Finite Group Databasebeta

← #26138929

Subgroups — #26138929

Order 12032.

2006 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093981C211yesmaximal, characteristic
2C2×#15689018C212yesmaximal, complemented
3C2×#15688982C211yesmaximal, characteristic
4C2×#15688970C211yesmaximal, characteristic
5#20241632C212yesmaximal
6#20241630C212yesmaximal
7Dic47⋊#95C211yesmaximal, characteristic, complemented
8Dic47⋊#95C212yesmaximal, complemented
9#20241615C211yesmaximal, characteristic, complemented
10#20241614C211yesmaximal, characteristic, complemented
11C47×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C2×#13093981—28no—
14C2×#13093981C418yescomplemented
15C22×Dic47⋊C4C2212yes—
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2211yescharacteristic
21C22×(C2×D47)⋊C4C2211yescharacteristic
22C22×(C2×D47)⋊C4C2212yes—
23C22×C4×Dic47C2212yes—
24C22×C4×Dic47C2211yescharacteristic
25#15689018—28no—
26#15688982—24no—
27#15688970—24no—
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2212yes—
36C22×C42×C47C2211yescharacteristic
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C24×D47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46C22×C4×D47—24no—
47C22×C4×D47—24no—
48C22×C4×D47C2×C414yes—
49C22×C4×D47D414yes—
50C22×C4×D47Q814yes—
51#13093981—232no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—24no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—42no—
59C2×Dic47⋊C4C2×C414yes—
60C2×Dic47⋊C4C2×C414yes—
61C2×Dic47⋊C4D418yes—
62C2×(C2×D47)⋊C4—24no—
63C2×(C2×D47)⋊C4—24no—
64C2×(C2×D47)⋊C4—24no—
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4—24no—
67C2×(C2×D47)⋊C4—24no—
68C2×(C2×D47)⋊C4—42no—
69C2×(C2×D47)⋊C4—42no—
70C2×(C2×D47)⋊C4C2×C414yes—
71C2×(C2×D47)⋊C4C2×C414yes—
72C2×C4×Dic47—24no—
73C2×C4×Dic47—24no—
74C2×C4×Dic47—24no—
75C2×C4×Dic47—24no—
76C2×C4×Dic47C2×C414yes—
77C47⋊#95—22no—
78C47⋊#95—22no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—24no—
82C47⋊#95—24no—
83C47⋊#95—42no—
84C23×C4×C47C2312yes—
85C23×C4×C47C2311yescharacteristic
86C23×C4×C47C2312yes—
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47—24no—
90C2×C4⋊C4×C47—24no—
91C2×C4⋊C4×C47—42no—
92C2×C42×C47—24no—
93C2×C42×C47C2×C414yes—
94#95×C47—22no—
95#95×C47—22no—
96#95×C47—42no—
97C24×C47C2411yescharacteristic
98C23×D47—22no—
99C23×D47—22no—
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×Q812yes—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
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—21no—
123C22×C4×C47—21no—
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×C47C22×C412yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D414yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×Q812yes—
137C22×C4×C47#4114yes—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—21no—
148C22×Dic47—21no—
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×Dic47C22×C412yes—
159C22×Dic47C22×C412yes—
160C22×Dic47C22×C412yes—
161C22×Dic47C2×D414yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×Q812yes—
165C22×Dic47#4114yes—
166C22×Dic47#4114yes—
167C2×C4×D47—28no—
168C2×C4×D47—28no—
169C2×C4×D47—216no—
170C2×C4×D47—216no—
171C2×C4×D47C4⋊C4116yescomplemented
172C4⋊C4×C47—48no—
173C42×C47—28no—
174(C2×D47)⋊C4—216no—
175(C2×D47)⋊C4—28no—
176(C2×D47)⋊C4—28no—
177(C2×D47)⋊C4—48no—
178Dic47⋊C4—28no—
179Dic47⋊C4—48no—
180Dic47⋊C4—48no—
181Dic47⋊C4C22⋊C4116yescomplemented
182C4×Dic47—28no—
183C4×Dic47—48no—
184C23×C47C23×C411yescharacteristic
185C23×C47C22×D412yes—
186C23×C47C22×D411yescharacteristic
187C23×C47C22×Q811yescharacteristic
188C23×C47C2×#4112yes—
189C23×C47C2×#4112yes—
190C23×C47C2×#4112yes—
191C23×C47H(5,𝔽2)11yescharacteristic
192C23×C47H(5,𝔽2)11yescharacteristic
193C23×C472−1+411yescharacteristic
194C23×C472−1+411yescharacteristic
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—24no—
202C22×D47—24no—
203C22×D47—24no—
204C22×D47—24no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47—42no—
217C22×D47C2×C4⋊C414yes—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
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—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47C2×C22⋊C414yes—
246C2×C4×C47C2×C4⋊C414yes—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—42no—
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×Dic47C2×C22⋊C414yes—
271C2×Dic47C2×C22⋊C414yes—
272C2×Dic47C2×C4⋊C414yes—
273C2×Dic47#11714yes—
274C2×Dic47#12118yes—
275C2×Dic47C22⋊Q818yes—
276C4×D47—232no—
277#13951—471nomaximal, Sylow
278C22×C47C22×C22⋊C411yescharacteristic
279C22×C47C22×C4⋊C411yescharacteristic
280C22×C47C2×#11711yescharacteristic
281C22×C47C4×#4111yescharacteristic
282C22×C47C4×#4111yescharacteristic
283C22×C47#52011yescharacteristic
284C22×C47#52011yescharacteristic
285C22×C47C2×#12112yes—
286C22×C47C2×C22⋊Q812yes—
287C22×C47#52512yes—
288C22×C47#53312yes—
289C22×C47#53511yescharacteristic
290C22×C47#53611yescharacteristic
291C22×C47#54411yescharacteristic
292C22×C47#54411yescharacteristic
293C22×C47#54612yes—
294C22×C47#54712yes—
295C22×C47#54812yes—
296C22×C47#55012yes—
297C22×C47#55212yes—
298C22×C47#55312yes—
299C22×C47#55512yes—
300C22×C47#55612yes—
301C2×D47—24no—
302C2×D47—24no—
303C2×D47—24no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C2×D47—42no—
308C2×D47—44no—
309C2×D47—44no—
310C2×D47—44no—
311C2×D47—44no—
312C4×C47—24no—
313C4×C47—24no—
314C4×C47—42no—
315C4×C47—44no—
316C4×C47—82no—
317Dic47—24no—
318Dic47—42no—
319Dic47—44no—
320Dic47—82no—
321Dic47#38018yes—
322C22×#117—471no—
323C2×#389—472no—
324C2×#380—471no—
325C2×C4×C22⋊C4—471no—
326#1439—472no—
327#1439—472no—
328C4⋊#95—472no—
329C4⋊#95—471no—
330C4⋊#95D4711yescharacteristic, complemented
331#1434—471no—
332#1434—471no—
333C2×C47C2×#38011yescharacteristic
334C2×C47#229611yescharacteristic
335C2×C47#231711yescharacteristic
336C2×C47#233911yescharacteristic
337C2×C47#234111yescharacteristic
338C2×C47#234611yescharacteristic
339C2×C47#234711yescharacteristic
340C2×C47#241412yes—
341C2×C47#241512yes—
342C2×C47#243912yes—
343C2×C47#244112yes—
344D47—44no—
345C24×C4—471no—
346C2×#117—478no—
347C2×#117—948no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4—471no—
350C22×C4⋊C4—472no—
351C22×C4⋊C4—472no—
352C22×C4⋊C4C2×D4712yes—
353C22×C22⋊C4—471no—
354C22×C22⋊C4—471no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4—472no—
357C22×C42—471no—
358C22×C42—472no—
359C22×C42C2×D4711yescharacteristic
360#389—948no—
361#380—944no—
362C4×C22⋊C4—944no—
363C2×#95—471no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95—471no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95C2×D4711yescharacteristic
371C2×#95C2×D4711yescharacteristic
372C2×#95C2×D4712yes—
373C47#1395111yescharacteristic, complemented, Sylow
374C25—471no—
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—474no—
378C23×C4—474no—
379C23×C4—474no—
380C23×C4—471no—
381C23×C4—472no—
382C23×C4—472no—
383C23×C4—944no—
384C23×C4—944no—
385C23×C4C22×D4712yes—
386C23×C4C22×D4711yescharacteristic
387C23×C4C22×D4712yes—
388C23×C4C22×D4711yescharacteristic
389C23×C4C22×D4711yescharacteristic
390#117—9432no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—42no—
394C2×C4⋊C4—474no—
395C2×C4⋊C4—478no—
396C2×C4⋊C4—474no—
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—1882no—
404C2×C22⋊C4—474no—
405C2×C22⋊C4—474no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—944no—
408C2×C22⋊C4—944no—
409C2×C22⋊C4—944no—
410C2×C22⋊C4—944no—
411C2×C22⋊C4—944no—
412C2×C22⋊C4—1882no—
413C2×C22⋊C4—1882no—
414C2×C42—24no—
415C2×C42—474no—
416C2×C42—944no—
417C2×C42—944no—
418C2×C42—944no—
419C2×C42—944no—
420C2×C42C4×D4714yes—
421#95—22no—
422#95—22no—
423#95—42no—
424#95—944no—
425#95—944no—
426#95—944no—
427#95—942no—
428#95—942no—
429#95—944no—
430#95—1882no—
431C24—472no—
432C24—472no—
433C24—472no—
434C24—942no—
435C24—942no—
436C24—942no—
437C24—942no—
438C24—941no—
439C24—941no—
440C24—941no—
441C24—941no—
442C24C23×D4711yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—472no—
466C22×C4—472no—
467C22×C4—474no—
468C22×C4—472no—
469C22×C4—474no—
470C22×C4—472no—
471C22×C4—474no—
472C22×C4—4716no—
473C22×C4—472no—
474C22×C4—474no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—9416no—
478C22×C4—9416no—
479C22×C4—942no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—941no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4C2×C4×D4712yes—
500C22×C4C2×C4×D4712yes—
501C22×C4C2×C4×D4712yes—
502C22×C4#1062535014yes—
503C22×C4D4×D4714yes—
504C22×C4D4×D4712yes—
505C22×C4Q8×D4712yes—
506C4⋊C4—48no—
507C4⋊C4—4716no—
508C4⋊C4—948no—
509C4⋊C4—1888no—
510C4⋊C4—1888no—
511C22⋊C4—948no—
512C22⋊C4—9416no—
513C22⋊C4—948no—
514C22⋊C4—1888no—
515C42—28no—
516C42—948no—
517C42—1888no—
518C23—474no—
519C23—942no—
520C23—942no—
521C23—942no—
522C23—942no—
523C23—942no—
524C23—942no—
525C23—942no—
526C23—942no—
527C23—944no—
528C23—944no—
529C23—944no—
530C23—944no—
531C23—942no—
532C23—942no—
533C23—1882no—
534C23—1882no—
535C23—1882no—
536C23—1882no—
537C23—1882no—
538C23—1882no—
539C23—1882no—
540C23—1882no—
541C23C22×C4×D4711yescharacteristic
542C23C2×D4×D4712yes—
543C23C2×D4×D4711yescharacteristic
544C23C2×Q8×D4711yescharacteristic
545C23#41×D4712yes—
546C23#41×D4712yes—
547C23C2×#1062535012yes—
548C23#1309406911yescharacteristic
549C23#1309406911yescharacteristic
550C23#1309407211yescharacteristic
551C23#1309407211yescharacteristic
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
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—42no—
578C2×C4—42no—
579C2×C4—474no—
580C2×C4—474no—
581C2×C4—474no—
582C2×C4—478no—
583C2×C4—478no—
584C2×C4—474no—
585C2×C4—9432no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—944no—
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—1882no—
608C2×C4—1882no—
609C2×C4C22⋊C4×D4714yes—
610C2×C4C4⋊C4×D4714yes—
611C22—944no—
612C22—944no—
613C22—944no—
614C22—1882no—
615C22—1884no—
616C22—1884no—
617C22—1884no—
618C22—1884no—
619C22—1882no—
620C22—1882no—
621C22—1882no—
622C22C2×C22⋊C4×D4711yescharacteristic
623C22C2×C4⋊C4×D4711yescharacteristic
624C22#117×D4711yescharacteristic
625C22#1568959111yescharacteristic
626C22C4×#1062535011yescharacteristic
627C22#1568960411yescharacteristic
628C22#1568960411yescharacteristic
629C22#121×D4712yes—
630C22C22⋊Q8×D4712yes—
631C22#1568963212yes—
632C22#1568967712yes—
633C22#1568968611yescharacteristic
634C22#1568969811yescharacteristic
635C22#1568975011yescharacteristic
636C22#1568975011yescharacteristic
637C22#1568976312yes—
638C22#1568977412yes—
639C22#1568978312yes—
640C22#1568979112yes—
641C22#1568980512yes—
642C22#1568982212yes—
643C22#1568983512yes—
644C22#1568984512yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—82no—
650C4—478no—
651C4—944no—
652C4—3762no—
653C4—1884no—
654C4—1882no—
655C2—1884no—
656C2#380×D4711yescharacteristic
657C2#2024586811yescharacteristic
658C2#2024600911yescharacteristic
659C2#2024621511yescharacteristic
660C2#2024622511yescharacteristic
661C2#2024626811yescharacteristic
662C2#2024628211yescharacteristic
663C2#2024680312yes—
664C2#2024681612yes—
665C2#2024707612yes—
666C2#2024709812yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C4⋊#95C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC4⋊#95D47
hypocenterC47#13951
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#15689018C2semidirect
7. Dic47⋊#95C2semidirect
8. Dic47⋊#95C2semidirect
9. #20241615C2semidirect
10. #20241614C2semidirect
11. C47×C4⋊#95C2semidirect
14. C2×#13093981C4semidirect
171. C2×C4×D47C4⋊C4semidirect
181. Dic47⋊C4C22⋊C4semidirect
330. C4⋊#95D47semidirect
373. C47#13951semidirect

Maximal subgroups

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

subgroupindex
C22×#130939812
C2×#156890182
C2×#156889822
C2×#156889702
#202416322
#202416302
Dic47⋊#952
#202416152
#202416142
C47×C4⋊#952
#1395147