Finite Group Databasebeta

← #26182187

Subgroups — #26182187

Order 12032.

1210 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689604C211yesmaximal, characteristic
2#20252414C212yesmaximal, complemented
3#20241730C212yesmaximal, complemented
4#20241725C212yesmaximal, complemented
5#20241721C212yesmaximal, complemented
6#20241716C212yesmaximal, complemented
7#20241704C212yesmaximal, complemented
8C2×#15688849C211yesmaximal, characteristic, complemented
9C2×C47×#324C211yesmaximal, characteristic, complemented
10C22×#10625350C411yescharacteristic
11#15689604—24no—
12#15689604—24no—
13C2×#13093989C411yescharacteristic
14C2×#13093989C2211yescharacteristic
15C2×#13093986C411yescharacteristic
16C2×#13093986C2211yescharacteristic
17C2×C4×C22⋊D47C411yescharacteristic
18C2×C4×C22⋊D47C411yescharacteristic
19C2×C4×C22⋊D47C2211yescharacteristic
20C2×C4×C22⋊D47C2211yescharacteristic
21C2×#13093983C411yescharacteristic
22C2×#13093982C411yescharacteristic
23C2×#13093982C2211yescharacteristic
24C2×C4⋊C4×D47C411yescharacteristic
25C2×C4⋊C4×D47C2211yescharacteristic
26#15689194—24no—
27#15689191—24no—
28#15689177C2214yes—
29#15689085C2214yes—
30C2×C22⋊(C47⋊C8)C2212yes—
31#15688849C2212yes—
32#15688849C2212yes—
33#15688848C2214yes—
34#15688846C2214yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C2×C22⋊C8×C47C2212yes—
37C47×#324C2212yes—
38C47×#324C2212yes—
39C2×#10625350—24no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—24no—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47C2×C411yescharacteristic
46C22×C22⋊D47C2×C411yescharacteristic
47C22×C4×D47C2×C411yescharacteristic
48C22×C4×D47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2311yescharacteristic
53C2×#10625324—24no—
54C22×C47⋊C8—22no—
55#13093989—24no—
56#13093989—24no—
57#13093986—24no—
58#13093986—24no—
59C4×C22⋊D47—22no—
60C4×C22⋊D47—22no—
61C4×C22⋊D47—22no—
62C4×C22⋊D47—22no—
63C4×C22⋊D47—22no—
64C4×C22⋊D47—22no—
65C4×C22⋊D47D414yes—
66C4×C22⋊D47D414yes—
67#13093983—22no—
68#13093983—22no—
69#13093982—24no—
70#13093982—24no—
71C2×C4⋊Dic47C2×C411yescharacteristic
72C2×C4⋊Dic47C2×C411yescharacteristic
73C2×Dic47⋊C4C2×C411yescharacteristic
74C2×Dic47⋊C4C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2311yescharacteristic
77C4⋊C4×D47—24no—
78C4⋊C4×D47—24no—
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
85C2×C4×Dic47C2×C411yescharacteristic
86C2×C4×Dic47C2×C411yescharacteristic
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—
96C23×C4×C47C2×C411yescharacteristic
97C23×C4×C47C2×C411yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47—22no—
100C2×C4⋊C4×C47—22no—
101C2×C4⋊C4×C47—22no—
102C2×C4⋊C4×C47C2×C411yescharacteristic
103C2×C4⋊C4×C47C2×C411yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C22⋊C8×C47—22no—
107C22⋊C8×C47C2314yes—
108C24×C47C22×C411yescharacteristic
109C23×D47C22×C411yescharacteristic
110C23×D47C22×C411yescharacteristic
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—81no—
117C22×C4×C47—81no—
118C22×C4×C47M1614yescomplemented
119C22×C4×C47C22×C411yescharacteristic
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C22×C411yescharacteristic
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2411yescharacteristic
127C2×C22⋊D47—21no—
128C2×C22⋊D47—21no—
129C2×C22⋊D47—22no—
130C2×C22⋊D47—22no—
131C2×C22⋊D47—22no—
132C2×C22⋊D47C22⋊C412yes—
133C2×C22⋊D47C22⋊C412yes—
134C2×C22⋊D47M1614yescomplemented
135C22×Dic47C22×C411yescharacteristic
136C22×Dic47C22×C411yescharacteristic
137C22×Dic47C22×C411yescharacteristic
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141#10625350—216no—
142#10625350—216no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—81no—
156C2×C4×D47—81no—
157C2×C4×D47—81no—
158C2×C4×D47—81no—
159C2×C4×D47C2×D414yes—
160C2×Dic94—22no—
161C2×Dic94—22no—
162C2×Dic94—22no—
163C2×C8×C47—24no—
164C2×C8×C47—82no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C22⋊Dic47—21no—
170C22⋊Dic47—21no—
171C22⋊Dic47C22⋊C412yes—
172C22⋊Dic47C22⋊C412yes—
173(C2×D47)⋊C4—22no—
174(C2×D47)⋊C4—22no—
175(C2×D47)⋊C4—22no—
176(C2×D47)⋊C4C22⋊C414yes—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179Dic47⋊C4—22no—
180Dic47⋊C4—22no—
181Dic47⋊C4—22no—
182Dic47⋊C4C22⋊C414yes—
183C4×Dic47—22no—
184C4×Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C4×Dic47C2×D414yes—
189C2×C47⋊C8—24no—
190C2×C47⋊C8—82no—
191#10625324—84no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47C2×C22⋊C411yescharacteristic
195C23×C47C2×C22⋊C411yescharacteristic
196C23×C47C2×M1612yes—
197C23×C47C2×M1612yes—
198C23×C47C23×C411yescharacteristic
199C22×D47—21no—
200C22×D47—21no—
201C22×D47—21no—
202C22×D47—21no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47C2×C22⋊C412yes—
208C22×D47C2×M1612yes—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—81no—
224C2×C4×C47—81no—
225C2×C4×C47C2×C22⋊C412yes—
226C2×C4×C47C2×M1612yes—
227C2×C4×C47C2×M1612yes—
228C2×C4×C47C22×D412yes—
229C22⋊D47—24no—
230C22⋊D47—24no—
231C22⋊D47—24no—
232C22⋊D47—24no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47C2×C22⋊C412yes—
244C2×Dic47C2×C22⋊C412yes—
245C2×Dic47C2×C22⋊C412yes—
246C2×Dic47C2×M1612yes—
247D188—28no—
248C4×D47—28no—
249C4×D47—28no—
250C4×D47—44no—
251C4×D47—82no—
252C4×D47—82no—
253C4×D47—82no—
254Dic94—28no—
255C8×C47—44no—
256C47⋊C8—44no—
257#17368—471nomaximal, Sylow
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—21no—
261C22×C47—21no—
262C22×C47—21no—
263C22×C47—21no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47#40712yes—
267C22×C47#40712yes—
268C22×C47C22×C22⋊C411yescharacteristic
269C22×C47C22×M1611yescharacteristic
270C22×C47C22×M1611yescharacteristic
271C2×D47—22no—
272C2×D47—22no—
273C2×D47—22no—
274C2×D47—22no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—41no—
279C2×D47—41no—
280C2×D47—42no—
281C4×C47—22no—
282C4×C47—22no—
283C4×C47—22no—
284C4×C47—22no—
285C4×C47—41no—
286C4×C47—41no—
287Dic47—22no—
288Dic47—22no—
289Dic47—22no—
290Dic47—22no—
291Dic47—41no—
292Dic47—41no—
293C2×#520—471no—
294#2973—472no—
295#1467—472no—
296#1465—472no—
297#1464—472no—
298#1463—472no—
299#1459—472no—
300C2×#324—471no—
301C2×#324D4711yescharacteristic, complemented
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—22no—
305C2×C47—22no—
306C2×C47C2×#40711yescharacteristic
307C2×C47#288311yescharacteristic
308C2×C47#288911yescharacteristic
309D47—42no—
310D47—42no—
311C22×#41—471no—
312#520—944no—
313#520—944no—
314C2×C4×Q8—471no—
315C2×C4×Q8—471no—
316C2×C4×D4—471no—
317C2×C4×D4—471no—
318C2×C4×D4—471no—
319C2×C4×D4—471no—
320C2×C4×D4—471no—
321C2×C4×D4—471no—
322C2×#117—471no—
323C2×#117—471no—
324C2×#117—471no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4—471no—
327C22×C4⋊C4C2×D4711yescharacteristic
328#435—944no—
329#435—944no—
330#432—474no—
331#407—474no—
332C2×C22⋊C8—472no—
333C2×C22⋊C8C2×D4712yes—
334#324—472no—
335#324—472no—
336#324—474no—
337#324C2×D4712yes—
338#324C2×D4712yes—
339#323—474no—
340C47#1736811yescharacteristic, complemented, Sylow
341C2×#41—942no—
342C2×#41—942no—
343C2×#41—944no—
344C2×#41—944no—
345C22×Q8—471no—
346C22×D4—471no—
347C22×D4—471no—
348C22×D4—471no—
349C23×C4—471no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4C4×D4711yescharacteristic
356C23×C4C4×D4711yescharacteristic
357C23×C4C22×D4711yescharacteristic
358C2×M16—944no—
359C22×C8—22no—
360C22×C8—942no—
361C4×Q8—944no—
362C4×Q8—944no—
363C4×D4—474no—
364C4×D4—474no—
365C4×D4—944no—
366C4×D4—942no—
367C4×D4—942no—
368C4×D4—944no—
369C4×D4—942no—
370C4×D4—942no—
371C4×D4—942no—
372C4×D4—942no—
373#117—944no—
374#117—944no—
375#117—942no—
376#117—942no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4C4×D4711yescharacteristic
388C2×C4⋊C4C4×D4711yescharacteristic
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C22⋊C4—471no—
392C2×C22⋊C4—471no—
393C2×C22⋊C4—471no—
394C2×C22⋊C4—471no—
395C2×C22⋊C4—471no—
396C2×C22⋊C4—471no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—471no—
403C4⋊C8—944no—
404C22⋊C8—22no—
405C22⋊C8—474no—
406C22⋊C8—944no—
407C22⋊C8—942no—
408C22⋊C8C22×D4714yes—
409#97—944no—
410C24—471no—
411C24—471no—
412C24C2×C4×D4711yescharacteristic
413#41—9416no—
414#41—9416no—
415C2×Q8—942no—
416C2×Q8—942no—
417C2×Q8—942no—
418C2×D4—472no—
419C2×D4—472no—
420C2×D4—474no—
421C2×D4—942no—
422C2×D4—942no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—942no—
426C2×D4—942no—
427C2×D4—942no—
428C2×D4—942no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—471no—
440C22×C4—474no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—3761no—
453C22×C4—3761no—
454C22×C4—3761no—
455C22×C4—3761no—
456C22×C4#1062532414yescomplemented
457C22×C4C2×C4×D4711yescharacteristic
458C22×C4C2×C4×D4711yescharacteristic
459C22×C4C2×C4×D4711yescharacteristic
460C22×C4C2×C4×D4711yescharacteristic
461C22×C4C2×C4×D4711yescharacteristic
462C22×C4D4×D4712yes—
463C22×C4D4×D4712yes—
464C22×C4C23×D4711yescharacteristic
465M16—3764no—
466C2×C8—24no—
467C2×C8—82no—
468C2×C8—944no—
469C2×C8—3762no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—474no—
475C4⋊C4—942no—
476C4⋊C4—942no—
477C4⋊C4—942no—
478C4⋊C4—942no—
479C4⋊C4—942no—
480C22⋊C4—472no—
481C22⋊C4—472no—
482C22⋊C4—474no—
483C22⋊C4—942no—
484C22⋊C4—942no—
485C22⋊C4—941no—
486C22⋊C4—941no—
487C22⋊C4—942no—
488C42—474no—
489C42—942no—
490C42—942no—
491C42—942no—
492C42—942no—
493C42—942no—
494C23—41no—
495C23—41no—
496C23—472no—
497C23—472no—
498C23—941no—
499C23—941no—
500C23—941no—
501C23—941no—
502C23—1881no—
503C23—1881no—
504C23—1881no—
505C23—1881no—
506C23C22⋊C4×D4711yescharacteristic
507C23C22⋊C4×D4711yescharacteristic
508C23M16×D4712yes—
509C23C2×#1062532412yes—
510C23C22×C4×D4711yescharacteristic
511Q8—948no—
512D4—948no—
513D4—944no—
514D4—944no—
515D4—944no—
516D4—944no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—948no—
538C2×C4—948no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—3762no—
546C2×C4—3762no—
547C2×C4—3762no—
548C2×C4—1884no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4C22⋊C4×D4712yes—
554C2×C4C2×#1062532412yes—
555C2×C4C2×#1062532412yes—
556C2×C4C2×D4×D4712yes—
557C8—44no—
558C8—1884no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—942no—
568C22—942no—
569C22—942no—
570C22—942no—
571C22—1882no—
572C22—1882no—
573C22—1882no—
574C22—1882no—
575C22—1881no—
576C22—1881no—
577C22#1568908512yes—
578C22#1568908512yes—
579C22C2×C22⋊C4×D4711yescharacteristic
580C22C2×M16×D4711yescharacteristic
581C22C22×#1062532411yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—942no—
589C4—942no—
590C4—942no—
591C4—942no—
592C4—1881no—
593C4—1881no—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—1882no—
599C2—1882no—
600C2C2×#1568908511yescharacteristic
601C2#2025184611yescharacteristic
602C2#2025186811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#324C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625350C4
cosocleC22×C4×C47C24
hypercenterC2×#324D47
hypocenterC47#17368
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20252414C2semidirect
3. #20241730C2semidirect
4. #20241725C2semidirect
5. #20241721C2semidirect
6. #20241716C2semidirect
7. #20241704C2semidirect
8. C2×#15688849C2semidirect
9. C2×C47×#324C2semidirect
118. C22×C4×C47M16semidirect
134. C2×C22⋊D47M16semidirect
301. C2×#324D47semidirect
340. C47#17368semidirect
456. C22×C4#10625324semidirect

Maximal subgroups

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

subgroupindex
C2×#156896042
#202524142
#202417302
#202417252
#202417212
#202417162
#202417042
C2×#156888492
C2×C47×#3242
#1736847