Finite Group Databasebeta

← #26218021

Subgroups — #26218021

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
1C2×#15689701C211yesmaximal, characteristic
2C2×#15689125C211yesmaximal, characteristic, complemented
3C2×#15689120C211yesmaximal, characteristic, complemented
4#20245980C211yesmaximal, characteristic, complemented
5#20243826C212yesmaximal, complemented
6#20243821C212yesmaximal, complemented
7#20243816C212yesmaximal, complemented
8#20243811C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10C2×#15688886C211yesmaximal, characteristic, complemented
11C22×#131×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13#15689701—24no—
14#15689701C418yescomplemented
15C2×#13094003C2212yes—
16C2×#13094000—21no—
17C2×#13093998C2212yes—
18C2×#13093996—22no—
19C2×#13093995—21no—
20C2×#13093983C2211yescharacteristic
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689125—24no—
24#15689120—24no—
25C2×#13093949C2211yescharacteristic
26C2×#13093949C2212yes—
27C2×#13093949C2211yescharacteristic
28C2×#13093946C2211yescharacteristic
29C2×#13093946C2212yes—
30C2×#13093946C2211yescharacteristic
31#15689030—21no—
32#15689026—21no—
33#15689003—22no—
34#15688983C2212yes—
35#15688983C2212yes—
36#15688886C2212yes—
37#15688886C2212yes—
38#15688886C2212yes—
39#15688886C2212yes—
40C22×#41×C47C2211yescharacteristic
41C2×#131×C47—24no—
42C2×#131×C47—24no—
43C2×#131×C47—24no—
44C22×M16×C47C2212yes—
45C22×M16×C47C2211yescharacteristic
46C23×C8×C47C2212yes—
47C23×C8×C47C2211yescharacteristic
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350—22no—
52C2×#10625350—22no—
53C2×#10625350C2×C414yes—
54C22×Dic94C2311yescharacteristic
55C22×D188C2311yescharacteristic
56C22×C22⋊D47—21no—
57C22×C22⋊D47—22no—
58C23×Dic47—21no—
59C23×Dic47—21no—
60C22×C4×D47—21no—
61#13094003—24no—
62#13094003C2×C418yescomplemented
63#13094000—28no—
64#13093998—24no—
65#13093998C2×C418yescomplemented
66#13093996—216no—
67#13093995—28no—
68#13093983—22no—
69#13093983—22no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47—22no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2×C414yes—
76C2×C4⋊Dic47C2312yes—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2312yes—
80C2×Dic47⋊C4—22no—
81C2×C22⋊Dic47—22no—
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47—22no—
84C2×C22⋊Dic47—22no—
85C2×(C2×D47)⋊C4—22no—
86C2×C4×Dic47—21no—
87#13093949—22no—
88#13093949—22no—
89#13093949—24no—
90#13093946—22no—
91#13093946—22no—
92#13093946—24no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C2×#41×C47—22no—
96C2×#41×C47—22no—
97C2×#41×C47—22no—
98C2×#41×C47—22no—
99C2×#41×C47C2×C414yes—
100C2×#41×C47D414yes—
101C22×Q8×C47D411yescharacteristic
102C22×D4×C47D412yes—
103C22×D4×C47D411yescharacteristic
104C23×C4×C47C2312yes—
105C23×C4×C47C2311yescharacteristic
106#131×C47—232no—
107C2×M16×C47—24no—
108C2×M16×C47—22no—
109C2×M16×C47—22no—
110C2×M16×C47—24no—
111C2×M16×C47—24no—
112C2×M16×C47D414yescomplemented
113C22×C8×C47—24no—
114C22×C8×C47—22no—
115C22×C8×C47—22no—
116C22×C8×C47—24no—
117C22×C8×C47—24no—
118C22×C8×C47D414yescomplemented
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2312yes—
121C22×C8×C47C2311yescharacteristic
122C24×C47C2×D412yes—
123C24×C47C2×D411yescharacteristic
124C23×D47—21no—
125#41×C47—28no—
126#41×C47—28no—
127#41×C47—28no—
128#41×C47—28no—
129C2×Q8×C47—22no—
130C2×Q8×C47—22no—
131C2×Q8×C47C22⋊C414yes—
132C2×D4×C47—24no—
133C2×D4×C47—24no—
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47C22⋊C414yes—
137C2×D4×C47C22⋊C418yes—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47C22×C414yes—
143C22×C4×C47C22×C412yes—
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2×D414yes—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47#4112yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C2×C22⋊D47—24no—
154C2×C22⋊D47—28no—
155C2×C22⋊D47—42no—
156C2×C22⋊D47—42no—
157C2×C22⋊D47—82no—
158C2×C22⋊D47—82no—
159C22×Dic47—22no—
160C22×Dic47—21no—
161C22×Dic47—24no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47—22no—
165C22×Dic47—42no—
166C22×Dic47—81no—
167C22×Dic47—81no—
168C22×Dic47—81no—
169C22×Dic47—81no—
170#10625350—28no—
171#10625350—28no—
172#10625350—44no—
173#10625350—44no—
174C2×D188—21no—
175C2×D188—21no—
176C2×D188—21no—
177C2×D188—21no—
178C2×D188—21no—
179C2×D188C22×C412yes—
180C2×C4×D47—24no—
181C2×C4×D47—42no—
182C2×C4×D47—42no—
183C2×Dic94—21no—
184C2×Dic94—21no—
185C2×Dic94—21no—
186C2×Dic94—21no—
187C2×Dic94—21no—
188C2×Dic94C22×C412yes—
189M16×C47—216no—
190M16×C47—28no—
191C2×C8×C47—216no—
192C2×C8×C47—28no—
193C2×C8×C47—21no—
194C2×C8×C47—21no—
195C2×C8×C47—21no—
196C2×C8×C47—22no—
197C2×C8×C47—22no—
198C2×C8×C47—21no—
199C2×C8×C47C2×D412yes—
200C2×C8×C47C2×D414yes—
201C2×C8×C47C2×D412yes—
202C22⋊Dic47—28no—
203C22⋊Dic47—42no—
204C22⋊Dic47—82no—
205C22⋊Dic47—82no—
206(C2×D47)⋊C4—28no—
207C4⋊Dic47—24no—
208C4⋊Dic47—22no—
209C4⋊Dic47—22no—
210C4⋊Dic47—21no—
211C4⋊Dic47—21no—
212C4⋊Dic47—22no—
213C4⋊Dic47—21no—
214C4⋊Dic47—21no—
215C4⋊Dic47—42no—
216C4⋊Dic47C22×C414yes—
217Dic47⋊C4—28no—
218C4×Dic47—42no—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47—22no—
228C23×C47—22no—
229C23×C47C2×C22⋊C414yes—
230C23×C47C2×C22⋊C412yes—
231C23×C47#12112yes—
232C23×C47#12112yes—
233C23×C47C22×D411yescharacteristic
234C22×D47—22no—
235C22×D47—41no—
236C22×D47—41no—
237C22×D47—41no—
238C22×D47—41no—
239C22×D47—41no—
240C22×D47—41no—
241Q8×C47—28no—
242D4×C47—216no—
243D4×C47—28no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—24no—
250C2×C4×C47C2×C22⋊C414yes—
251C2×C4×C47C2×C22⋊C412yes—
252C2×C4×C47C4×D414yes—
253C2×C4×C47C4×D414yes—
254C2×C4×C47#12014yes—
255C2×C4×C47#12112yes—
256C2×C4×C47#12112yes—
257C2×C4×C47#12314yes—
258C2×C4×C47C23×C411yescharacteristic
259C2×C4×C47C22×D411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C22⋊D47—48no—
265C22⋊D47—88no—
266C2×Dic47—22no—
267C2×Dic47—24no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—42no—
275C2×Dic47—41no—
276C2×Dic47—42no—
277C2×Dic47—41no—
278C2×Dic47—42no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—84no—
282C2×Dic47—84no—
283D188—22no—
284D188—22no—
285D188—41no—
286D188—41no—
287C4×D47—48no—
288Dic94—22no—
289Dic94—22no—
290Dic94—41no—
291Dic94—41no—
292C8×C47—22no—
293C8×C47—24no—
294C8×C47—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—24no—
303C22×C47—24no—
304C22×C47—24no—
305C22×C47—42no—
306C22×C47—42no—
307C22×C47—42no—
308C22×C47—42no—
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—42no—
318C2×D47—42no—
319C2×D47—42no—
320C2×D47—81no—
321C2×D47—81no—
322C2×D47—81no—
323C2×D47—81no—
324C4×C47—24no—
325C4×C47#38614yes—
326C4×C47C2×C4×D411yescharacteristic
327C4×C47C2×C4×D411yescharacteristic
328C4×C47C2×#12011yescharacteristic
329C4×C47C2×#12311yescharacteristic
330Dic47—44no—
331Dic47—42no—
332Dic47—81no—
333Dic47—81no—
334Dic47—82no—
335C22×#131D4711yescharacteristic, complemented
336C2×#537—471no—
337C2×#417—471no—
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—44no—
348C2×C47C2×#38611yescharacteristic
349C2×C47#289711yescharacteristic
350C2×C47#289811yescharacteristic
351C2×C47#306011yescharacteristic
352C2×C47#306111yescharacteristic
353C2×C47#306211yescharacteristic
354C2×C47#306311yescharacteristic
355D47—82no—
356C22×#41—471no—
357C22×#41C2×D4711yescharacteristic
358C2×#131—24no—
359C2×#131—24no—
360C2×#131—24no—
361C22×M16C2×D4712yes—
362C22×M16C2×D4711yescharacteristic
363C23×C8C2×D4712yes—
364C23×C8C2×D4711yescharacteristic
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⋊C4—471no—
375#417—944no—
376#416—944no—
377C2×#103—471no—
378C2×#103—471no—
379C2×#103—472no—
380C2×#102—471no—
381C2×#102—471no—
382C2×#102—472no—
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—474no—
398C2×#41—942no—
399C2×#41—942no—
400C2×#41—942no—
401C2×#41—942no—
402C2×#41—942no—
403C2×#41C4×D4714yes—
404C2×#41C22⋊D4714yes—
405C22×Q8—471no—
406C22×Q8D18811yescharacteristic
407C22×D4—471no—
408C22×D4—942no—
409C22×D4—941no—
410C22×D4D18812yes—
411C22×D4D18811yescharacteristic
412C23×C4—941no—
413C23×C4—941no—
414C23×C4—941no—
415C23×C4C22×D4712yes—
416C23×C4C22×D4711yescharacteristic
417#131—232no—
418C2×M16—24no—
419C2×M16—22no—
420C2×M16—22no—
421C2×M16—24no—
422C2×M16—24no—
423C2×M16C22⋊D4714yescomplemented
424C22×C8—24no—
425C22×C8—22no—
426C22×C8—22no—
427C22×C8—24no—
428C22×C8—24no—
429C22×C8C22⋊D4714yescomplemented
430C22×C8C22×D4711yescharacteristic
431C22×C8C22×D4712yes—
432C22×C8C22×D4711yescharacteristic
433C22⋊Q8—478no—
434C22⋊Q8—948no—
435C22⋊Q8—944no—
436#121—478no—
437#121—948no—
438#121—9416no—
439#121—944no—
440#117—942no—
441#117—942no—
442C2×C4⋊C4—474no—
443C2×C4⋊C4—472no—
444C2×C4⋊C4—472no—
445C2×C4⋊C4—471no—
446C2×C4⋊C4—471no—
447C2×C4⋊C4—941no—
448C2×C4⋊C4—942no—
449C2×C4⋊C4—942no—
450C2×C4⋊C4—942no—
451C2×C4⋊C4—942no—
452C2×C4⋊C4—941no—
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—942no—
460#103—942no—
461#103—944no—
462#102—942no—
463#102—942no—
464#102—944no—
465#95—942no—
466#95—942no—
467C24—941no—
468C24C2×D18812yes—
469C24C2×D18811yescharacteristic
470#41—28no—
471#41—28no—
472#41—28no—
473#41—28no—
474#41—948no—
475#41—948no—
476#41—1884no—
477#41—1884no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—472no—
481C2×Q8—941no—
482C2×Q8—941no—
483C2×Q8—941no—
484C2×Q8—941no—
485C2×Q8—941no—
486C2×Q8(C2×D47)⋊C414yes—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—472no—
492C2×D4—948no—
493C2×D4—944no—
494C2×D4—941no—
495C2×D4—941no—
496C2×D4—941no—
497C2×D4—941no—
498C2×D4—941no—
499C2×D4—3762no—
500C2×D4—3762no—
501C2×D4—1882no—
502C2×D4—1882no—
503C2×D4(C2×D47)⋊C418yes—
504C2×D4(C2×D47)⋊C414yes—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—944no—
510C22×C4—942no—
511C22×C4—941no—
512C22×C4—944no—
513C22×C4—941no—
514C22×C4—941no—
515C22×C4—942no—
516C22×C4—3761no—
517C22×C4—3761no—
518C22×C4—3761no—
519C22×C4—3761no—
520C22×C4—1882no—
521C22×C4—1882no—
522C22×C4—1882no—
523C22×C4C2×C4×D4714yes—
524C22×C4C2×C4×D4712yes—
525C22×C4C2×D18812yes—
526C22×C4C2×D18811yescharacteristic
527C22×C4#1062535012yes—
528C22×C4#1062535012yes—
529C22×C4C2×C22⋊D4714yes—
530C22×C4C2×C22⋊D4712yes—
531C22×C4C2×C22⋊D4712yes—
532C22×C4C2×C22⋊D4712yes—
533C22×C4C23×D4711yescharacteristic
534M16—216no—
535M16—28no—
536C2×C8—216no—
537C2×C8—28no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—21no—
544C2×C8C2×C22⋊D4712yes—
545C2×C8C2×C22⋊D4714yes—
546C2×C8C2×C22⋊D4712yes—
547C4⋊C4—474no—
548C4⋊C4—948no—
549C4⋊C4—944no—
550C4⋊C4—942no—
551C4⋊C4—942no—
552C4⋊C4—941no—
553C4⋊C4—941no—
554C4⋊C4—942no—
555C4⋊C4—941no—
556C4⋊C4—941no—
557C4⋊C4—1882no—
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—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—942no—
575C23—1881no—
576C23—1881no—
577C23—1881no—
578C23—1881no—
579C23—1881no—
580C23—1881no—
581C23C2×(C2×D47)⋊C414yes—
582C23C2×(C2×D47)⋊C412yes—
583C23#1309399812yes—
584C23#1309399812yes—
585C23C22×D18811yescharacteristic
586Q8—28no—
587Q8—942no—
588Q8—942no—
589Q8—1881no—
590Q8—1881no—
591D4—216no—
592D4—28no—
593D4—942no—
594D4—942no—
595D4—3768no—
596D4—1888no—
597D4—1881no—
598D4—1881no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—944no—
606C2×C4—942no—
607C2×C4—3764no—
608C2×C4—3764no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1882no—
612C2×C4—1881no—
613C2×C4—1882no—
614C2×C4—1881no—
615C2×C4—1882no—
616C2×C4—1881no—
617C2×C4—1881no—
618C2×C4—1881no—
619C2×C4—1881no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1888no—
623C2×C4C2×(C2×D47)⋊C414yes—
624C2×C4C2×(C2×D47)⋊C412yes—
625C2×C4C4×C22⋊D4714yes—
626C2×C4C4×C22⋊D4714yes—
627C2×C4#1309399314yes—
628C2×C4#1309399812yes—
629C2×C4#1309399812yes—
630C2×C4#1309400714yes—
631C2×C4C22×C4×D4711yescharacteristic
632C2×C4C22×C22⋊D4711yescharacteristic
633C2×C4C22×C22⋊D4711yescharacteristic
634C2×C4C22×C22⋊D4711yescharacteristic
635C2×C4C2×#1062535011yescharacteristic
636C2×C4C2×#1062535011yescharacteristic
637C8—22no—
638C8—24no—
639C8—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—3761no—
654C22—3761no—
655C22—3761no—
656C22—3761no—
657C22—1882no—
658C22—1882no—
659C22—1882no—
660C22#1568900214yes—
661C22C22×(C2×D47)⋊C411yescharacteristic
662C22C2×#1309399811yescharacteristic
663C22C2×#1309399811yescharacteristic
664C22#1568992811yescharacteristic
665C22#1568993511yescharacteristic
666C22#1568993511yescharacteristic
667C22#1568994011yescharacteristic
668C4—24no—
669C4—3761no—
670C4—3761no—
671C4—3762no—
672C4—1884no—
673C4—1882no—
674C4#1568900214yes—
675C4C2×C4×C22⋊D4711yescharacteristic
676C4C2×C4×C22⋊D4711yescharacteristic
677C4C2×#1309399311yescharacteristic
678C4C2×#1309400711yescharacteristic
679C2—24no—
680C2—44no—
681C2—3762no—
682C2C2×#1568900211yescharacteristic
683C2#2025191511yescharacteristic
684C2#2025192211yescharacteristic
685C2#2025305011yescharacteristic
686C2#2025305811yescharacteristic
687C2#2025306811yescharacteristic
688C2#2025307911yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#131×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689701C2
cosocleC22×C4×C47C24
hypercenterC22×#131D47
hypocenterC47#20976
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C22×#131

Complements

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

NH ≅ G/Ntype
2. C2×#15689125C2semidirect
3. C2×#15689120C2semidirect
4. #20245980C2semidirect
5. #20243826C2semidirect
6. #20243821C2semidirect
7. #20243816C2semidirect
8. #20243811C2semidirect
9. #20243108C2semidirect
10. C2×#15688886C2semidirect
11. C22×#131×C47C2semidirect
14. #15689701C4semidirect
62. #13094003C2×C4semidirect
65. #13093998C2×C4semidirect
112. C2×M16×C47D4semidirect
118. C22×C8×C47D4semidirect
335. C22×#131D47semidirect
392. C47#20976semidirect
423. C2×M16C22⋊D47semidirect
429. C22×C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897012
C2×#156891252
C2×#156891202
#202459802
#202438262
#202438212
#202438162
#202438112
#202431082
C2×#156888862
C22×#131×C472
#2097647