Finite Group Databasebeta

← #26217985

Subgroups — #26217985

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×#13094000C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4#20245767C211yesmaximal, characteristic
5#20243818C214yesmaximal, complemented
6#20243808C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C2×C47×#419C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C22×#10625324C2212yes—
13C2×M16×D47—24no—
14C2×M16×D47—24no—
15C2×M16×D47—24no—
16C22×C8×D47C2212yes—
17C2×#13094000—24no—
18C2×#13094000—21no—
19C2×#13094000—21no—
20C2×#13094000C418yescomplemented
21C2×#13094000C2212yes—
22C2×#13094000C2212yes—
23C2×#13093983C2211yescharacteristic
24C22×C4⋊Dic47C2211yescharacteristic
25C22×Dic47⋊C4—21no—
26C22×(C2×D47)⋊C4—21no—
27#15689132—24no—
28C2×#13093948C2214yescomplemented
29#15688984—21no—
30#15688980C2212yes—
31#15688980C2212yes—
32#15688978—21no—
33C4×(C2×D47)⋊C4—21no—
34C4×(C2×D47)⋊C4—21no—
35#15688886C2214yescomplemented
36#15688885C2214yescomplemented
37C23×Q8×C47C2211yescharacteristic
38C22×M16×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C47×#419—24no—
41C2×#103×C47C2214yescomplemented
42C24×D47D411yescharacteristic
43C23×Dic47—21no—
44C23×Dic47D411yescharacteristic
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47C2×C414yes—
50C22×C4×D47D411yescharacteristic
51C22×C4×D47D411yescharacteristic
52C22×C4×D47D414yes—
53C22×C4×D47C2311yescharacteristic
54C22×C4×D47C2311yescharacteristic
55C2×#10625329—22no—
56C2×#10625329—22no—
57C2×#10625329D414yes—
58C2×#10625324—24no—
59C2×#10625324—24no—
60C2×#10625324—24no—
61M16×D47—232no—
62C22×C47⋊C8C2312yes—
63C2×C8×D47—24no—
64C2×C8×D47—24no—
65C2×C8×D47—24no—
66#13094000—28no—
67#13094000—28no—
68#13094000—24no—
69#13094000—24no—
70#13094000C2×C418yes—
71#13094000C2×C418yes—
72#13093983—22no—
73#13093983—22no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—22no—
76C2×C4⋊Dic47C2×C414yes—
77C2×C4⋊Dic47C2312yes—
78C2×C4⋊Dic47C2312yes—
79C2×Dic47⋊C4—28no—
80C2×Dic47⋊C4—22no—
81C2×Dic47⋊C4—22no—
82C2×C22⋊Dic47—21no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—22no—
85C2×(C2×D47)⋊C4—22no—
86C2×(C2×D47)⋊C4—22no—
87C2×(C2×D47)⋊C4—22no—
88C2×C4×Dic47—21no—
89#13093948—28no—
90C47⋊#95—22no—
91C47⋊#95—22no—
92C22×Q8×C47—22no—
93C22×Q8×C47—21no—
94C22×Q8×C47—22no—
95C22×Q8×C47—22no—
96C22×Q8×C47—22no—
97C22×Q8×C47—22no—
98C22×Q8×C47C2×C414yes—
99C22×Q8×C47C2312yes—
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102C2×M16×C47—22no—
103C2×M16×C47—22no—
104C2×M16×C47D414yes—
105C22×C8×C47C2312yes—
106#117×C47—22no—
107#117×C47—22no—
108C2×C4⋊C4×C47C2312yes—
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111#103×C47—28no—
112C24×C47C2×D411yescharacteristic
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—24no—
116C23×D47—42no—
117C23×D47C22⋊C414yes—
118C23×D47C2×D411yescharacteristic
119C23×D47C2×D411yescharacteristic
120C2×Q8×C47—24no—
121C2×Q8×C47—22no—
122C2×Q8×C47—28no—
123C2×Q8×C47—28no—
124C2×Q8×C47—22no—
125C2×Q8×C47—22no—
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C2×Q8×C47—44no—
129C2×Q8×C47—42no—
130C2×Q8×C47—42no—
131C2×Q8×C47—44no—
132C2×Q8×C47C22×C414yes—
133C22×C4×C47—24no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47#4112yes—
144C22×C4×C47#4112yes—
145C22×C4×C47C2411yescharacteristic
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—44no—
152C22×Dic47C22⋊C414yes—
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C2×C4×D47—28no—
156C2×C4×D47—28no—
157C2×C4×D47—28no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—28no—
163C2×C4×D47—21no—
164C2×C4×D47—21no—
165C2×C4×D47—21no—
166C2×C4×D47—21no—
167C2×C4×D47—21no—
168C2×C4×D47—21no—
169C2×C4×D47—21no—
170C2×C4×D47—21no—
171C2×C4×D47C22⋊C414yes—
172C2×C4×D47C22⋊C414yes—
173C2×C4×D47C22×C412yes—
174C2×C4×D47C22×C412yes—
175C2×C4×D47C2×D412yes—
176C2×C4×D47C2×D412yes—
177M16×C47—28no—
178C2×C8×C47—22no—
179C2×C8×C47—22no—
180C2×C8×C47C2×D414yes—
181C4⋊C4×C47—22no—
182C4⋊C4×C47—22no—
183C22⋊C4×C47—42no—
184C42×C47—42no—
185C22⋊Dic47—42no—
186(C2×D47)⋊C4—28no—
187(C2×D47)⋊C4—28no—
188(C2×D47)⋊C4—44no—
189(C2×D47)⋊C4—44no—
190C4⋊Dic47—24no—
191C4⋊Dic47—22no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47C22×C414yes—
195Dic47⋊C4—28no—
196Dic47⋊C4—28no—
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—
225Q8×C47—24no—
226Q8×C47—24no—
227Q8×C47—42no—
228Q8×C47—42no—
229Q8×C47—48no—
230C2×C4×C47—24no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—48no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47C2×C22⋊C412yes—
238C2×C4×C47C4×D414yes—
239C2×C4×C47C4×D414yes—
240C2×C4×C47#12014yes—
241C2×C4×C47#12112yes—
242C2×C4×C47#12112yes—
243C2×C4×C47#12314yes—
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—28no—
255C2×Dic47—24no—
256C2×Dic47—48no—
257C2×Dic47—44no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47C2×C22⋊C412yes—
262C2×Dic47C2×C22⋊C412yes—
263C4×D47—22no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—22no—
267C4×D47—22no—
268C4×D47—22no—
269C4×D47—22no—
270C4×D47—22no—
271C4×D47—28no—
272C4×D47—28no—
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×#419D4711yescharacteristic, complemented
312#2280—471no—
313#1894—474no—
314#1892—474no—
315#1736—471no—
316#1736—471no—
317C2×C47—24no—
318C2×C47C2×#38611yescharacteristic
319C2×C47C2×#41911yescharacteristic
320C2×C47#289611yescharacteristic
321C2×C47#305612yes—
322C2×C47#305612yes—
323D47—24no—
324D47—42no—
325C23×Q8C2×D4711yescharacteristic
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—471no—
343C22×C4⋊C4—941no—
344C22×C22⋊C4—941no—
345#419—24no—
346#419—944no—
347C2×#103—474no—
348C2×#103C2×D4714yescomplemented
349#381—941no—
350#380—472no—
351#380—472no—
352#379—941no—
353C4×C22⋊C4—941no—
354C4×C22⋊C4—941no—
355#340—474no—
356#340—474no—
357C47#2097211yescharacteristic, complemented, Sylow
358C25—471no—
359C22×Q8—22no—
360C22×Q8—21no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—22no—
365C22×Q8C4×D4714yes—
366C22×Q8C22×D4712yes—
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×M16C22⋊D4714yes—
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—474no—
407C2×C4⋊C4—472no—
408C2×C4⋊C4—472no—
409C2×C4⋊C4—948no—
410C2×C4⋊C4—942no—
411C2×C4⋊C4—942no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—942no—
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—28no—
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—24no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—28no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—44no—
448C2×Q8C2×C4×D4714yes—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—474no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—472no—
462C22×C4—472no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—948no—
466C22×C4—948no—
467C22×C4—948no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—948no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—944no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—1884no—
485C22×C4C2×C4×D4712yes—
486C22×C4#1062535012yes—
487C22×C4D4×D4711yescharacteristic
488C22×C4D4×D4712yes—
489C22×C4D4×D4712yes—
490C22×C4#1062535212yes—
491C22×C4C2×C22⋊D4712yes—
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×C22⋊D4714yes—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—474no—
506C4⋊C4—948no—
507C4⋊C4—948no—
508C4⋊C4—944no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4—942no—
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#1309399512yes—
539C23#1309399712yes—
540C23C2×D4×D4711yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—48no—
544Q8—42no—
545Q8—42no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—948no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—948no—
561C2×C4—948no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—944no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1888no—
575C2×C4—1884no—
576C2×C4C22⋊C4×D4712yes—
577C2×C4(C22⋊D47)⋊C414yes—
578C2×C4C4×C22⋊D4714yes—
579C2×C4#1309399214yes—
580C2×C4#1309399512yes—
581C2×C4#1309399712yes—
582C2×C4#1309400614yes—
583C2×C4C22×C4×D4711yescharacteristic
584C2×C4C2×D4×D4711yescharacteristic
585C2×C4C2×D4×D4711yescharacteristic
586C2×C4C22×C22⋊D4711yescharacteristic
587C2×C4C2×#1062535211yescharacteristic
588C2×C4C2×#1062535011yescharacteristic
589C8—24no—
590C8—944no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—22no—
595C22—474no—
596C22—942no—
597C22—942no—
598C22—944no—
599C22—944no—
600C22—942no—
601C22—942no—
602C22—942no—
603C22—18816no—
604C22—1882no—
605C22—1882no—
606C22#1568899914yes—
607C22C2×C22⋊C4×D4711yescharacteristic
608C22C2×#1309399511yescharacteristic
609C22C2×#1309399711yescharacteristic
610C22#137×D4712yes—
611C22#1568992212yes—
612C4—44no—
613C4—82no—
614C4—942no—
615C4—942no—
616C4—3762no—
617C4—1884no—
618C4#1568899914yes—
619C4C2×(C22⋊D47)⋊C411yescharacteristic
620C4C2×C4×C22⋊D4711yescharacteristic
621C4C2×#1309399211yescharacteristic
622C4C2×#1309400611yescharacteristic
623C2—24no—
624C2—944no—
625C2—1882no—
626C2C2×#1568899911yescharacteristic
627C2#419×D4711yescharacteristic
628C2#2025190611yescharacteristic
629C2#2025301212yes—
630C2#2025301612yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#419C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC2×#419D47
hypocenterC47#20972
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×#15689132C2semidirect
5. #20243818C2semidirect
6. #20243808C2semidirect
7. #20243109C2semidirect
8. #20243108C2semidirect
9. C2×C47×#419C2semidirect
20. C2×#13094000C4semidirect
28. C2×#13093948C22semidirect
35. #15688886C22semidirect
36. #15688885C22semidirect
41. C2×#103×C47C22semidirect
296. C2×D47#419semidirect
311. C2×#419D47semidirect
348. C2×#103C2×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×#130940002
C2×#156891322
#202457672
#202438182
#202438082
#202431092
#202431082
C2×C47×#4192
#2097247