⋊C4—472no—
407C2×C4⋊C4—948no—
408C2×C4⋊C4—9416no—
409C2×C4⋊C4—944no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—948no—
416C2×C4⋊C4C22⋊D4714yescomplemented
417C2×C4⋊C4C22×D4712yes—
418C2×C22⋊C4—948no—
419C2×C22⋊C4—944no—
420#102—28no—
421#102—948no—
422C22⋊C8—948no—
423C24—24no—
424C24—22no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—474no—
432C24—474no—
433C24—474no—
434C24—472no—
435C24—942no—
436C24—942no—
437C24—942no—
438C24—942no—
439C24D4×D4711yescharacteristic
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—9416no—
453C2×D4—18832no—
454C2×D4—1888no—
455C2×D4—1888no—
456C2×D4C2×C4×D4714yescomplemented
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—474no—
462C22×C4—472no—
463C22×C4—478no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—9416no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—948no—
470C22×C4—948no—
471C22×C4—944no—
472C22×C4—948no—
473C22×C4—948no—
474C22×C4—944no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—944no—
480C22×C4—944no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—1888no—
484C22×C4—18832no—
485C22×C4—1888no—
486C22×C4—1882no—
487C22×C4—1882no—
488C22×C4—1882no—
489C22×C4—1882no—
490C22×C4C2×C4×D4712yes—
491C22×C4#1062535012yes—
492C22×C4D4×D4711yescharacteristic
493C22×C4D4×D4712yes—
494C22×C4D4×D4712yes—
495C22×C4#1062535212yes—
496C22×C4C2×C22⋊D4712yes—
497C22×C4C23×D4711yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—942no—
503C2×C8—942no—
504C2×C8—942no—
505C2×C8—1884no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—474no—
509C4⋊C4—474no—
510C4⋊C4—9416no—
511C4⋊C4—944no—
512C4⋊C4—944no—
513C4⋊C4—942no—
514C4⋊C4—942no—
515C4⋊C4C2×C22⋊D4714yescomplemented
516C22⋊C4—9416no—
517C23—24no—
518C23—48no—
519C23—42no—
520C23—48no—
521C23—42no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—474no—
532C23—4716no—
533C23—474no—
534C23—474no—
535C23—948no—
536C23—948no—
537C23—948no—
538C23—942no—
539C23—942no—
540C23—948no—
541C23—942no—
542C23—942no—
543C23—948no—
544C23—948no—
545C23C22⋊C4×D4712yes—
546C23#1309399212yes—
547C23#1309400612yes—
548C23D8×D4712yes—
549C23SD16×D4712yes—
550C23#1309404812yes—
551C23#1309405312yes—
552C23C2×D4×D4711yescharacteristic
553D4—24no—
554D4—24no—
555D4—48no—
556D4—42no—
557D4—42no—
558D4—18832no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—474no—
564C2×C4—948no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—9416no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—18832no—
575C2×C4—1888no—
576C2×C4—1888no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4C22⋊C4×D4712yes—
582C2×C4(C22⋊D47)⋊C414yes—
583C2×C4C4×C22⋊D4714yes—
584C2×C4#1309399212yes—
585C2×C4#1309399514yes—
586C2×C4#1309399714yes—
587C2×C4#1309400612yes—
588C2×C4C22×C4×D4711yescharacteristic
589C2×C4C2×D4×D4711yescharacteristic
590C2×C4C2×D4×D4711yescharacteristic
591C2×C4C22×C22⋊D4711yescharacteristic
592C2×C4C2×#1062535211yescharacteristic
593C2×C4C2×#1062535011yescharacteristic
594C8—42no—
595C8—1882no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—816no—
604C22—478no—
605C22—944no—
606C22—944no—
607C22—944no—
608C22—944no—
609C22—944no—
610C22—9432no—
611C22—944no—
612C22#1568899914yes—
613C22#102×D4714yes—
614C22#1568912114yes—
615C22#1568925514yes—
616C22#1568926014yes—
617C22#1568943514yes—
618C22#1568944014yes—
619C22C2×C22⋊C4×D4711yescharacteristic
620C22C2×#1309399211yescharacteristic
621C22C2×#1309400611yescharacteristic
622C22C2×D8×D4711yescharacteristic
623C22C2×SD16×D4711yescharacteristic
624C22C2×#1309404811yescharacteristic
625C22C2×#1309405311yescharacteristic
626C4—44no—
627C4—944no—
628C4—1884no—
629C4—1884no—
630C4#1568899914yes—
631C4C2×(C22⋊D47)⋊C411yescharacteristic
632C4C2×C4×C22⋊D4711yescharacteristic
633C4C2×#1309399511yescharacteristic
634C4C2×#1309399711yescharacteristic
635C2—84no—
636C2—948no—
637C2#2024370218yesdirect product factor
638C2C2×#1568899911yescharacteristic
639C2C2×#102×D4711yescharacteristic
640C2C2×#1568912111yescharacteristic
641C2C2×#1568925511yescharacteristic
642C2C2×#1568926011yescharacteristic
643C2C2×#1568943511yescharacteristic
644C2C2×#1568944011yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#102×C47C2
socleC24×C47C2×D4
prime radicalC22×#13093995C2
cosocleC22×C4×C47C24
hypercenterC22×#102D47
hypocenterC47C2×#1874
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D47C2semidirect
3. C22×#13093945C2semidirect
4. C22×(C2×D47)⋊C8C2semidirect
5. #20243702C2direct (N × H)
6. C2×#15688886C2semidirect
7. C2×#15688885C2semidirect
8. C22×#102×C47C2semidirect
15. C2×#13093995C4semidirect
23. C2×C4⋊C4×D47C22semidirect
26. C2×#13093945C22semidirect
28. C2×(C2×D47)⋊C8C22semidirect
29. #15688886C22semidirect
30. #15688885C22semidirect
35. C2×#102×C47C22semidirect
62. #13093995C2×C4semidirect
67. C2×C4⋊Dic47C2×C4semidirect
68. C2×C4⋊Dic47D4semidirect
87. C22×D4×C47C2×C4semidirect
97. C2×C4⋊C4×C47D4semidirect
129. C2×D4×C47C22×C4semidirect
181. C4⋊C4×C47C2×D4semidirect
187. C4⋊Dic47C22×C4semidirect
188. C4⋊Dic47C2×D4semidirect
226. C22×D47#102semidirect
303. C2×D47C2×#102semidirect
316. C22×#102D47semidirect
322. C2×C47#1874semidirect
353. C2×#102C2×D47semidirect
358. C47C2×#1874semidirect
371. C22×D4C4×D47semidirect
416. C2×C4⋊C4C22⋊D47semidirect
456. C2×D4C2×C4×D47semidirect
515. C4⋊C4C2×C22⋊D47semidirect
637. C2#20243702direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939952
C22×C4⋊C4×D472
C22×#130939452
C22×(C2×D47)⋊C82
#202437022
C2×#156888862
C2×#156888852
C22×#102×C472
C2×#187447