Finite Group Databasebeta

← C2×#20243668 #26217358

Subgroups — C2×#20243668 #26217358

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
1C2×#15689598C211yesmaximal, characteristic
2C22×#13093939C211yesmaximal, characteristic
3C2×#15688996C212yesmaximal, complemented
4#20243668C218yesmaximal, direct product factor
5C2×#15688889C212yesmaximal, complemented
6C2×C47×#386C211yesmaximal, characteristic
7#15689598—24no—
8#15689598—22no—
9#15689598C2214yes—
10C2×D4×Dic47—22no—
11C2×D4×Dic47—22no—
12C2×#13093983—21no—
13C22×Dic47⋊C4—22no—
14C22×C22⋊Dic47—21no—
15C22×C22⋊Dic47C2212yes—
16C22×C22⋊Dic47C2212yes—
17C2×#13093939C418yescomplemented
18C2×#13093939C2214yes—
19#15688996—24no—
20#15688996C2218yescomplemented
21C2×C47⋊#95—22no—
22#15688889C2218yescomplemented
23C23×D4×C47C2211yescharacteristic
24C24×C4×C47C2211yescharacteristic
25C22×C22⋊C4×C47—21no—
26C22×C22⋊C4×C47C2211yescharacteristic
27C47×#386—22no—
28C47×#386C2214yes—
29C2×#95×C47—21no—
30C23×Dic47—22no—
31C23×Dic47—21no—
32C23×Dic47—21no—
33C23×Dic47D412yes—
34C23×Dic47D412yes—
35D4×Dic47—28no—
36D4×Dic47—28no—
37D4×Dic47—28no—
38D4×Dic47—28no—
39#13093983—24no—
40#13093983—24no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×Dic47⋊C4—28no—
44C2×Dic47⋊C4—48no—
45C2×C22⋊Dic47—22no—
46C2×C22⋊Dic47—24no—
47C2×C22⋊Dic47—22no—
48C2×C22⋊Dic47—22no—
49C2×C22⋊Dic47—24no—
50C2×C22⋊Dic47—22no—
51C2×C22⋊Dic47—22no—
52C2×C22⋊Dic47—22no—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47C2×C418yescomplemented
55C2×C22⋊Dic47D414yes—
56C2×C22⋊Dic47C2314yes—
57C2×C22⋊Dic47C2314yes—
58C2×C4×Dic47—22no—
59#13093939C2×C4116yescomplemented
60C47⋊#95—28no—
61C25×C47D411yescharacteristic
62C25×C47C2311yescharacteristic
63C22×D4×C47—22no—
64C22×D4×C47—22no—
65C22×D4×C47—24no—
66C22×D4×C47C2×C414yes—
67C22×D4×C47D412yes—
68C22×D4×C47Q814yes—
69C22×D4×C47C2312yes—
70C23×C4×C47—22no—
71C23×C4×C47—21no—
72C23×C4×C47—22no—
73C23×C4×C47—21no—
74C23×C4×C47—21no—
75C23×C4×C47—44no—
76C23×C4×C47D411yescharacteristic
77C23×C4×C47C2312yes—
78C2×C22⋊C4×C47—24no—
79C2×C22⋊C4×C47—24no—
80C2×C22⋊C4×C47—21no—
81C2×C22⋊C4×C47—44no—
82C2×C22⋊C4×C47C2312yes—
83#95×C47—24no—
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—24no—
116C22×C4×C47—28no—
117C22×C4×C47—22no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—44no—
121C22×C4×C47—48no—
122C22×C4×C47—44no—
123C22×C4×C47—44no—
124C22×C4×C47—42no—
125C22×C4×C47—41no—
126C22×C4×C47—42no—
127C22×C4×C47—41no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47C4⋊C414yescomplemented
132C22×C4×C47C2×D412yes—
133C22×Dic47—28no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—24no—
137C22×Dic47—24no—
138C22×Dic47—24no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—44no—
142C22×Dic47—44no—
143C22×Dic47—44no—
144C22×Dic47—44no—
145C22×Dic47—82no—
146C22×Dic47—82no—
147C22×Dic47C2×D414yes—
148C22×Dic47C2×D414yes—
149C22⋊C4×C47—24no—
150C22⋊C4×C47—48no—
151C22⋊C4×C47—42no—
152C22⋊Dic47—28no—
153C22⋊Dic47—28no—
154C22⋊Dic47—24no—
155C22⋊Dic47—24no—
156C22⋊Dic47—24no—
157C22⋊Dic47—24no—
158C22⋊Dic47—24no—
159C22⋊Dic47C22×C418yescomplemented
160C4⋊Dic47—24no—
161C4⋊Dic47—24no—
162Dic47⋊C4—416no—
163C4×Dic47—28no—
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×D412yes—
203C23×C47C22×D411yescharacteristic
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—28no—
214C2×C4×C47—24no—
215C2×C4×C47—44no—
216C2×C4×C47—48no—
217C2×C4×C47—42no—
218C2×C4×C47—44no—
219C2×C4×C47—42no—
220C2×C4×C47—44no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—84no—
225C2×C4×C47C2×C4⋊C414yescomplemented
226C2×Dic47—28no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—44no—
230C2×Dic47—416no—
231C2×Dic47—44no—
232C2×Dic47—44no—
233C2×Dic47—44no—
234C2×Dic47—44no—
235C2×Dic47—44no—
236C2×Dic47—82no—
237C2×Dic47—82no—
238C2×Dic47—84no—
239C2×Dic47—84no—
240C2×#1867—471nomaximal, Sylow
241C22×C47—28no—
242C22×C47—24no—
243C22×C47—24no—
244C22×C47—28no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—21no—
248C22×C47—21no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—21no—
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—82no—
280Dic47—48no—
281Dic47—44no—
282Dic47—44no—
283C2×#518—471no—
284C22×#99—471no—
285C2×#386D4711yescharacteristic
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×C4C2×D4711yescharacteristic
302#518—474no—
303#518—944no—
304#518—942no—
305C2×C4×D4—942no—
306C2×C4×D4—942no—
307C2×#117—941no—
308C22×C4⋊C4—942no—
309C22×C22⋊C4—21no—
310C22×C22⋊C4—472no—
311C22×C22⋊C4—472no—
312C22×C22⋊C4—941no—
313C22×C22⋊C4C2×D4711yescharacteristic
314C2×#99—478no—
315C2×#99—474no—
316#386—22no—
317#386C2×D4714yes—
318#385—478no—
319#385—944no—
320C2×#95—21no—
321C2×#95—942no—
322#342—478no—
323C47C2×#186711yescharacteristic, complemented, Sylow
324C25C22⋊D4711yescharacteristic
325C25C22×D4711yescharacteristic
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—24no—
329C22×D4Dic9414yes—
330C22×D4C4×D4714yes—
331C22×D4C22⋊D4712yes—
332C22×D4C22×D4712yes—
333C23×C4—22no—
334C23×C4—21no—
335C23×C4—22no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—44no—
339C23×C4—472no—
340C23×C4—472no—
341C23×C4—941no—
342C23×C4—941no—
343C23×C4—942no—
344C23×C4C22⋊D4711yescharacteristic
345C23×C4C22×D4712yes—
346C4×D4—948no—
347C4×D4—948no—
348C4×D4—948no—
349C4×D4—948no—
350#117—944no—
351#117—944no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—948no—
355C2×C4⋊C4—1888no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4—478no—
361C2×C22⋊C4—474no—
362C2×C22⋊C4—474no—
363C2×C22⋊C4—474no—
364C2×C22⋊C4—942no—
365C2×C22⋊C4—944no—
366C2×C22⋊C4—942no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—944no—
370C2×C22⋊C4—942no—
371C2×C22⋊C4—942no—
372C2×C22⋊C4—944no—
373C2×C22⋊C4C22×D4712yes—
374C2×C42—942no—
375#99—4716no—
376#95—24no—
377#95—948no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24Dic47⋊C414yescomplemented
386C24C2×Dic9412yes—
387C24C2×C4×D4712yes—
388C24D4×D4712yes—
389C24D4×D4712yes—
390C24#1062535214yes—
391C24C2×C22⋊D4711yescharacteristic
392C24C2×C22⋊D4712yes—
393C24C2×C22⋊D4711yescharacteristic
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—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—474no—
426C22×C4—474no—
427C22×C4—944no—
428C22×C4—944no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—948no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—944no—
435C22×C4—3762no—
436C22×C4—3762no—
437C22×C4—1884no—
438C22×C4—1884no—
439C22×C4—1884no—
440C22×C4—1884no—
441C22×C4Dic47⋊C414yescomplemented
442C22×C4C2×C22⋊D4712yes—
443C4⋊C4—944no—
444C4⋊C4—944no—
445C4⋊C4—18816no—
446C22⋊C4—24no—
447C22⋊C4—48no—
448C22⋊C4—42no—
449C22⋊C4—478no—
450C22⋊C4—948no—
451C22⋊C4—948no—
452C22⋊C4—944no—
453C22⋊C4—944no—
454C22⋊C4—944no—
455C22⋊C4—944no—
456C22⋊C4—944no—
457C42—948no—
458C23—22no—
459C23—24no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—48no—
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
486C23(C22⋊D47)⋊C414yes—
487C23(C22⋊D47)⋊C414yes—
488C23#1309399211yescharacteristic
489C23#1309399212yes—
490C23#1309399211yescharacteristic
491C23#1309400114yes—
492C23#1309400114yes—
493C23#1309400512yes—
494C23#1309400512yes—
495C23C22×C4×D4711yescharacteristic
496C23C2×D4×D4712yes—
497C23C22×C22⋊D4711yescharacteristic
498C23C22×Dic9411yescharacteristic
499C23C2×#1062535212yes—
500D4—216no—
501D4—28no—
502D4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—948no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—3762no—
523C2×C4—3762no—
524C2×C4—3764no—
525C2×C4—3764no—
526C2×C4—1884no—
527C2×C4—1884no—
528C2×C4—18816no—
529C2×C4—1884no—
530C2×C4—1884no—
531C2×C4—1884no—
532C2×C4—1884no—
533C2×C4C2×Dic47⋊C414yescomplemented
534C22—21no—
535C22—22no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
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#1568899512yes—
561C22#1568899514yes—
562C22#1568899512yes—
563C22#1568930912yes—
564C22#1568931212yes—
565C22C22×Dic47⋊C411yescharacteristic
566C22C2×(C22⋊D47)⋊C412yes—
567C22C2×#1309399211yescharacteristic
568C22C2×#1309400112yes—
569C22C2×#1309400511yescharacteristic
570C4—24no—
571C4—44no—
572C4—82no—
573C4—1888no—
574C4—1884no—
575C4—1884no—
576C2—21no—
577C2—22no—
578C2—21no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—44no—
583C2#2024366814yesdirect product factor
584C2C2×#1568899511yescharacteristic
585C2C2×#1568930911yescharacteristic
586C2C2×#1568931211yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#386C2
socleC23×C47#120
prime radicalC23×D4×C47C22
cosocleC24×C47C24
hypercenterC2×#386D47
hypocenterC47C2×#1867
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C25 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
3. C2×#15688996C2semidirect
4. #20243668C2direct (N × H)
5. C2×#15688889C2semidirect
17. C2×#13093939C4semidirect
20. #15688996C22semidirect
22. #15688889C22semidirect
54. C2×C22⋊Dic47C2×C4semidirect
59. #13093939C2×C4semidirect
91. C24×C47C4⋊C4semidirect
109. C2×D4×C47C4⋊C4semidirect
131. C22×C4×C47C4⋊C4semidirect
159. C22⋊Dic47C22×C4semidirect
191. C23×C47C2×C4⋊C4semidirect
225. C2×C4×C47C2×C4⋊C4semidirect
296. C2×C47#1867semidirect
323. C47C2×#1867semidirect
385. C24Dic47⋊C4semidirect
403. C2×D4Dic47⋊C4semidirect
441. C22×C4Dic47⋊C4semidirect
485. C23C2×Dic47⋊C4semidirect
533. C2×C4C2×Dic47⋊C4semidirect
583. C2#20243668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156895982
C22×#130939392
C2×#156889962
#202436682
C2×#156888892
C2×C47×#3862
C2×#186747