Finite Group Databasebeta

← #26138981

Subgroups — #26138981

Order 12032.

2078 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093981C211yesmaximal, characteristic
2C2×#15688985C211yesmaximal, characteristic
3C2×#95×D47C211yesmaximal, characteristic
4#20241654C212yesmaximal, complemented
5#20241654C212yesmaximal, complemented
6#20241615C211yesmaximal, characteristic, complemented
7#20241612C211yesmaximal, characteristic, complemented
8#20241612C211yesmaximal, characteristic, complemented
9(C2×D47)⋊#95C212yesmaximal
10(C2×D47)⋊#95C212yesmaximal
11C47×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D47C2212yes—
13C23×C4×D47C2211yescharacteristic
14C2×#13093981—24no—
15C2×#13093981—24no—
16C2×#13093981C418yescomplemented
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2211yescharacteristic
21C22×(C2×D47)⋊C4C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23#15688985—24no—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34#95×D47—24no—
35C22×C42×C47C2211yescharacteristic
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C24×D47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2312yes—
45C23×Dic47C2311yescharacteristic
46C22×C4×D47—24no—
47C22×C4×D47—24no—
48C22×C4×D47—24no—
49C22×C4×D47—24no—
50C22×C4×D47—24no—
51C22×C4×D47—24no—
52C22×C4×D47—24no—
53C22×C4×D47—42no—
54C22×C4×D47—42no—
55C22×C4×D47C2×C414yes—
56C22×C4×D47D414yes—
57C22×C4×D47D414yes—
58#13093981—232no—
59C2×Dic47⋊C4—22no—
60C2×Dic47⋊C4—22no—
61C2×Dic47⋊C4—22no—
62C2×Dic47⋊C4—22no—
63C2×Dic47⋊C4C2×C414yescomplemented
64C2×Dic47⋊C4C2×C414yescomplemented
65C2×(C2×D47)⋊C4—22no—
66C2×(C2×D47)⋊C4—22no—
67C2×(C2×D47)⋊C4—22no—
68C2×(C2×D47)⋊C4—22no—
69C2×(C2×D47)⋊C4—24no—
70C2×(C2×D47)⋊C4—24no—
71C2×(C2×D47)⋊C4—24no—
72C2×(C2×D47)⋊C4C2×C414yes—
73C2×(C2×D47)⋊C4C2×C414yes—
74C2×C4×Dic47—22no—
75C2×C4×Dic47—22no—
76C2×C4×Dic47C2×C414yescomplemented
77C47⋊#95—22no—
78C47⋊#95—22no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—24no—
82C47⋊#95—24no—
83C47⋊#95—24no—
84C47⋊#95—42no—
85C47⋊#95—42no—
86C47⋊#95—42no—
87C23×C4×C47C2312yes—
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2312yes—
91C23×C4×C47C2311yescharacteristic
92C2×C42×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47C2×C414yescomplemented
95#95×C47—22no—
96#95×C47—22no—
97#95×C47—24no—
98#95×C47—42no—
99C24×C47C2411yescharacteristic
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—21no—
103C23×D47—21no—
104C23×D47—21no—
105C23×D47—21no—
106C23×D47C22×C412yes—
107C23×D47C2×D412yes—
108C23×D47C2×D412yes—
109C23×D47#4114yes—
110C23×D47#4114yes—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
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—21no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47C22×C412yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47#4114yes—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47C22×C412yes—
168C22×Dic47C22×C412yes—
169C22×Dic47C22×C412yes—
170C22×Dic47C2×D412yes—
171C22×Dic47C2×D412yes—
172C22×Dic47#4114yes—
173C2×C4×D47—28no—
174C2×C4×D47—28no—
175C2×C4×D47—216no—
176C2×C4×D47—216no—
177C2×C4×D47—216no—
178C2×C4×D47—48no—
179C2×C4×D47—48no—
180C2×C4×D47—48no—
181C2×C4×D47—48no—
182C2×C4×D47—48no—
183C2×C4×D47—48no—
184C2×C4×D47C22⋊C4116yes—
185C42×C47—28no—
186(C2×D47)⋊C4—28no—
187(C2×D47)⋊C4—28no—
188(C2×D47)⋊C4—48no—
189Dic47⋊C4—28no—
190Dic47⋊C4—28no—
191C4×Dic47—28no—
192C23×C47C23×C411yescharacteristic
193C23×C47C22×D411yescharacteristic
194C23×C47C22×D411yescharacteristic
195C23×C47C2×#4112yes—
196C23×C47C2×#4112yes—
197C23×C47C2×#4112yes—
198C23×C47C2×#4112yes—
199C23×C47H(5,𝔽2)11yescharacteristic
200C23×C47H(5,𝔽2)11yescharacteristic
201C23×C472−1+411yescharacteristic
202C23×C472−1+411yescharacteristic
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47—24no—
221C22×D47C2×C22⋊C414yes—
222C22×D47#11714yes—
223C22×D47#11714yes—
224C22×D47#12318yes—
225C22×D47#12318yes—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47C2×C22⋊C414yes—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47—42no—
285C2×Dic47—42no—
286C2×Dic47—42no—
287C2×Dic47C2×C22⋊C414yes—
288C4×D47—232no—
289C4×D47—432no—
290#13958—471nomaximal, Sylow
291C22×C47C22×C22⋊C411yescharacteristic
292C22×C47C2×#11711yescharacteristic
293C22×C47C2×#11711yescharacteristic
294C22×C47C4×#4111yescharacteristic
295C22×C47C4×#4111yescharacteristic
296C22×C47#52011yescharacteristic
297C22×C47#52011yescharacteristic
298C22×C47C2×#12312yes—
299C22×C47C2×#12312yes—
300C22×C47#52512yes—
301C22×C47#53212yes—
302C22×C47#53511yescharacteristic
303C22×C47#53611yescharacteristic
304C22×C47#53711yescharacteristic
305C22×C47#53711yescharacteristic
306C22×C47#55112yes—
307C22×C47#55112yes—
308C22×C47#55112yes—
309C22×C47#55112yes—
310C22×C47#55212yes—
311C22×C47#55212yes—
312C22×C47#55212yes—
313C22×C47#55212yes—
314C2×D47—24no—
315C2×D47—24no—
316C2×D47—24no—
317C2×D47—24no—
318C2×D47—24no—
319C2×D47—28no—
320C2×D47—28no—
321C2×D47—28no—
322C2×D47—28no—
323C2×D47—24no—
324C2×D47#38118yes—
325C4×C47—24no—
326C4×C47—42no—
327C4×C47—44no—
328C4×C47—42no—
329C4×C47—82no—
330Dic47—24no—
331Dic47—42no—
332Dic47—44no—
333Dic47—42no—
334Dic47—82no—
335C22×#117—471no—
336C2×#381—471no—
337C22×#95—471no—
338#1445—472no—
339#1445—472no—
340#1434—471no—
341#1434—471no—
342#1434—471no—
343#1434D4711yescharacteristic, complemented
344C22⋊#95—472no—
345C22⋊#95—472no—
346C2×C47C2×#38111yescharacteristic
347C2×C47#229711yescharacteristic
348C2×C47#231111yescharacteristic
349C2×C47#234311yescharacteristic
350C2×C47#234311yescharacteristic
351C2×C47#234311yescharacteristic
352C2×C47#234311yescharacteristic
353C2×C47#248212yes—
354C2×C47#248412yes—
355C2×C47#249812yes—
356C2×C47#249812yes—
357D47—28no—
358C24×C4—471no—
359C24×C4—472no—
360C2×#117—478no—
361C2×#117—944no—
362C2×#117—944no—
363C22×C4⋊C4—471no—
364C22×C4⋊C4—471no—
365C22×C22⋊C4—471no—
366C22×C22⋊C4—471no—
367C22×C22⋊C4—472no—
368C22×C42—471no—
369C22×C42C2×D4711yescharacteristic
370#381—944no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95—472no—
374C2×#95—471no—
375C2×#95—471no—
376C2×#95—472no—
377C2×#95—472no—
378C2×#95—472no—
379C2×#95—472no—
380C2×#95—472no—
381C2×#95—944no—
382C2×#95C2×D4712yes—
383C2×#95C2×D4711yescharacteristic
384C2×#95C2×D4711yescharacteristic
385C2×#95C2×D4712yes—
386C47#1395811yescharacteristic, complemented, Sylow
387C25—471no—
388C23×C4—471no—
389C23×C4—474no—
390C23×C4—474no—
391C23×C4—474no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—472no—
395C23×C4—472no—
396C23×C4—944no—
397C23×C4—944no—
398C23×C4—944no—
399C23×C4—944no—
400C23×C4—944no—
401C23×C4—944no—
402C23×C4—944no—
403C23×C4—1882no—
404C23×C4—1882no—
405C23×C4C22×D4712yes—
406C23×C4C22×D4711yescharacteristic
407C23×C4C22×D4711yescharacteristic
408C23×C4C22×D4712yes—
409C23×C4C22×D4711yescharacteristic
410#117—9432no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—474no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—942no—
416C2×C4⋊C4—942no—
417C2×C22⋊C4—474no—
418C2×C22⋊C4—474no—
419C2×C22⋊C4—942no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4—942no—
423C2×C22⋊C4—944no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—944no—
426C2×C42—22no—
427C2×C42—22no—
428C2×C42—474no—
429C2×C42—942no—
430C2×C42—942no—
431C2×C42C4×D4714yescomplemented
432#95—24no—
433#95—22no—
434#95—22no—
435#95—42no—
436#95—944no—
437#95—944no—
438#95—944no—
439#95—944no—
440#95—944no—
441#95—942no—
442#95—942no—
443#95—1882no—
444#95—1882no—
445#95—1882no—
446C24—472no—
447C24—472no—
448C24—472no—
449C24—474no—
450C24—474no—
451C24—942no—
452C24—942no—
453C24—941no—
454C24—941no—
455C24—941no—
456C24—941no—
457C24C23×D4711yescharacteristic
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—472no—
484C22×C4—472no—
485C22×C4—472no—
486C22×C4—4716no—
487C22×C4—472no—
488C22×C4—472no—
489C22×C4—474no—
490C22×C4—948no—
491C22×C4—948no—
492C22×C4—9416no—
493C22×C4—9416no—
494C22×C4—9416no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—941no—
508C22×C4—942no—
509C22×C4—941no—
510C22×C4—942no—
511C22×C4—941no—
512C22×C4—941no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—1888no—
521C22×C4—1888no—
522C22×C4—1888no—
523C22×C4—1888no—
524C22×C4—1888no—
525C22×C4—1888no—
526C22×C4C2×C4×D4712yes—
527C22×C4C2×C4×D4712yes—
528C22×C4C2×C4×D4712yes—
529C22×C4#1062535014yes—
530C22×C4D4×D4712yes—
531C22×C4D4×D4712yes—
532C4⋊C4—948no—
533C4⋊C4—948no—
534C22⋊C4—948no—
535C22⋊C4—948no—
536C22⋊C4—1888no—
537C42—28no—
538C42—948no—
539C23—474no—
540C23—474no—
541C23—474no—
542C23—478no—
543C23—478no—
544C23—944no—
545C23—944no—
546C23—944no—
547C23—944no—
548C23—944no—
549C23—944no—
550C23—944no—
551C23—944no—
552C23—942no—
553C23—942no—
554C23—942no—
555C23—942no—
556C23—944no—
557C23—944no—
558C23—942no—
559C23—942no—
560C23—942no—
561C23—942no—
562C23C22×C4×D4711yescharacteristic
563C23C2×D4×D4711yescharacteristic
564C23C2×D4×D4711yescharacteristic
565C23#41×D4712yes—
566C23#41×D4712yes—
567C23#41×D4712yes—
568C23C2×#1062535012yes—
569C23#1309406911yescharacteristic
570C23#1309407011yescharacteristic
571C23#1309407111yescharacteristic
572C23#1309407211yescharacteristic
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—474no—
604C2×C4—9432no—
605C2×C4—942no—
606C2×C4—942no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—942no—
612C2×C4—942no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—942no—
616C2×C4—942no—
617C2×C4—944no—
618C2×C4—18832no—
619C2×C4—1882no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1882no—
623C2×C4—1882no—
624C2×C4—1882no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4—1882no—
629C2×C4—1882no—
630C2×C4—1882no—
631C2×C4—1881no—
632C2×C4—1882no—
633C2×C4—1881no—
634C2×C4—1881no—
635C2×C4—1881no—
636C2×C4C22⋊C4×D4714yes—
637C22—478no—
638C22—948no—
639C22—948no—
640C22—948no—
641C22—944no—
642C22—948no—
643C22—944no—
644C22—944no—
645C22—944no—
646C22—944no—
647C22—944no—
648C22C2×C22⋊C4×D4711yescharacteristic
649C22#117×D4711yescharacteristic
650C22#117×D4711yescharacteristic
651C22#1568959111yescharacteristic
652C22C4×#1062535011yescharacteristic
653C22#1568960311yescharacteristic
654C22#1568960411yescharacteristic
655C22#123×D4712yes—
656C22#123×D4712yes—
657C22#1568963212yes—
658C22#1568967012yes—
659C22#1568968911yescharacteristic
660C22#1568969611yescharacteristic
661C22#1568970211yescharacteristic
662C22#1568970311yescharacteristic
663C22#1568979712yes—
664C22#1568979812yes—
665C22#1568980212yes—
666C22#1568980312yes—
667C22#1568980512yes—
668C22#1568980912yes—
669C22#1568981212yes—
670C22#1568981412yes—
671C4—24no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—944no—
677C4—3762no—
678C4—1884no—
679C4—1882no—
680C4—1882no—
681C2—948no—
682C2#381×D4711yescharacteristic
683C2#2024587411yescharacteristic
684C2#2024597711yescharacteristic
685C2#2024624111yescharacteristic
686C2#2024624511yescharacteristic
687C2#2024624911yescharacteristic
688C2#2024625111yescharacteristic
689C2#2024766012yes—
690C2#2024767312yes—
691C2#2024782912yes—
692C2#2024783112yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1434C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1434D47
hypocenterC47#13958
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #20241654C2semidirect
5. #20241654C2semidirect
6. #20241615C2semidirect
7. #20241612C2semidirect
8. #20241612C2semidirect
11. C47×#1434C2semidirect
16. C2×#13093981C4semidirect
63. C2×Dic47⋊C4C2×C4semidirect
64. C2×Dic47⋊C4C2×C4semidirect
76. C2×C4×Dic47C2×C4semidirect
94. C2×C42×C47C2×C4semidirect
343. #1434D47semidirect
386. C47#13958semidirect
431. C2×C42C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939812
C2×#156889852
C2×#95×D472
#202416542
#202416152
#202416122
(C2×D47)⋊#952
C47×#14342
#1395847