Finite Group Databasebeta

← C2×#20248862 #26386789

Subgroups — C2×#20248862 #26386789

Order 12032.

3438 conjugacy classes of subgroups.

The subgroup lattice has 588 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
3#20248862C2116yesmaximal, direct product factor
4C2×#15689063C211yesmaximal, characteristic, complemented
5C2×#15689062C211yesmaximal, characteristic, complemented
6C2×#15689037C211yesmaximal, characteristic, complemented
7C2×#15689016C212yesmaximal, complemented
8C2×#15689011C212yesmaximal, complemented
9C2×#15688999C212yesmaximal, complemented
10C2×#15688999C212yesmaximal, complemented
11C2×#15688970C211yesmaximal, characteristic, complemented
12C2×C47×#394C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2212yes—
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C2×#13093995—28no—
18C2×#13093995C2218yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2211yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C2×C4⋊C4×D47—28no—
24C2×C4⋊C4×D47C2218yescomplemented
25C22×C22⋊Dic47C2212yes—
26C22×C22⋊Dic47C2212yes—
27C22×C22⋊Dic47C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×C4×Dic47C2212yes—
30#15689063C2218yescomplemented
31#15689062C2218yescomplemented
32#15689037C2218yescomplemented
33#15689016C22116yescomplemented
34#15689011C22116yescomplemented
35#15688999C22116yescomplemented
36#15688999C22116yescomplemented
37#15688970C2218yescomplemented
38C2×C47⋊#95C2212yes—
39C2×C47⋊#95C2212yes—
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C23×D4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C22⋊C4×C47C2212yes—
46C22×C22⋊C4×C47C2212yes—
47C47×#394C2218yescomplemented
48C2×#95×C47C2211yescharacteristic
49C24×D47C2311yescharacteristic
50C22×C22⋊D47—28no—
51C22×C22⋊D47—28no—
52C22×C22⋊D47—28no—
53C22×C22⋊D47C2318yes—
54C23×Dic47C2312yes—
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58C23×Dic47C2311yescharacteristic
59C22×C4×D47—24no—
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47—24no—
63C22×C4×D47—44no—
64C22×C4×D47—44no—
65C22×C4×D47D418yes—
66C22×C4×D47C2314yes—
67C22×C4×D47C2314yes—
68C22×C4×D47C2314yes—
69#13093995—232no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47—24no—
72C2×C4⋊Dic47C2314yes—
73C2×C4⋊Dic47C2314yes—
74C2×Dic47⋊C4—24no—
75C2×Dic47⋊C4—24no—
76C2×Dic47⋊C4C2314yes—
77C2×Dic47⋊C4C2314yes—
78C4⋊C4×D47—232no—
79C2×C22⋊Dic47—28no—
80C2×C22⋊Dic47—28no—
81C2×C22⋊Dic47—44no—
82C2×C22⋊Dic47C2318yes—
83C2×C22⋊Dic47C2318yes—
84C2×C22⋊Dic47C2318yes—
85C2×(C2×D47)⋊C4—28no—
86C2×(C2×D47)⋊C4C2318yes—
87C2×C4×Dic47—44no—
88C2×C4×Dic47C2318yes—
89C47⋊#95C2314yes—
90C47⋊#95C2318yes—
91C47⋊#95C2318yes—
92C47⋊#95C2318yes—
93C47⋊#95C2318yes—
94C25×C47C2312yes—
95C22×D4×C47—24no—
96C22×D4×C47—44no—
97C22×D4×C47C2314yes—
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47C2314yes—
104C2×C22⋊C4×C47—28no—
105C2×C22⋊C4×C47—44no—
106C2×C22⋊C4×C47C2318yes—
107C2×C22⋊C4×C47C2318yes—
108#95×C47C2314yes—
109C24×C47—24no—
110C24×C47—24no—
111C24×C47—24no—
112C24×C47—24no—
113C24×C47—24no—
114C24×C47—24no—
115C24×C47—24no—
116C24×C47C2414yes—
117C24×C47C2411yescharacteristic
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—24no—
121C23×D47C2×D414yes—
122C23×D47#4118yes—
123C23×D47C2412yes—
124C2×D4×C47—28no—
125C2×D4×C47—48no—
126C2×D4×C47—416no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—22no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47C2×D414yes—
136C22×C4×C47#4114yes—
137C22×C4×C47C2412yes—
138C22×C4×C47C2414yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C2×C22⋊D47—216no—
142C2×C22⋊D47—216no—
143C2×C22⋊D47—216no—
144C2×C22⋊D47—432no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47C2×D414yes—
157C22×Dic47#4114yes—
158C22×Dic47C2414yes—
159C22×Dic47C2414yes—
160C22×Dic47C2412yes—
161C22×Dic47C2412yes—
162C22×Dic47C2412yes—
163C2×C4×D47—232no—
164C2×C4×D47—28no—
165C2×C4×D47—28no—
166C2×C4×D47—28no—
167C2×C4×D47—28no—
168C2×C4×D47—416no—
169C2×C4×D47—48no—
170C2×C4×D47—48no—
171C2×C4×D47—416no—
172C2×C4×D47C2×D4116yes—
173C4⋊C4×C47—28no—
174C22⋊C4×C47—216no—
175C22⋊C4×C47—48no—
176C22⋊Dic47—216no—
177C22⋊Dic47—216no—
178C22⋊Dic47—48no—
179(C2×D47)⋊C4—216no—
180C4⋊Dic47—28no—
181C4⋊Dic47—28no—
182Dic47⋊C4—28no—
183Dic47⋊C4—28no—
184C4×Dic47—48no—
185C23×C47—24no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—24no—
189C23×C47—24no—
190C23×C47—24no—
191C23×C47—24no—
192C23×C47—48no—
193C23×C47—48no—
194C23×C47—48no—
195C23×C47—48no—
196C23×C47—48no—
197C23×C47—48no—
198C23×C47—48no—
199C23×C47C22×D412yes—
200C23×C47C2×#4112yes—
201C23×C47C2×#4114yes—
202C23×C47C2×#4112yes—
203C23×C47H(5,𝔽2)14yes—
204C23×C47C2511yescharacteristic
205C22×D47—28no—
206C22×D47—28no—
207C22×D47—216no—
208C22×D47—216no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—24no—
212C22×D47#124116yes—
213C22×D47C22×D414yes—
214C22×D47C2×#4118yes—
215D4×C47—416no—
216C2×C4×C47—28no—
217C2×C4×C47—28no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—44no—
227C2×C4×C47—48no—
228C2×C4×C47C22×D414yes—
229C2×C4×C47C2×#4114yes—
230C22⋊D47—432no—
231C2×Dic47—28no—
232C2×Dic47—28no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—22no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—48no—
245C2×Dic47—44no—
246C2×Dic47—48no—
247C2×Dic47C22×D414yes—
248C2×Dic47C2×#4114yes—
249C4×D47—232no—
250C4×D47—416no—
251C4×D47—416no—
252C2×#2591—471nomaximal, Sylow
253C22×C47—44no—
254C22×C47—44no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—44no—
258C22×C47—44no—
259C22×C47—44no—
260C22×C47—816no—
261C22×C47#52514yes—
262C22×C47C2×#12414yes—
263C22×C47#53414yes—
264C22×C47#55318yes—
265C22×C47#55318yes—
266C22×C47C23×D411yescharacteristic
267C22×C47C22×#4111yescharacteristic
268C22×C47C22×#4112yes—
269C22×C47C22×#4111yescharacteristic
270C22×C47C2×H(5,𝔽2)12yes—
271C2×D47—232no—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47—28no—
275C2×D47—28no—
276C2×D47C2×#12418yes—
277C4×C47—24no—
278C4×C47—24no—
279C4×C47—42no—
280C4×C47—44no—
281Dic47—24no—
282Dic47—24no—
283Dic47—44no—
284Dic47—42no—
285Dic47—44no—
286C22×#121—471no—
287C23×C4⋊C4—471no—
288#2591—4716no—
289C2×#400—471no—
290C2×#400—471no—
291C2×#394—471no—
292C2×#394D4711yescharacteristic, complemented
293C2×#388—472no—
294C2×#388—472no—
295C2×#386—472no—
296C2×#386—472no—
297C2×C4×C22⋊C4—471no—
298C2×C47—84no—
299C2×C47#259118yescomplemented
300C2×C47C2×#52511yescharacteristic
301C2×C47C22×#12411yescharacteristic
302C2×C47C2×#53411yescharacteristic
303C2×C47C2×#55312yes—
304C2×C47C2×#55312yes—
305D47—28no—
306C23×D4—472no—
307C23×D4C2×D4711yescharacteristic
308C24×C4—471no—
309C24×C4—471no—
310C24×C4—471no—
311C2×#121—478no—
312C2×#121—948no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4—471no—
315C22×C4⋊C4—471no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—478no—
318C22×C4⋊C4—948no—
319C22×C4⋊C4C2×D4711yescharacteristic
320C22×C22⋊C4—472no—
321C22×C22⋊C4—472no—
322C22×C22⋊C4—472no—
323C22×C22⋊C4—472no—
324C22×C22⋊C4C2×D4712yes—
325C22×C22⋊C4C2×D4712yes—
326C22×C42—472no—
327#400—478no—
328#400—478no—
329#394—478no—
330#394C2×D4718yescomplemented
331#388—4716no—
332#388—4716no—
333#386—4716no—
334#386—4716no—
335C4×C22⋊C4—478no—
336C2×#95—471no—
337C2×#95—472no—
338C2×#95—472no—
339C2×#95—472no—
340C2×#95—472no—
341C2×#95C2×D4711yescharacteristic
342C47C2×#259111yescharacteristic, complemented, Sylow
343C25—471no—
344C25C22×D4712yes—
345C22×D4—24no—
346C22×D4—44no—
347C22×D4—478no—
348C22×D4—948no—
349C22×D4—948no—
350C22×D4—948no—
351C22×D4C22×D4714yes—
352C23×C4—471no—
353C23×C4—471no—
354C23×C4—478no—
355C23×C4—474no—
356C23×C4—472no—
357C23×C4—471no—
358C23×C4—472no—
359C23×C4—474no—
360C23×C4—474no—
361C23×C4—944no—
362C23×C4—944no—
363C23×C4—944no—
364C23×C4—944no—
365C23×C4—1884no—
366C23×C4—1884no—
367C23×C4C22×D4712yes—
368C23×C4C22×D4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C23×C4C22×D4711yescharacteristic
371#121—9432no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—474no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—474no—
377C2×C4⋊C4—9432no—
378C2×C4⋊C4—944no—
379C2×C4⋊C4—944no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4C22×D4714yes—
383C2×C22⋊C4—28no—
384C2×C22⋊C4—44no—
385C2×C22⋊C4—478no—
386C2×C22⋊C4—478no—
387C2×C22⋊C4—478no—
388C2×C22⋊C4—478no—
389C2×C22⋊C4—948no—
390C2×C22⋊C4—948no—
391C2×C22⋊C4—948no—
392C2×C22⋊C4—1884no—
393C2×C22⋊C4C22×D4718yes—
394C2×C22⋊C4C22×D4718yes—
395C2×C42—478no—
396C2×C42—1884no—
397#95—474no—
398#95—478no—
399#95—478no—
400#95—478no—
401#95—478no—
402#95C22×D4714yes—
403C24—24no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—24no—
408C24—24no—
409C24—24no—
410C24—474no—
411C24—478no—
412C24—472no—
413C24—942no—
414C24—942no—
415C24—944no—
416C24C23×D4714yes—
417C24C23×D4711yescharacteristic
418C2×D4—28no—
419C2×D4—48no—
420C2×D4—416no—
421C2×D4—9416no—
422C2×D4—9416no—
423C2×D4—9416no—
424C2×D4—18832no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—474no—
434C22×C4—474no—
435C22×C4—472no—
436C22×C4—472no—
437C22×C4—4716no—
438C22×C4—474no—
439C22×C4—474no—
440C22×C4—472no—
441C22×C4—948no—
442C22×C4—948no—
443C22×C4—9432no—
444C22×C4—948no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—944no—
448C22×C4—948no—
449C22×C4—944no—
450C22×C4—942no—
451C22×C4—944no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—18816no—
458C22×C4—1888no—
459C22×C4—1888no—
460C22×C4—18816no—
461C22×C4D4×D4714yes—
462C22×C4#1062535214yes—
463C22×C4C23×D4714yes—
464C22×C4C23×D4712yes—
465C22×C4C23×D4712yes—
466C22×C4C23×D4712yes—
467C4⋊C4—28no—
468C4⋊C4—948no—
469C4⋊C4—948no—
470C4⋊C4—948no—
471C4⋊C4—948no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—9416no—
475C22⋊C4—9416no—
476C22⋊C4—9416no—
477C22⋊C4—1888no—
478C42—1888no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—48no—
493C23—4716no—
494C23—474no—
495C23—478no—
496C23—9416no—
497C23—9416no—
498C23—948no—
499C23—948no—
500C23—942no—
501C23—942no—
502C23—944no—
503C23C2×D4×D4712yes—
504C23#41×D4712yes—
505C23#41×D4714yes—
506C23C2×#1062535212yes—
507C23#1309406914yes—
508C23C24×D4711yescharacteristic
509D4—416no—
510D4—18832no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—474no—
524C2×C4—474no—
525C2×C4—9432no—
526C2×C4—948no—
527C2×C4—948no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—942no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—18816no—
540C2×C4—18816no—
541C2×C4—1888no—
542C2×C4—1888no—
543C2×C4—1884no—
544C2×C4C2×D4×D4714yes—
545C2×C4C2×#1062535214yes—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—478no—
555C22—9432no—
556C22—944no—
557C22—948no—
558C22—948no—
559C22—944no—
560C22#1568963314yes—
561C22#124×D4714yes—
562C22#1568968314yes—
563C22#1568981818yes—
564C22#1568982018yes—
565C22C22×D4×D4711yescharacteristic
566C22C2×#41×D4711yescharacteristic
567C22C2×#41×D4712yes—
568C22C22×#1062535211yescharacteristic
569C22C2×#1309406912yes—
570C4—24no—
571C4—24no—
572C4—44no—
573C4—42no—
574C4—944no—
575C4—944no—
576C4—1884no—
577C4—1884no—
578C4—1882no—
579C2—84no—
580C2—948no—
581C2#2024886218yesdirect product factor
582C2C2×#1568963311yescharacteristic
583C2C2×#124×D4711yescharacteristic
584C2C2×#1568968311yescharacteristic
585C2C2×#1568981812yes—
586C2C2×#1568982012yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#394C2
socleC24×C47C24
prime radicalC22×#13093995C2
hypercenterC2×#394D47
hypocenterC47C2×#2591
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D47C2semidirect
3. #20248862C2direct (N × H)
4. C2×#15689063C2semidirect
5. C2×#15689062C2semidirect
6. C2×#15689037C2semidirect
7. C2×#15689016C2semidirect
8. C2×#15689011C2semidirect
9. C2×#15688999C2semidirect
10. C2×#15688999C2semidirect
11. C2×#15688970C2semidirect
12. C2×C47×#394C2semidirect
24. C2×C4⋊C4×D47C22semidirect
30. #15689063C22semidirect
31. #15689062C22semidirect
32. #15689037C22semidirect
33. #15689016C22semidirect
34. #15689011C22semidirect
35. #15688999C22semidirect
36. #15688999C22semidirect
37. #15688970C22semidirect
47. C47×#394C22semidirect
292. C2×#394D47semidirect
299. C2×C47#2591semidirect
330. #394C2×D47semidirect
342. C47C2×#2591semidirect
581. C2#20248862direct (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
#202488622
C2×#156890632
C2×#156890622
C2×#156890372
C2×#156890162
C2×#156890112
C2×#156889992
C2×#156889702
C2×C47×#3942
C2×#259147