Finite Group Databasebeta

← #26138243

Subgroups — #26138243

Order 12032.

2222 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C2×#15688985C211yesmaximal, characteristic
3C2×#15688982C212yesmaximal
4C2×#95×D47C211yesmaximal, characteristic
5#20241652C211yesmaximal, characteristic, complemented
6#20241651C211yesmaximal, characteristic, complemented
7#20241645C211yesmaximal, characteristic, complemented
8#20241623C212yesmaximal, complemented
9#20241619C212yesmaximal, complemented
10(C2×D47)⋊#95C212yesmaximal
11C47×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D47C2212yes—
13C23×C4×D47C2211yescharacteristic
14C2×#13093982—24no—
15C2×#13093982—28no—
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4C2212yes—
18C22×Dic47⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2211yescharacteristic
20C22×(C2×D47)⋊C4C2211yescharacteristic
21C22×(C2×D47)⋊C4C2212yes—
22C22×C4×Dic47C2212yes—
23#15688985—24no—
24#15688982—28no—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2211yescharacteristic
34#95×D47C418yescomplemented
35C22×C4⋊C4×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2211yescharacteristic
40C24×D47C2311yescharacteristic
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2312yes—
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—42no—
53C22×C4×D47—42no—
54C22×C4×D47C2×C418yes—
55C22×C4×D47C2×C414yes—
56C22×C4×D47D414yes—
57C22×C4×D47Q814yes—
58#13093982—232no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47—24no—
61C2×Dic47⋊C4—24no—
62C2×Dic47⋊C4—24no—
63C2×Dic47⋊C4—24no—
64C2×Dic47⋊C4—24no—
65C2×Dic47⋊C4—42no—
66C2×Dic47⋊C4—42no—
67C2×(C2×D47)⋊C4—24no—
68C2×(C2×D47)⋊C4—24no—
69C2×(C2×D47)⋊C4—24no—
70C2×(C2×D47)⋊C4—22no—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—24no—
73C2×(C2×D47)⋊C4D418yes—
74C2×C4×Dic47—24no—
75C2×C4×Dic47—24no—
76C2×C4×Dic47—24no—
77C47⋊#95—22no—
78C47⋊#95—24no—
79C47⋊#95—22no—
80C47⋊#95—22no—
81C47⋊#95—24no—
82C47⋊#95—24no—
83C47⋊#95—42no—
84C47⋊#95C2×C418yes—
85C47⋊#95C2×C414yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2312yes—
89C23×C4×C47C2312yes—
90C23×C4×C47C2311yescharacteristic
91C2×C4⋊C4×C47—24no—
92C2×C4⋊C4×C47—22no—
93#95×C47—24no—
94#95×C47—22no—
95#95×C47—42no—
96#95×C47C2×C414yes—
97C24×C47C2411yescharacteristic
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—21no—
101C23×D47—21no—
102C23×D47—21no—
103C23×D47—21no—
104C23×D47C22×C412yes—
105C23×D47C2×D412yes—
106C23×D47C2×D414yes—
107C23×D47C2×Q812yes—
108C23×D47#4114yes—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
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—22no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×Q812yes—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47C22×C412yes—
161C22×Dic47C22×C414yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×Q812yes—
165C22×Dic47#4114yes—
166C22×Dic47#4114yes—
167C2×C4×D47—216no—
168C2×C4×D47—28no—
169C2×C4×D47—216no—
170C2×C4×D47—28no—
171C2×C4×D47—216no—
172C2×C4×D47—216no—
173C2×C4×D47—48no—
174C2×C4×D47—48no—
175C2×C4×D47—48no—
176C2×C4×D47—48no—
177C2×C4×D47—48no—
178C2×C4×D47C4⋊C4116yescomplemented
179C4⋊C4×C47—28no—
180(C2×D47)⋊C4—216no—
181(C2×D47)⋊C4—44no—
182(C2×D47)⋊C4—44no—
183C4⋊Dic47—28no—
184Dic47⋊C4—48no—
185Dic47⋊C4—48no—
186C4×Dic47—216no—
187C23×C47C23×C411yescharacteristic
188C23×C47C22×D411yescharacteristic
189C23×C47C22×D412yes—
190C23×C47C22×Q811yescharacteristic
191C23×C47C2×#4112yes—
192C23×C47C2×#4112yes—
193C23×C47C2×#4112yes—
194C23×C47H(5,𝔽2)11yescharacteristic
195C23×C47H(5,𝔽2)11yescharacteristic
196C23×C472−1+411yescharacteristic
197C23×C472−1+411yescharacteristic
198C22×D47—24no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—24no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47C2×C4⋊C414yes—
217C22×D47C4×D418yes—
218C22×D47#12014yes—
219C22×D47C22⋊Q818yes—
220C22×D47#12314yes—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
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—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47C2×C4⋊C414yes—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47C2×C4⋊C414yes—
278C2×Dic47#11718yescomplemented
279C4×D47—232no—
280C4×D47—432no—
281#13878—471nomaximal, Sylow
282C22×C47C22×C4⋊C411yescharacteristic
283C22×C47C2×#11712yes—
284C22×C47C2×C4×D412yes—
285C22×C47#51811yescharacteristic
286C22×C47#51911yescharacteristic
287C22×C47C2×#12011yescharacteristic
288C22×C47C2×C22⋊Q812yes—
289C22×C47C2×#12311yescharacteristic
290C22×C47#53211yescharacteristic
291C22×C47#53211yescharacteristic
292C22×C47#53312yes—
293C22×C47#53711yescharacteristic
294C22×C47#53711yescharacteristic
295C22×C47#54411yescharacteristic
296C22×C47#54411yescharacteristic
297C22×C47#54612yes—
298C22×C47#54612yes—
299C22×C47#54712yes—
300C22×C47#54712yes—
301C22×C47#55112yes—
302C22×C47#55112yes—
303C22×C47#55212yes—
304C22×C47#55212yes—
305C2×D47—24no—
306C2×D47—28no—
307C2×D47—28no—
308C2×D47—24no—
309C2×D47—24no—
310C2×D47—28no—
311C2×D47—28no—
312C2×D47—24no—
313C2×D47—24no—
314C2×D47—24no—
315C2×D47#38518yes—
316C4×C47—24no—
317C4×C47—44no—
318C4×C47—44no—
319C4×C47—81no—
320C4×C47—81no—
321Dic47—28no—
322Dic47—24no—
323Dic47—44no—
324Dic47—81no—
325Dic47—81no—
326C22×#117—471no—
327C2×#381—471no—
328C2×#380—472no—
329C22×#95—471no—
330#1444—471no—
331#1444—471no—
332#1442—471no—
333#1442D4711yescharacteristic, complemented
334#1436—472no—
335#1436—472no—
336C22⋊#95—472no—
337C2×C47#228511yescharacteristic
338C2×C47C2×#38511yescharacteristic
339C2×C47#231611yescharacteristic
340C2×C47#234212yes—
341C2×C47#234412yes—
342C2×C47#243011yescharacteristic
343C2×C47#243011yescharacteristic
344C2×C47#243812yes—
345C2×C47#243812yes—
346C2×C47#249811yescharacteristic
347C2×C47#249811yescharacteristic
348D47—28no—
349C24×C4—471no—
350C24×C4—472no—
351C2×#117—948no—
352C2×#117—944no—
353C22×C4⋊C4—471no—
354C22×C4⋊C4—472no—
355C22×C4⋊C4—472no—
356C22×C4⋊C4C2×D4711yescharacteristic
357C22×C22⋊C4—472no—
358C22×C22⋊C4—471no—
359C22×C22⋊C4—471no—
360C22×C42—472no—
361#381—944no—
362#380—948no—
363C2×#95—472no—
364C2×#95—471no—
365C2×#95—478no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—471no—
369C2×#95—471no—
370C2×#95—472no—
371C2×#95—471no—
372C2×#95—472no—
373C2×#95C2×D4712yes—
374C2×#95C2×D4711yescharacteristic
375C2×#95C2×D4712yes—
376C2×#95C2×D4711yescharacteristic
377C47#1387811yescharacteristic, complemented, Sylow
378C25—471no—
379C23×C4—472no—
380C23×C4—471no—
381C23×C4—474no—
382C23×C4—474no—
383C23×C4—474no—
384C23×C4—472no—
385C23×C4—478no—
386C23×C4—471no—
387C23×C4—471no—
388C23×C4—944no—
389C23×C4—944no—
390C23×C4—944no—
391C23×C4—944no—
392C23×C4—944no—
393C23×C4—944no—
394C23×C4—1882no—
395C23×C4—1882no—
396C23×C4C22×D4711yescharacteristic
397C23×C4C22×D4711yescharacteristic
398C23×C4C22×D4712yes—
399C23×C4C22×D4712yes—
400C23×C4C22×D4711yescharacteristic
401#117—9432no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—944no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—944no—
407C2×C4⋊C4—944no—
408C2×C4⋊C4—944no—
409C2×C4⋊C4—944no—
410C2×C4⋊C4—1882no—
411C2×C4⋊C4—1882no—
412C2×C22⋊C4—478no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—944no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—944no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—942no—
419C2×C42—944no—
420C2×C42—944no—
421C2×C42—944no—
422#95—24no—
423#95—22no—
424#95—42no—
425#95—478no—
426#95—474no—
427#95—944no—
428#95—944no—
429#95—944no—
430#95—942no—
431#95—942no—
432#95—942no—
433#95—1882no—
434#95C4×D4714yes—
435C24—472no—
436C24—474no—
437C24—472no—
438C24—472no—
439C24—474no—
440C24—942no—
441C24—942no—
442C24—941no—
443C24—941no—
444C24—941no—
445C24—941no—
446C24C23×D4711yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
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—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—472no—
472C22×C4—474no—
473C22×C4—472no—
474C22×C4—472no—
475C22×C4—474no—
476C22×C4—474no—
477C22×C4—4716no—
478C22×C4—474no—
479C22×C4—9416no—
480C22×C4—948no—
481C22×C4—9416no—
482C22×C4—9416no—
483C22×C4—948no—
484C22×C4—9416no—
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—941no—
498C22×C4—941no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4—941no—
506C22×C4—941no—
507C22×C4—1888no—
508C22×C4—1888no—
509C22×C4—1888no—
510C22×C4—1888no—
511C22×C4—1888no—
512C22×C4C2×C4×D4714yes—
513C22×C4C2×C4×D4712yes—
514C22×C4D4×D4714yes—
515C22×C4D4×D4712yes—
516C22×C4Q8×D4712yes—
517C4⋊C4—28no—
518C4⋊C4—948no—
519C4⋊C4—1888no—
520C4⋊C4—1888no—
521C22⋊C4—9416no—
522C22⋊C4—1884no—
523C22⋊C4—1884no—
524C42—9416no—
525C23—474no—
526C23—478no—
527C23—474no—
528C23—478no—
529C23—474no—
530C23—942no—
531C23—942no—
532C23—944no—
533C23—944no—
534C23—944no—
535C23—944no—
536C23—944no—
537C23—944no—
538C23—944no—
539C23—944no—
540C23—944no—
541C23—942no—
542C23—942no—
543C23—942no—
544C23—942no—
545C23—944no—
546C23—942no—
547C23—942no—
548C23C22×C4×D4711yescharacteristic
549C23C2×D4×D4712yes—
550C23C2×D4×D4711yescharacteristic
551C23C2×Q8×D4711yescharacteristic
552C23#41×D4712yes—
553C23#41×D4712yes—
554C23#41×D4712yes—
555C23#1309406911yescharacteristic
556C23#1309407011yescharacteristic
557C23#1309407111yescharacteristic
558C23#1309407211yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
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—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—474no—
588C2×C4—478no—
589C2×C4—9432no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—944no—
593C2×C4—944no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—942no—
598C2×C4—944no—
599C2×C4—942no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—18832no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4C4⋊C4×D4714yes—
619C22—478no—
620C22—948no—
621C22—944no—
622C22—948no—
623C22—944no—
624C22—944no—
625C22—944no—
626C22—944no—
627C22—948no—
628C22—948no—
629C22—944no—
630C22C2×C4⋊C4×D4711yescharacteristic
631C22#117×D4712yes—
632C22C4×D4×D4712yes—
633C22#1568959511yescharacteristic
634C22#1568959911yescharacteristic
635C22#120×D4711yescharacteristic
636C22C22⋊Q8×D4712yes—
637C22#123×D4711yescharacteristic
638C22#1568967211yescharacteristic
639C22#1568967211yescharacteristic
640C22#1568967912yes—
641C22#1568970211yescharacteristic
642C22#1568970311yescharacteristic
643C22#1568975011yescharacteristic
644C22#1568975311yescharacteristic
645C22#1568976212yes—
646C22#1568976612yes—
647C22#1568977112yes—
648C22#1568977512yes—
649C22#1568979812yes—
650C22#1568980112yes—
651C22#1568980512yes—
652C22#1568980912yes—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—944no—
659C4—948no—
660C4—3761no—
661C4—3761no—
662C4—1884no—
663C2—948no—
664C2#2024579411yescharacteristic
665C2#385×D4711yescharacteristic
666C2#2024600311yescharacteristic
667C2#2024623612yes—
668C2#2024625412yes—
669C2#2024699311yescharacteristic
670C2#2024700111yescharacteristic
671C2#2024705912yes—
672C2#2024706712yes—
673C2#2024782711yescharacteristic
674C2#2024783511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1442C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1442D47
hypocenterC47#13878
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
5. #20241652C2semidirect
6. #20241651C2semidirect
7. #20241645C2semidirect
8. #20241623C2semidirect
9. #20241619C2semidirect
11. C47×#1442C2semidirect
34. #95×D47C4semidirect
178. C2×C4×D47C4⋊C4semidirect
278. C2×Dic47#117semidirect
333. #1442D47semidirect
377. C47#13878semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C2×#156889852
C2×#156889822
C2×#95×D472
#202416522
#202416512
#202416452
#202416232
#202416192
(C2×D47)⋊#952
C47×#14422
#1387847