Finite Group Databasebeta

← #26182183

Subgroups — #26182183

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689595C211yesmaximal, characteristic
2#20252414C212yesmaximal, complemented
3#20241727C212yesmaximal, complemented
4#20241725C212yesmaximal, complemented
5#20241721C212yesmaximal, complemented
6#20241713C212yesmaximal, complemented
7#20241706C211yesmaximal, characteristic, complemented
8C2×#15688846C212yesmaximal, complemented
9C47×#1459C211yesmaximal, characteristic, complemented
10C22×D4×D47C411yescharacteristic
11#15689595—24no—
12#15689595—24no—
13C2×C4×D188C411yescharacteristic
14C2×D4×Dic47C411yescharacteristic
15C2×#13093986C2211yescharacteristic
16C2×C4×C22⋊D47C411yescharacteristic
17C2×C4×C22⋊D47C2211yescharacteristic
18C2×(C22⋊D47)⋊C4C411yescharacteristic
19C2×(C22⋊D47)⋊C4C2211yescharacteristic
20C2×#13093982C2211yescharacteristic
21C2×#13093981C411yescharacteristic
22C22×(C2×D47)⋊C4C411yescharacteristic
23C22×(C2×D47)⋊C4C2211yescharacteristic
24C2×C22⋊C4×D47C411yescharacteristic
25C2×C22⋊C4×D47C2211yescharacteristic
26#15689194—24no—
27#15689191—24no—
28#15689177C2214yescomplemented
29#15689085C2214yescomplemented
30C2×C22⋊(C47⋊C8)C2212yescomplemented
31#15688849C2212yescomplemented
32#15688847C2212yescomplemented
33#15688846C2214yes—
34#15688846C2214yescomplemented
35C2×C4×D4×C47C2211yescharacteristic
36C2×C22⋊C8×C47C2212yescomplemented
37C47×#324C2212yescomplemented
38C47×#323C2212yescomplemented
39C24×D47C2×C411yescharacteristic
40C24×D47C2×C411yescharacteristic
41C22×D188C2×C411yescharacteristic
42C22×C22⋊D47C2×C411yescharacteristic
43C22×C22⋊D47C2×C411yescharacteristic
44C2×D4×D47—24no—
45C2×D4×D47—24no—
46C2×D4×D47—24no—
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2311yescharacteristic
54C2×#10625324—24no—
55C22×C47⋊C8—22no—
56C4×D188—22no—
57C4×D188—22no—
58D4×Dic47—24no—
59#13093986—24no—
60C4×C22⋊D47—22no—
61C4×C22⋊D47—22no—
62C4×C22⋊D47D414yes—
63C4×C22⋊D47D414yes—
64(C22⋊D47)⋊C4—24no—
65(C22⋊D47)⋊C4—24no—
66#13093982—24no—
67#13093981—22no—
68#13093981—22no—
69C2×C4⋊Dic47C2×C411yescharacteristic
70C2×Dic47⋊C4C2×C411yescharacteristic
71C2×Dic47⋊C4C2×C411yescharacteristic
72C2×C22⋊Dic47C2×C411yescharacteristic
73C2×C22⋊Dic47C2×C411yescharacteristic
74C2×(C2×D47)⋊C4—22no—
75C2×(C2×D47)⋊C4—22no—
76C2×(C2×D47)⋊C4—22no—
77C2×(C2×D47)⋊C4—22no—
78C2×(C2×D47)⋊C4C2×C411yescharacteristic
79C2×(C2×D47)⋊C4C2×C411yescharacteristic
80C2×(C2×D47)⋊C4C2×C411yescharacteristic
81C2×(C2×D47)⋊C4C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2311yescharacteristic
85C2×(C2×D47)⋊C4C2311yescharacteristic
86C22⋊C4×D47—24no—
87C22⋊C4×D47—24no—
88C2×C4×Dic47C2×C411yescharacteristic
89C2×C4×Dic47C2×C411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91Dic47⋊C8—24no—
92C22⋊(C47⋊C8)—22no—
93C22⋊(C47⋊C8)C2314yes—
94(C2×D47)⋊C8—24no—
95#13093934—24no—
96C22×D4×C47C2×C411yescharacteristic
97C23×C4×C47C2×C411yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47—22no—
100C4×D4×C47—22no—
101C4×D4×C47—22no—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47C2×C411yescharacteristic
104C2×C22⋊C4×C47C2311yescharacteristic
105C2×C42×C47C2×C411yescharacteristic
106C22⋊C8×C47—22no—
107C22⋊C8×C47C2314yes—
108C24×C47C22×C411yescharacteristic
109C24×C47C22×C411yescharacteristic
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—81no—
116C23×D47—81no—
117C23×D47—81no—
118C23×D47—81no—
119C23×D47M1614yescomplemented
120C23×D47C22×C411yescharacteristic
121C23×D47C22×C411yescharacteristic
122C23×D47C22×C411yescharacteristic
123C2×D4×C47—22no—
124C2×D4×C47—22no—
125C2×D4×C47M1614yescomplemented
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—81no—
129C22×C4×C47—81no—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C22×C411yescharacteristic
132C22×C4×C47C22×C411yescharacteristic
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2411yescharacteristic
137C2×C22⋊D47—22no—
138C2×C22⋊D47—22no—
139C2×C22⋊D47—22no—
140C2×C22⋊D47—22no—
141C2×C22⋊D47—22no—
142C2×C22⋊D47C22⋊C414yes—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—81no—
146C22×Dic47—81no—
147C22×Dic47C22×C411yescharacteristic
148C22×Dic47C22×C411yescharacteristic
149C22×Dic47C22×C411yescharacteristic
150C22×Dic47C22×C411yescharacteristic
151C22×Dic47C22×C411yescharacteristic
152D4×D47—232no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—81no—
163C2×C4×D47—81no—
164C2×C4×D47—81no—
165C2×C4×D47—81no—
166C2×C4×D47C2×D414yes—
167C2×C8×C47—24no—
168C2×C8×C47—82no—
169C4⋊C4×C47—22no—
170C22⋊C4×C47—22no—
171C22⋊C4×C47—22no—
172C42×C47—21no—
173C42×C47—21no—
174C22⋊Dic47—22no—
175C22⋊Dic47C22⋊C414yes—
176(C2×D47)⋊C4—22no—
177(C2×D47)⋊C4—21no—
178(C2×D47)⋊C4—21no—
179(C2×D47)⋊C4—21no—
180(C2×D47)⋊C4—21no—
181(C2×D47)⋊C4—22no—
182(C2×D47)⋊C4—22no—
183(C2×D47)⋊C4—22no—
184(C2×D47)⋊C4—22no—
185(C2×D47)⋊C4C22⋊C412yes—
186(C2×D47)⋊C4C22⋊C412yes—
187C4⋊Dic47—22no—
188Dic47⋊C4—22no—
189Dic47⋊C4C22⋊C412yes—
190Dic47⋊C4C22⋊C412yes—
191C4×Dic47—22no—
192C4×Dic47—22no—
193C4×Dic47C2×D414yes—
194C2×C47⋊C8—24no—
195C2×C47⋊C8—82no—
196#10625324—84no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47C2×C22⋊C412yes—
204C23×C47C2×M1612yes—
205C23×C47C2×M1612yes—
206C23×C47C2×M1612yes—
207C23×C47C23×C411yescharacteristic
208C22×D47—21no—
209C22×D47—21no—
210C22×D47—21no—
211C22×D47—21no—
212C22×D47—21no—
213C22×D47—21no—
214C22×D47—28no—
215C22×D47—28no—
216C22×D47—41no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—82no—
223C22×D47—82no—
224C22×D47—82no—
225C22×D47—82no—
226C22×D47—84no—
227C22×D47—84no—
228C22×D47—84no—
229C22×D47—84no—
230C22×D47C2×C22⋊C412yes—
231C22×D47C2×M1612yes—
232C22×D47C2×M1612yes—
233D4×C47—28no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—81no—
245C2×C4×C47—81no—
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C2×M1612yes—
249C2×C4×C47C22×D412yes—
250C22⋊D47—28no—
251C22⋊D47—28no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—82no—
261C2×Dic47—82no—
262C2×Dic47C2×C22⋊C412yes—
263C2×Dic47C2×C22⋊C412yes—
264C2×Dic47C2×C22⋊C412yes—
265D188—28no—
266C4×D47—28no—
267C4×D47—44no—
268C4×D47—82no—
269C4×D47—82no—
270C4×D47—82no—
271C8×C47—44no—
272C47⋊C8—44no—
273#17367—471nomaximal, Sylow
274C22×C47—21no—
275C22×C47—22no—
276C22×C47—21no—
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47#40714yes—
286C22×C47C22×C22⋊C411yescharacteristic
287C22×C47C22×M1611yescharacteristic
288C22×C47C22×M1611yescharacteristic
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47—22no—
294C2×D47—42no—
295C2×D47—42no—
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—41no—
299C2×D47—41no—
300C2×D47—41no—
301C2×D47—41no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—88no—
305C2×D47—88no—
306C4×C47—22no—
307C4×C47—22no—
308C4×C47—21no—
309C4×C47—21no—
310C4×C47—41no—
311C4×C47—41no—
312Dic47—22no—
313Dic47—22no—
314Dic47—22no—
315Dic47—41no—
316Dic47—41no—
317C2×#518—471no—
318#2973—472no—
319#1466—472no—
320#1465—472no—
321#1464—472no—
322#1462—472no—
323#1459—471no—
324#1459D4711yescharacteristic, complemented
325C2×#323—472no—
326C2×C47—21no—
327C2×C47—21no—
328C2×C47—24no—
329C2×C47—42no—
330C2×C47C2×#40711yescharacteristic
331C2×C47#288311yescharacteristic
332C2×C47#288811yescharacteristic
333D47—42no—
334D47—42no—
335D47—42no—
336C23×D4—471no—
337#518—944no—
338#518—944no—
339C2×C4×D4—471no—
340C2×C4×D4—471no—
341C2×C4×D4—471no—
342C2×C4×D4—471no—
343C2×C4×D4—471no—
344C2×C4×D4—471no—
345C2×C4×D4—471no—
346C2×C4×D4C2×D4711yescharacteristic
347C2×#117—471no—
348C2×#117—471no—
349C22×C22⋊C4—471no—
350C22×C22⋊C4—471no—
351C22×C22⋊C4—471no—
352C22×C22⋊C4—471no—
353#435—944no—
354#435—944no—
355#432—474no—
356#407—474no—
357C2×C22⋊C8—472no—
358C2×C22⋊C8C2×D4712yescomplemented
359#324—472no—
360#324C2×D4712yescomplemented
361#323—472no—
362#323—474no—
363#323—474no—
364#323C2×D4712yescomplemented
365C47#1736711yescharacteristic, complemented, Sylow
366C25—471no—
367C25—471no—
368C22×D4—471no—
369C22×D4—471no—
370C22×D4—471no—
371C22×D4—944no—
372C22×D4—944no—
373C22×D4—944no—
374C22×D4C4×D4711yescharacteristic
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—471no—
381C23×C4—471no—
382C23×C4C4×D4711yescharacteristic
383C23×C4C22×D4711yescharacteristic
384C2×M16—944no—
385C22×C8—22no—
386C22×C8—942no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—474no—
390C4×D4—474no—
391C4×D4—942no—
392C4×D4—942no—
393C4×D4—944no—
394C4×D4—944no—
395C4×D4—944no—
396C4×D4—944no—
397C4×D4—942no—
398C4×D4—942no—
399#117—942no—
400#117—942no—
401#117—944no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4C22×D4711yescharacteristic
406C2×C22⋊C4—471no—
407C2×C22⋊C4—471no—
408C2×C22⋊C4—471no—
409C2×C22⋊C4—471no—
410C2×C22⋊C4—471no—
411C2×C22⋊C4—471no—
412C2×C22⋊C4—471no—
413C2×C22⋊C4—471no—
414C2×C22⋊C4—471no—
415C2×C22⋊C4—471no—
416C2×C22⋊C4—944no—
417C2×C22⋊C4—944no—
418C2×C22⋊C4—942no—
419C2×C22⋊C4—942no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4C4×D4711yescharacteristic
423C2×C22⋊C4C22×D4711yescharacteristic
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42C4×D4711yescharacteristic
428C4⋊C8—944no—
429C22⋊C8—22no—
430C22⋊C8—474no—
431C22⋊C8—944no—
432C22⋊C8—942no—
433C22⋊C8C22×D4714yes—
434#97—944no—
435C24—474no—
436C24—471no—
437C24—471no—
438C24—471no—
439C24—942no—
440C24—942no—
441C24—942no—
442C24—942no—
443C24—942no—
444C24—3761no—
445C24—3761no—
446C24—3761no—
447C24—3761no—
448C24C2×C4×D4711yescharacteristic
449C24C2×C4×D4711yescharacteristic
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—474no—
453C2×D4—942no—
454C2×D4—942no—
455C2×D4—942no—
456C2×D4—9432no—
457C2×D4—942no—
458C2×D4—942no—
459C2×D4—942no—
460C2×D4—942no—
461C2×D4—942no—
462C2×D4#1062532414yescomplemented
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—471no—
468C22×C4—471no—
469C22×C4—471no—
470C22×C4—474no—
471C22×C4—471no—
472C22×C4—471no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—3761no—
482C22×C4—3761no—
483C22×C4—3761no—
484C22×C4—3761no—
485C22×C4—3761no—
486C22×C4—3761no—
487C22×C4C2×C4×D4711yescharacteristic
488C22×C4C2×C4×D4711yescharacteristic
489C22×C4C2×C4×D4711yescharacteristic
490C22×C4C2×C4×D4711yescharacteristic
491C22×C4D4×D4712yes—
492C22×C4D4×D4712yes—
493C22×C4C23×D4711yescharacteristic
494M16—3764no—
495C2×C8—24no—
496C2×C8—82no—
497C2×C8—944no—
498C2×C8—3762no—
499C4⋊C4—22no—
500C4⋊C4—472no—
501C4⋊C4—472no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—474no—
507C22⋊C4—472no—
508C22⋊C4—472no—
509C22⋊C4—942no—
510C22⋊C4—942no—
511C22⋊C4—942no—
512C22⋊C4—942no—
513C22⋊C4—941no—
514C22⋊C4—941no—
515C22⋊C4—941no—
516C22⋊C4—941no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C42—21no—
520C42—21no—
521C42—474no—
522C42—942no—
523C42—942no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—472no—
531C23—472no—
532C23—472no—
533C23—948no—
534C23—948no—
535C23—941no—
536C23—941no—
537C23—941no—
538C23—941no—
539C23—941no—
540C23—941no—
541C23—3762no—
542C23—3762no—
543C23—3762no—
544C23—3762no—
545C23—3764no—
546C23—3764no—
547C23—3764no—
548C23—3764no—
549C23—1881no—
550C23—1881no—
551C23—1881no—
552C23—1881no—
553C23—1881no—
554C23—1881no—
555C23C22⋊C4×D4712yes—
556C23M16×D4712yes—
557C23C2×#1062532412yes—
558C23C2×#1062532412yes—
559C23C22×C4×D4711yescharacteristic
560D4—28no—
561D4—948no—
562D4—948no—
563D4—948no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—472no—
577C2×C4—472no—
578C2×C4—472no—
579C2×C4—948no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—3762no—
585C2×C4—3762no—
586C2×C4—3762no—
587C2×C4—3762no—
588C2×C4—3762no—
589C2×C4—1884no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4C22⋊C4×D4711yescharacteristic
595C2×C4C22⋊C4×D4711yescharacteristic
596C2×C4C2×#1062532412yes—
597C2×C4C2×D4×D4712yes—
598C8—44no—
599C8—1884no—
600C22—21no—
601C22—22no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—41no—
610C22—41no—
611C22—942no—
612C22—942no—
613C22—942no—
614C22—942no—
615C22—942no—
616C22—3768no—
617C22—3768no—
618C22—1882no—
619C22—1882no—
620C22—1882no—
621C22—1882no—
622C22—1882no—
623C22—1882no—
624C22—1881no—
625C22—1881no—
626C22—1881no—
627C22—1881no—
628C22#1568908514yes—
629C22C2×C22⋊C4×D4711yescharacteristic
630C22C2×M16×D4711yescharacteristic
631C22C22×#1062532411yescharacteristic
632C4—22no—
633C4—22no—
634C4—21no—
635C4—21no—
636C4—41no—
637C4—41no—
638C4—942no—
639C4—942no—
640C4—942no—
641C4—1881no—
642C4—1881no—
643C2—21no—
644C2—21no—
645C2—24no—
646C2—42no—
647C2—1882no—
648C2—1882no—
649C2—1882no—
650C2C2×#1568908511yescharacteristic
651C2#2025184611yescharacteristic
652C2#2025186511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1459C2
socleC22×C47C22×C22⋊C4
prime radicalC22×D4×D47C4
cosocleC22×C4×C47C24
hypercenter#1459D47
hypocenterC47#17367
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #20252414C2semidirect
3. #20241727C2semidirect
4. #20241725C2semidirect
5. #20241721C2semidirect
6. #20241713C2semidirect
7. #20241706C2semidirect
8. C2×#15688846C2semidirect
9. C47×#1459C2semidirect
28. #15689177C22semidirect
29. #15689085C22semidirect
30. C2×C22⋊(C47⋊C8)C22semidirect
31. #15688849C22semidirect
32. #15688847C22semidirect
34. #15688846C22semidirect
36. C2×C22⋊C8×C47C22semidirect
37. C47×#324C22semidirect
38. C47×#323C22semidirect
119. C23×D47M16semidirect
125. C2×D4×C47M16semidirect
324. #1459D47semidirect
358. C2×C22⋊C8C2×D47semidirect
360. #324C2×D47semidirect
364. #323C2×D47semidirect
365. C47#17367semidirect
462. C2×D4#10625324semidirect

Maximal subgroups

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

subgroupindex
C2×#156895952
#202524142
#202417272
#202417252
#202417212
#202417132
#202417062
C2×#156888462
C47×#14592
#1736747