Finite Group Databasebeta

← #398×Dic47 #26138117

Subgroups — #398×Dic47 #26138117

Order 12032.

1574 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic47C212yesmaximal
2C2×C4⋊C4×Dic47C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20241656C211yesmaximal, characteristic
5#20241653C211yesmaximal, characteristic
6#20241645C212yesmaximal
7#20241641C211yesmaximal, characteristic
8#20241641C212yesmaximal
9#95×Dic47C211yesmaximal, characteristic
10#95×Dic47C212yesmaximal
11C2×C47×#398C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×C4⋊Dic47C2212yes—
15C22×C4×Dic47C2211yescharacteristic
16C22×C4×Dic47C2212yes—
17C22×C4×Dic47C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19C22×C4×Dic47C2212yes—
20C4⋊C4×Dic47—28no—
21C4⋊C4×Dic47—24no—
22C4⋊C4×Dic47—24no—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2211yescharacteristic
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2211yescharacteristic
33C22×C4⋊C4×C47C2212yes—
34C22×C4⋊C4×C47C2211yescharacteristic
35C22×C4⋊C4×C47C2211yescharacteristic
36C47×#398C418yescomplemented
37C2×#95×C47C2212yes—
38C2×#95×C47C2211yescharacteristic
39C23×Dic47C2311yescharacteristic
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—24no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—24no—
50C2×C4⋊Dic47—22no—
51C2×C4⋊Dic47—24no—
52C2×C4⋊Dic47—24no—
53C2×C4×Dic47—22no—
54C2×C4×Dic47—24no—
55C2×C4×Dic47—22no—
56C2×C4×Dic47—24no—
57C2×C4×Dic47—24no—
58C2×C4×Dic47—24no—
59C2×C4×Dic47—24no—
60C2×C4×Dic47—22no—
61C2×C4×Dic47—24no—
62C2×C4×Dic47—24no—
63C2×C4×Dic47D414yes—
64C2×C4×Dic47D414yes—
65C2×C4×Dic47Q818yes—
66C47⋊#95—24no—
67C47⋊#95—22no—
68C47⋊#95—22no—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—22no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C23×C4×C47C2312yes—
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2312yes—
81C2×C4⋊C4×C47—24no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47C2×C414yes—
88C2×C4⋊C4×C47C2×C418yes—
89C2×C4⋊C4×C47C2×C414yes—
90#95×C47C2×C418yes—
91#95×C47C2×C414yes—
92C24×C47C2411yescharacteristic
93C22×C4×C47—21no—
94C22×C4×C47—22no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—22no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47C22×C414yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C22×C414yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×Q814yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C22×C4×C47#4114yes—
124C22×Dic47—21no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—21no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—21no—
147C22×Dic47—22no—
148C22×Dic47C2×D412yes—
149C22×Dic47C2×D412yes—
150C22×Dic47C2×D412yes—
151C22×Dic47C2×D412yes—
152C22×Dic47C2×Q814yes—
153C22×Dic47C2×Q814yes—
154C22×Dic47#4112yes—
155C22×Dic47#4114yes—
156C22×Dic47#4112yes—
157C22×Dic47#4114yes—
158C4⋊C4×C47—28no—
159C4⋊C4×C47—216no—
160C4⋊C4×C47—28no—
161C4⋊Dic47—28no—
162C4⋊Dic47—216no—
163C4⋊Dic47—28no—
164C4×Dic47—28no—
165C4×Dic47—216no—
166C4×Dic47—28no—
167C4×Dic47—48no—
168C4×Dic47—44no—
169C23×C47C23×C411yescharacteristic
170C23×C47C22×D411yescharacteristic
171C23×C47C22×D411yescharacteristic
172C23×C47C22×Q812yes—
173C23×C47C2×#4111yescharacteristic
174C23×C47C2×#4112yes—
175C23×C47C2×#4112yes—
176C23×C47C2×#4111yescharacteristic
177C23×C47C2×#4111yescharacteristic
178C23×C47C2×#4111yescharacteristic
179C23×C47C2×#4112yes—
180C2×C4×C47—22no—
181C2×C4×C47—24no—
182C2×C4×C47—22no—
183C2×C4×C47—24no—
184C2×C4×C47—24no—
185C2×C4×C47—22no—
186C2×C4×C47—24no—
187C2×C4×C47—22no—
188C2×C4×C47—22no—
189C2×C4×C47—22no—
190C2×C4×C47—24no—
191C2×C4×C47—24no—
192C2×C4×C47—22no—
193C2×C4×C47—24no—
194C2×C4×C47—22no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—22no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47C4×D414yes—
209C2×C4×C47C4×D414yes—
210C2×C4×C47C4×Q818yes—
211C2×Dic47—22no—
212C2×Dic47—24no—
213C2×Dic47—22no—
214C2×Dic47—22no—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—22no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—24no—
221C2×Dic47—22no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—42no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47—41no—
240C2×Dic47#12114yes—
241C2×Dic47C22⋊Q818yes—
242C2×Dic47#12314yes—
243C2×Dic47#12518yes—
244C2×Dic47C4⋊Q814yes—
245C4×#398—471nomaximal, Sylow
246C22×C47C2×C4×D411yescharacteristic
247C22×C47C2×C4×D411yescharacteristic
248C22×C47C2×C4×Q812yes—
249C22×C47C4×#4111yescharacteristic
250C22×C47C4×#4112yes—
251C22×C47C2×#12111yescharacteristic
252C22×C47C2×C22⋊Q812yes—
253C22×C47C2×#12311yescharacteristic
254C22×C47#52511yescharacteristic
255C22×C47#52511yescharacteristic
256C22×C47#52512yes—
257C22×C47C2×#12512yes—
258C22×C47#52912yes—
259C22×C47#52911yescharacteristic
260C22×C47#52912yes—
261C22×C47#52912yes—
262C22×C47#52912yes—
263C22×C47C2×C4⋊Q811yescharacteristic
264C22×C47#53211yescharacteristic
265C22×C47#53211yescharacteristic
266C22×C47#53312yes—
267C22×C47#53312yes—
268C22×C47#53312yes—
269C4×C47—24no—
270C4×C47—24no—
271C4×C47—28no—
272C4×C47—44no—
273C4×C47—42no—
274Dic47—42no—
275Dic47—44no—
276Dic47—42no—
277Dic47—42no—
278Dic47—44no—
279Dic47#39818yesdirect product factor
280C2×#398D4711yescharacteristic
281C2×C4×C4⋊C4—471no—
282C2×C4×C4⋊C4—471no—
283C2×C4×C4⋊C4—472no—
284#1445—471no—
285#1444—471no—
286#1442—472no—
287#1441—472no—
288#1441—471no—
289C4×#95—471no—
290C4×#95—472no—
291C2×C47C4×#12111yescharacteristic
292C2×C47C4×#12311yescharacteristic
293C2×C47C4×C22⋊Q812yes—
294C2×C47C4×#12512yes—
295C2×C47C4×C4⋊Q811yescharacteristic
296C2×C47C2×#39811yescharacteristic
297C2×C47#239211yescharacteristic
298C2×C47#239511yescharacteristic
299C2×C47#239612yes—
300C2×C47#239911yescharacteristic
301C2×C47#240212yes—
302C22×C4⋊C4—471no—
303C22×C4⋊C4—471no—
304C22×C4⋊C4—472no—
305C22×C4⋊C4C2×D4712yes—
306C22×C4⋊C4C2×D4711yescharacteristic
307C22×C4⋊C4C2×D4711yescharacteristic
308C22×C42—472no—
309C22×C42—471no—
310C22×C42—471no—
311C22×C42—471no—
312C22×C42—472no—
313#398Dic4718yesdirect product factor
314C4×C4⋊C4—944no—
315C4×C4⋊C4—944no—
316C4×C4⋊C4—948no—
317C2×#95—472no—
318C2×#95—472no—
319C2×#95—472no—
320C2×#95—472no—
321C2×#95—471no—
322C2×#95—472no—
323C2×#95—472no—
324C2×#95—471no—
325C2×#95—472no—
326C2×#95—471no—
327C2×#95C2×D4712yes—
328C2×#95C2×D4711yescharacteristic
329C47C4×#39811yescharacteristic, complemented, Sylow
330C23×C4—471no—
331C23×C4—471no—
332C23×C4—472no—
333C23×C4—471no—
334C23×C4—471no—
335C23×C4—472no—
336C23×C4C22×D4712yes—
337C23×C4C22×D4711yescharacteristic
338C23×C4C22×D4711yescharacteristic
339C23×C4C22×D4711yescharacteristic
340C23×C4C22×D4712yes—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—942no—
348C2×C4⋊C4—944no—
349C2×C4⋊C4—944no—
350C2×C4⋊C4—942no—
351C2×C4⋊C4—944no—
352C2×C4⋊C4—942no—
353C2×C4⋊C4—942no—
354C2×C4⋊C4—944no—
355C2×C4⋊C4C2×Dic4714yes—
356C2×C4⋊C4C2×Dic4714yes—
357C2×C4⋊C4C2×Dic4718yes—
358C2×C42—474no—
359C2×C42—474no—
360C2×C42—478no—
361C2×C42—944no—
362C2×C42—942no—
363C2×C42—944no—
364C2×C42—944no—
365C2×C42—942no—
366C2×C42—942no—
367C2×C42—944no—
368C2×C42—944no—
369C2×C42—944no—
370C2×C42—944no—
371#95—944no—
372#95—944no—
373#95—942no—
374#95—944no—
375#95—944no—
376#95—944no—
377#95—942no—
378#95—942no—
379#95—944no—
380#95—944no—
381#95C2×Dic4718yes—
382#95C2×Dic4714yes—
383C24C23×D4711yescharacteristic
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—472no—
405C22×C4—472no—
406C22×C4—474no—
407C22×C4—472no—
408C22×C4—474no—
409C22×C4—472no—
410C22×C4—472no—
411C22×C4—474no—
412C22×C4—472no—
413C22×C4—474no—
414C22×C4—941no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—941no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—941no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—941no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4D4×D4712yes—
439C22×C4D4×D4712yes—
440C22×C4#1062535212yes—
441C22×C4#1062535212yes—
442C22×C4Q8×D4714yes—
443C22×C4#1062535414yes—
444C22×C4C22×Dic4712yes—
445C22×C4C22×Dic4714yes—
446C22×C4C22×Dic4712yes—
447C22×C4C22×Dic4712yes—
448C22×C4C22×Dic4714yes—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—948no—
453C4⋊C4—948no—
454C4⋊C4—9416no—
455C42—948no—
456C42—9416no—
457C42—948no—
458C42—1888no—
459C42—1884no—
460C23C23×Dic4711yescharacteristic
461C23C2×D4×D4711yescharacteristic
462C23C2×D4×D4711yescharacteristic
463C23C2×Q8×D4712yes—
464C23#41×D4711yescharacteristic
465C23#41×D4712yes—
466C23#41×D4712yes—
467C23#41×D4711yescharacteristic
468C23C2×#1062535211yescharacteristic
469C23C2×#1062535211yescharacteristic
470C23C2×#1062535412yes—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—474no—
500C2×C4—478no—
501C2×C4—474no—
502C2×C4—478no—
503C2×C4—474no—
504C2×C4—942no—
505C2×C4—944no—
506C2×C4—942no—
507C2×C4—944no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—944no—
513C2×C4—944no—
514C2×C4—942no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1881no—
533C2×C4D4×Dic4714yes—
534C2×C4D4×Dic4714yes—
535C2×C4Q8×Dic4718yes—
536C22C2×D4×Dic4711yescharacteristic
537C22C2×D4×Dic4711yescharacteristic
538C22C2×Q8×Dic4712yes—
539C22#41×Dic4711yescharacteristic
540C22#41×Dic4712yes—
541C22#121×D4711yescharacteristic
542C22C22⋊Q8×D4712yes—
543C22#123×D4711yescharacteristic
544C22#1568963111yescharacteristic
545C22#1568963311yescharacteristic
546C22#1568963612yes—
547C22#125×D4712yes—
548C22#1568965012yes—
549C22#1568965411yescharacteristic
550C22#1568965812yes—
551C22#1568965812yes—
552C22#1568965912yes—
553C22C4⋊Q8×D4711yescharacteristic
554C22#1568966811yescharacteristic
555C22#1568967411yescharacteristic
556C22#1568967512yes—
557C22#1568967812yes—
558C22#1568967912yes—
559C4—24no—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—42no—
564C4—478no—
565C4—1882no—
566C4—1884no—
567C4—1882no—
568C4—1882no—
569C4—1884no—
570C2#121×Dic4711yescharacteristic
571C2#123×Dic4711yescharacteristic
572C2C22⋊Q8×Dic4712yes—
573C2#125×Dic4712yes—
574C2C4⋊Q8×Dic4711yescharacteristic
575C2#398×D4711yescharacteristic
576C2#2024662911yescharacteristic
577C2#2024665211yescharacteristic
578C2#2024666512yes—
579C2#2024668011yescharacteristic
580C2#2024669212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#398C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×#398D47
hypocenterC47C4×#398
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
36. C47×#398C4semidirect
279. Dic47#398direct (N × H)
313. #398Dic47direct (N × H)
329. C47C4×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic472
#202416562
#202416532
#202416452
#202416412
#95×Dic472
C2×C47×#3982
C4×#39847