Finite Group Databasebeta

← #26184123

Subgroups — #26184123

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252417C211yesmaximal, characteristic, complemented
2#20246328C211yesmaximal, characteristic, complemented
3#20241944C212yesmaximal, complemented
4#20241939C211yesmaximal, characteristic, complemented
5#20241935C211yesmaximal, characteristic, complemented
6#20241931C212yesmaximal, complemented
7#20241900C211yesmaximal, characteristic, complemented
8#20241896C211yesmaximal, characteristic, complemented
9#20241892C212yesmaximal, complemented
10#20241879C212yesmaximal, complemented
11C47×#3506C211yesmaximal, characteristic
12#15689195—22no—
13#15689195—22no—
14#15689180C2211yescharacteristic
15#15689180C2211yescharacteristic
16#15689087C2212yes—
17C2×C22⋊(C47⋊C8)C2212yes—
18#15689016—21no—
19#15689009—21no—
20#15689003—21no—
21#15688991C2211yescharacteristic
22#15688986C2211yescharacteristic
23#15688983C2211yescharacteristic
24C22⋊C4×Dic47—21no—
25C4×C22⋊Dic47C2211yescharacteristic
26#15688969C2211yescharacteristic
27C2×C47⋊#95C2211yescharacteristic
28#15688860C2212yes—
29#15688860C2212yes—
30#15688860C2212yes—
31#15688860C2212yes—
32#15688857C2212yes—
33#15688857C2212yes—
34#15688857C2212yes—
35#15688857C2212yes—
36C47×#559—22no—
37C47×#556C412yescomplemented
38C47×#555C412yescomplemented
39C47×#553—22no—
40C47×#550C412yescomplemented
41C47×#547C412yescomplemented
42C47×#538—21no—
43C47×#538—21no—
44C47×#535—21no—
45C47×#533C2211yescharacteristic
46C47×#532C2211yescharacteristic
47C47×#525C2212yes—
48C47×#525C2212yes—
49C47×#520—21no—
50C2×C4×D4×C47C2211yescharacteristic
51C23×Dic47—21no—
52C2×#10625329—22no—
53C22×C47⋊C8—22no—
54C2×C4⋊Dic47—21no—
55C2×C22⋊Dic47—21no—
56C2×C22⋊Dic47—21no—
57C2×C22⋊Dic47—21no—
58C2×C4×Dic47—21no—
59C2×C4×Dic47—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)—21no—
62C22⋊(C47⋊C8)—22no—
63C22⋊(C47⋊C8)—22no—
64C22⋊(C47⋊C8)C2312yes—
65C22⋊(C47⋊C8)C2312yes—
66#13093935—21no—
67#13093935—21no—
68C47⋊#95—21no—
69C47⋊#95—21no—
70C47⋊#95—21no—
71C47⋊#95—21no—
72C47⋊#95—21no—
73C47⋊#95C2311yescharacteristic
74C47⋊#95C2311yescharacteristic
75C47⋊#95C2311yescharacteristic
76C47⋊#95C2311yescharacteristic
77C2×#41×C47—21no—
78C2×#41×C47C2×C412yes—
79C2×#41×C47C2×C412yescomplemented
80C22×D4×C47D411yescharacteristic
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C4⋊Q8×C47C2×C411yescharacteristic, complemented
84C4⋊Q8×C47C2×C411yescharacteristic, complemented
85C4⋊Q8×C47C2×C411yescharacteristic, complemented
86#127×C47—22no—
87#127×C47C2×C411yescharacteristic, complemented
88#126×C47—22no—
89#126×C47—22no—
90#125×C47—21no—
91#124×C47—22no—
92#124×C47—22no—
93#124×C47—22no—
94#124×C47—21no—
95#123×C47—22no—
96#123×C47—22no—
97#123×C47—22no—
98#123×C47—22no—
99C22⋊Q8×C47—21no—
100C22⋊Q8×C47C2×C412yes—
101C22⋊Q8×C47C2×C412yescomplemented
102C22⋊Q8×C47C2312yes—
103#121×C47—22no—
104#121×C47—22no—
105#121×C47—21no—
106#121×C47—22no—
107#121×C47—22no—
108#121×C47C2×C412yes—
109#121×C47C2×C412yescomplemented
110#121×C47C2312yes—
111#120×C47—22no—
112#120×C47—22no—
113C4×Q8×C47—22no—
114C4×Q8×C47—21no—
115C4×Q8×C47C2×C412yescomplemented
116C4×D4×C47—22no—
117C4×D4×C47—22no—
118C4×D4×C47—22no—
119C4×D4×C47—22no—
120C4×D4×C47—22no—
121C4×D4×C47—21no—
122C4×D4×C47C2×C412yes—
123C4×D4×C47C2×C412yes—
124C4×D4×C47C2×C412yescomplemented
125C4×D4×C47D412yes—
126C4×D4×C47D412yes—
127#117×C47—21no—
128#117×C47—22no—
129#117×C47C2×C412yes—
130#117×C47C2×C412yescomplemented
131C2×C4⋊C4×C47—22no—
132C2×C4⋊C4×C47D411yescharacteristic
133C2×C22⋊C4×C47D411yescharacteristic
134C2×C22⋊C4×C47D411yescharacteristic
135C2×C42×C47C2311yescharacteristic
136C24×C47C2×D411yescharacteristic
137C24×C47C2×D411yescharacteristic
138#41×C47—42no—
139#41×C47—42no—
140#41×C47—82no—
141#41×C47—82no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47C22×C412yes—
145C2×D4×C47—22no—
146C2×D4×C47—22no—
147C2×D4×C47—22no—
148C2×D4×C47—22no—
149C2×D4×C47—22no—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47—41no—
153C2×D4×C47—41no—
154C2×D4×C47C22⋊C412yes—
155C2×D4×C47C22⋊C412yes—
156C2×D4×C47C22×C412yes—
157C22×C4×C47—22no—
158C22×C4×C47—22no—
159C22×C4×C47—22no—
160C22×C4×C47—81no—
161C22×C4×C47—81no—
162C22×C4×C47C22×C412yes—
163C22×C4×C47C22×C412yes—
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D411yescharacteristic
167C22×C4×C47C2×D411yescharacteristic
168C22×C4×C47C2×D412yes—
169C22×C4×C47C2×D412yes—
170C22×C4×C47C2411yescharacteristic
171C22×Dic47—21no—
172C22×Dic47—21no—
173C22×Dic47—21no—
174C22×Dic47—21no—
175C22×Dic47—42no—
176C22×Dic47—42no—
177C22×Dic47—81no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47—22no—
186C4⋊C4×C47C22⋊C412yes—
187C4⋊C4×C47C22⋊C412yes—
188C4⋊C4×C47C22×C412yes—
189C4⋊C4×C47C22×C412yes—
190C22⋊C4×C47—22no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—22no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—22no—
196C22⋊C4×C47—22no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47—22no—
199C22⋊C4×C47C22⋊C412yes—
200C22⋊C4×C47C22⋊C412yes—
201C22⋊C4×C47C22⋊C412yes—
202C22⋊C4×C47C22⋊C412yes—
203C42×C47—22no—
204C42×C47—22no—
205C42×C47—22no—
206C42×C47C22×C411yescharacteristic
207C42×C47C22×C411yescharacteristic
208C42×C47C2×D411yescharacteristic
209C42×C47C2×D411yescharacteristic
210C22⋊Dic47—42no—
211C22⋊Dic47—42no—
212C22⋊Dic47—81no—
213C4⋊Dic47—81no—
214C4×Dic47—42no—
215C4×Dic47—81no—
216#10625329—82no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—82no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—22no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47—41no—
229C23×C47C2×C22⋊C412yes—
230C23×C47C2×C22⋊C412yes—
231C23×C47C22×D411yescharacteristic
232Q8×C47—42no—
233Q8×C47—42no—
234Q8×C47—81no—
235D4×C47—42no—
236D4×C47—42no—
237D4×C47—42no—
238D4×C47—42no—
239D4×C47—42no—
240D4×C47—42no—
241D4×C47—81no—
242D4×C47—82no—
243D4×C47—82no—
244D4×C47—82no—
245D4×C47—82no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—42no—
258C2×C4×C47—42no—
259C2×C4×C47—44no—
260C2×C4×C47—82no—
261C2×C4×C47—82no—
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C2×C22⋊C412yes—
264C2×C4×C47C2×C22⋊C412yes—
265C2×C4×C47C2×C22⋊C411yescharacteristic
266C2×C4×C47C2×C22⋊C411yescharacteristic
267C2×C4×C47C2×C22⋊C411yescharacteristic
268C2×C4×C47C23×C411yescharacteristic
269C2×C4×C47C22×D411yescharacteristic
270C2×Dic47—22no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—84no—
283C47⋊C8—42no—
284C47⋊C8—42no—
285#17629—471nomaximal, Sylow
286C22×C47—21no—
287C22×C47—21no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—22no—
293C22×C47—22no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—81no—
300C22×C47—81no—
301C22×C47C22×C22⋊C411yescharacteristic
302C22×C47#57511yescharacteristic
303C22×C47#57511yescharacteristic
304C4×C47—21no—
305C4×C47—21no—
306C4×C47—22no—
307C4×C47—22no—
308C4×C47—21no—
309C4×C47—21no—
310C4×C47—21no—
311C4×C47—21no—
312C4×C47—42no—
313C4×C47—42no—
314C4×C47—42no—
315Dic47—42no—
316Dic47—81no—
317Dic47—81no—
318Dic47—81no—
319#3506D4711yescharacteristic
320#2973—471no—
321#2353—471no—
322#1516—472no—
323#1515—471no—
324#1514—471no—
325#1513—472no—
326#1506—471no—
327#1505—471no—
328#1504—472no—
329#1501—472no—
330C2×C47—21no—
331C2×C47—21no—
332C2×C47—42no—
333C2×C47—42no—
334C2×C47—82no—
335C2×C47#288311yescharacteristic
336C2×C47#289611yescharacteristic
337C2×C47#290111yescharacteristic
338#559—22no—
339#556Dic4712yescomplemented
340#555Dic4712yescomplemented
341#553—22no—
342#550Dic4712yescomplemented
343#547Dic4712yescomplemented
344#538—21no—
345#538—21no—
346#535—21no—
347#533C2×D4711yescharacteristic
348#532C2×D4711yescharacteristic
349#525C2×D4712yes—
350#525C2×D4712yes—
351#520—21no—
352C2×C4×D4C2×D4711yescharacteristic
353#435—942no—
354#435—942no—
355#432—471no—
356#432—471no—
357#407—472no—
358C2×C22⋊C8—472no—
359#388—941no—
360#387—941no—
361#386—941no—
362#383—471no—
363#381—471no—
364#380—471no—
365C4×C22⋊C4—471no—
366C4×C22⋊C4—941no—
367#376—471no—
368C2×#95—471no—
369#328—472no—
370#328—472no—
371#328—472no—
372#328—472no—
373#327—472no—
374#327—472no—
375#327—472no—
376#327—472no—
377C47#1762911yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C2×#41C2×Dic4712yes—
380C2×#41C2×Dic4712yescomplemented
381C22×D4C22⋊D4711yescharacteristic
382C23×C4—941no—
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C2×M16—942no—
386C22×C8—942no—
387C4⋊Q8C2×Dic4711yescharacteristic, complemented
388C4⋊Q8C2×Dic4711yescharacteristic, complemented
389C4⋊Q8C2×Dic4711yescharacteristic, complemented
390#127—22no—
391#127C2×Dic4711yescharacteristic, complemented
392#126—22no—
393#126—22no—
394#125—21no—
395#124—22no—
396#124—22no—
397#124—22no—
398#124—21no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403C22⋊Q8—21no—
404C22⋊Q8C2×Dic4712yes—
405C22⋊Q8C2×Dic4712yescomplemented
406C22⋊Q8C22×D4712yes—
407#121—22no—
408#121—22no—
409#121—21no—
410#121—22no—
411#121—22no—
412#121C2×Dic4712yes—
413#121C2×Dic4712yescomplemented
414#121C22×D4712yes—
415#120—22no—
416#120—22no—
417C4×Q8—22no—
418C4×Q8—21no—
419C4×Q8C2×Dic4712yescomplemented
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—22no—
425C4×D4—21no—
426C4×D4C2×Dic4712yes—
427C4×D4C2×Dic4712yes—
428C4×D4C2×Dic4712yescomplemented
429C4×D4C22⋊D4712yes—
430C4×D4C22⋊D4712yes—
431#117—21no—
432#117—22no—
433#117C2×Dic4712yes—
434#117C2×Dic4712yescomplemented
435C2×C4⋊C4—22no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4C22⋊D4711yescharacteristic
438C2×C22⋊C4—941no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C22⋊C4C22⋊D4711yescharacteristic
442C2×C22⋊C4C22⋊D4711yescharacteristic
443C2×C42—941no—
444C2×C42—941no—
445C2×C42C22×D4711yescharacteristic
446C4⋊C8—941no—
447C4⋊C8—941no—
448C22⋊C8—472no—
449C22⋊C8—472no—
450C22⋊C8—942no—
451C22⋊C8—942no—
452#97—941no—
453#97—941no—
454#95—471no—
455#95—471no—
456#95—471no—
457#95—471no—
458#95—941no—
459#95—941no—
460#95—941no—
461#95—941no—
462#95—941no—
463C24C2×C22⋊D4711yescharacteristic
464C24C2×C22⋊D4711yescharacteristic
465#41—42no—
466#41—42no—
467#41—82no—
468#41—82no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8C22×Dic4712yes—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4C22⋊Dic4712yes—
482C2×D4C22⋊Dic4712yes—
483C2×D4C22×Dic4712yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—3761no—
494C22×C4—1882no—
495C22×C4—1882no—
496C22×C4C22×Dic4712yes—
497C22×C4C22×Dic4712yes—
498C22×C4C2×C22⋊D4711yescharacteristic
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4712yes—
503C22×C4C2×C22⋊D4712yes—
504C22×C4C23×D4711yescharacteristic
505M16—3762no—
506C2×C8—942no—
507C2×C8—942no—
508C2×C8—3762no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—3761no—
518C4⋊C4C22⋊Dic4712yes—
519C4⋊C4C22⋊Dic4712yes—
520C4⋊C4C22×Dic4712yes—
521C4⋊C4C22×Dic4712yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—3761no—
532C22⋊C4—1882no—
533C22⋊C4—1882no—
534C22⋊C4C22⋊Dic4712yes—
535C22⋊C4C22⋊Dic4712yes—
536C22⋊C4C22⋊Dic4712yes—
537C22⋊C4C22⋊Dic4712yes—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—3761no—
542C42—1882no—
543C42C22×Dic4711yescharacteristic
544C42C22×Dic4711yescharacteristic
545C42C2×C22⋊D4711yescharacteristic
546C42C2×C22⋊D4711yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×C22⋊Dic4712yes—
557C23C2×C22⋊Dic4712yes—
558C23C22×C22⋊D4711yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—81no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—82no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—942no—
590C2×C4—3764no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4C2×C22⋊Dic4711yescharacteristic
603C2×C4C2×C22⋊Dic4712yes—
604C2×C4C2×C22⋊Dic4712yes—
605C2×C4C2×C22⋊Dic4711yescharacteristic
606C2×C4C2×C22⋊Dic4711yescharacteristic
607C2×C4C2×C22⋊Dic4711yescharacteristic
608C2×C4C23×Dic4711yescharacteristic
609C2×C4C22×C22⋊D4711yescharacteristic
610C8—1882no—
611C8—1882no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22C22×C22⋊Dic4711yescharacteristic
628C22#1568992111yescharacteristic
629C22#1568992111yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—3761no—
642C4—3761no—
643C4—3761no—
644C4—1882no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—82no—
650C2#2025184911yescharacteristic
651C2#2025190711yescharacteristic
652C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3506C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenter#3506D47
hypocenterC47#17629
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #20252417C2semidirect
2. #20246328C2semidirect
3. #20241944C2semidirect
4. #20241939C2semidirect
5. #20241935C2semidirect
6. #20241931C2semidirect
7. #20241900C2semidirect
8. #20241896C2semidirect
9. #20241892C2semidirect
10. #20241879C2semidirect
37. C47×#556C4semidirect
38. C47×#555C4semidirect
40. C47×#550C4semidirect
41. C47×#547C4semidirect
79. C2×#41×C47C2×C4semidirect
83. C4⋊Q8×C47C2×C4semidirect
84. C4⋊Q8×C47C2×C4semidirect
85. C4⋊Q8×C47C2×C4semidirect
87. #127×C47C2×C4semidirect
101. C22⋊Q8×C47C2×C4semidirect
109. #121×C47C2×C4semidirect
115. C4×Q8×C47C2×C4semidirect
124. C4×D4×C47C2×C4semidirect
130. #117×C47C2×C4semidirect
339. #556Dic47semidirect
340. #555Dic47semidirect
342. #550Dic47semidirect
343. #547Dic47semidirect
377. C47#17629semidirect
380. C2×#41C2×Dic47semidirect
387. C4⋊Q8C2×Dic47semidirect
388. C4⋊Q8C2×Dic47semidirect
389. C4⋊Q8C2×Dic47semidirect
391. #127C2×Dic47semidirect
405. C22⋊Q8C2×Dic47semidirect
413. #121C2×Dic47semidirect
419. C4×Q8C2×Dic47semidirect
428. C4×D4C2×Dic47semidirect
434. #117C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202463282
#202419442
#202419392
#202419352
#202419312
#202419002
#202418962
#202418922
#202418792
C47×#35062
#1762947