Finite Group Databasebeta

← #26182179

Subgroups — #26182179

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689596C211yesmaximal, characteristic
2#20252414C212yesmaximal, complemented
3#20241727C212yesmaximal, complemented
4#20241723C212yesmaximal, complemented
5#20241719C212yesmaximal, complemented
6#20241713C212yesmaximal, complemented
7#20241704C212yesmaximal, complemented
8C2×#15688847C211yesmaximal, characteristic, complemented
9C2×C47×#323C211yesmaximal, characteristic, complemented
10C23×C22⋊D47C411yescharacteristic
11#15689596—24no—
12#15689596—24no—
13C2×C4×C22⋊D47C411yescharacteristic
14C2×C4×C22⋊D47C411yescharacteristic
15C2×C4×C22⋊D47C2211yescharacteristic
16C2×C4×C22⋊D47C2211yescharacteristic
17C2×(C22⋊D47)⋊C4C411yescharacteristic
18C2×(C22⋊D47)⋊C4C411yescharacteristic
19C2×(C22⋊D47)⋊C4C2211yescharacteristic
20C2×(C22⋊D47)⋊C4C2211yescharacteristic
21C2×#13093980C411yescharacteristic
22C2×#13093980C2211yescharacteristic
23C22×C22⋊Dic47C411yescharacteristic
24C2×C22⋊C4×D47C411yescharacteristic
25C2×C22⋊C4×D47C2211yescharacteristic
26#15689194—24no—
27#15689191—24no—
28#15689177C2214yescomplemented
29#15689085C2214yescomplemented
30C2×C22⋊(C47⋊C8)C2212yescomplemented
31#15688848C2214yescomplemented
32#15688847C2212yes—
33#15688847C2212yescomplemented
34#15688846C2214yescomplemented
35C22×C22⋊C4×C47C2211yescharacteristic
36C2×C22⋊C8×C47C2212yescomplemented
37C47×#323C2212yes—
38C47×#323C2212yescomplemented
39C24×D47C2×C411yescharacteristic
40C22×C22⋊D47—24no—
41C22×C22⋊D47—22no—
42C22×C22⋊D47—22no—
43C22×C22⋊D47—24no—
44C22×C22⋊D47C2×C411yescharacteristic
45C22×C22⋊D47C2×C411yescharacteristic
46C22×C22⋊D47C2×C411yescharacteristic
47C22×C22⋊D47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C23×Dic47C2×C411yescharacteristic
50C23×Dic47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2×C411yescharacteristic
54C22×C4×D47C2311yescharacteristic
55C2×#10625324—24no—
56C22×C47⋊C8—22no—
57C4×C22⋊D47—22no—
58C4×C22⋊D47—22no—
59C4×C22⋊D47—22no—
60C4×C22⋊D47—22no—
61C4×C22⋊D47—22no—
62C4×C22⋊D47—22no—
63C4×C22⋊D47D414yes—
64C4×C22⋊D47D414yes—
65(C22⋊D47)⋊C4—24no—
66(C22⋊D47)⋊C4—24no—
67(C22⋊D47)⋊C4—24no—
68(C22⋊D47)⋊C4—24no—
69#13093980—24no—
70#13093980—24no—
71C2×Dic47⋊C4C2×C411yescharacteristic
72C2×Dic47⋊C4C2×C411yescharacteristic
73C2×Dic47⋊C4C2×C411yescharacteristic
74C2×Dic47⋊C4C2311yescharacteristic
75C2×C22⋊Dic47—22no—
76C2×C22⋊Dic47—22no—
77C2×C22⋊Dic47C2×C411yescharacteristic
78C2×C22⋊Dic47C2×C411yescharacteristic
79C2×C22⋊Dic47C2×C411yescharacteristic
80C2×C22⋊Dic47C2×C411yescharacteristic
81C2×(C2×D47)⋊C4C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2311yescharacteristic
85C22⋊C4×D47—24no—
86C22⋊C4×D47—24no—
87C2×C4×Dic47C2×C411yescharacteristic
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—
96C25×C47C2×C411yescharacteristic
97C23×C4×C47C2×C411yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47—22no—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47—22no—
102C2×C22⋊C4×C47C2×C411yescharacteristic
103C2×C22⋊C4×C47C2×C411yescharacteristic
104C2×C22⋊C4×C47C2311yescharacteristic
105C2×C22⋊C4×C47C2311yescharacteristic
106C22⋊C8×C47—22no—
107C22⋊C8×C47C2314yes—
108C24×C47—21no—
109C24×C47—21no—
110C24×C47—22no—
111C24×C47—81no—
112C24×C47—81no—
113C24×C47M1614yescomplemented
114C24×C47C22×C411yescharacteristic
115C24×C47C22×C411yescharacteristic
116C24×C47C22×C411yescharacteristic
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—81no—
121C23×D47—81no—
122C23×D47C22×C411yescharacteristic
123C23×D47C22×C411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—81no—
127C22×C4×C47—81no—
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—216no—
135C2×C22⋊D47—216no—
136C2×C22⋊D47—21no—
137C2×C22⋊D47—21no—
138C2×C22⋊D47—22no—
139C2×C22⋊D47—22no—
140C2×C22⋊D47—22no—
141C2×C22⋊D47—22no—
142C2×C22⋊D47—22no—
143C2×C22⋊D47—22no—
144C2×C22⋊D47—22no—
145C2×C22⋊D47—22no—
146C2×C22⋊D47—22no—
147C2×C22⋊D47C22⋊C412yes—
148C2×C22⋊D47C22⋊C412yes—
149C2×C22⋊D47M1614yescomplemented
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—81no—
156C22×Dic47—81no—
157C22×Dic47C22×C411yescharacteristic
158C22×Dic47C22×C411yescharacteristic
159C22×Dic47C22×C411yescharacteristic
160C22×Dic47C22×C411yescharacteristic
161C22×Dic47C22×C411yescharacteristic
162C22×Dic47C22×C411yescharacteristic
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×C4×D47—81no—
167C2×C4×D47—81no—
168C2×C4×D47—81no—
169C2×C4×D47—81no—
170C2×C4×D47C2×D414yes—
171C2×C8×C47—24no—
172C2×C8×C47—82no—
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—22no—
177C22⋊Dic47—21no—
178C22⋊Dic47—21no—
179C22⋊Dic47—22no—
180C22⋊Dic47—22no—
181C22⋊Dic47C22⋊C412yes—
182C22⋊Dic47C22⋊C412yes—
183(C2×D47)⋊C4—22no—
184(C2×D47)⋊C4—22no—
185(C2×D47)⋊C4—22no—
186(C2×D47)⋊C4C22⋊C414yes—
187Dic47⋊C4—22no—
188Dic47⋊C4—22no—
189Dic47⋊C4—22no—
190Dic47⋊C4C22⋊C414yes—
191C4×Dic47—22no—
192C4×Dic47—22no—
193C4×Dic47—22no—
194C4×Dic47C2×D414yes—
195C2×C47⋊C8—24no—
196C2×C47⋊C8—82no—
197#10625324—84no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47—81no—
211C23×C47—81no—
212C23×C47—81no—
213C23×C47—81no—
214C23×C47—84no—
215C23×C47—84no—
216C23×C47C2×C22⋊C411yescharacteristic
217C23×C47C2×C22⋊C411yescharacteristic
218C23×C47C2×M1612yes—
219C23×C47C2×M1612yes—
220C23×C47C2×M1612yes—
221C23×C47C2×M1612yes—
222C23×C47C23×C411yescharacteristic
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—21no—
227C22×D47—28no—
228C22×D47—41no—
229C22×D47—41no—
230C22×D47—41no—
231C22×D47—41no—
232C22×D47—82no—
233C22×D47—82no—
234C22×D47—84no—
235C22×D47—84no—
236C22×D47C2×C22⋊C412yes—
237C22×D47C2×M1612yes—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—81no—
247C2×C4×C47—81no—
248C2×C4×C47C2×C22⋊C412yes—
249C2×C4×C47C22×D412yes—
250C22⋊D47—28no—
251C22⋊D47—28no—
252C22⋊D47—24no—
253C22⋊D47—24no—
254C22⋊D47—24no—
255C22⋊D47—24no—
256C2×Dic47—28no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—82no—
268C2×Dic47—82no—
269C2×Dic47C2×C22⋊C412yes—
270C2×Dic47C2×C22⋊C412yes—
271C2×Dic47C2×C22⋊C412yes—
272C2×Dic47C2×M1612yes—
273C4×D47—44no—
274C4×D47—82no—
275C4×D47—82no—
276C4×D47—82no—
277C8×C47—44no—
278C47⋊C8—44no—
279#17367—471nomaximal, Sylow
280C22×C47—21no—
281C22×C47—21no—
282C22×C47—21no—
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—21no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—41no—
295C22×C47—41no—
296C22×C47—41no—
297C22×C47—41no—
298C22×C47—84no—
299C22×C47—84no—
300C22×C47#40712yes—
301C22×C47#40712yes—
302C22×C47C22×C22⋊C411yescharacteristic
303C22×C47C22×M1611yescharacteristic
304C22×C47C22×M1611yescharacteristic
305C2×D47—22no—
306C2×D47—22no—
307C2×D47—22no—
308C2×D47—22no—
309C2×D47—42no—
310C2×D47—42no—
311C2×D47—42no—
312C2×D47—41no—
313C2×D47—41no—
314C2×D47—42no—
315C2×D47—88no—
316C4×C47—22no—
317C4×C47—22no—
318C4×C47—41no—
319C4×C47—41no—
320Dic47—22no—
321Dic47—22no—
322Dic47—22no—
323Dic47—22no—
324Dic47—41no—
325Dic47—41no—
326C2×#518—471no—
327#2973—472no—
328#1466—472no—
329#1465—472no—
330#1464—472no—
331#1462—472no—
332#1459—472no—
333C2×#323—471no—
334C2×#323D4711yescharacteristic, complemented
335C2×C47—21no—
336C2×C47—21no—
337C2×C47—22no—
338C2×C47—22no—
339C2×C47—42no—
340C2×C47—42no—
341C2×C47C2×#40711yescharacteristic
342C2×C47#288311yescharacteristic
343C2×C47#288811yescharacteristic
344D47—42no—
345D47—42no—
346C23×D4—471no—
347#518—944no—
348#518—944no—
349C2×C4×D4—471no—
350C2×C4×D4—471no—
351C2×C4×D4—471no—
352C2×C4×D4—471no—
353C2×C4×D4—471no—
354C2×C4×D4—471no—
355C2×C4×D4—471no—
356C2×C4×D4—471no—
357C2×#117—471no—
358C2×#117—471no—
359C22×C22⋊C4—471no—
360C22×C22⋊C4—471no—
361C22×C22⋊C4—471no—
362C22×C22⋊C4C2×D4711yescharacteristic
363#435—944no—
364#435—944no—
365#432—474no—
366#407—474no—
367C2×C22⋊C8—472no—
368C2×C22⋊C8C2×D4712yescomplemented
369#324—474no—
370#323—472no—
371#323—472no—
372#323—474no—
373#323C2×D4712yes—
374#323C2×D4712yescomplemented
375C47#1736711yescharacteristic, complemented, Sylow
376C25—471no—
377C25C4×D4711yescharacteristic
378C22×D4—471no—
379C22×D4—471no—
380C22×D4—471no—
381C22×D4—471no—
382C22×D4—942no—
383C22×D4—942no—
384C22×D4—944no—
385C22×D4—944no—
386C23×C4—471no—
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—471no—
393C23×C4C4×D4711yescharacteristic
394C23×C4C22×D4711yescharacteristic
395C2×M16—944no—
396C22×C8—22no—
397C22×C8—942no—
398C4×D4—474no—
399C4×D4—474no—
400C4×D4—944no—
401C4×D4—944no—
402C4×D4—944no—
403C4×D4—944no—
404C4×D4—942no—
405C4×D4—942no—
406C4×D4—942no—
407C4×D4—942no—
408C4×D4—942no—
409C4×D4—942no—
410#117—944no—
411#117—944no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—471no—
419C2×C22⋊C4—471no—
420C2×C22⋊C4—471no—
421C2×C22⋊C4—471no—
422C2×C22⋊C4—471no—
423C2×C22⋊C4—471no—
424C2×C22⋊C4—471no—
425C2×C22⋊C4—471no—
426C2×C22⋊C4—944no—
427C2×C22⋊C4—944no—
428C2×C22⋊C4—942no—
429C2×C22⋊C4—942no—
430C2×C22⋊C4C4×D4711yescharacteristic
431C2×C22⋊C4C4×D4711yescharacteristic
432C2×C22⋊C4C22×D4711yescharacteristic
433C2×C22⋊C4C22×D4711yescharacteristic
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—471no—
438C4⋊C8—944no—
439C22⋊C8—22no—
440C22⋊C8—474no—
441C22⋊C8—944no—
442C22⋊C8—942no—
443C22⋊C8C22×D4714yes—
444#97—944no—
445C24—21no—
446C24—21no—
447C24—22no—
448C24—81no—
449C24—81no—
450C24—471no—
451C24—471no—
452C24—942no—
453C24—942no—
454C24—942no—
455C24—3761no—
456C24—3761no—
457C24#1062532414yescomplemented
458C24C2×C4×D4711yescharacteristic
459C24C2×C4×D4711yescharacteristic
460C24C2×C4×D4711yescharacteristic
461C2×D4—472no—
462C2×D4—472no—
463C2×D4—474no—
464C2×D4—942no—
465C2×D4—942no—
466C2×D4—942no—
467C2×D4—942no—
468C2×D4—9416no—
469C2×D4—9416no—
470C2×D4—941no—
471C2×D4—941no—
472C2×D4—942no—
473C2×D4—942no—
474C2×D4—942no—
475C2×D4—942no—
476C2×D4—942no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—474no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—3761no—
497C22×C4—3761no—
498C22×C4—3761no—
499C22×C4—3761no—
500C22×C4—3761no—
501C22×C4—3761no—
502C22×C4C2×C4×D4711yescharacteristic
503C22×C4C2×C4×D4711yescharacteristic
504C22×C4C2×C4×D4711yescharacteristic
505C22×C4D4×D4712yes—
506C22×C4D4×D4712yes—
507C22×C4C23×D4711yescharacteristic
508M16—3764no—
509C2×C8—24no—
510C2×C8—82no—
511C2×C8—944no—
512C2×C8—3762no—
513C4⋊C4—474no—
514C4⋊C4—942no—
515C4⋊C4—942no—
516C4⋊C4—942no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—472no—
522C22⋊C4—472no—
523C22⋊C4—474no—
524C22⋊C4—942no—
525C22⋊C4—942no—
526C22⋊C4—942no—
527C22⋊C4—942no—
528C22⋊C4—941no—
529C22⋊C4—941no—
530C22⋊C4—942no—
531C42—474no—
532C42—942no—
533C42—942no—
534C42—942no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—24no—
540C23—24no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—81no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—84no—
552C23—84no—
553C23—472no—
554C23—472no—
555C23—948no—
556C23—941no—
557C23—941no—
558C23—941no—
559C23—941no—
560C23—3762no—
561C23—3762no—
562C23—3764no—
563C23—3764no—
564C23—1881no—
565C23—1881no—
566C23—1881no—
567C23—1881no—
568C23C22⋊C4×D4711yescharacteristic
569C23C22⋊C4×D4711yescharacteristic
570C23M16×D4712yes—
571C23C2×#1062532412yes—
572C23C2×#1062532412yes—
573C23C2×#1062532412yes—
574C23C22×C4×D4711yescharacteristic
575D4—948no—
576D4—948no—
577D4—944no—
578D4—944no—
579D4—944no—
580D4—944no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—472no—
592C2×C4—472no—
593C2×C4—472no—
594C2×C4—472no—
595C2×C4—948no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—941no—
602C2×C4—3762no—
603C2×C4—3762no—
604C2×C4—3762no—
605C2×C4—3762no—
606C2×C4—3762no—
607C2×C4—1884no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4C22⋊C4×D4712yes—
613C2×C4C2×D4×D4712yes—
614C8—44no—
615C8—1884no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—84no—
635C22—84no—
636C22—942no—
637C22—942no—
638C22—942no—
639C22—942no—
640C22—3768no—
641C22—1882no—
642C22—1882no—
643C22—1882no—
644C22—1882no—
645C22—1881no—
646C22—1881no—
647C22#1568908512yes—
648C22#1568908512yes—
649C22C2×C22⋊C4×D4711yescharacteristic
650C22C2×M16×D4711yescharacteristic
651C22C22×#1062532411yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—942no—
657C4—942no—
658C4—942no—
659C4—942no—
660C4—1881no—
661C4—1881no—
662C2—21no—
663C2—21no—
664C2—22no—
665C2—22no—
666C2—42no—
667C2—42no—
668C2—1882no—
669C2—1882no—
670C2C2×#1568908511yescharacteristic
671C2#2025184611yescharacteristic
672C2#2025186411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#323C2
socleC22×C47C22×C22⋊C4
prime radicalC23×C22⋊D47C4
cosocleC22×C4×C47C24
hypercenterC2×#323D47
hypocenterC47#17367
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #20252414C2semidirect
3. #20241727C2semidirect
4. #20241723C2semidirect
5. #20241719C2semidirect
6. #20241713C2semidirect
7. #20241704C2semidirect
8. C2×#15688847C2semidirect
9. C2×C47×#323C2semidirect
28. #15689177C22semidirect
29. #15689085C22semidirect
30. C2×C22⋊(C47⋊C8)C22semidirect
31. #15688848C22semidirect
33. #15688847C22semidirect
34. #15688846C22semidirect
36. C2×C22⋊C8×C47C22semidirect
38. C47×#323C22semidirect
113. C24×C47M16semidirect
149. C2×C22⋊D47M16semidirect
334. C2×#323D47semidirect
368. C2×C22⋊C8C2×D47semidirect
374. #323C2×D47semidirect
375. C47#17367semidirect
457. C24#10625324semidirect

Maximal subgroups

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

subgroupindex
C2×#156895962
#202524142
#202417272
#202417232
#202417192
#202417132
#202417042
C2×#156888472
C2×C47×#3232
#1736747