Finite Group Databasebeta

← #26217984

Subgroups — #26217984

Order 12032.

1894 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625324C211yesmaximal, characteristic
2C2×C22⋊Q8×D47C211yesmaximal, characteristic
3C2×#15689130C212yesmaximal, complemented
4#20245770C211yesmaximal, characteristic
5#20243820C212yesmaximal, complemented
6#20243818C212yesmaximal, complemented
7#20243812C211yesmaximal, characteristic, complemented
8#20243811C211yesmaximal, characteristic, complemented
9#20243809C211yesmaximal, characteristic, complemented
10#20243106C212yesmaximal, complemented
11C47×#1892C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C23×C4×D47C2211yescharacteristic
14C22×#10625324—24no—
15C22×#10625324—28no—
16C22×#10625324C2212yes—
17C22×#10625324C2212yes—
18C23×C47⋊C8C2211yescharacteristic
19C2×#13094004C2211yescharacteristic
20C2×#13094003C2211yescharacteristic
21C2×#13094002—21no—
22C2×#13094001—21no—
23C2×#13094000C2211yescharacteristic
24C22⋊Q8×D47—24no—
25C22⋊Q8×D47C418yescomplemented
26C2×#13093982C2212yes—
27C2×C4⋊C4×D47—21no—
28C2×C4⋊C4×D47C2211yescharacteristic
29C2×C22⋊C4×D47—21no—
30#15689130—28no—
31C2×#13093949C2212yescomplemented
32C2×#13093948C2212yescomplemented
33C2×#13093947C2212yescomplemented
34#15688985—21no—
35#15688983C2211yescharacteristic
36#15688980C2212yes—
37#15688978—21no—
38C22⋊C4×Dic47—21no—
39#15688970—21no—
40#15688886C2212yescomplemented
41#15688885C2212yescomplemented
42#15688885C2212yescomplemented
43C23×C8×C47C2211yescharacteristic
44C2×C22⋊Q8×C47C2211yescharacteristic
45C2×#103×C47C2212yescomplemented
46C47×#380C2211yescharacteristic
47C47×#340C2212yescomplemented
48C24×D47D411yescharacteristic
49C22×Dic94—21no—
50C22×Dic94C2311yescharacteristic
51C2×Q8×D47—22no—
52C2×Q8×D47—24no—
53C2×Q8×D47—24no—
54C2×Q8×D47C2×C414yes—
55C23×Dic47D411yescharacteristic
56C22×C4×D47—24no—
57C22×C4×D47—24no—
58C22×C4×D47—21no—
59C22×C4×D47—21no—
60C22×C4×D47C2×C414yes—
61C22×C4×D47D411yescharacteristic
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D414yes—
64C22×C4×D47C2311yescharacteristic
65C22×C4×D47C2311yescharacteristic
66C2×#10625324—232no—
67C2×#10625324—24no—
68C2×#10625324—24no—
69C2×#10625324—24no—
70C2×#10625324—24no—
71C2×#10625324—24no—
72C2×#10625324D418yes—
73C22×C47⋊C8—22no—
74C22×C47⋊C8—24no—
75C22×C47⋊C8C2312yes—
76#13094004—22no—
77#13094004C2×C414yes—
78#13094003—22no—
79#13094003C2×C414yes—
80#13094002—28no—
81#13094001—28no—
82#13094000—22no—
83#13094000C2×C414yes—
84#13093982—24no—
85#13093982—24no—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47C2312yes—
88C2×C4⋊Dic47C2311yescharacteristic
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C4⋊C4×D47—28no—
93C4⋊C4×D47—22no—
94C4⋊C4×D47C2×C414yes—
95C2×C22⋊Dic47—21no—
96C2×C22⋊Dic47—21no—
97C2×(C2×D47)⋊C4—21no—
98C2×(C2×D47)⋊C4—21no—
99C2×(C2×D47)⋊C4—22no—
100C2×(C2×D47)⋊C4—22no—
101C22⋊C4×D47—28no—
102C2×C4×Dic47—22no—
103#13093949—24no—
104#13093948—24no—
105#13093947—24no—
106C47⋊#95—21no—
107C47⋊#95—22no—
108C22×Q8×C47C2311yescharacteristic
109C23×C4×C47C2311yescharacteristic
110C22×C8×C47—22no—
111C22×C8×C47—24no—
112C22×C8×C47C2312yes—
113C22⋊Q8×C47—22no—
114C22⋊Q8×C47C2×C414yes—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47C2312yes—
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C22⋊C4×C47—21no—
120#103×C47—24no—
121#95×C47—21no—
122C24×C47C2×D411yescharacteristic
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—24no—
126C23×D47—42no—
127C23×D47C22⋊C414yes—
128C23×D47C2×D411yescharacteristic
129C23×D47C2×D411yescharacteristic
130C2×Q8×C47—21no—
131C2×Q8×C47—22no—
132C2×Q8×C47—22no—
133C2×Q8×C47C22×C412yes—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—22no—
139C22×C4×C47C22×C412yes—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—24no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—22no—
147C22×Dic47C22⋊C414yes—
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150Q8×D47—216no—
151Q8×D47—44no—
152Q8×D47—44no—
153C2×C4×D47—24no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—24no—
157C2×C4×D47—216no—
158C2×C4×D47—28no—
159C2×C4×D47—28no—
160C2×C4×D47—24no—
161C2×C4×D47—24no—
162C2×C4×D47—44no—
163C2×C4×D47—44no—
164C2×C4×D47C22⋊C414yes—
165C2×C4×D47C22⋊C414yes—
166C2×C4×D47C22×C412yes—
167C2×C4×D47C22×C412yes—
168C2×C4×D47C2×D412yes—
169C2×C4×D47C2×D414yes—
170C2×C4×D47C2×D412yes—
171C2×C4×D47#4114yes—
172C2×Dic94—22no—
173C2×Dic94—22no—
174C2×Dic94—21no—
175C2×Dic94—24no—
176C2×Dic94—44no—
177C2×Dic94C22×C412yes—
178C2×C8×C47—28no—
179C2×C8×C47—22no—
180C2×C8×C47—22no—
181C2×C8×C47C2×D414yes—
182C4⋊C4×C47—24no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47—22no—
186C4⋊C4×C47C22×C412yes—
187C22⋊C4×C47—24no—
188C22⋊C4×C47—42no—
189C22⋊Dic47—24no—
190C22⋊Dic47—42no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—44no—
194(C2×D47)⋊C4—44no—
195C4⋊Dic47—21no—
196C4⋊Dic47—24no—
197C4⋊Dic47—22no—
198C4⋊Dic47—22no—
199C4⋊Dic47C22×C412yes—
200Dic47⋊C4—24no—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203C4×Dic47—44no—
204C2×C47⋊C8—28no—
205C2×C47⋊C8—22no—
206C2×C47⋊C8—22no—
207C2×C47⋊C8C2×D414yes—
208#10625324—216no—
209#10625324—216no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47C2×C22⋊C412yes—
213C23×C47#13712yes—
214C23×C47#13712yes—
215C23×C47C22×D411yescharacteristic
216C22×D47—21no—
217C22×D47—21no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—22no—
223C22×D47—28no—
224C22×D47—28no—
225C22×D47—44no—
226C22×D47—48no—
227C22×D47—48no—
228C22×D47C2×C22⋊C412yes—
229C22×D47C2×C22⋊C412yes—
230C22×D47#12114yes—
231Q8×C47—24no—
232Q8×C47—41no—
233Q8×C47—41no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C22×D412yes—
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C2×#4112yes—
250C2×Dic47—22no—
251C2×Dic47—28no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47C2×C22⋊C412yes—
260C2×Dic47C2×C22⋊C412yes—
261C2×Dic47#12114yes—
262C4×D47—28no—
263C4×D47—24no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—28no—
267C4×D47—48no—
268C4×D47—48no—
269C4×D47C4×D418yes—
270C4×D47#12014yes—
271C4×D47#12314yes—
272Dic94—24no—
273Dic94—41no—
274Dic94—41no—
275Dic94—48no—
276C8×C47—24no—
277C47⋊C8—24no—
278#20972—471nomaximal, Sylow
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—24no—
282C22×C47—24no—
283C22×C47#41914yescomplemented
284C22×C47C22×C22⋊C411yescharacteristic
285C22×C47C2×#12112yes—
286C22×C47C2×#13711yescharacteristic
287C22×C47C2×#13711yescharacteristic
288C22×C47#57512yes—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—24no—
292C2×D47—24no—
293C2×D47—22no—
294C2×D47—22no—
295C2×D47—22no—
296C2×D47—416no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47#38614yes—
300C4×C47—22no—
301C4×C47—42no—
302C4×C47—42no—
303C4×C47—82no—
304C4×C47C2×C4×D412yes—
305C4×C47C2×#12011yescharacteristic
306C4×C47C2×#12311yescharacteristic
307Dic47—22no—
308Dic47—42no—
309Dic47—42no—
310Dic47—82no—
311Dic47#38614yes—
312C23×M16—471no—
313C22×C22⋊Q8—471no—
314C2×#419—472no—
315#2280—471no—
316#1894—472no—
317#1894—472no—
318#1892—471no—
319#1892—471no—
320#1892—471no—
321#1892D4711yescharacteristic, complemented
322#1736—472no—
323C2×C47—24no—
324C2×C47C2×#38611yescharacteristic
325C2×C47C2×#41911yescharacteristic
326C2×C47#289611yescharacteristic
327C2×C47#305612yes—
328C2×C47#305612yes—
329D47—24no—
330D47—42no—
331C23×Q8—471no—
332C24×C4—471no—
333C22×M16—472no—
334C22×M16—472no—
335C22×M16—948no—
336C22×M16—944no—
337C23×C8—471no—
338C23×C8C2×D4711yescharacteristic
339C2×C22⋊Q8—471no—
340C2×C22⋊Q8—471no—
341C2×C22⋊Q8—478no—
342C2×C22⋊Q8—471no—
343C2×C22⋊Q8—941no—
344C2×C22⋊Q8—941no—
345C2×C22⋊Q8—944no—
346C2×C22⋊Q8C2×D4711yescharacteristic
347C2×#117—472no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4—941no—
350C22×C22⋊C4—941no—
351#419—948no—
352C2×#103—472no—
353C2×#103—472no—
354C2×#103—472no—
355C2×#103C2×D4712yescomplemented
356#381—941no—
357#380—471no—
358#380—472no—
359#380C2×D4711yescharacteristic
360#379—941no—
361C4×C22⋊C4—941no—
362C4×C22⋊C4—941no—
363#340—472no—
364#340—472no—
365#340—472no—
366#340C2×D4712yescomplemented
367C47#2097211yescharacteristic, complemented, Sylow
368C25—471no—
369C22×Q8—474no—
370C22×Q8—471no—
371C22×Q8—941no—
372C22×Q8—942no—
373C22×Q8—944no—
374C22×Q8—944no—
375C22×Q8C22×D4711yescharacteristic
376C23×C4—471no—
377C23×C4—471no—
378C23×C4—474no—
379C23×C4—471no—
380C23×C4—474no—
381C23×C4—471no—
382C23×C4—471no—
383C23×C4—944no—
384C23×C4—944no—
385C23×C4—941no—
386C23×C4—941no—
387C23×C4C22×D4711yescharacteristic
388C2×M16—478no—
389C2×M16—944no—
390C2×M16—944no—
391C2×M16—944no—
392C2×M16—944no—
393C2×M16—9432no—
394C2×M16—944no—
395C22×C8—22no—
396C22×C8—24no—
397C22×C8—472no—
398C22×C8—944no—
399C22×C8—942no—
400C22×C8C22×D4712yes—
401C22⋊Q8—22no—
402C22⋊Q8—474no—
403C22⋊Q8—474no—
404C22⋊Q8—474no—
405C22⋊Q8—948no—
406C22⋊Q8—948no—
407C22⋊Q8—942no—
408C22⋊Q8—942no—
409C22⋊Q8—942no—
410C22⋊Q8C4×D4714yes—
411#117—944no—
412#117—944no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—474no—
416C2×C4⋊C4—472no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—948no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4C22×D4712yes—
424C2×C4⋊C4C22×D4711yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—948no—
428C2×C22⋊C4—941no—
429C2×C22⋊C4—941no—
430C2×C22⋊C4—941no—
431C2×C22⋊C4—942no—
432C2×C22⋊C4—942no—
433C2×C22⋊C4—941no—
434C2×C42—942no—
435#103—24no—
436#103—944no—
437#103—944no—
438#103—944no—
439#95—21no—
440#95—942no—
441#95—941no—
442C24—471no—
443C24—471no—
444C24—474no—
445C24—944no—
446C24—942no—
447C24—942no—
448C24—1882no—
449C24D4×D4711yescharacteristic
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—472no—
454C2×Q8—944no—
455C2×Q8—9416no—
456C2×Q8—941no—
457C2×Q8—942no—
458C2×Q8—942no—
459C2×Q8—1884no—
460C2×Q8—1884no—
461C2×Q8—1884no—
462C2×Q8C2×C4×D4712yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—471no—
469C22×C4—472no—
470C22×C4—472no—
471C22×C4—474no—
472C22×C4—474no—
473C22×C4—474no—
474C22×C4—471no—
475C22×C4—474no—
476C22×C4—474no—
477C22×C4—472no—
478C22×C4—472no—
479C22×C4—948no—
480C22×C4—944no—
481C22×C4—9416no—
482C22×C4—948no—
483C22×C4—944no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—942no—
492C22×C4—1884no—
493C22×C4—1884no—
494C22×C4C2×C4×D4712yes—
495C22×C4D4×D4711yescharacteristic
496C22×C4D4×D4712yes—
497C22×C4C23×D4711yescharacteristic
498M16—9416no—
499M16—9416no—
500C2×C8—28no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—474no—
504C2×C8—948no—
505C2×C8—942no—
506C2×C8—942no—
507C2×C8D4×D4714yes—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—472no—
513C4⋊C4—944no—
514C4⋊C4—944no—
515C4⋊C4—944no—
516C4⋊C4—944no—
517C4⋊C4—941no—
518C4⋊C4—942no—
519C4⋊C4—942no—
520C4⋊C4C2×C4×D4712yes—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—944no—
524C22⋊C4—944no—
525C22⋊C4—944no—
526C22⋊C4—1884no—
527C22⋊C4—1884no—
528C22⋊C4—1882no—
529C42—1884no—
530C23—22no—
531C23—22no—
532C23—472no—
533C23—474no—
534C23—472no—
535C23—948no—
536C23—941no—
537C23—941no—
538C23—942no—
539C23—942no—
540C23—948no—
541C23—941no—
542C23—941no—
543C23—942no—
544C23—1884no—
545C23—1888no—
546C23—1888no—
547C23C22⋊C4×D4712yes—
548C23#1309405312yes—
549C23#1309405612yes—
550C23C2×D4×D4711yescharacteristic
551Q8—24no—
552Q8—41no—
553Q8—41no—
554Q8—944no—
555Q8—1888no—
556Q8—1881no—
557Q8—1881no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—472no—
570C2×C4—474no—
571C2×C4—478no—
572C2×C4—474no—
573C2×C4—474no—
574C2×C4—472no—
575C2×C4—948no—
576C2×C4—948no—
577C2×C4—948no—
578C2×C4—942no—
579C2×C4—944no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—1888no—
585C2×C4—1888no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4C22⋊C4×D4712yes—
592C2×C4C22×C4×D4711yescharacteristic
593C2×C4C2×D4×D4712yes—
594C2×C4C2×D4×D4711yescharacteristic
595C2×C4#41×D4712yes—
596C8—24no—
597C8—944no—
598C22—22no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—474no—
603C22—944no—
604C22—944no—
605C22—942no—
606C22—942no—
607C22—942no—
608C22—942no—
609C22—942no—
610C22—18816no—
611C22—1882no—
612C22—1882no—
613C22#1568913014yescomplemented
614C22C2×C22⋊C4×D4711yescharacteristic
615C22#121×D4712yes—
616C22C2×#1309405311yescharacteristic
617C22C2×#1309405611yescharacteristic
618C22#1568992012yes—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—82no—
623C4—474no—
624C4—942no—
625C4—3762no—
626C4—1882no—
627C4—1882no—
628C4C4×D4×D4712yes—
629C4#120×D4711yescharacteristic
630C4#123×D4711yescharacteristic
631C2—24no—
632C2—944no—
633C2—1882no—
634C2#386×D4711yescharacteristic
635C2C2×#1568913011yescharacteristic
636C2#2025190311yescharacteristic
637C2#2025301112yes—
638C2#2025301512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1892C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1892D47
hypocenterC47#20972
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. C2×#15689130C2semidirect
5. #20243820C2semidirect
6. #20243818C2semidirect
7. #20243812C2semidirect
8. #20243811C2semidirect
9. #20243809C2semidirect
10. #20243106C2semidirect
11. C47×#1892C2semidirect
25. C22⋊Q8×D47C4semidirect
31. C2×#13093949C22semidirect
32. C2×#13093948C22semidirect
33. C2×#13093947C22semidirect
40. #15688886C22semidirect
41. #15688885C22semidirect
42. #15688885C22semidirect
45. C2×#103×C47C22semidirect
47. C47×#340C22semidirect
283. C22×C47#419semidirect
321. #1892D47semidirect
355. C2×#103C2×D47semidirect
366. #340C2×D47semidirect
367. C47#20972semidirect
613. C22#15689130semidirect

Maximal subgroups

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

subgroupindex
C23×#106253242
C2×C22⋊Q8×D472
C2×#156891302
#202457702
#202438202
#202438182
#202438122
#202438112
#202438092
#202431062
C47×#18922
#2097247