Finite Group Databasebeta

← #26184512

Subgroups — #26184512

Order 12032.

896 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
1#20252408C211yesmaximal, characteristic, complemented
2#20252402C211yesmaximal, characteristic, complemented
3#20242023C211yesmaximal, characteristic, complemented
4#20242019C212yesmaximal, complemented
5#20242008C212yesmaximal, complemented
6#20241995C212yesmaximal, complemented
7#20241990C211yesmaximal, characteristic, complemented
8#20241990C211yesmaximal, characteristic, complemented
9#20241980C211yesmaximal, characteristic, complemented
10#20241976C212yesmaximal, complemented
11C47×#3512C211yesmaximal, characteristic
12#15689254—21no—
13Q8×C47⋊C8—21no—
14#15689201—22no—
15#15689201—21no—
16#15689195—22no—
17D4×C47⋊C8—21no—
18#15689180C2211yescharacteristic
19#15689180C2211yescharacteristic
20#15689180C2212yes—
21#15689180C2212yes—
22C4×#10625329C2212yes—
23C4×#10625329C2211yescharacteristic
24C4×#10625329C2211yescharacteristic
25C2×#13093935C2212yes—
26#15688865C2212yes—
27#15688865C2212yes—
28#15688865C2212yes—
29#15688865C2212yes—
30#15688863C2211yescharacteristic
31#15688863C2212yes—
32#15688863C2211yescharacteristic
33#15688863C2211yescharacteristic
34#15688863C2211yescharacteristic
35#15688863C2212yes—
36C47×#553—22no—
37C47×#553—22no—
38C47×#553—21no—
39C47×#553—21no—
40C47×#552—21no—
41C47×#552—21no—
42C47×#541C412yes—
43C47×#540C412yes—
44C47×#537C411yescharacteristic
45C47×#537C411yescharacteristic
46C47×#532C2211yescharacteristic
47C47×#529C2212yes—
48C47×#529C2212yes—
49C2×#125×C47C2211yescharacteristic
50C47×#520C411yescharacteristic
51C47×#520C411yescharacteristic
52C4×#41×C47C2211yescharacteristic
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—22no—
56C22×C47⋊C8—22no—
57C4⋊(C47⋊C8)—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—22no—
61C22⋊(C47⋊C8)—22no—
62C22⋊(C47⋊C8)—21no—
63C22⋊(C47⋊C8)—21no—
64C22⋊(C47⋊C8)—22no—
65#13093935—22no—
66#13093935C2311yescharacteristic
67#13093935C2312yes—
68#13093935C2311yescharacteristic
69#13093935C2312yes—
70#13093935C2311yescharacteristic
71#13093935C2311yescharacteristic
72C4×C47⋊C8—22no—
73C4×C47⋊C8—21no—
74C4×C47⋊C8—21no—
75C2×#41×C47C2×C411yescharacteristic
76C2×#41×C47C2×C411yescharacteristic
77C2×#41×C47C2×C411yescharacteristic
78C4⋊Q8×C47C2×C411yescharacteristic
79#127×C47C2×C411yescharacteristic
80#126×C47—22no—
81#126×C47—22no—
82#126×C47—22no—
83#126×C47—22no—
84#125×C47—22no—
85#125×C47—22no—
86#125×C47—21no—
87#125×C47—21no—
88#125×C47C2×C412yes—
89#125×C47C2312yes—
90#124×C47C2311yescharacteristic
91#124×C47C2311yescharacteristic
92#123×C47—22no—
93#123×C47—22no—
94#123×C47—22no—
95#123×C47—22no—
96C22⋊Q8×C47—21no—
97C22⋊Q8×C47—21no—
98C22⋊Q8×C47C2×C412yes—
99C22⋊Q8×C47C2×C412yes—
100#121×C47—21no—
101#121×C47—21no—
102#121×C47—22no—
103#121×C47—22no—
104#121×C47—21no—
105#121×C47—21no—
106#121×C47—22no—
107#121×C47—22no—
108#121×C47C2×C412yes—
109#121×C47C2×C412yes—
110C4×Q8×C47C2×C412yes—
111C4×Q8×C47C2×C412yes—
112C4×Q8×C47D411yescharacteristic
113C4×Q8×C47D411yescharacteristic
114C4×D4×C47—22no—
115C4×D4×C47—22no—
116C4×D4×C47—22no—
117C4×D4×C47—22no—
118C4×D4×C47—21no—
119C4×D4×C47—21no—
120C4×D4×C47C2×C412yes—
121C4×D4×C47C2×C412yes—
122C4×D4×C47D411yescharacteristic
123C4×D4×C47D412yes—
124C4×D4×C47D412yes—
125C4×D4×C47D411yescharacteristic
126#117×C47—21no—
127#117×C47—21no—
128#117×C47—22no—
129#117×C47—22no—
130#117×C47C2×C411yescharacteristic
131#117×C47C2×C412yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47—21no—
135C2×C4⋊C4×C47—21no—
136C2×C4⋊C4×C47C2×C411yescharacteristic
137C2×C4⋊C4×C47C2×C411yescharacteristic
138C2×C42×C47C2311yescharacteristic
139C2×C42×C47C2312yes—
140#41×C47—42no—
141#41×C47—42no—
142#41×C47—42no—
143C2×Q8×C47C22⋊C411yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47—22no—
149C2×D4×C47—22no—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47C22⋊C411yescharacteristic
153C2×D4×C47C22⋊C412yes—
154C2×D4×C47C22×C411yescharacteristic
155C2×D4×C47C22×C411yescharacteristic
156C22×C4×C47—21no—
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47—21no—
160C22×C4×C47—22no—
161C22×C4×C47—22no—
162C22×C4×C47C22×C411yescharacteristic
163C22×C4×C47C22×C412yes—
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D412yes—
167C22×C4×C47C2×D412yes—
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—22no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47C22⋊C411yescharacteristic
181C4⋊C4×C47C22⋊C411yescharacteristic
182C4⋊C4×C47C22⋊C412yes—
183C4⋊C4×C47C22⋊C412yes—
184C4⋊C4×C47C22×C412yes—
185C4⋊C4×C47C22×C412yes—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—22no—
188C22⋊C4×C47—22no—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—21no—
193C22⋊C4×C47—21no—
194C22⋊C4×C47—21no—
195C22⋊C4×C47—21no—
196C22⋊C4×C47C22⋊C412yes—
197C22⋊C4×C47C22⋊C412yes—
198C22⋊C4×C47C22⋊C412yes—
199C42×C47—22no—
200C42×C47—22no—
201C42×C47C22×C412yes—
202C42×C47C22×C411yescharacteristic
203C42×C47C2×D412yes—
204C42×C47C2×D412yes—
205C42×C47C2×D411yescharacteristic
206C42×C47C2×D411yescharacteristic
207C42×C47C2411yescharacteristic
208#10625329—42no—
209#10625329—44no—
210#10625329—42no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—22no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—44no—
218C23×C47—21no—
219C23×C47—21no—
220C23×C47C2×C22⋊C412yes—
221C23×C47C2×C22⋊C411yescharacteristic
222Q8×C47—22no—
223Q8×C47—41no—
224Q8×C47—41no—
225D4×C47—22no—
226D4×C47—42no—
227D4×C47—41no—
228D4×C47—41no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—81no—
232D4×C47—81no—
233D4×C47—82no—
234D4×C47—82no—
235C2×C4×C47—22no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—82no—
248C2×C4×C47—82no—
249C2×C4×C47—81no—
250C2×C4×C47—81no—
251C2×C4×C47C2×C22⋊C412yes—
252C2×C4×C47C2×C22⋊C412yes—
253C2×C4×C47C2×C22⋊C411yescharacteristic
254C2×C4×C47C2×C22⋊C411yescharacteristic
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C2×C22⋊C411yescharacteristic
257C2×C4×C47C23×C411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C22×D411yescharacteristic
260C47⋊C8—22no—
261C47⋊C8—22no—
262C47⋊C8—42no—
263C47⋊C8—41no—
264C47⋊C8—42no—
265C47⋊C8—41no—
266#17687—471nomaximal, Sylow
267C22×C47—21no—
268C22×C47—22no—
269C22×C47—21no—
270C22×C47—21no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47C22×C22⋊C411yescharacteristic
280C4×C47—21no—
281C4×C47—22no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—21no—
287C4×C47—21no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47#41312yes—
293#3512D4711yescharacteristic
294#2971—471no—
295#2970—471no—
296#1536—471no—
297#1535—472no—
298#1532—472no—
299#1529—472no—
300#1528—471no—
301#1528—471no—
302#1525—471no—
303#1524—472no—
304C2×C47—41no—
305C2×C47—42no—
306C2×C47—81no—
307C2×C47—81no—
308C2×C47C2×#41311yescharacteristic
309C2×C47#290111yescharacteristic
310C2×C47#290111yescharacteristic
311#553—22no—
312#553—22no—
313#553—21no—
314#553—21no—
315#552—21no—
316#552—21no—
317#541Dic4712yes—
318#540Dic4712yes—
319#537Dic4711yescharacteristic
320#537Dic4711yescharacteristic
321#532C2×D4711yescharacteristic
322#529C2×D4712yes—
323#529C2×D4712yes—
324C2×#125C2×D4711yescharacteristic
325#520Dic4711yescharacteristic
326#520Dic4711yescharacteristic
327C4×#41C2×D4711yescharacteristic
328#446—941no—
329C8×Q8—941no—
330#436—942no—
331#436—941no—
332#435—942no—
333C8×D4—941no—
334#432—471no—
335#432—471no—
336#432—472no—
337#432—472no—
338C4×M16—471no—
339C4×M16—472no—
340C4×M16—471no—
341C2×#97—472no—
342#330—472no—
343#330—472no—
344#330—472no—
345#330—472no—
346#329—472no—
347#329—472no—
348#329—471no—
349#329—471no—
350#329—471no—
351#329—471no—
352C47#1768711yescharacteristic, complemented, Sylow
353C2×#41C2×Dic4711yescharacteristic
354C2×#41C2×Dic4711yescharacteristic
355C2×#41C2×Dic4711yescharacteristic
356C2×M16—941no—
357C2×M16—941no—
358C2×M16—942no—
359C22×C8—942no—
360C4⋊Q8C2×Dic4711yescharacteristic
361#127C2×Dic4711yescharacteristic
362#126—22no—
363#126—22no—
364#126—22no—
365#126—22no—
366#125—22no—
367#125—22no—
368#125—21no—
369#125—21no—
370#125C2×Dic4712yes—
371#125C22×D4712yes—
372#124C22×D4711yescharacteristic
373#124C22×D4711yescharacteristic
374#123—22no—
375#123—22no—
376#123—22no—
377#123—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8C2×Dic4712yes—
381C22⋊Q8C2×Dic4712yes—
382#121—21no—
383#121—21no—
384#121—22no—
385#121—22no—
386#121—21no—
387#121—21no—
388#121—22no—
389#121—22no—
390#121C2×Dic4712yes—
391#121C2×Dic4712yes—
392C4×Q8C2×Dic4712yes—
393C4×Q8C2×Dic4712yes—
394C4×Q8C22⋊D4711yescharacteristic
395C4×Q8C22⋊D4711yescharacteristic
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4C2×Dic4712yes—
403C4×D4C2×Dic4712yes—
404C4×D4C22⋊D4711yescharacteristic
405C4×D4C22⋊D4712yes—
406C4×D4C22⋊D4712yes—
407C4×D4C22⋊D4711yescharacteristic
408#117—21no—
409#117—21no—
410#117—22no—
411#117—22no—
412#117C2×Dic4711yescharacteristic
413#117C2×Dic4712yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4C2×Dic4711yescharacteristic
419C2×C4⋊C4C2×Dic4711yescharacteristic
420C2×C42C22×D4711yescharacteristic
421C2×C42C22×D4712yes—
422C4⋊C8—941no—
423C4⋊C8—942no—
424C4⋊C8—942no—
425C4⋊C8—941no—
426C22⋊C8—942no—
427C22⋊C8—941no—
428C22⋊C8—941no—
429C22⋊C8—942no—
430#97—471no—
431#97—472no—
432#97—471no—
433#97—472no—
434#97—471no—
435#97—471no—
436#97—942no—
437C4×C8—941no—
438C4×C8—942no—
439C4×C8—941no—
440#41—42no—
441#41—42no—
442#41—42no—
443C2×Q8C22⋊Dic4711yescharacteristic
444C2×Q8C22×Dic4711yescharacteristic
445C2×Q8C22×Dic4711yescharacteristic
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C22⋊Dic4711yescharacteristic
453C2×D4C22⋊Dic4712yes—
454C2×D4C22×Dic4711yescharacteristic
455C2×D4C22×Dic4711yescharacteristic
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4C22×Dic4711yescharacteristic
463C22×C4C22×Dic4712yes—
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4712yes—
467C22×C4C2×C22⋊D4712yes—
468M16—1882no—
469M16—1884no—
470M16—1882no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—942no—
475C2×C8—941no—
476C2×C8—942no—
477C2×C8—1884no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4C22⋊Dic4711yescharacteristic
491C4⋊C4C22⋊Dic4711yescharacteristic
492C4⋊C4C22⋊Dic4712yes—
493C4⋊C4C22⋊Dic4712yes—
494C4⋊C4C22×Dic4712yes—
495C4⋊C4C22×Dic4712yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4C22⋊Dic4712yes—
507C22⋊C4C22⋊Dic4712yes—
508C22⋊C4C22⋊Dic4712yes—
509C42—22no—
510C42—22no—
511C42C22×Dic4712yes—
512C42C22×Dic4711yescharacteristic
513C42C2×C22⋊D4712yes—
514C42C2×C22⋊D4711yescharacteristic
515C42C2×C22⋊D4711yescharacteristic
516C42C2×C22⋊D4712yes—
517C42C23×D4711yescharacteristic
518C23—21no—
519C23—21no—
520C23C2×C22⋊Dic4712yes—
521C23C2×C22⋊Dic4711yescharacteristic
522Q8—22no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—42no—
527D4—41no—
528D4—41no—
529D4—42no—
530D4—42no—
531D4—81no—
532D4—81no—
533D4—82no—
534D4—82no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4C2×C22⋊Dic4712yes—
552C2×C4C2×C22⋊Dic4712yes—
553C2×C4C2×C22⋊Dic4711yescharacteristic
554C2×C4C2×C22⋊Dic4711yescharacteristic
555C2×C4C2×C22⋊Dic4712yes—
556C2×C4C2×C22⋊Dic4711yescharacteristic
557C2×C4C23×Dic4711yescharacteristic
558C2×C4C22×C22⋊D4711yescharacteristic
559C2×C4C22×C22⋊D4711yescharacteristic
560C8—942no—
561C8—942no—
562C8—1882no—
563C8—1881no—
564C8—1882no—
565C8—1881no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—21no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C22×C22⋊Dic4711yescharacteristic
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—41no—
591C4#1568910712yes—
592C2—41no—
593C2—42no—
594C2—81no—
595C2—81no—
596C2C2×#1568910711yescharacteristic
597C2#2025193511yescharacteristic
598C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3512C2
socleC22×C47C22×C22⋊C4
cosocleC42×C47C24
hypercenter#3512D47
hypocenterC47#17687
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #20252408C2semidirect
2. #20252402C2semidirect
3. #20242023C2semidirect
4. #20242019C2semidirect
5. #20242008C2semidirect
6. #20241995C2semidirect
7. #20241990C2semidirect
8. #20241990C2semidirect
9. #20241980C2semidirect
10. #20241976C2semidirect
352. C47#17687semidirect

Maximal subgroups

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

subgroupindex
#202524082
#202524022
#202420232
#202420192
#202420082
#202419952
#202419902
#202419802
#202419762
C47×#35122
#1768747