Finite Group Databasebeta

← #26218026

Subgroups — #26218026

Order 12032.

1818 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094032C211yesmaximal, characteristic, complemented
2C2×#15689701C211yesmaximal, characteristic
3C2×#15689119C211yesmaximal, characteristic, complemented
4#20245981C211yesmaximal, characteristic, complemented
5#20243823C212yesmaximal, complemented
6#20243818C212yesmaximal, complemented
7#20243813C212yesmaximal, complemented
8#20243808C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C2×C47×#417C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13094032—24no—
14C2×#13094032—24no—
15C2×#13094032—24no—
16C22×#10625324C2212yes—
17C23×C47⋊C8C2211yescharacteristic
18C22×C8×D47C2212yes—
19#15689701—24no—
20#15689701C418yescomplemented
21C2×#13094003—21no—
22C2×#13094000C2212yes—
23C2×#13093998—21no—
24C2×#13093996—22no—
25C2×#13093995C2212yes—
26C2×#13093983C2211yescharacteristic
27C22×C4⋊Dic47C2211yescharacteristic
28#15689119—24no—
29C2×#13093948C2211yescharacteristic
30C2×#13093948C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33#15689028—21no—
34#15689024—21no—
35#15689002—22no—
36#15688980C2212yes—
37#15688980C2212yes—
38#15688886C2212yes—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C22×#41×C47C2211yescharacteristic
43C22×M16×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C47×#417—24no—
46C2×#103×C47C2212yes—
47C2×#102×C47C2212yes—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350—22no—
52C2×#10625350C2×C414yes—
53C2×#10625350D414yes—
54C22×Dic94D411yescharacteristic
55C22×D188D411yescharacteristic
56C22×C22⋊D47—22no—
57C22×C22⋊D47D412yes—
58C23×Dic47—21no—
59C22×C4×D47C2312yes—
60#13094032—232no—
61C2×#10625324—24no—
62C2×#10625324—24no—
63C2×#10625324—24no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8—22no—
66C22×C47⋊C8D414yescomplemented
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C2×C8×D47—24no—
70C2×C8×D47—24no—
71C2×C8×D47—24no—
72#13094003—28no—
73#13094000—24no—
74#13094000C2×C418yescomplemented
75#13093998—28no—
76#13093996—216no—
77#13093995—24no—
78#13093995C2×C418yescomplemented
79#13093983—22no—
80#13093983—22no—
81C2×C4⋊Dic47—21no—
82C2×C4⋊Dic47—22no—
83C2×C4⋊Dic47C2×C414yes—
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2312yes—
87C2×Dic47⋊C4—22no—
88C2×C22⋊Dic47—22no—
89C2×C22⋊Dic47—21no—
90C2×(C2×D47)⋊C4—22no—
91C2×(C2×D47)⋊C4—22no—
92C2×(C2×D47)⋊C4—22no—
93C2×C4×Dic47—21no—
94#13093948—22no—
95#13093948—22no—
96#13093945—22no—
97#13093945—22no—
98C47⋊#95—22no—
99C47⋊#95—22no—
100C2×#41×C47—22no—
101C2×#41×C47—22no—
102C2×#41×C47—22no—
103C2×#41×C47—22no—
104C2×#41×C47—22no—
105C2×#41×C47C2×C414yes—
106C22×Q8×C47C2311yescharacteristic
107C22×D4×C47—21no—
108C22×D4×C47C2311yescharacteristic
109C23×C4×C47—21no—
110C23×C4×C47—21no—
111C23×C4×C47C2311yescharacteristic
112C2×M16×C47—22no—
113C2×M16×C47—22no—
114C2×M16×C47D414yescomplemented
115C22×C8×C47C2312yes—
116C2×C4⋊C4×C47—22no—
117C2×C4⋊C4×C47—22no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47C2312yes—
120#103×C47—24no—
121#102×C47—24no—
122C24×C47—21no—
123C24×C47C2×D411yescharacteristic
124C23×D47C2×D412yes—
125#41×C47—28no—
126#41×C47—28no—
127#41×C47—44no—
128#41×C47—44no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47C22×C412yes—
135C2×D4×C47—24no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—21no—
141C2×D4×C47—42no—
142C2×D4×C47—42no—
143C2×D4×C47C22×C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—24no—
146C22×C4×C47—22no—
147C22×C4×C47—42no—
148C22×C4×C47—42no—
149C22×C4×C47—42no—
150C22×C4×C47—81no—
151C22×C4×C47—81no—
152C22×C4×C47C22×C412yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2×D412yes—
157C22×C4×C47#4112yes—
158C22×C4×C47#4112yes—
159C22×C4×C47C2411yescharacteristic
160C2×C22⋊D47—24no—
161C2×C22⋊D47—24no—
162C2×C22⋊D47—28no—
163C2×C22⋊D47—82no—
164C2×C22⋊D47—82no—
165C2×C22⋊D47C22⋊C418yes—
166C22×Dic47—22no—
167C22×Dic47—24no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C22×Dic47—81no—
171C22×Dic47—81no—
172C22×Dic47C2×D412yes—
173#10625350—28no—
174#10625350—28no—
175#10625350—28no—
176#10625350—28no—
177C2×D188—22no—
178C2×D188—22no—
179C2×D188C22⋊C414yes—
180C2×C4×D47—22no—
181C2×C4×D47—22no—
182C2×C4×D47—22no—
183C2×C4×D47—22no—
184C2×C4×D47C22×C414yes—
185C2×C4×D47C2×D414yes—
186C2×Dic94—22no—
187C2×Dic94—22no—
188C2×Dic94C22⋊C414yes—
189M16×C47—28no—
190C2×C8×C47—22no—
191C2×C8×C47—22no—
192C2×C8×C47C2×D414yes—
193C4⋊C4×C47—22no—
194C4⋊C4×C47—22no—
195C4⋊C4×C47—42no—
196C22⋊Dic47—28no—
197C22⋊Dic47—42no—
198(C2×D47)⋊C4—28no—
199(C2×D47)⋊C4—82no—
200(C2×D47)⋊C4—82no—
201C4⋊Dic47—24no—
202C4⋊Dic47—22no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47—21no—
206C4⋊Dic47—21no—
207C4⋊Dic47C22×C414yes—
208Dic47⋊C4—28no—
209C4×Dic47—42no—
210C2×C47⋊C8—28no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8C2×D412yes—
216C2×C47⋊C8C2×D412yes—
217#10625324—216no—
218C8×D47—216no—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—22no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47—41no—
229C23×C47—41no—
230C23×C47C2×C22⋊C412yes—
231C23×C47#12112yes—
232C23×C47#12112yes—
233C23×C47C22×D411yescharacteristic
234C22×D47—22no—
235C22×D47—22no—
236C22×D47—22no—
237C22×D47—22no—
238C22×D47—22no—
239C22×D47—22no—
240C22×D47C2×C22⋊C414yes—
241Q8×C47—22no—
242Q8×C47—22no—
243Q8×C47—41no—
244Q8×C47—41no—
245D4×C47—22no—
246D4×C47—22no—
247D4×C47—48no—
248D4×C47—41no—
249D4×C47—41no—
250C2×C4×C47—22no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—48no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47—84no—
258C2×C4×C47C2×C22⋊C412yes—
259C2×C4×C47C4×D414yes—
260C2×C4×C47C4×D414yes—
261C2×C4×C47#12014yes—
262C2×C4×C47#12112yes—
263C2×C4×C47#12112yes—
264C2×C4×C47#12314yes—
265C2×C4×C47C23×C411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C22⋊D47—216no—
272C22⋊D47—88no—
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—24no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—84no—
285C2×Dic47C2×C22⋊C414yes—
286D188—28no—
287C4×D47—24no—
288C4×D47—24no—
289C4×D47—24no—
290C4×D47—24no—
291Dic94—28no—
292C8×C47—24no—
293C47⋊C8—22no—
294C47⋊C8—22no—
295#20976—471nomaximal, Sylow
296C22×C47—22no—
297C22×C47—22no—
298C22×C47—22no—
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—22no—
302C22×C47—42no—
303C22×C47—42no—
304C22×C47—42no—
305C22×C47—81no—
306C22×C47—81no—
307C22×C47—81no—
308C22×C47—81no—
309C22×C47#38614yes—
310C22×C47C22×C22⋊C411yescharacteristic
311C22×C47C2×#12111yescharacteristic
312C22×C47C2×#12111yescharacteristic
313C22×C47#57611yescharacteristic
314C22×C47#57711yescharacteristic
315C22×C47#57711yescharacteristic
316C22×C47#57811yescharacteristic
317C2×D47—24no—
318C2×D47—24no—
319C2×D47—24no—
320C2×D47—42no—
321C2×D47—42no—
322C2×D47—42no—
323C2×D47—42no—
324C4×C47—42no—
325C4×C47—82no—
326C4×C47#38614yes—
327C4×C47C2×C4×D411yescharacteristic
328C4×C47C2×C4×D411yescharacteristic
329C4×C47C2×#12011yescharacteristic
330C4×C47C2×#12311yescharacteristic
331Dic47—24no—
332Dic47—44no—
333Dic47—81no—
334Dic47—81no—
335C22×#131—471no—
336C2×#537—471no—
337C2×#417D4711yescharacteristic, complemented
338C2×#416—471no—
339#2311—471no—
340#1895—472no—
341#1894—472no—
342#1893—472no—
343#1892—472no—
344#1736—471no—
345C2×#340—471no—
346C2×C47—24no—
347C2×C47—82no—
348C2×C47C2×#38611yescharacteristic
349C2×C47#289711yescharacteristic
350C2×C47#289811yescharacteristic
351C2×C47#306011yescharacteristic
352C2×C47#306111yescharacteristic
353C2×C47#306211yescharacteristic
354C2×C47#306311yescharacteristic
355D47—44no—
356C22×#41—471no—
357C22×#41C2×D4711yescharacteristic
358C2×#131—944no—
359C2×#131—944no—
360C2×#131—944no—
361C22×M16—472no—
362C22×M16C2×D4711yescharacteristic
363C23×C8—471no—
364C23×C8—472no—
365#537—478no—
366#537—944no—
367C2×C22⋊Q8—472no—
368C2×C22⋊Q8—941no—
369C2×#121—472no—
370C2×#121—941no—
371C2×#121—942no—
372C2×#117—471no—
373C22×C4⋊C4—471no—
374C22×C4⋊C4C2×D4711yescharacteristic
375#417—24no—
376#416—944no—
377C2×#103—471no—
378C2×#103—471no—
379C2×#103C2×D4712yes—
380C2×#102—471no—
381C2×#102—471no—
382C2×#102C2×D4712yes—
383#391—941no—
384#390—941no—
385#386—942no—
386#380—472no—
387#380—472no—
388#340—472no—
389#340—472no—
390#340—472no—
391#340—472no—
392C47#2097611yescharacteristic, complemented, Sylow
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—474no—
399C2×#41—474no—
400C2×#41—942no—
401C2×#41—942no—
402C2×#41—942no—
403C2×#41—942no—
404C2×#41C4×D4714yes—
405C22×Q8—471no—
406C22×Q8C22×D4711yescharacteristic
407C22×D4—21no—
408C22×D4—472no—
409C22×D4—471no—
410C22×D4—942no—
411C22×D4C22×D4711yescharacteristic
412C23×C4—21no—
413C23×C4—21no—
414C23×C4—472no—
415C23×C4—941no—
416C23×C4C22×D4711yescharacteristic
417#131—9432no—
418C2×M16—22no—
419C2×M16—22no—
420C2×M16—944no—
421C2×M16—944no—
422C2×M16—944no—
423C2×M16C22⋊D4714yescomplemented
424C22×C8—474no—
425C22×C8—471no—
426C22×C8—471no—
427C22×C8—944no—
428C22×C8—944no—
429C22×C8—942no—
430C22×C8—942no—
431C22×C8—944no—
432C22×C8C22×D4712yes—
433C22⋊Q8—478no—
434C22⋊Q8—948no—
435C22⋊Q8—944no—
436#121—478no—
437#121—948no—
438#121—944no—
439#121—9416no—
440#117—942no—
441#117—942no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—474no—
446C2×C4⋊C4—472no—
447C2×C4⋊C4—471no—
448C2×C4⋊C4—471no—
449C2×C4⋊C4—942no—
450C2×C4⋊C4—941no—
451C2×C4⋊C4—942no—
452C2×C4⋊C4C22×D4712yes—
453C2×C22⋊C4—942no—
454C2×C22⋊C4—942no—
455C2×C22⋊C4—942no—
456C2×C22⋊C4—942no—
457C2×C22⋊C4—941no—
458C2×C42—941no—
459#103—24no—
460#103—942no—
461#103—942no—
462#102—24no—
463#102—942no—
464#102—942no—
465#95—942no—
466#95—942no—
467C24—21no—
468C24—472no—
469C24D4×D4711yescharacteristic
470#41—28no—
471#41—28no—
472#41—44no—
473#41—44no—
474#41—948no—
475#41—948no—
476#41—948no—
477#41—948no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—474no—
484C2×Q8—942no—
485C2×Q8—942no—
486C2×Q8C2×C4×D4712yes—
487C2×D4—24no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—478no—
496C2×D4—474no—
497C2×D4—944no—
498C2×D4—944no—
499C2×D4—948no—
500C2×D4—942no—
501C2×D4—942no—
502C2×D4—3762no—
503C2×D4—3762no—
504C2×D4C2×C4×D4712yes—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—81no—
512C22×C4—81no—
513C22×C4—472no—
514C22×C4—474no—
515C22×C4—474no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—944no—
521C22×C4—942no—
522C22×C4—941no—
523C22×C4—941no—
524C22×C4—3761no—
525C22×C4—3761no—
526C22×C4C2×C4×D4712yes—
527C22×C4#1062535012yes—
528C22×C4D4×D4711yescharacteristic
529C22×C4D4×D4712yes—
530C22×C4D4×D4712yes—
531C22×C4#1062535212yes—
532C22×C4C2×C22⋊D4712yes—
533C22×C4C23×D4711yescharacteristic
534M16—28no—
535M16—9416no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—472no—
539C2×C8—472no—
540C2×C8—9416no—
541C2×C8—948no—
542C2×C8—941no—
543C2×C8—941no—
544C2×C8—941no—
545C2×C8—941no—
546C2×C8C2×C22⋊D4714yes—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—42no—
550C4⋊C4—474no—
551C4⋊C4—948no—
552C4⋊C4—944no—
553C4⋊C4—942no—
554C4⋊C4—941no—
555C4⋊C4—941no—
556C4⋊C4—941no—
557C4⋊C4—941no—
558C22⋊C4—948no—
559C22⋊C4—948no—
560C22⋊C4—3762no—
561C22⋊C4—3762no—
562C22⋊C4—1882no—
563C42—1882no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—474no—
576C23—942no—
577C23—942no—
578C23—942no—
579C23—942no—
580C23—942no—
581C23—942no—
582C23C22⋊C4×D4712yes—
583C23#1309399512yes—
584C23#1309399712yes—
585C23C2×D4×D4711yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—948no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4—41no—
595D4—41no—
596D4—9416no—
597D4—948no—
598D4—3768no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—474no—
608C2×C4—942no—
609C2×C4—942no—
610C2×C4—944no—
611C2×C4—944no—
612C2×C4—944no—
613C2×C4—944no—
614C2×C4—944no—
615C2×C4—3764no—
616C2×C4—1881no—
617C2×C4—1881no—
618C2×C4—1881no—
619C2×C4—1881no—
620C2×C4—1881no—
621C2×C4—1881no—
622C2×C4—1882no—
623C2×C4—1882no—
624C2×C4C22⋊C4×D4712yes—
625C2×C4(C22⋊D47)⋊C414yes—
626C2×C4C4×C22⋊D4714yes—
627C2×C4#1309399214yes—
628C2×C4#1309399512yes—
629C2×C4#1309399712yes—
630C2×C4#1309400614yes—
631C2×C4C22×C4×D4711yescharacteristic
632C2×C4C2×D4×D4711yescharacteristic
633C2×C4C2×D4×D4711yescharacteristic
634C2×C4C22×C22⋊D4711yescharacteristic
635C2×C4C2×#1062535211yescharacteristic
636C2×C4C2×#1062535011yescharacteristic
637C8—24no—
638C8—942no—
639C8—942no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—944no—
654C22—944no—
655C22—944no—
656C22—1882no—
657C22—1882no—
658C22—1882no—
659C22—1882no—
660C22#1568899914yes—
661C22C2×C22⋊C4×D4711yescharacteristic
662C22C2×#1309399511yescharacteristic
663C22C2×#1309399711yescharacteristic
664C22#1568992711yescharacteristic
665C22#1568993211yescharacteristic
666C22#1568993311yescharacteristic
667C22#1568993911yescharacteristic
668C4—42no—
669C4—82no—
670C4—944no—
671C4—3761no—
672C4—3761no—
673C4—1884no—
674C4#1568899914yes—
675C4C2×(C22⋊D47)⋊C411yescharacteristic
676C4C2×C4×C22⋊D4711yescharacteristic
677C4C2×#1309399211yescharacteristic
678C4C2×#1309400611yescharacteristic
679C2—24no—
680C2—82no—
681C2—1884no—
682C2C2×#1568899911yescharacteristic
683C2#2025191211yescharacteristic
684C2#2025191911yescharacteristic
685C2#2025304911yescharacteristic
686C2#2025305711yescharacteristic
687C2#2025306211yescharacteristic
688C2#2025307311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47C22×D4
prime radicalC2×#15689701C2
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#20976
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C22×#13094032C2semidirect
3. C2×#15689119C2semidirect
4. #20245981C2semidirect
5. #20243823C2semidirect
6. #20243818C2semidirect
7. #20243813C2semidirect
8. #20243808C2semidirect
9. #20243108C2semidirect
10. C2×#15688885C2semidirect
11. C2×C47×#417C2semidirect
20. #15689701C4semidirect
66. C22×C47⋊C8D4semidirect
74. #13094000C2×C4semidirect
78. #13093995C2×C4semidirect
114. C2×M16×C47D4semidirect
337. C2×#417D47semidirect
392. C47#20976semidirect
423. C2×M16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940322
C2×#156897012
C2×#156891192
#202459812
#202438232
#202438182
#202438132
#202438082
#202431082
C2×#156888852
C2×C47×#4172
#2097647