Finite Group Databasebeta

← #26182189

Subgroups — #26182189

Order 12032.

1210 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689602C211yesmaximal, characteristic
2#20252417C211yesmaximal, characteristic, complemented
3#20241731C212yesmaximal, complemented
4#20241724C212yesmaximal, complemented
5#20241720C212yesmaximal, complemented
6#20241717C212yesmaximal, complemented
7#20241704C212yesmaximal, complemented
8C2×#15688848C212yesmaximal, complemented
9C47×#2973C211yesmaximal, characteristic, complemented
10C22×#10625352C411yescharacteristic
11#15689602—24no—
12#15689602—24no—
13C2×C4×Dic94C2211yescharacteristic
14C2×#13093989C411yescharacteristic
15C2×D4×Dic47C411yescharacteristic
16C2×C4×C22⋊D47C2211yescharacteristic
17C2×C4×C22⋊D47C2211yescharacteristic
18C2×(C22⋊D47)⋊C4C411yescharacteristic
19C2×(C22⋊D47)⋊C4C411yescharacteristic
20C2×#13093981C2211yescharacteristic
21C2×#13093980C411yescharacteristic
22C2×#13093980C411yescharacteristic
23C22×Dic47⋊C4C2211yescharacteristic
24C22×Dic47⋊C4C2211yescharacteristic
25C2×C4⋊C4×D47C411yescharacteristic
26#15689195—24no—
27#15689180C2212yes—
28#15689087C2212yes—
29C2×C22⋊(C47⋊C8)C2212yes—
30#15688848C2214yes—
31#15688848C2214yes—
32#15688848C2214yes—
33#15688846C2214yes—
34C2×C4×D4×C47C2211yescharacteristic
35C47×#435—24no—
36C47×#432C2212yes—
37C47×#407C2212yes—
38C2×C22⋊C8×C47C2212yes—
39C2×#10625352—24no—
40C2×#10625352—24no—
41C2×#10625352—24no—
42C22×Dic94C2×C411yescharacteristic
43C22×C22⋊D47C2×C411yescharacteristic
44C22×C22⋊D47C2×C411yescharacteristic
45C23×Dic47C2×C411yescharacteristic
46C23×Dic47C2×C411yescharacteristic
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C2×#10625329—22no—
53C22×C47⋊C8—22no—
54C4×Dic94—22no—
55C4×Dic94—22no—
56#13093989—24no—
57D4×Dic47—24no—
58C4×C22⋊D47—22no—
59C4×C22⋊D47—22no—
60C4×C22⋊D47—22no—
61C4×C22⋊D47—22no—
62(C22⋊D47)⋊C4—24no—
63(C22⋊D47)⋊C4—24no—
64#13093981—22no—
65#13093981—22no—
66#13093980—24no—
67#13093980—24no—
68C2×C4⋊Dic47C2×C411yescharacteristic
69C2×Dic47⋊C4—22no—
70C2×Dic47⋊C4—22no—
71C2×Dic47⋊C4—22no—
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4C2×C411yescharacteristic
74C2×Dic47⋊C4C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×Dic47⋊C4C2311yescharacteristic
79C2×Dic47⋊C4C2311yescharacteristic
80C2×Dic47⋊C4C2311yescharacteristic
81C4⋊C4×D47—24no—
82C2×C22⋊Dic47C2×C411yescharacteristic
83C2×C22⋊Dic47C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2×C411yescharacteristic
85C2×(C2×D47)⋊C4C2311yescharacteristic
86C2×C4×Dic47C2×C411yescharacteristic
87C2×C4×Dic47C2×C411yescharacteristic
88C2×C4×Dic47C2×C411yescharacteristic
89C2×C4×Dic47C2×C411yescharacteristic
90C2×C4×Dic47C2×C411yescharacteristic
91C4⋊(C47⋊C8)—22no—
92C22⋊(C47⋊C8)—22no—
93C22⋊(C47⋊C8)—22no—
94C22⋊(C47⋊C8)C2314yes—
95#13093935—22no—
96C22×D4×C47C2×C411yescharacteristic
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C2×M16×C47—22no—
100C22×C8×C47—22no—
101C4×D4×C47D414yes—
102C4×D4×C47D414yes—
103C2×C4⋊C4×C47C2×C411yescharacteristic
104C2×C22⋊C4×C47C2×C411yescharacteristic
105C2×C22⋊C4×C47C2×C411yescharacteristic
106C2×C42×C47C2311yescharacteristic
107C4⋊C8×C47—22no—
108C22⋊C8×C47—22no—
109C22⋊C8×C47—22no—
110C22⋊C8×C47C2314yes—
111#97×C47—22no—
112C24×C47C22×C411yescharacteristic
113C24×C47C22×C411yescharacteristic
114C23×D47C22×C411yescharacteristic
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47C22⋊C414yes—
118C22×C4×C47—81no—
119C22×C4×C47—81no—
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×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—22no—
130C2×C22⋊D47—22no—
131C2×C22⋊D47—22no—
132C2×C22⋊D47—22no—
133C2×C22⋊D47—22no—
134C2×C22⋊D47M1614yescomplemented
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—81no—
143C22×Dic47—81no—
144C22×Dic47M1614yescomplemented
145C22×Dic47C22×C411yescharacteristic
146C22×Dic47C22×C411yescharacteristic
147C22×Dic47C22×C411yescharacteristic
148C22×Dic47C22×C411yescharacteristic
149C22×Dic47C22×C411yescharacteristic
150C22×Dic47C22×C411yescharacteristic
151C22×Dic47C22×C411yescharacteristic
152#10625352—232no—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—81no—
159C2×C4×D47—81no—
160C2×Dic94—22no—
161C2×Dic94—22no—
162C2×Dic94—22no—
163M16×C47—82no—
164C2×C8×C47—24no—
165C2×C8×C47—82no—
166C4⋊C4×C47C22⋊C414yes—
167C22⋊C4×C47C22⋊C414yes—
168C22⋊C4×C47C22⋊C414yes—
169C42×C47C2×D412yes—
170C42×C47C2×D412yes—
171C22⋊Dic47—22no—
172C22⋊Dic47—22no—
173(C2×D47)⋊C4—22no—
174(C2×D47)⋊C4—21no—
175(C2×D47)⋊C4—21no—
176C4⋊Dic47—22no—
177Dic47⋊C4—22no—
178Dic47⋊C4—22no—
179Dic47⋊C4—22no—
180Dic47⋊C4—22no—
181Dic47⋊C4—22no—
182Dic47⋊C4—21no—
183Dic47⋊C4—21no—
184Dic47⋊C4—21no—
185Dic47⋊C4—21no—
186Dic47⋊C4—21no—
187Dic47⋊C4—21no—
188C4×Dic47—22no—
189C4×Dic47—22no—
190C4×Dic47—22no—
191C4×Dic47—22no—
192C4×Dic47—22no—
193#10625329—82no—
194C2×C47⋊C8—24no—
195C2×C47⋊C8—82no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47C2×C22⋊C412yes—
203C23×C47C2×C22⋊C412yes—
204C23×C47C2×M1612yes—
205C23×C47C2×M1612yes—
206C23×C47C23×C411yescharacteristic
207C22×D47—21no—
208C22×D47—21no—
209C22×D47—41no—
210C22×D47—41no—
211C22×D47C2×M1612yes—
212D4×C47—28no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
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×C47C2×C22⋊C412yes—
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47C2×C22⋊C412yes—
226C2×C4×C47C2×C22⋊C412yes—
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C22⋊D47—28no—
230C22⋊D47—28no—
231C2×Dic47—28no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—28no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—82no—
248C2×Dic47—82no—
249C2×Dic47C2×M1612yes—
250C2×Dic47C2×M1612yes—
251C2×Dic47C2×M1612yes—
252C4×D47—28no—
253C4×D47—82no—
254C4×D47—82no—
255Dic94—28no—
256C8×C47—44no—
257C47⋊C8—44no—
258#17368—471nomaximal, Sylow
259C22×C47—21no—
260C22×C47—22no—
261C22×C47—21no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47#40714yes—
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47C22×M1611yescharacteristic
273C22×C47C22×M1611yescharacteristic
274C2×D47—22no—
275C2×D47—42no—
276C2×D47—41no—
277C2×D47—41no—
278C2×D47—42no—
279C4×C47—22no—
280C4×C47—21no—
281C4×C47—21no—
282C4×C47—22no—
283C4×C47—22no—
284C4×C47—22no—
285Dic47—22no—
286Dic47—22no—
287Dic47—22no—
288Dic47—41no—
289Dic47—41no—
290Dic47—41no—
291Dic47—41no—
292C2×#520—471no—
293#2973—471no—
294#2973D4711yescharacteristic, complemented
295#1467—472no—
296#1465—472no—
297#1464—472no—
298#1463—472no—
299#1459—472no—
300C2×#324—472no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—24no—
304C2×C47—42no—
305C2×C47C2×#40711yescharacteristic
306C2×C47#288311yescharacteristic
307C2×C47#288911yescharacteristic
308D47—42no—
309C22×#41—471no—
310#520—944no—
311#520—944no—
312C2×C4×Q8—471no—
313C2×C4×Q8—471no—
314C2×C4×D4—471no—
315C2×C4×D4—471no—
316C2×C4×D4—471no—
317C2×C4×D4—471no—
318C2×C4×D4—471no—
319C2×C4×D4C2×D4711yescharacteristic
320C2×#117—471no—
321C2×#117—471no—
322C2×#117—471no—
323C22×C4⋊C4—471no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4—471no—
326#435—24no—
327#435—944no—
328#432—472no—
329#432C2×D4712yes—
330#407—472no—
331#407C2×D4712yes—
332C2×C22⋊C8—472no—
333C2×C22⋊C8C2×D4712yes—
334#324—474no—
335#324—474no—
336#324—474no—
337#323—474no—
338C47#1736811yescharacteristic, complemented, Sylow
339C2×#41—944no—
340C2×#41—944no—
341C2×#41—944no—
342C22×Q8—471no—
343C22×D4—471no—
344C22×D4—471no—
345C22×D4C4×D4711yescharacteristic
346C23×C4—471no—
347C23×C4—471no—
348C23×C4—471no—
349C23×C4—471no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4C22×D4711yescharacteristic
354C23×C4C22×D4711yescharacteristic
355C2×M16—22no—
356C2×M16—942no—
357C22×C8—22no—
358C22×C8—942no—
359C4×Q8—944no—
360C4×Q8—942no—
361C4×Q8—942no—
362C4×D4—944no—
363C4×D4—942no—
364C4×D4—942no—
365C4×D4—942no—
366C4×D4—942no—
367C4×D4—944no—
368C4×D4—944no—
369C4×D4D18814yes—
370C4×D4C22⋊D4714yes—
371#117—944no—
372#117—944no—
373#117—942no—
374#117—942no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4C4×D4711yescharacteristic
390C2×C22⋊C4—471no—
391C2×C22⋊C4—471no—
392C2×C22⋊C4—471no—
393C2×C22⋊C4—471no—
394C2×C22⋊C4C4×D4711yescharacteristic
395C2×C22⋊C4C4×D4711yescharacteristic
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42C22×D4711yescharacteristic
402C4⋊C8—22no—
403C4⋊C8—942no—
404C22⋊C8—22no—
405C22⋊C8—22no—
406C22⋊C8—474no—
407C22⋊C8—942no—
408C22⋊C8—942no—
409C22⋊C8C22×D4714yes—
410#97—22no—
411#97—942no—
412C24—471no—
413C24C2×C4×D4711yescharacteristic
414C24C2×C4×D4711yescharacteristic
415#41—9432no—
416C2×Q8—942no—
417C2×Q8—942no—
418C2×Q8—942no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—474no—
422C2×D4—942no—
423C2×D4—942no—
424C2×D4—942no—
425C2×D4—942no—
426C2×D4—942no—
427C2×D4(C2×D47)⋊C414yes—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—474no—
431C22×C4—471no—
432C22×C4—471no—
433C22×C4—471no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—3761no—
451C22×C4—3761no—
452C22×C4—3761no—
453C22×C4—3761no—
454C22×C4C2×C4×D4711yescharacteristic
455C22×C4C2×C4×D4711yescharacteristic
456C22×C4C2×C4×D4711yescharacteristic
457C22×C4C2×C4×D4711yescharacteristic
458C22×C4C2×D18812yes—
459C22×C4C2×D18812yes—
460C22×C4C2×C22⋊D4712yes—
461C22×C4C2×C22⋊D4712yes—
462C22×C4C23×D4711yescharacteristic
463M16—82no—
464M16—3762no—
465C2×C8—24no—
466C2×C8—82no—
467C2×C8—944no—
468C2×C8—3762no—
469C4⋊C4—942no—
470C4⋊C4—942no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4—942no—
474C4⋊C4—942no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4(C2×D47)⋊C414yes—
482C22⋊C4—942no—
483C22⋊C4—942no—
484C22⋊C4—942no—
485C22⋊C4—941no—
486C22⋊C4—941no—
487C22⋊C4(C2×D47)⋊C414yes—
488C22⋊C4(C2×D47)⋊C414yes—
489C42—942no—
490C42—942no—
491C42—942no—
492C42—942no—
493C42—942no—
494C42C2×D18812yes—
495C42C2×C22⋊D4712yes—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—472no—
503C23—941no—
504C23—941no—
505C23—1881no—
506C23—1881no—
507C23C2×(C2×D47)⋊C412yes—
508C23C2×(C2×D47)⋊C412yes—
509C23M16×D4712yes—
510C23M16×D4712yes—
511C23C22×C4×D4711yescharacteristic
512Q8—948no—
513D4—28no—
514D4—948no—
515D4—948no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—472no—
527C2×C4—472no—
528C2×C4—472no—
529C2×C4—948no—
530C2×C4—948no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—948no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—3762no—
543C2×C4—3762no—
544C2×C4—3762no—
545C2×C4—3762no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4C2×(C2×D47)⋊C412yes—
551C2×C4C2×(C2×D47)⋊C412yes—
552C2×C4C2×(C2×D47)⋊C412yes—
553C2×C4C2×(C2×D47)⋊C412yes—
554C2×C4C22×C22⋊D4711yescharacteristic
555C2×C4C22×D18811yescharacteristic
556C8—44no—
557C8—1884no—
558C22—21no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—942no—
570C22—1882no—
571C22—1882no—
572C22—1881no—
573C22—1881no—
574C22#1568908614yes—
575C22C22×(C2×D47)⋊C411yescharacteristic
576C22C2×M16×D4711yescharacteristic
577C22C2×M16×D4711yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—942no—
585C4—942no—
586C4—942no—
587C4—1881no—
588C4—1881no—
589C4—1881no—
590C4—1881no—
591C2—21no—
592C2—21no—
593C2—24no—
594C2—42no—
595C2—1882no—
596C2C2×#1568908611yescharacteristic
597C2#2025184811yescharacteristic
598C2#2025187011yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2973C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625352C4
cosocleC22×C4×C47C24
hypercenter#2973D47
hypocenterC47#17368
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20252417C2semidirect
3. #20241731C2semidirect
4. #20241724C2semidirect
5. #20241720C2semidirect
6. #20241717C2semidirect
7. #20241704C2semidirect
8. C2×#15688848C2semidirect
9. C47×#2973C2semidirect
134. C2×C22⋊D47M16semidirect
144. C22×Dic47M16semidirect
294. #2973D47semidirect
338. C47#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#156896022
#202524172
#202417312
#202417242
#202417202
#202417172
#202417042
C2×#156888482
C47×#29732
#1736847