Finite Group Databasebeta

← C2×#20243669 #26217359

Subgroups — C2×#20243669 #26217359

Order 12032.

1994 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×#13093939C211yesmaximal, characteristic
2C2×#15689003C211yesmaximal, characteristic
3C2×#15688995C212yesmaximal, complemented
4#20243669C218yesmaximal, direct product factor
5C2×#15688890C212yesmaximal, complemented
6C2×C47×#518C211yesmaximal, characteristic
7C24×Dic47C2211yescharacteristic
8C22×Dic47⋊C4—22no—
9C22×C22⋊Dic47—21no—
10C22×C22⋊Dic47C2211yescharacteristic
11C22×C22⋊Dic47C2212yes—
12C2×#13093939C418yescomplemented
13C2×#13093939C2214yes—
14#15689003—22no—
15#15689003C2214yes—
16#15688995—24no—
17#15688995C2218yescomplemented
18C2×C47⋊#95—22no—
19C2×C47⋊#95—21no—
20#15688890C2218yescomplemented
21C23×D4×C47C2211yescharacteristic
22C47×#518—22no—
23C47×#518—24no—
24C47×#518C2214yes—
25C2×C4×D4×C47—22no—
26C2×C4×D4×C47—22no—
27C2×#117×C47—21no—
28C22×C22⋊C4×C47—21no—
29C22×C22⋊C4×C47C2212yes—
30C23×Dic47—21no—
31C23×Dic47—22no—
32C23×Dic47—22no—
33C23×Dic47—21no—
34C23×Dic47—22no—
35C23×Dic47—21no—
36C23×Dic47—44no—
37C23×Dic47C2312yes—
38C2×Dic47⋊C4—28no—
39C2×Dic47⋊C4—48no—
40C2×C22⋊Dic47—22no—
41C2×C22⋊Dic47—24no—
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—21no—
44C2×C22⋊Dic47—24no—
45C2×C22⋊Dic47—44no—
46C2×C22⋊Dic47C2×C418yescomplemented
47C2×C22⋊Dic47C2312yes—
48C2×C22⋊Dic47C2314yes—
49#13093939C2×C4116yescomplemented
50C47⋊#95—28no—
51C47⋊#95—24no—
52C25×C47D411yescharacteristic
53C25×C47C2311yescharacteristic
54C22×D4×C47—22no—
55C22×D4×C47—22no—
56C22×D4×C47—24no—
57C22×D4×C47C2×C414yes—
58C22×D4×C47D412yes—
59C22×D4×C47Q814yes—
60C22×D4×C47C2312yes—
61C23×C4×C47—21no—
62C23×C4×C47—21no—
63C23×C4×C47D411yescharacteristic
64C23×C4×C47D412yes—
65C23×C4×C47D412yes—
66C4×D4×C47—28no—
67C4×D4×C47—28no—
68C4×D4×C47—28no—
69C4×D4×C47—28no—
70#117×C47—24no—
71#117×C47—24no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47—21no—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47—24no—
76C2×C22⋊C4×C47—22no—
77C2×C22⋊C4×C47—24no—
78C2×C22⋊C4×C47—22no—
79C2×C22⋊C4×C47—22no—
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47D414yes—
82C2×C22⋊C4×C47C2314yes—
83C2×C42×C47—22no—
84C24×C47—21no—
85C24×C47—21no—
86C24×C47—22no—
87C24×C47—42no—
88C24×C47—42no—
89C24×C47—42no—
90C24×C47—42no—
91C24×C47C4⋊C414yescomplemented
92C24×C47C22×C412yes—
93C24×C47C2×D411yescharacteristic
94C24×C47C2×D412yes—
95C24×C47C2×D412yes—
96C24×C47C2×D411yescharacteristic
97C24×C47C2×D412yes—
98C24×C47C2×Q812yes—
99C24×C47#4114yes—
100C24×C47C2411yescharacteristic
101C2×D4×C47—28no—
102C2×D4×C47—28no—
103C2×D4×C47—24no—
104C2×D4×C47—28no—
105C2×D4×C47—28no—
106C2×D4×C47—24no—
107C2×D4×C47—24no—
108C2×D4×C47—24no—
109C2×D4×C47C4⋊C418yescomplemented
110C2×D4×C47C22×C414yes—
111C2×D4×C47C2×Q814yes—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—24no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—44no—
122C22×C4×C47—44no—
123C22×C4×C47—82no—
124C22×C4×C47—82no—
125C22×C4×C47C4⋊C414yescomplemented
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D414yes—
128C22×C4×C47C2×D414yes—
129C22×Dic47—24no—
130C22×Dic47—28no—
131C22×Dic47—28no—
132C22×Dic47—22no—
133C22×Dic47—24no—
134C22×Dic47—24no—
135C22×Dic47—22no—
136C22×Dic47—44no—
137C22×Dic47—48no—
138C22×Dic47—44no—
139C22×Dic47—44no—
140C22×Dic47—44no—
141C22×Dic47—44no—
142C22×Dic47—42no—
143C22×Dic47—41no—
144C22×Dic47—42no—
145C22×Dic47—41no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—24no—
151C22⋊C4×C47—28no—
152C22⋊C4×C47—24no—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—24no—
155C22⋊C4×C47—24no—
156C42×C47—28no—
157C22⋊Dic47—28no—
158C22⋊Dic47—24no—
159C22⋊Dic47—24no—
160C22⋊Dic47—48no—
161C22⋊Dic47—42no—
162C22⋊Dic47C22×C418yescomplemented
163Dic47⋊C4—416no—
164C23×C47—24no—
165C23×C47—22no—
166C23×C47—24no—
167C23×C47—22no—
168C23×C47—22no—
169C23×C47—24no—
170C23×C47—22no—
171C23×C47—24no—
172C23×C47—42no—
173C23×C47—41no—
174C23×C47—41no—
175C23×C47—44no—
176C23×C47—42no—
177C23×C47—41no—
178C23×C47—44no—
179C23×C47—48no—
180C23×C47—41no—
181C23×C47—42no—
182C23×C47—48no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—84no—
187C23×C47—81no—
188C23×C47—81no—
189C23×C47C2×C4⋊C412yes—
190C23×C47C2×C4⋊C412yes—
191C23×C47C2×C4⋊C414yescomplemented
192C23×C47C4×D414yes—
193C23×C47C4×D414yes—
194C23×C47#12011yescharacteristic
195C23×C47#12012yes—
196C23×C47#12011yescharacteristic
197C23×C47C22⋊Q814yes—
198C23×C47C22⋊Q814yes—
199C23×C47#12312yes—
200C23×C47#12312yes—
201C23×C47C23×C411yescharacteristic
202C23×C47C22×D411yescharacteristic
203C23×C47C22×D412yes—
204C23×C47C22×Q811yescharacteristic
205C23×C47C2×#4112yes—
206D4×C47—28no—
207D4×C47—28no—
208D4×C47—216no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—44no—
216C2×C4×C47—44no—
217C2×C4×C47—44no—
218C2×C4×C47—44no—
219C2×C4×C47—82no—
220C2×C4×C47—82no—
221C2×C4×C47—84no—
222C2×C4×C47—84no—
223C2×C4×C47C2×C4⋊C414yescomplemented
224C2×Dic47—28no—
225C2×Dic47—24no—
226C2×Dic47—28no—
227C2×Dic47—44no—
228C2×Dic47—416no—
229C2×Dic47—44no—
230C2×Dic47—44no—
231C2×Dic47—42no—
232C2×Dic47—48no—
233C2×Dic47—44no—
234C2×Dic47—42no—
235C2×Dic47—44no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—84no—
240C2×#1867—471nomaximal, Sylow
241C22×C47—28no—
242C22×C47—24no—
243C22×C47—24no—
244C22×C47—28no—
245C22×C47—22no—
246C22×C47—21no—
247C22×C47—22no—
248C22×C47—21no—
249C22×C47—22no—
250C22×C47—21no—
251C22×C47—22no—
252C22×C47—21no—
253C22×C47—24no—
254C22×C47—42no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—42no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—42no—
261C22×C47—44no—
262C22×C47—48no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—82no—
266C22×C47—84no—
267C22×C47#38512yes—
268C22×C47#38514yes—
269C22×C47#38512yes—
270C22×C47B(4,𝔽2)12yes—
271C22×C47#45812yes—
272C22×C47C22×C4⋊C411yescharacteristic
273C22×C47C2×C4×D412yes—
274C22×C47C2×#12011yescharacteristic
275C22×C47C2×C22⋊Q812yes—
276C22×C47C2×#12311yescharacteristic
277C4×C47—24no—
278C4×C47—44no—
279C4×C47—44no—
280Dic47—48no—
281Dic47—44no—
282Dic47—82no—
283C2×#518D4711yescharacteristic
284C22×#99—471no—
285C2×#386—471no—
286C2×#385—472no—
287#1867—478no—
288C2×#342—472no—
289C2×C47—22no—
290C2×C47—21no—
291C2×C47—21no—
292C2×C47—22no—
293C2×C47—22no—
294C2×C47—24no—
295C2×C47—44no—
296C2×C47#186714yescomplemented
297C2×C47C2×#38511yescharacteristic
298C2×C47C2×B(4,𝔽2)11yescharacteristic
299C2×C47C2×#45811yescharacteristic
300C23×D4C2×D4711yescharacteristic
301C24×C4—471no—
302#518—22no—
303#518—24no—
304#518C2×D4714yes—
305C2×C4×D4—22no—
306C2×C4×D4—22no—
307C2×#117—21no—
308C22×C4⋊C4—942no—
309C22×C22⋊C4—21no—
310C22×C22⋊C4—472no—
311C22×C22⋊C4—471no—
312C22×C22⋊C4—941no—
313C22×C22⋊C4C2×D4712yes—
314C2×#99—478no—
315C2×#99—474no—
316#386—474no—
317#386—942no—
318#385—478no—
319#385—944no—
320C2×#95—942no—
321C2×#95—941no—
322#342—478no—
323C47C2×#186711yescharacteristic, complemented, Sylow
324C25C22⋊D4711yescharacteristic
325C25C22×D4711yescharacteristic
326C22×D4—24no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4Dic9414yes—
330C22×D4C4×D4714yes—
331C22×D4C22⋊D4712yes—
332C22×D4C22×D4712yes—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—472no—
336C23×C4—942no—
337C23×C4—941no—
338C23×C4—942no—
339C23×C4—941no—
340C23×C4—942no—
341C23×C4—941no—
342C23×C4—1884no—
343C23×C4C22⋊D4711yescharacteristic
344C23×C4C22⋊D4712yes—
345C23×C4C22⋊D4712yes—
346C4×D4—28no—
347C4×D4—28no—
348C4×D4—28no—
349C4×D4—28no—
350#117—24no—
351#117—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—948no—
355C2×C4⋊C4—1888no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—478no—
364C2×C22⋊C4—474no—
365C2×C22⋊C4—472no—
366C2×C22⋊C4—944no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—944no—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—944no—
371C2×C22⋊C4—1884no—
372C2×C22⋊C4C22⋊D4714yes—
373C2×C22⋊C4C22×D4714yes—
374C2×C42—22no—
375#99—4716no—
376#95—948no—
377#95—944no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24Dic47⋊C414yescomplemented
386C24C2×Dic9412yes—
387C24C2×C4×D4712yes—
388C24#1062535014yes—
389C24C2×C22⋊D4711yescharacteristic
390C24C2×C22⋊D4712yes—
391C24C2×C22⋊D4712yes—
392C24C2×C22⋊D4711yescharacteristic
393C24C2×C22⋊D4712yes—
394C24C23×D4711yescharacteristic
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4Dic47⋊C418yescomplemented
404C2×D4C2×Dic9414yes—
405C2×D4C2×C4×D4714yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—944no—
420C22×C4—948no—
421C22×C4—948no—
422C22×C4—944no—
423C22×C4—942no—
424C22×C4—944no—
425C22×C4—942no—
426C22×C4—1884no—
427C22×C4—1884no—
428C22×C4—1888no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4—1882no—
432C22×C4—1884no—
433C22×C4—1884no—
434C22×C4—1884no—
435C22×C4—1882no—
436C22×C4—1881no—
437C22×C4—1882no—
438C22×C4—1881no—
439C22×C4Dic47⋊C414yescomplemented
440C22×C4C2×C22⋊D4712yes—
441C22×C4C2×C22⋊D4714yes—
442C22×C4C2×C22⋊D4714yes—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—18816no—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—478no—
452C22⋊C4—948no—
453C22⋊C4—944no—
454C22⋊C4—944no—
455C22⋊C4—1888no—
456C22⋊C4—1882no—
457C42—28no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—44no—
470C23—44no—
471C23—48no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—81no—
481C23—81no—
482C23—84no—
483C23C2×Dic47⋊C412yes—
484C23C2×Dic47⋊C412yes—
485C23C2×Dic47⋊C414yescomplemented
486C23C4×C22⋊D4714yes—
487C23C4×C22⋊D4714yes—
488C23#1309399311yescharacteristic
489C23#1309399312yes—
490C23#1309399311yescharacteristic
491C23#1309400314yes—
492C23#1309400314yes—
493C23#1309400712yes—
494C23#1309400712yes—
495C23C22×C4×D4711yescharacteristic
496C23C22×C22⋊D4711yescharacteristic
497C23C22×C22⋊D4712yes—
498C23C22×Dic9411yescharacteristic
499C23C2×#1062535012yes—
500D4—216no—
501D4—28no—
502D4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—944no—
518C2×C4—948no—
519C2×C4—948no—
520C2×C4—3764no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1884no—
525C2×C4—1884no—
526C2×C4—18816no—
527C2×C4—1884no—
528C2×C4—1888no—
529C2×C4—1882no—
530C2×C4—1884no—
531C2×C4—1884no—
532C2×C4—1882no—
533C2×C4C2×Dic47⋊C414yescomplemented
534C22—22no—
535C22—22no—
536C22—21no—
537C22—21no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—24no—
543C22—28no—
544C22—24no—
545C22—28no—
546C22—24no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—48no—
558C22—82no—
559C22—84no—
560C22#1568899612yes—
561C22#1568899614yes—
562C22#1568899612yes—
563C22#1568931012yes—
564C22#1568931312yes—
565C22C22×Dic47⋊C411yescharacteristic
566C22C2×C4×C22⋊D4712yes—
567C22C2×#1309399311yescharacteristic
568C22C2×#1309400312yes—
569C22C2×#1309400711yescharacteristic
570C4—24no—
571C4—44no—
572C4—44no—
573C4—3762no—
574C4—1888no—
575C4—1884no—
576C2—21no—
577C2—22no—
578C2—21no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—44no—
583C2#2024366914yesdirect product factor
584C2C2×#1568899611yescharacteristic
585C2C2×#1568931011yescharacteristic
586C2C2×#1568931311yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#518C2
socleC23×C47#120
prime radicalC23×D4×C47C22
cosocleC24×C47C24
hypercenterC2×#518D47
hypocenterC47C2×#1867
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
3. C2×#15688995C2semidirect
4. #20243669C2direct (N × H)
5. C2×#15688890C2semidirect
12. C2×#13093939C4semidirect
17. #15688995C22semidirect
20. #15688890C22semidirect
46. C2×C22⋊Dic47C2×C4semidirect
49. #13093939C2×C4semidirect
91. C24×C47C4⋊C4semidirect
109. C2×D4×C47C4⋊C4semidirect
125. C22×C4×C47C4⋊C4semidirect
162. C22⋊Dic47C22×C4semidirect
191. C23×C47C2×C4⋊C4semidirect
223. C2×C4×C47C2×C4⋊C4semidirect
296. C2×C47#1867semidirect
323. C47C2×#1867semidirect
385. C24Dic47⋊C4semidirect
403. C2×D4Dic47⋊C4semidirect
439. C22×C4Dic47⋊C4semidirect
485. C23C2×Dic47⋊C4semidirect
533. C2×C4C2×Dic47⋊C4semidirect
583. C2#20243669direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939392
C2×#156890032
C2×#156889952
#202436692
C2×#156888902
C2×C47×#5182
C2×#186747