Finite Group Databasebeta

← #26153042

Subgroups — #26153042

Order 12032.

1710 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689051C212yesmaximal
2C2×#15689048C211yesmaximal, characteristic
3C2×#15688992C211yesmaximal, characteristic, complemented
4C2×#15688981C212yesmaximal
5C2×C22⋊C4×Dic47C211yesmaximal, characteristic
6#20241657C211yesmaximal, characteristic, complemented
7#20241656C211yesmaximal, characteristic, complemented
8#20241654C212yesmaximal, complemented
9Dic47⋊#95C212yesmaximal, complemented
10Dic47⋊#95C211yesmaximal, characteristic, complemented
11C2×C47×#397C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16C22×Dic47⋊C4C2212yes—
17C22×C22⋊Dic47C2212yes—
18C22×C22⋊Dic47C2211yescharacteristic
19C22×C4×Dic47C2211yescharacteristic
20C22×C4×Dic47C2211yescharacteristic
21#15689051—28no—
22#15689048—24no—
23#15688992—24no—
24#15688981—28no—
25C22⋊C4×Dic47—24no—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2211yescharacteristic
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2212yes—
37C22×C22⋊C4×C47C2211yescharacteristic
38C47×#397—24no—
39C2×#95×C47C2212yes—
40C2×#95×C47C2211yescharacteristic
41C23×Dic47—22no—
42C23×Dic47—22no—
43C23×Dic47—24no—
44C23×Dic47—24no—
45C23×Dic47D414yes—
46C23×Dic47C2312yes—
47C23×Dic47C2312yes—
48C23×Dic47C2311yescharacteristic
49C23×Dic47C2311yescharacteristic
50C23×Dic47C2311yescharacteristic
51C23×Dic47C2311yescharacteristic
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—22no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4—24no—
60C2×Dic47⋊C4—24no—
61C2×Dic47⋊C4—24no—
62C2×Dic47⋊C4—24no—
63C2×Dic47⋊C4—24no—
64C2×C22⋊Dic47—22no—
65C2×C22⋊Dic47—22no—
66C2×C22⋊Dic47—24no—
67C2×C22⋊Dic47—24no—
68C2×C22⋊Dic47—24no—
69C2×C22⋊Dic47D414yes—
70C2×C4×Dic47—24no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—24no—
73C2×C4×Dic47D414yes—
74C47⋊#95—22no—
75C47⋊#95—22no—
76C47⋊#95—24no—
77C47⋊#95—22no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—24no—
82C47⋊#95—24no—
83C25×C47C2311yescharacteristic
84C23×C4×C47C2312yes—
85C23×C4×C47C2312yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C2×C4⋊C4×C47—22no—
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—24no—
94C2×C22⋊C4×C47—24no—
95C2×C22⋊C4×C47—24no—
96C2×C22⋊C4×C47D414yes—
97#95×C47—24no—
98#95×C47—22no—
99C24×C47—21no—
100C24×C47—21no—
101C24×C47—21no—
102C24×C47—21no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47—22no—
106C24×C47—22no—
107C24×C47C2×D412yes—
108C24×C47C2×D412yes—
109C24×C47C2×D412yes—
110C24×C47C2411yescharacteristic
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47#4112yes—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—28no—
146C22×Dic47—216no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—44no—
162C22×Dic47—44no—
163C22×Dic47—48no—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×D412yes—
166C22×Dic47C2×D412yes—
167C22×Dic47C2×D412yes—
168C22×Dic47C2×D412yes—
169C22×Dic47C2×D412yes—
170C22×Dic47#4112yes—
171C22×Dic47#4112yes—
172C22×Dic47#4114yes—
173C22×Dic47#4114yes—
174C4⋊C4×C47—44no—
175C22⋊C4×C47—28no—
176C22⋊C4×C47—48no—
177C22⋊Dic47—28no—
178C22⋊Dic47—48no—
179C4⋊Dic47—44no—
180Dic47⋊C4—48no—
181Dic47⋊C4—48no—
182Dic47⋊C4—48no—
183C4×Dic47—28no—
184C4×Dic47—44no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—24no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—24no—
193C23×C47—24no—
194C23×C47—24no—
195C23×C47—24no—
196C23×C47—24no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—42no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47#12714yes—
208C23×C47C22×D411yescharacteristic
209C23×C47C22×D411yescharacteristic
210C23×C47C22×D411yescharacteristic
211C23×C47C22×D411yescharacteristic
212C23×C47C2×#4111yescharacteristic
213C23×C47C2×#4111yescharacteristic
214C23×C47C2×#4112yes—
215C23×C47C2×#4112yes—
216C23×C47C2×#4111yescharacteristic
217C23×C472−1+412yes—
218C23×C472−1+412yes—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47#12114yes—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—42no—
255C2×Dic47—41no—
256C2×Dic47—42no—
257C2×Dic47—416no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47#12114yes—
273C2×Dic47#12114yes—
274C2×Dic47#12314yes—
275C2×Dic47#12318yes—
276C2×Dic47#12414yes—
277C2×Dic47#12618yes—
278#15150—471nomaximal, Sylow
279C22×C47—24no—
280C22×C47—28no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—44no—
286C22×C47—44no—
287C22×C47—44no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47C2×#12111yescharacteristic
291C22×C47C2×#12111yescharacteristic
292C22×C47C2×#12111yescharacteristic
293C22×C47C2×#12311yescharacteristic
294C22×C47C2×#12312yes—
295C22×C47C2×#12411yescharacteristic
296C22×C47C2×#12612yes—
297C22×C47C2×#12711yescharacteristic
298C22×C47#53211yescharacteristic
299C22×C47#53211yescharacteristic
300C22×C47#53211yescharacteristic
301C22×C47#53611yescharacteristic
302C22×C47#53611yescharacteristic
303C22×C47#53612yes—
304C22×C47#54112yes—
305C22×C47#54712yes—
306C22×C47#54712yes—
307C22×C47#54712yes—
308C22×C47#54712yes—
309C22×C47#55212yes—
310C22×C47#55212yes—
311C22×C47#55612yes—
312C22×C47#55612yes—
313C4×C47—24no—
314C4×C47—42no—
315C4×C47—82no—
316C4×C47—82no—
317Dic47—24no—
318Dic47—42no—
319Dic47—42no—
320Dic47—82no—
321Dic47—82no—
322Dic47#39718yescomplemented
323C2×#397—471no—
324C2×#397—472no—
325C2×#397D4711yescharacteristic
326C2×#384—471no—
327C2×#380—472no—
328C2×C4×C22⋊C4—471no—
329#1445—472no—
330#1445—471no—
331#1445—471no—
332C4⋊#95—471no—
333C4⋊#95—472no—
334C2×C47—44no—
335C2×C47C2×#39711yescharacteristic
336C2×C47#248611yescharacteristic
337C2×C47#248611yescharacteristic
338C2×C47#249612yes—
339C2×C47#249612yes—
340C2×C47#249912yes—
341C2×C47#250112yes—
342C2×C47#254711yescharacteristic
343C2×C47#254711yescharacteristic
344C2×C47#262111yescharacteristic
345C2×C47#285611yescharacteristic
346C24×C4—471no—
347C22×C4⋊C4—472no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4—472no—
350C22×C4⋊C4—472no—
351C22×C4⋊C4C2×D4711yescharacteristic
352C22×C22⋊C4—471no—
353C22×C22⋊C4—472no—
354C22×C22⋊C4C2×D4712yes—
355C22×C22⋊C4C2×D4711yescharacteristic
356C22×C42—471no—
357C22×C42—471no—
358#397—24no—
359#397—948no—
360#397—944no—
361#384—944no—
362#380—948no—
363C4×C22⋊C4—944no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—471no—
369C2×#95—471no—
370C2×#95—471no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95C2×D4712yes—
374C2×#95C2×D4711yescharacteristic
375C47#1515011yescharacteristic, complemented, Sylow
376C25C22×D4711yescharacteristic
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—474no—
380C23×C4—471no—
381C23×C4—471no—
382C23×C4—472no—
383C23×C4—472no—
384C23×C4—942no—
385C23×C4—944no—
386C23×C4—944no—
387C23×C4—942no—
388C23×C4C22×D4712yes—
389C23×C4C22×D4712yes—
390C23×C4C22×D4711yescharacteristic
391C23×C4C22×D4711yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4—944no—
397C2×C4⋊C4—944no—
398C2×C4⋊C4—944no—
399C2×C4⋊C4—944no—
400C2×C4⋊C4—944no—
401C2×C4⋊C4—944no—
402C2×C4⋊C4—944no—
403C2×C4⋊C4—944no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—942no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—474no—
413C2×C22⋊C4—942no—
414C2×C22⋊C4—942no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—944no—
417C2×C22⋊C4—944no—
418C2×C22⋊C4C22⋊D4714yes—
419C2×C42—474no—
420C2×C42—944no—
421C2×C42—942no—
422C2×C42—944no—
423#95—22no—
424#95—24no—
425#95—942no—
426#95—944no—
427#95—944no—
428#95—944no—
429#95—944no—
430#95—944no—
431#95—942no—
432#95—942no—
433#95—944no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—22no—
438C24—21no—
439C24—21no—
440C24—22no—
441C24—22no—
442C24D4×D4712yes—
443C24D4×D4712yes—
444C24C2×C22⋊D4712yes—
445C24C23×D4711yescharacteristic
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—472no—
467C22×C4—472no—
468C22×C4—474no—
469C22×C4—474no—
470C22×C4—472no—
471C22×C4—472no—
472C22×C4—472no—
473C22×C4—472no—
474C22×C4—472no—
475C22×C4—472no—
476C22×C4—948no—
477C22×C4—9416no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—942no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—1884no—
503C22×C4—1884no—
504C22×C4—1888no—
505C22×C4D4×D4712yes—
506C22×C4#1062535212yes—
507C22×C4C2×C22⋊D4712yes—
508C22×C4C2×C22⋊D4712yes—
509C4⋊C4—44no—
510C4⋊C4—1888no—
511C4⋊C4—1888no—
512C4⋊C4—1888no—
513C4⋊C4—1884no—
514C22⋊C4—28no—
515C22⋊C4—48no—
516C22⋊C4—948no—
517C22⋊C4—1888no—
518C42—948no—
519C42—1884no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23#1309402014yes—
543C23C2×D4×D4711yescharacteristic
544C23C2×D4×D4711yescharacteristic
545C23C2×D4×D4711yescharacteristic
546C23C22×C22⋊D4711yescharacteristic
547C23#41×D4711yescharacteristic
548C23#41×D4712yes—
549C23#41×D4712yes—
550C23#41×D4711yescharacteristic
551C23C2×#1062535211yescharacteristic
552C23#1309407112yes—
553C23#1309407112yes—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—474no—
578C2×C4—474no—
579C2×C4—478no—
580C2×C4—474no—
581C2×C4—478no—
582C2×C4—474no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—944no—
591C2×C4—18816no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1882no—
611C2×C4—1881no—
612C2×C4#1309399614yes—
613C22—28no—
614C22—24no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22#121×D4711yescharacteristic
625C22#121×D4711yescharacteristic
626C22C2×#1309399611yescharacteristic
627C22#123×D4712yes—
628C22#123×D4711yescharacteristic
629C22#124×D4711yescharacteristic
630C22#126×D4712yes—
631C22C2×#1309402011yescharacteristic
632C22#1568966811yescharacteristic
633C22#1568966811yescharacteristic
634C22#1568966911yescharacteristic
635C22#1568969311yescharacteristic
636C22#1568969512yes—
637C22#1568969511yescharacteristic
638C22#1568973012yes—
639C22#1568977112yes—
640C22#1568977112yes—
641C22#1568977212yes—
642C22#1568977212yes—
643C22#1568980912yes—
644C22#1568981012yes—
645C22#1568984012yes—
646C22#1568984212yes—
647C4—24no—
648C4—42no—
649C4—82no—
650C4—82no—
651C4—478no—
652C4—944no—
653C4—3762no—
654C4—3762no—
655C4—1882no—
656C4—1882no—
657C2—44no—
658C2#397×D4711yescharacteristic
659C2#2024768911yescharacteristic
660C2#2024769311yescharacteristic
661C2#2024780312yes—
662C2#2024780712yes—
663C2#2024783612yes—
664C2#2024785712yes—
665C2#2024837911yescharacteristic
666C2#2024838111yescharacteristic
667C2#2024915311yescharacteristic
668C2#2025173111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutator, socle, cosocleC24×C47C24
FittingC2×C47×#397C2
prime radicalC25×C47C23
hypercenterC2×#397D47
hypocenterC47#15150
LCSG ▷ C24×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. C2×#15688992C2semidirect
6. #20241657C2semidirect
7. #20241656C2semidirect
8. #20241654C2semidirect
9. Dic47⋊#95C2semidirect
10. Dic47⋊#95C2semidirect
322. Dic47#397semidirect
375. C47#15150semidirect

Maximal subgroups

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

subgroupindex
C2×#156890512
C2×#156890482
C2×#156889922
C2×#156889812
C2×C22⋊C4×Dic472
#202416572
#202416562
#202416542
Dic47⋊#952
C2×C47×#3972
#1515047