Finite Group Databasebeta

← C2×#20248864 #26386793

Subgroups — C2×#20248864 #26386793

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×#13093999C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
3#20248864C2116yesmaximal, direct product factor
4C2×#15689061C211yesmaximal, characteristic, complemented
5C2×#15689041C211yesmaximal, characteristic, complemented
6C2×#15689040C211yesmaximal, characteristic, complemented
7C2×#15689015C212yesmaximal, complemented
8C2×#15689012C212yesmaximal, complemented
9C2×#15689001C212yesmaximal, complemented
10C2×#15689001C212yesmaximal, complemented
11C2×#15688970C211yesmaximal, characteristic, complemented
12C2×C47×#400C211yesmaximal, characteristic, complemented
13C23×D188C2212yes—
14C23×D188C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2211yescharacteristic
17C23×C4×D47C2211yescharacteristic
18C2×#13093999—28no—
19C2×#13093999C2218yes—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2211yescharacteristic
22C22×Dic47⋊C4C2211yescharacteristic
23C2×C4⋊C4×D47—28no—
24C2×C4⋊C4×D47C2218yescomplemented
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×C4×Dic47C2212yes—
32#15689061C2218yescomplemented
33#15689041C2218yescomplemented
34#15689040C2218yescomplemented
35#15689015C22116yescomplemented
36#15689012C22116yescomplemented
37#15689001C22116yescomplemented
38#15689001C22116yescomplemented
39#15688970C2218yescomplemented
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C47×#400C2218yescomplemented
46C2×#95×C47C2212yes—
47C2×#95×C47C2212yes—
48C2×#95×C47C2211yescharacteristic
49C24×D47C2312yes—
50C24×D47C2311yescharacteristic
51C22×D188—28no—
52C22×D188—28no—
53C22×D188—28no—
54C22×D188—24no—
55C22×D188—44no—
56C22×D188C2318yes—
57C22×D188C2314yes—
58C23×Dic47C2312yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2311yescharacteristic
61C23×Dic47C2311yescharacteristic
62C22×C4×D47—24no—
63C22×C4×D47—24no—
64C22×C4×D47—24no—
65C22×C4×D47—24no—
66C22×C4×D47—44no—
67C22×C4×D47—44no—
68C22×C4×D47D418yes—
69C22×C4×D47C2314yes—
70C22×C4×D47C2314yes—
71C22×C4×D47C2314yes—
72#13093999—232no—
73C2×C4⋊Dic47—24no—
74C2×C4⋊Dic47C2314yes—
75C2×Dic47⋊C4—24no—
76C2×Dic47⋊C4—24no—
77C2×Dic47⋊C4C2314yes—
78C2×Dic47⋊C4C2314yes—
79C4⋊C4×D47—232no—
80C2×(C2×D47)⋊C4—28no—
81C2×(C2×D47)⋊C4—28no—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—44no—
85C2×(C2×D47)⋊C4—44no—
86C2×(C2×D47)⋊C4C2318yes—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×C4×Dic47—44no—
93C2×C4×Dic47C2318yes—
94C47⋊#95C2314yes—
95C47⋊#95C2318yes—
96C47⋊#95C2318yes—
97C23×C4×C47C2312yes—
98C23×C4×C47C2311yescharacteristic
99C23×C4×C47C2312yes—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47—24no—
104C2×C4⋊C4×C47C2314yes—
105C2×C4⋊C4×C47C2314yes—
106#95×C47C2314yes—
107#95×C47C2318yes—
108#95×C47C2318yes—
109C24×C47C2411yescharacteristic
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—24no—
113C23×D47—24no—
114C23×D47—24no—
115C23×D47—24no—
116C23×D47—24no—
117C23×D47—24no—
118C23×D47—24no—
119C23×D47—24no—
120C23×D47C2×D414yes—
121C23×D47#4118yes—
122C23×D47C2414yes—
123C23×D47C2412yes—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—24no—
127C22×C4×C47—24no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
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×C47C2414yes—
138C22×C4×C47C2412yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—22no—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47C2×D414yes—
151C22×Dic47#4114yes—
152C22×Dic47C2414yes—
153C22×Dic47C2412yes—
154C22×Dic47C2412yes—
155C22×Dic47C2412yes—
156C2×D188—216no—
157C2×D188—216no—
158C2×D188—216no—
159C2×D188—28no—
160C2×D188—48no—
161C2×D188—416no—
162C2×D188—432no—
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—
174C4⋊C4×C47—28no—
175(C2×D47)⋊C4—216no—
176(C2×D47)⋊C4—216no—
177(C2×D47)⋊C4—216no—
178(C2×D47)⋊C4—216no—
179(C2×D47)⋊C4—48no—
180(C2×D47)⋊C4—48no—
181C4⋊Dic47—28no—
182Dic47⋊C4—28no—
183Dic47⋊C4—28no—
184C4×Dic47—48no—
185C23×C47C22×D412yes—
186C23×C47C2×#4112yes—
187C23×C47C2×#4114yes—
188C23×C47C2×#4112yes—
189C23×C47H(5,𝔽2)14yes—
190C23×C47C2511yescharacteristic
191C22×D47—28no—
192C22×D47—28no—
193C22×D47—216no—
194C22×D47—216no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—24no—
202C22×D47—24no—
203C22×D47—24no—
204C22×D47—24no—
205C22×D47—48no—
206C22×D47—48no—
207C22×D47—48no—
208C22×D47—48no—
209C22×D47—48no—
210C22×D47—48no—
211C22×D47—48no—
212C22×D47#124116yes—
213C22×D47C22×D414yes—
214C22×D47C2×#4118yes—
215C2×C4×C47—28no—
216C2×C4×C47—28no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—48no—
229C2×C4×C47—44no—
230C2×C4×C47—48no—
231C2×C4×C47C22×D414yes—
232C2×C4×C47C2×#4114yes—
233C2×Dic47—28no—
234C2×Dic47—28no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—24no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—44no—
244C2×Dic47—48no—
245C2×Dic47C22×D414yes—
246C2×Dic47C2×#4114yes—
247D188—416no—
248D188—432no—
249C4×D47—232no—
250C4×D47—416no—
251C4×D47—416no—
252C2×#2591—471nomaximal, Sylow
253C22×C47#52514yes—
254C22×C47C2×#12414yes—
255C22×C47#53414yes—
256C22×C47#55318yes—
257C22×C47#55318yes—
258C22×C47C23×D411yescharacteristic
259C22×C47C22×#4111yescharacteristic
260C22×C47C22×#4112yes—
261C22×C47C22×#4111yescharacteristic
262C22×C47C2×H(5,𝔽2)12yes—
263C2×D47—232no—
264C2×D47—24no—
265C2×D47—24no—
266C2×D47—28no—
267C2×D47—28no—
268C2×D47—44no—
269C2×D47—44no—
270C2×D47—44no—
271C2×D47—44no—
272C2×D47—44no—
273C2×D47—44no—
274C2×D47—44no—
275C2×D47—816no—
276C2×D47C2×#12418yes—
277C4×C47—24no—
278C4×C47—24no—
279C4×C47—44no—
280C4×C47—42no—
281C4×C47—44no—
282Dic47—24no—
283Dic47—24no—
284Dic47—42no—
285Dic47—44no—
286C22×#121—471no—
287C23×C4⋊C4—471no—
288#2591—4716no—
289C2×#400—471no—
290C2×#400D4711yescharacteristic, complemented
291C2×#394—471no—
292C2×#394—471no—
293C2×#388—472no—
294C2×#388—472no—
295C2×#386—472no—
296C2×#386—472no—
297C2×C4×C22⋊C4—471no—
298C2×C47#259118yescomplemented
299C2×C47C2×#52511yescharacteristic
300C2×C47C22×#12411yescharacteristic
301C2×C47C2×#53411yescharacteristic
302C2×C47C2×#55312yes—
303C2×C47C2×#55312yes—
304D47—28no—
305D47—84no—
306C23×D4—472no—
307C23×D4—471no—
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—478no—
317C22×C4⋊C4—948no—
318C22×C4⋊C4C2×D4711yescharacteristic
319C22×C4⋊C4C2×D4711yescharacteristic
320C22×C22⋊C4—472no—
321C22×C22⋊C4—472no—
322C22×C22⋊C4—472no—
323C22×C22⋊C4—472no—
324C22×C22⋊C4—472no—
325C22×C22⋊C4—472no—
326C22×C42—472no—
327#400—478no—
328#400C2×D4718yescomplemented
329#394—478no—
330#394—478no—
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×#95C2×D4711yescharacteristic
340C2×#95C2×D4712yes—
341C2×#95C2×D4712yes—
342C47C2×#259111yescharacteristic, complemented, Sylow
343C25—471no—
344C25—472no—
345C22×D4—478no—
346C22×D4—474no—
347C22×D4—948no—
348C22×D4—948no—
349C22×D4—948no—
350C22×D4—944no—
351C22×D4—1884no—
352C23×C4—471no—
353C23×C4—471no—
354C23×C4—478no—
355C23×C4—474no—
356C23×C4—472no—
357C23×C4—471no—
358C23×C4—474no—
359C23×C4—474no—
360C23×C4—944no—
361C23×C4—944no—
362C23×C4—944no—
363C23×C4—944no—
364C23×C4—1884no—
365C23×C4—1884no—
366C23×C4C22×D4712yes—
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4712yes—
369C23×C4C22×D4711yescharacteristic
370C23×C4C22×D4711yescharacteristic
371#121—9432no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—24no—
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⋊C4C22×D4714yes—
382C2×C4⋊C4C22×D4714yes—
383C2×C22⋊C4—478no—
384C2×C22⋊C4—478no—
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—948no—
393C2×C22⋊C4—1884no—
394C2×C22⋊C4—1884no—
395C2×C42—478no—
396C2×C42—1884no—
397#95—474no—
398#95—478no—
399#95—478no—
400#95C22×D4714yes—
401#95C22×D4718yes—
402#95C22×D4718yes—
403C24—474no—
404C24—478no—
405C24—472no—
406C24—474no—
407C24—942no—
408C24—942no—
409C24—944no—
410C24—944no—
411C24—944no—
412C24—944no—
413C24—944no—
414C24—944no—
415C24—944no—
416C24—944no—
417C24C23×D4711yescharacteristic
418C2×D4—9416no—
419C2×D4—9416no—
420C2×D4—9416no—
421C2×D4—948no—
422C2×D4—1888no—
423C2×D4—18816no—
424C2×D4—18832no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—474no—
437C22×C4—474no—
438C22×C4—472no—
439C22×C4—472no—
440C22×C4—4716no—
441C22×C4—474no—
442C22×C4—472no—
443C22×C4—948no—
444C22×C4—948no—
445C22×C4—9432no—
446C22×C4—948no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—944no—
450C22×C4—948no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—18816no—
457C22×C4—1888no—
458C22×C4—1888no—
459C22×C4—18816no—
460C22×C4D4×D4714yes—
461C22×C4#1062535414yes—
462C22×C4C23×D4714yes—
463C22×C4C23×D4714yes—
464C22×C4C23×D4712yes—
465C22×C4C23×D4712yes—
466C22×C4C23×D4712yes—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—948no—
470C4⋊C4—948no—
471C4⋊C4—948no—
472C22⋊C4—9416no—
473C22⋊C4—9416no—
474C22⋊C4—9416no—
475C22⋊C4—9416no—
476C22⋊C4—1888no—
477C22⋊C4—1888no—
478C42—1888no—
479C23—4716no—
480C23—474no—
481C23—478no—
482C23—9416no—
483C23—9416no—
484C23—948no—
485C23—948no—
486C23—942no—
487C23—942no—
488C23—944no—
489C23—944no—
490C23—944no—
491C23—944no—
492C23—944no—
493C23—944no—
494C23—944no—
495C23—944no—
496C23—1888no—
497C23—1888no—
498C23—1888no—
499C23—1888no—
500C23—1888no—
501C23—1888no—
502C23—1888no—
503C23C2×D4×D4712yes—
504C23#41×D4712yes—
505C23#41×D4714yes—
506C23C2×#1062535412yes—
507C23#1309407014yes—
508C23C24×D4711yescharacteristic
509D4—18816no—
510D4—18832no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—474no—
528C2×C4—474no—
529C2×C4—9432no—
530C2×C4—948no—
531C2×C4—948no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—18816no—
541C2×C4—18816no—
542C2×C4—1888no—
543C2×C4—1884no—
544C2×C4C2×D4×D4714yes—
545C2×C4C2×#1062535414yes—
546C22—478no—
547C22—9432no—
548C22—944no—
549C22—948no—
550C22—948no—
551C22—944no—
552C22—37616no—
553C22—1884no—
554C22—1884no—
555C22—1884no—
556C22—1884no—
557C22—1884no—
558C22—1884no—
559C22—1884no—
560C22#1568963614yes—
561C22#124×D4714yes—
562C22#1568968414yes—
563C22#1568982418yes—
564C22#1568982618yes—
565C22C22×D4×D4711yescharacteristic
566C22C2×#41×D4711yescharacteristic
567C22C2×#41×D4712yes—
568C22C22×#1062535411yescharacteristic
569C22C2×#1309407012yes—
570C4—24no—
571C4—24no—
572C4—44no—
573C4—44no—
574C4—42no—
575C4—944no—
576C4—944no—
577C4—1884no—
578C4—1882no—
579C2—948no—
580C2—3764no—
581C2#2024886418yesdirect product factor
582C2C2×#1568963611yescharacteristic
583C2C2×#124×D4711yescharacteristic
584C2C2×#1568968411yescharacteristic
585C2C2×#1568982412yes—
586C2C2×#1568982612yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#400C2
socleC24×C47C24
prime radicalC22×#13093999C2
hypercenterC2×#400D47
hypocenterC47C2×#2591
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D47C2semidirect
3. #20248864C2direct (N × H)
4. C2×#15689061C2semidirect
5. C2×#15689041C2semidirect
6. C2×#15689040C2semidirect
7. C2×#15689015C2semidirect
8. C2×#15689012C2semidirect
9. C2×#15689001C2semidirect
10. C2×#15689001C2semidirect
11. C2×#15688970C2semidirect
12. C2×C47×#400C2semidirect
24. C2×C4⋊C4×D47C22semidirect
32. #15689061C22semidirect
33. #15689041C22semidirect
34. #15689040C22semidirect
35. #15689015C22semidirect
36. #15689012C22semidirect
37. #15689001C22semidirect
38. #15689001C22semidirect
39. #15688970C22semidirect
45. C47×#400C22semidirect
290. C2×#400D47semidirect
298. C2×C47#2591semidirect
328. #400C2×D47semidirect
342. C47C2×#2591semidirect
581. C2#20248864direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939992
C22×C4⋊C4×D472
#202488642
C2×#156890612
C2×#156890412
C2×#156890402
C2×#156890152
C2×#156890122
C2×#156890012
C2×#156889702
C2×C47×#4002
C2×#259147