Finite Group Databasebeta

← #26217988

Subgroups — #26217988

Order 12032.

1894 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D47C211yesmaximal, characteristic
2C22×#13094004C211yesmaximal, characteristic
3C2×#15689133C211yesmaximal, characteristic, complemented
4C2×#15689131C211yesmaximal, characteristic, complemented
5#20245767C211yesmaximal, characteristic
6#20243820C214yesmaximal, complemented
7#20243810C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C47×#1736C211yesmaximal, characteristic, complemented
10C23×Dic94C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C22×#10625324C2212yes—
14C2×M16×D47—24no—
15C2×M16×D47—24no—
16C2×M16×D47—24no—
17C22×C8×D47C2212yes—
18C2×#13094004—24no—
19C2×#13094004—21no—
20C2×#13094004—21no—
21C2×#13094004C418yescomplemented
22C2×#13094004C2212yes—
23C2×#13094004C2212yes—
24C2×#13093983C2211yescharacteristic
25C22×C4⋊Dic47—21no—
26C22×(C2×D47)⋊C4—21no—
27#15689133—24no—
28#15689131—24no—
29C2×#13093949C2214yescomplemented
30C2×#13093947C2214yescomplemented
31#15688984—21no—
32#15688980C2212yes—
33#15688980C2212yes—
34#15688978—21no—
35C4×(C2×D47)⋊C4—21no—
36C4×(C2×D47)⋊C4—21no—
37#15688885C2214yescomplemented
38C22×M16×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C47×#340C2214yescomplemented
42C24×D47D411yescharacteristic
43C22×Dic94—22no—
44C22×Dic94—22no—
45C22×Dic94—22no—
46C22×Dic94—22no—
47C22×Dic94—21no—
48C22×Dic94—22no—
49C22×Dic94C2×C414yes—
50C22×Dic94C2312yes—
51C23×Dic47—21no—
52C23×Dic47D411yescharacteristic
53C22×C4×D47—22no—
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47—22no—
57C22×C4×D47C2×C414yes—
58C22×C4×D47D411yescharacteristic
59C22×C4×D47D411yescharacteristic
60C22×C4×D47D414yes—
61C22×C4×D47C2311yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329D414yes—
66C2×#10625324—24no—
67C2×#10625324—24no—
68C2×#10625324—24no—
69M16×D47—232no—
70C22×C47⋊C8C2312yes—
71C2×C8×D47—24no—
72C2×C8×D47—24no—
73C2×C8×D47—24no—
74#13094004—28no—
75#13094004—28no—
76#13094004—24no—
77#13094004—24no—
78#13094004C2×C418yes—
79#13094004C2×C418yes—
80#13093983—22no—
81#13093983—22no—
82C2×C4⋊Dic47—28no—
83C2×C4⋊Dic47—22no—
84C2×C4⋊Dic47—22no—
85C2×C4⋊Dic47C2312yes—
86C2×C22⋊Dic47—21no—
87C2×(C2×D47)⋊C4—28no—
88C2×(C2×D47)⋊C4—22no—
89C2×(C2×D47)⋊C4—22no—
90C2×(C2×D47)⋊C4—22no—
91C2×(C2×D47)⋊C4—22no—
92C2×C4×Dic47—21no—
93#13093949—28no—
94#13093947—28no—
95C47⋊#95—22no—
96C47⋊#95—22no—
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C2×M16×C47—22no—
100C2×M16×C47—22no—
101C2×M16×C47D414yes—
102C22×C8×C47C2312yes—
103#117×C47—22no—
104#117×C47—22no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—22no—
107C2×C4⋊C4×C47C2×C414yes—
108C2×C4⋊C4×C47C2312yes—
109C2×C4⋊C4×C47C2312yes—
110C2×C22⋊C4×C47—21no—
111C2×C42×C47—21no—
112C24×C47C2×D411yescharacteristic
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—24no—
116C23×D47—42no—
117C23×D47C22⋊C414yes—
118C23×D47C2×D411yescharacteristic
119C23×D47C2×D411yescharacteristic
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—44no—
124C22×C4×C47C22×C412yes—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47#4112yes—
130C22×C4×C47#4112yes—
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—24no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47C22⋊C414yes—
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C2×C4×D47—28no—
143C2×C4×D47—28no—
144C2×C4×D47—28no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—28no—
150C2×C4×D47—21no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47—21no—
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×C4×D47C22⋊C414yes—
159C2×C4×D47C22⋊C414yes—
160C2×C4×D47C22×C412yes—
161C2×C4×D47C22×C412yes—
162C2×C4×D47C2×D412yes—
163C2×C4×D47C2×D412yes—
164C2×Dic94—28no—
165C2×Dic94—28no—
166C2×Dic94—22no—
167C2×Dic94—22no—
168C2×Dic94—22no—
169C2×Dic94—22no—
170C2×Dic94—24no—
171C2×Dic94—22no—
172C2×Dic94—44no—
173C2×Dic94—42no—
174C2×Dic94—42no—
175C2×Dic94—44no—
176C2×Dic94C22×C414yes—
177M16×C47—28no—
178C2×C8×C47—22no—
179C2×C8×C47—22no—
180C2×C8×C47C2×D414yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47—22no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47C22×C414yes—
186C22⋊C4×C47—42no—
187C42×C47—42no—
188C22⋊Dic47—42no—
189(C2×D47)⋊C4—28no—
190(C2×D47)⋊C4—28no—
191(C2×D47)⋊C4—44no—
192(C2×D47)⋊C4—44no—
193C4⋊Dic47—28no—
194C4⋊Dic47—28no—
195C4⋊Dic47—22no—
196C4⋊Dic47—22no—
197C4×Dic47—42no—
198#10625329—28no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8C2×D414yes—
202#10625324—216no—
203C8×D47—216no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47C2×C22⋊C412yes—
207C23×C47#12112yes—
208C23×C47#12112yes—
209C23×C47C22×D411yescharacteristic
210C22×D47—21no—
211C22×D47—21no—
212C22×D47—22no—
213C22×D47—21no—
214C22×D47—21no—
215C22×D47—22no—
216C22×D47—22no—
217C22×D47—28no—
218C22×D47—28no—
219C22×D47—44no—
220C22×D47—48no—
221C22×D47—48no—
222C22×D47C2×C22⋊C412yes—
223C22×D47C2×C22⋊C412yes—
224C22×D47#13714yes—
225C2×C4×C47—24no—
226C2×C4×C47—48no—
227C2×C4×C47—44no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47C2×C22⋊C412yes—
232C2×C4×C47C4×D414yes—
233C2×C4×C47C4×D414yes—
234C2×C4×C47#12014yes—
235C2×C4×C47#12112yes—
236C2×C4×C47#12112yes—
237C2×C4×C47#12314yes—
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—28no—
249C2×Dic47—24no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—48no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47C2×C22⋊C412yes—
257C2×Dic47C2×C22⋊C412yes—
258C4×D47—22no—
259C4×D47—22no—
260C4×D47—22no—
261C4×D47—22no—
262C4×D47—22no—
263C4×D47—22no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—28no—
267C4×D47—28no—
268Dic94—24no—
269Dic94—24no—
270Dic94—42no—
271Dic94—42no—
272Dic94—48no—
273C8×C47—24no—
274C47⋊C8—24no—
275#20972—471nomaximal, Sylow
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—24no—
279C22×C47—24no—
280C22×C47#38614yes—
281C22×C47C22×C22⋊C411yescharacteristic
282C22×C47C2×#12111yescharacteristic
283C22×C47C2×#12111yescharacteristic
284C22×C47C2×#13712yes—
285C22×C47#57512yes—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—24no—
289C2×D47—24no—
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47—416no—
294C2×D47—42no—
295C2×D47—42no—
296C2×D47#41914yescomplemented
297C4×C47—44no—
298C4×C47—82no—
299C4×C47#38614yes—
300C4×C47C2×C4×D411yescharacteristic
301C4×C47C2×C4×D411yescharacteristic
302C4×C47C2×#12011yescharacteristic
303C4×C47C2×#12311yescharacteristic
304Dic47—22no—
305Dic47—22no—
306Dic47—44no—
307Dic47—82no—
308C23×M16—471no—
309C22×C22⋊Q8—471no—
310C2×#419—471no—
311C2×#419—471no—
312#2280—471no—
313#1894—474no—
314#1892—474no—
315#1736—471no—
316#1736D4711yescharacteristic, complemented
317C2×C47—24no—
318C2×C47C2×#38611yescharacteristic
319C2×C47C2×#41911yescharacteristic
320C2×C47#289611yescharacteristic
321C2×C47#305612yes—
322C2×C47#305612yes—
323D47—24no—
324D47—42no—
325C23×Q8—471no—
326C24×C4—471no—
327C22×M16—471no—
328C22×M16—472no—
329C22×M16—944no—
330C22×M16—944no—
331C22×M16—944no—
332C22×M16C2×D4711yescharacteristic
333C23×C8—472no—
334C2×C22⋊Q8—478no—
335C2×C22⋊Q8—472no—
336C2×C22⋊Q8—472no—
337C2×C22⋊Q8—944no—
338C2×C22⋊Q8—941no—
339C2×C22⋊Q8—941no—
340C2×#117—471no—
341C2×#117C2×D4711yescharacteristic
342C22×C4⋊C4—941no—
343C22×C4⋊C4C2×D4711yescharacteristic
344C22×C22⋊C4—941no—
345#419—944no—
346#419—944no—
347C2×#103—474no—
348C2×#103—474no—
349#381—941no—
350#380—472no—
351#380—472no—
352#379—941no—
353C4×C22⋊C4—941no—
354C4×C22⋊C4—941no—
355#340—474no—
356#340C2×D4714yescomplemented
357C47#2097211yescharacteristic, complemented, Sylow
358C25—471no—
359C22×Q8—474no—
360C22×Q8—472no—
361C22×Q8—942no—
362C22×Q8—941no—
363C22×Q8—942no—
364C22×Q8—942no—
365C22×Q8—942no—
366C22×Q8—942no—
367C23×C4—21no—
368C23×C4—471no—
369C23×C4—474no—
370C23×C4—474no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C23×C4—942no—
376C23×C4—942no—
377C23×C4—942no—
378C23×C4—942no—
379C23×C4—941no—
380C23×C4C22×D4711yescharacteristic
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—474no—
384C2×M16—944no—
385C2×M16—944no—
386C2×M16—9432no—
387C2×M16—942no—
388C2×M16—942no—
389C2×M16—944no—
390C2×M16D18814yes—
391C22×C8—472no—
392C22×C8—944no—
393C22×C8—944no—
394C22×C8—944no—
395C22×C8C22×D4712yes—
396C22⋊Q8—478no—
397C22⋊Q8—478no—
398C22⋊Q8—948no—
399C22⋊Q8—948no—
400C22⋊Q8—944no—
401C22⋊Q8—944no—
402#117—22no—
403#117—22no—
404#117—942no—
405#117—942no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—472no—
409C2×C4⋊C4—948no—
410C2×C4⋊C4—942no—
411C2×C4⋊C4—942no—
412C2×C4⋊C4C4×D4714yes—
413C2×C4⋊C4C22×D4712yes—
414C2×C4⋊C4C22×D4712yes—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—948no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—942no—
419C2×C22⋊C4—942no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—941no—
422C2×C42—21no—
423C2×C42—941no—
424#103—948no—
425#103—948no—
426#95—942no—
427#95—942no—
428C24—471no—
429C24—471no—
430C24—474no—
431C24—944no—
432C24—942no—
433C24—942no—
434C24—1882no—
435C24D4×D4711yescharacteristic
436C2×Q8—474no—
437C2×Q8—944no—
438C2×Q8—942no—
439C2×Q8—948no—
440C2×Q8—948no—
441C2×Q8—942no—
442C2×Q8—942no—
443C2×Q8—942no—
444C2×Q8—942no—
445C2×Q8—1884no—
446C2×Q8—1882no—
447C2×Q8—1882no—
448C2×Q8—1884no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—474no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—472no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—948no—
465C22×C4—948no—
466C22×C4—948no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—948no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—944no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—1882no—
484C22×C4—1882no—
485C22×C4C2×C4×D4712yes—
486C22×C4C2×D18812yes—
487C22×C4#1062535012yes—
488C22×C4D4×D4711yescharacteristic
489C22×C4D4×D4712yes—
490C22×C4D4×D4712yes—
491C22×C4#1062535412yes—
492C22×C4C23×D4711yescharacteristic
493M16—28no—
494M16—9416no—
495M16—948no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—474no—
499C2×C8—9416no—
500C2×C8—942no—
501C2×C8—942no—
502C2×C8C2×D18814yes—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—948no—
508C4⋊C4—948no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4C2×C4×D4714yes—
512C22⋊C4—42no—
513C22⋊C4—948no—
514C22⋊C4—948no—
515C22⋊C4—1884no—
516C22⋊C4—1884no—
517C22⋊C4—1882no—
518C42—42no—
519C42—1882no—
520C23—22no—
521C23—22no—
522C23—472no—
523C23—474no—
524C23—472no—
525C23—948no—
526C23—941no—
527C23—941no—
528C23—942no—
529C23—948no—
530C23—941no—
531C23—941no—
532C23—942no—
533C23—942no—
534C23—1884no—
535C23—1888no—
536C23—1888no—
537C23C22⋊C4×D4712yes—
538C23#1309399712yes—
539C23#1309399912yes—
540C23C2×D4×D4711yescharacteristic
541Q8—944no—
542Q8—944no—
543Q8—1888no—
544Q8—1882no—
545Q8—1882no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—948no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—948no—
560C2×C4—948no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—944no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1888no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4C22⋊C4×D4712yes—
577C2×C4#1309398614yes—
578C2×C4C4×D18814yes—
579C2×C4#1309399414yes—
580C2×C4#1309399712yes—
581C2×C4#1309399912yes—
582C2×C4#1309400914yes—
583C2×C4C22×C4×D4711yescharacteristic
584C2×C4C2×D4×D4711yescharacteristic
585C2×C4C2×D4×D4711yescharacteristic
586C2×C4C22×D18811yescharacteristic
587C2×C4C2×#1062535011yescharacteristic
588C2×C4C2×#1062535411yescharacteristic
589C8—24no—
590C8—944no—
591C22—24no—
592C22—24no—
593C22—22no—
594C22—22no—
595C22—474no—
596C22—942no—
597C22—944no—
598C22—944no—
599C22—942no—
600C22—942no—
601C22—942no—
602C22—942no—
603C22—18816no—
604C22—1882no—
605C22—1882no—
606C22#1568900114yes—
607C22C2×C22⋊C4×D4711yescharacteristic
608C22C2×#1309399711yescharacteristic
609C22C2×#1309399911yescharacteristic
610C22#137×D4712yes—
611C22#1568991912yes—
612C4—44no—
613C4—82no—
614C4—942no—
615C4—942no—
616C4—3762no—
617C4—1884no—
618C4#1568900114yes—
619C4C2×#1309398611yescharacteristic
620C4C2×C4×D18811yescharacteristic
621C4C2×#1309399411yescharacteristic
622C4C2×#1309400911yescharacteristic
623C2—24no—
624C2—944no—
625C2—1882no—
626C2C2×#1568900111yescharacteristic
627C2#419×D4711yescharacteristic
628C2#2025190211yescharacteristic
629C2#2025301912yes—
630C2#2025302112yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20972
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×#15689133C2semidirect
4. C2×#15689131C2semidirect
6. #20243820C2semidirect
7. #20243810C2semidirect
8. #20243107C2semidirect
9. C47×#1736C2semidirect
21. C2×#13094004C4semidirect
29. C2×#13093949C22semidirect
30. C2×#13093947C22semidirect
37. #15688885C22semidirect
41. C47×#340C22semidirect
296. C2×D47#419semidirect
316. #1736D47semidirect
356. #340C2×D47semidirect
357. C47#20972semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D472
C22×#130940042
C2×#156891332
C2×#156891312
#202457672
#202438202
#202438102
#202431072
C47×#17362
#2097247