Finite Group Databasebeta

← #26183853

Subgroups — #26183853

Order 12032.

882 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252477C211yesmaximal, characteristic, complemented
2#20246039C211yesmaximal, characteristic, complemented
3#20241965C212yesmaximal
4#20241952C212yesmaximal, complemented
5#20241939C212yesmaximal, complemented
6#20241926C212yesmaximal
7#20241913C211yesmaximal, characteristic
8#20241900C211yesmaximal, characteristic, complemented
9#20241887C211yesmaximal, characteristic, complemented
10#20241874C211yesmaximal, characteristic
11C47×#3523C211yesmaximal, characteristic
12#15689254—21no—
13#15689254—22no—
14Q8×C47⋊C8—21no—
15#15689180C2212yes—
16#15689180C2211yescharacteristic
17#15689176C2212yes—
18#15689176C2211yescharacteristic
19#15689030—21no—
20#15689022—21no—
21#15689009—22no—
22#15688991C2212yes—
23#15688989C2212yes—
24C4×C4⋊Dic47C2211yescharacteristic
25#15688969C2211yescharacteristic
26#15688860C2212yes—
27#15688860C2212yes—
28#15688860C2211yescharacteristic
29#15688860C2212yes—
30#15688860C2212yes—
31#15688860C2211yescharacteristic
32#15688860C2212yes—
33#15688857C2211yescharacteristic
34#15688857C2212yes—
35#15688857C2211yescharacteristic
36C47×#557C411yescharacteristic, complemented
37C47×#557C412yescomplemented
38C47×#557C411yescharacteristic, complemented
39C47×#556C412yescomplemented
40C47×#556C411yescharacteristic, complemented
41C47×#552—22no—
42C47×#552—21no—
43D4×Q8×C47—21no—
44D4×Q8×C47C411yescharacteristic, complemented
45C47×#541—21no—
46C47×#541—22no—
47C47×#533C2212yes—
48C47×#533C2211yescharacteristic
49C47×#529C2212yes—
50C47×#529C2211yescharacteristic
51C47×#519—21no—
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—22no—
55C2×C4×Dic47—21no—
56C2×C4×Dic47—21no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—22no—
59C4⋊(C47⋊C8)—22no—
60C4⋊(C47⋊C8)—21no—
61C22⋊(C47⋊C8)C2311yescharacteristic
62C22⋊(C47⋊C8)C2311yescharacteristic
63C22⋊(C47⋊C8)C2312yes—
64#13093935—22no—
65#13093935—21no—
66C4×C47⋊C8—21no—
67C4×C47⋊C8—22no—
68C47⋊#95—21no—
69C47⋊#95—22no—
70C47⋊#95—22no—
71C47⋊#95—22no—
72C47⋊#95C2311yescharacteristic
73C47⋊#95C2311yescharacteristic
74C47⋊#95C2312yes—
75C22×Q8×C47—21no—
76C22×Q8×C47D411yescharacteristic
77C4⋊Q8×C47—21no—
78C4⋊Q8×C47—22no—
79C4⋊Q8×C47C2×C412yes—
80C4⋊Q8×C47C2×C411yescharacteristic
81C4⋊Q8×C47C2×C412yes—
82C4⋊Q8×C47C2×C411yescharacteristic
83#126×C47—22no—
84#126×C47—22no—
85#126×C47—22no—
86#126×C47—22no—
87#126×C47—21no—
88#125×C47—22no—
89#125×C47—22no—
90#125×C47—22no—
91#125×C47—22no—
92#125×C47—21no—
93#125×C47—22no—
94#125×C47—21no—
95#125×C47C2×C412yes—
96#125×C47C2×C411yescharacteristic
97#124×C47C2×C412yes—
98#124×C47C2×C411yescharacteristic
99#123×C47—21no—
100#123×C47—22no—
101C22⋊Q8×C47—21no—
102C22⋊Q8×C47—22no—
103C22⋊Q8×C47—21no—
104C22⋊Q8×C47—22no—
105C22⋊Q8×C47—22no—
106C22⋊Q8×C47—21no—
107C22⋊Q8×C47C2312yes—
108C22⋊Q8×C47C2311yescharacteristic
109#121×C47C2311yescharacteristic
110C4×Q8×C47—21no—
111C4×Q8×C47—22no—
112C4×Q8×C47—21no—
113C4×Q8×C47—21no—
114C4×Q8×C47—22no—
115C4×Q8×C47C2×C412yes—
116C4×Q8×C47C2×C411yescharacteristic
117C4×Q8×C47C2×C412yes—
118C4×Q8×C47C2×C412yes—
119C4×Q8×C47C2×C412yes—
120C4×Q8×C47C2×C411yescharacteristic
121C4×Q8×C47C2×C411yescharacteristic
122C4×Q8×C47C2×C412yes—
123C4×Q8×C47D411yescharacteristic
124C4×Q8×C47D411yescharacteristic
125C4×Q8×C47D412yes—
126C4×D4×C47—21no—
127C4×D4×C47—22no—
128C4×D4×C47C2×C412yes—
129C4×D4×C47C2×C411yescharacteristic
130#117×C47—21no—
131#117×C47—22no—
132#117×C47—22no—
133#117×C47—21no—
134#117×C47—21no—
135#117×C47—22no—
136C2×C4⋊C4×C47D412yes—
137C2×C4⋊C4×C47D411yescharacteristic
138C2×C42×C47C2311yescharacteristic
139C2×C42×C47C2312yes—
140C2×Q8×C47—22no—
141C2×Q8×C47—21no—
142C2×Q8×C47—42no—
143C2×Q8×C47—44no—
144C2×Q8×C47C22⋊C411yescharacteristic
145C2×Q8×C47C22⋊C411yescharacteristic
146C2×Q8×C47C22⋊C412yes—
147C2×Q8×C47C22×C412yes—
148C2×Q8×C47C22×C411yescharacteristic
149C2×D4×C47—21no—
150C2×D4×C47C22×C411yescharacteristic
151C22×C4×C47—22no—
152C22×C4×C47—21no—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2411yescharacteristic
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47C22⋊C412yes—
179C4⋊C4×C47C22⋊C411yescharacteristic
180C4⋊C4×C47C22⋊C412yes—
181C4⋊C4×C47C22⋊C411yescharacteristic
182C4⋊C4×C47C22⋊C412yes—
183C4⋊C4×C47C22⋊C412yes—
184C4⋊C4×C47C22⋊C412yes—
185C4⋊C4×C47C22×C412yes—
186C4⋊C4×C47C22×C411yescharacteristic
187C4⋊C4×C47C22×C411yescharacteristic
188C22⋊C4×C47—22no—
189C22⋊C4×C47—21no—
190C22⋊C4×C47—21no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—21no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—21no—
196C22⋊C4×C47—21no—
197C42×C47—22no—
198C42×C47—22no—
199C42×C47—22no—
200C42×C47—22no—
201C42×C47—21no—
202C42×C47C22×C412yes—
203C42×C47C22×C411yescharacteristic
204C42×C47C22×C411yescharacteristic
205C42×C47C22×C412yes—
206C42×C47C2×D411yescharacteristic
207C42×C47C2×D412yes—
208C42×C47C2×D412yes—
209C42×C47C2×D411yescharacteristic
210C4⋊Dic47—42no—
211C4⋊Dic47—82no—
212C4×Dic47—42no—
213C4×Dic47—81no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—22no—
217C23×C47—21no—
218C23×C47C22×D411yescharacteristic
219Q8×C47—22no—
220Q8×C47—22no—
221Q8×C47—42no—
222Q8×C47—42no—
223Q8×C47—41no—
224Q8×C47—44no—
225Q8×C47—42no—
226D4×C47—41no—
227D4×C47—41no—
228D4×C47—41no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—21no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—22no—
238C2×C4×C47—21no—
239C2×C4×C47—42no—
240C2×C4×C47—41no—
241C2×C4×C47—44no—
242C2×C4×C47—42no—
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C2×C22⋊C412yes—
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C412yes—
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—42no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C47⋊C8—41no—
263C47⋊C8—41no—
264C47⋊C8—42no—
265#17601—471nomaximal, Sylow
266C22×C47—21no—
267C22×C47—21no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47#57711yescharacteristic
273C22×C47#57811yescharacteristic
274C4×C47—21no—
275C4×C47—21no—
276C4×C47—21no—
277C4×C47—21no—
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—21no—
281C4×C47—22no—
282C4×C47—22no—
283C4×C47—42no—
284C4×C47—41no—
285C4×C47—41no—
286C4×C47—42no—
287C4×C47—41no—
288C4×C47#42112yes—
289Dic47—81no—
290Dic47—82no—
291Dic47—81no—
292#3523D4711yescharacteristic
293#2984—471no—
294#2320—471no—
295#1521—472no—
296#1518—472no—
297#1515—472no—
298#1512—472no—
299#1509—471no—
300#1506—471no—
301#1503—471no—
302#1500—471no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—81no—
306C2×C47#288411yescharacteristic
307C2×C47#289811yescharacteristic
308C2×C47C2×#42111yescharacteristic
309#557Dic4711yescharacteristic, complemented
310#557Dic4711yescharacteristic, complemented
311#557Dic4712yescomplemented
312#556Dic4711yescharacteristic, complemented
313#556Dic4712yescomplemented
314#552—22no—
315#552—21no—
316D4×Q8—21no—
317D4×Q8Dic4711yescharacteristic, complemented
318#541—21no—
319#541—22no—
320#533C2×D4711yescharacteristic
321#533C2×D4712yes—
322#529C2×D4711yescharacteristic
323#529C2×D4712yes—
324#519—21no—
325C2×C4×Q8C2×D4711yescharacteristic
326#446—941no—
327#446—942no—
328C8×Q8—941no—
329#432—471no—
330#432—472no—
331#431—471no—
332#431—472no—
333#391—941no—
334#389—941no—
335#387—942no—
336#383—472no—
337#382—472no—
338C4×C4⋊C4—471no—
339#376—471no—
340#328—471no—
341#328—471no—
342#328—472no—
343#328—472no—
344#328—472no—
345#328—472no—
346#328—472no—
347#327—471no—
348#327—471no—
349#327—472no—
350C47#1760111yescharacteristic, complemented, Sylow
351C22×Q8—21no—
352C22×Q8C22⋊D4711yescharacteristic
353C4⋊Q8—21no—
354C4⋊Q8—22no—
355C4⋊Q8C2×Dic4711yescharacteristic
356C4⋊Q8C2×Dic4712yes—
357C4⋊Q8C2×Dic4711yescharacteristic
358C4⋊Q8C2×Dic4712yes—
359#126—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#126—21no—
364#125—22no—
365#125—22no—
366#125—22no—
367#125—22no—
368#125—21no—
369#125—22no—
370#125—21no—
371#125C2×Dic4711yescharacteristic
372#125C2×Dic4712yes—
373#124C2×Dic4711yescharacteristic
374#124C2×Dic4712yes—
375#123—21no—
376#123—22no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383C22⋊Q8C22×D4711yescharacteristic
384C22⋊Q8C22×D4712yes—
385#121C22×D4711yescharacteristic
386C4×Q8—21no—
387C4×Q8—22no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—22no—
391C4×Q8C2×Dic4711yescharacteristic
392C4×Q8C2×Dic4712yes—
393C4×Q8C2×Dic4712yes—
394C4×Q8C2×Dic4712yes—
395C4×Q8C2×Dic4712yes—
396C4×Q8C2×Dic4711yescharacteristic
397C4×Q8C2×Dic4711yescharacteristic
398C4×Q8C2×Dic4712yes—
399C4×Q8C22⋊D4711yescharacteristic
400C4×Q8C22⋊D4711yescharacteristic
401C4×Q8C22⋊D4712yes—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4C2×Dic4711yescharacteristic
405C4×D4C2×Dic4712yes—
406#117—21no—
407#117—22no—
408#117—22no—
409#117—21no—
410#117—21no—
411#117—22no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4C22⋊D4712yes—
415C2×C4⋊C4C22⋊D4711yescharacteristic
416C2×C42—941no—
417C2×C42—941no—
418C2×C42C22×D4711yescharacteristic
419C2×C42C22×D4712yes—
420C4⋊C8—942no—
421C4⋊C8—941no—
422C4⋊C8—941no—
423C4⋊C8—942no—
424C22⋊C8—471no—
425C22⋊C8—471no—
426C22⋊C8—472no—
427#97—942no—
428#97—941no—
429C4×C8—941no—
430C4×C8—942no—
431#95—471no—
432#95—471no—
433#95—472no—
434#95—941no—
435#95—942no—
436#95—942no—
437#95—942no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8C22⋊Dic4711yescharacteristic
443C2×Q8C22⋊Dic4711yescharacteristic
444C2×Q8C22⋊Dic4712yes—
445C2×Q8C22×Dic4711yescharacteristic
446C2×Q8C22×Dic4712yes—
447C2×D4—21no—
448C2×D4C22×Dic4711yescharacteristic
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—942no—
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C2×C22⋊D4712yes—
456C22×C4C2×C22⋊D4711yescharacteristic
457C22×C4C2×C22⋊D4712yes—
458C22×C4C23×D4711yescharacteristic
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—942no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—3762no—
480C4⋊C4—1882no—
481C4⋊C4C22⋊Dic4712yes—
482C4⋊C4C22⋊Dic4711yescharacteristic
483C4⋊C4C22⋊Dic4712yes—
484C4⋊C4C22⋊Dic4711yescharacteristic
485C4⋊C4C22⋊Dic4712yes—
486C4⋊C4C22⋊Dic4712yes—
487C4⋊C4C22⋊Dic4712yes—
488C4⋊C4C22×Dic4711yescharacteristic
489C4⋊C4C22×Dic4712yes—
490C4⋊C4C22×Dic4711yescharacteristic
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—3761no—
506C42—1882no—
507C42C22×Dic4712yes—
508C42C22×Dic4711yescharacteristic
509C42C22×Dic4711yescharacteristic
510C42C22×Dic4712yes—
511C42C2×C22⋊D4711yescharacteristic
512C42C2×C22⋊D4712yes—
513C42C2×C22⋊D4712yes—
514C42C2×C22⋊D4711yescharacteristic
515C23—21no—
516C23C22×C22⋊D4711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—42no—
520Q8—41no—
521Q8—42no—
522Q8—44no—
523Q8—42no—
524D4—41no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1882no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4C2×C22⋊Dic4711yescharacteristic
551C2×C4C2×C22⋊Dic4712yes—
552C2×C4C2×C22⋊Dic4712yes—
553C2×C4C2×C22⋊Dic4712yes—
554C2×C4C2×C22⋊Dic4711yescharacteristic
555C2×C4C2×C22⋊Dic4711yescharacteristic
556C2×C4C2×C22⋊Dic4712yes—
557C2×C4C2×C22⋊Dic4711yescharacteristic
558C2×C4C23×Dic4711yescharacteristic
559C2×C4C22×C22⋊D4711yescharacteristic
560C8—1881no—
561C8—1881no—
562C8—1882no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22C22×C22⋊Dic4711yescharacteristic
569C22#1568993111yescharacteristic
570C22#1568993811yescharacteristic
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—3761no—
586C4—3762no—
587C4—3761no—
588C4#1568914012yes—
589C2—21no—
590C2—21no—
591C2—81no—
592C2#2025185211yescharacteristic
593C2#2025192111yescharacteristic
594C2C2×#1568914011yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#3523C2
socleC22×C47C22×C22⋊C4
prime radical#121×C47C23
cosocleC22×C4×C47C24
hypercenter#3523D47
hypocenterC47#17601
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3523

Complements

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

NH ≅ G/Ntype
1. #20252477C2semidirect
2. #20246039C2semidirect
4. #20241952C2semidirect
5. #20241939C2semidirect
8. #20241900C2semidirect
9. #20241887C2semidirect
36. C47×#557C4semidirect
37. C47×#557C4semidirect
38. C47×#557C4semidirect
39. C47×#556C4semidirect
40. C47×#556C4semidirect
44. D4×Q8×C47C4semidirect
309. #557Dic47semidirect
310. #557Dic47semidirect
311. #557Dic47semidirect
312. #556Dic47semidirect
313. #556Dic47semidirect
317. D4×Q8Dic47semidirect
350. C47#17601semidirect

Maximal subgroups

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

subgroupindex
#202524772
#202460392
#202419652
#202419522
#202419392
#202419262
#202419132
#202419002
#202418872
#202418742
C47×#35232
#1760147