Finite Group Databasebeta

← #26183842

Subgroups — #26183842

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252423C211yesmaximal, characteristic, complemented
2#20246031C211yesmaximal, characteristic, complemented
3#20241960C212yesmaximal, complemented
4#20241944C212yesmaximal, complemented
5#20241931C212yesmaximal, complemented
6#20241921C212yesmaximal, complemented
7#20241913C211yesmaximal, characteristic, complemented
8#20241900C211yesmaximal, characteristic, complemented
9#20241887C211yesmaximal, characteristic, complemented
10#20241874C211yesmaximal, characteristic, complemented
11C47×#3520C211yesmaximal, characteristic
12#15689201—21no—
13#15689195—22no—
14D4×C47⋊C8—21no—
15#15689180C2211yescharacteristic
16#15689176C2211yescharacteristic
17#15689087C2212yes—
18C2×C22⋊(C47⋊C8)C2212yes—
19#15689026—21no—
20#15689022—21no—
21#15689016—22no—
22#15688986C2212yes—
23#15688983C2212yes—
24C4×C4⋊Dic47C2211yescharacteristic
25#15688969C2211yescharacteristic
26#15688860C2211yescharacteristic
27#15688860C2212yes—
28#15688860C2212yes—
29#15688860C2212yes—
30#15688860C2212yes—
31#15688860C2212yes—
32#15688860C2211yescharacteristic
33#15688857C2211yescharacteristic
34#15688857C2212yes—
35#15688857C2211yescharacteristic
36C47×#556C411yescharacteristic, complemented
37C47×#554—21no—
38C47×#552C412yescomplemented
39C47×#551—22no—
40D4×Q8×C47C411yescharacteristic, complemented
41C47×#548C412yescomplemented
42C47×#547C411yescharacteristic, complemented
43C47×#547C411yescharacteristic, complemented
44C47×#546—21no—
45C47×#542—21no—
46C47×#539—22no—
47C47×#533C2211yescharacteristic
48C47×#529C2211yescharacteristic
49C47×#525C2212yes—
50C2×C22⋊Q8×C47C2212yes—
51C47×#520—21no—
52C2×C4×D4×C47C2211yescharacteristic
53C2×#10625329—22no—
54C22×C47⋊C8—22no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—22no—
57C2×C22⋊Dic47—22no—
58C2×C22⋊Dic47—22no—
59C2×C4×Dic47—21no—
60C2×C4×Dic47—21no—
61C4⋊(C47⋊C8)—21no—
62C4⋊(C47⋊C8)—21no—
63C22⋊(C47⋊C8)—22no—
64C22⋊(C47⋊C8)—22no—
65C22⋊(C47⋊C8)C2311yescharacteristic
66C22⋊(C47⋊C8)C2312yes—
67C22⋊(C47⋊C8)C2311yescharacteristic
68#13093935—21no—
69C4×C47⋊C8—21no—
70C47⋊#95—21no—
71C47⋊#95—22no—
72C47⋊#95C2311yescharacteristic
73C47⋊#95C2311yescharacteristic
74C47⋊#95C2312yes—
75C2×#41×C47—21no—
76C2×#41×C47C2×C412yes—
77C22×Q8×C47C2×C412yes—
78C22×D4×C47D411yescharacteristic
79C23×C4×C47C2312yes—
80C4⋊Q8×C47—21no—
81C4⋊Q8×C47C2×C411yescharacteristic
82C4⋊Q8×C47C2×C411yescharacteristic
83#126×C47—22no—
84#126×C47—22no—
85#126×C47—21no—
86#125×C47—22no—
87#125×C47—21no—
88#125×C47C2×C411yescharacteristic
89#124×C47—21no—
90#124×C47—22no—
91#124×C47C2×C411yescharacteristic
92#123×C47—22no—
93#123×C47—22no—
94#123×C47—21no—
95#123×C47—22no—
96#123×C47—22no—
97C22⋊Q8×C47—22no—
98C22⋊Q8×C47—22no—
99C22⋊Q8×C47—21no—
100C22⋊Q8×C47—22no—
101C22⋊Q8×C47—21no—
102C22⋊Q8×C47—22no—
103C22⋊Q8×C47C2×C412yes—
104C22⋊Q8×C47C2×C412yes—
105C22⋊Q8×C47C2×C412yes—
106C22⋊Q8×C47C2311yescharacteristic
107C22⋊Q8×C47C2312yes—
108#121×C47—22no—
109#121×C47—21no—
110#121×C47C2×C412yes—
111#121×C47C2311yescharacteristic
112#120×C47—22no—
113C4×Q8×C47—21no—
114C4×Q8×C47—21no—
115C4×Q8×C47C2×C411yescharacteristic
116C4×Q8×C47C2×C412yes—
117C4×D4×C47—21no—
118C4×D4×C47—22no—
119C4×D4×C47—22no—
120C4×D4×C47—21no—
121C4×D4×C47—22no—
122C4×D4×C47C2×C411yescharacteristic
123C4×D4×C47C2×C411yescharacteristic
124C4×D4×C47C2×C411yescharacteristic
125C4×D4×C47C2×C412yes—
126C4×D4×C47D411yescharacteristic
127C4×D4×C47D412yes—
128C4×D4×C47D411yescharacteristic
129#117×C47—21no—
130#117×C47—22no—
131#117×C47—21no—
132#117×C47C2×C412yes—
133C2×C4⋊C4×C47—22no—
134C2×C4⋊C4×C47—21no—
135C2×C4⋊C4×C47C2×C412yes—
136C2×C4⋊C4×C47D411yescharacteristic
137C2×C22⋊C4×C47—22no—
138C2×C22⋊C4×C47D412yes—
139C2×C42×C47C2311yescharacteristic
140C24×C47C2×D412yes—
141#41×C47—44no—
142#41×C47—82no—
143C2×Q8×C47—21no—
144C2×Q8×C47—22no—
145C2×Q8×C47—82no—
146C2×Q8×C47C22×C411yescharacteristic
147C2×Q8×C47C22×C412yes—
148C2×D4×C47—22no—
149C2×D4×C47—21no—
150C2×D4×C47—22no—
151C2×D4×C47—42no—
152C2×D4×C47C22⋊C411yescharacteristic
153C2×D4×C47C22⋊C411yescharacteristic
154C2×D4×C47C22⋊C412yes—
155C2×D4×C47C22×C411yescharacteristic
156C22×C4×C47—22no—
157C22×C4×C47—22no—
158C22×C4×C47—21no—
159C22×C4×C47—22no—
160C22×C4×C47—82no—
161C22×C4×C47C22×C412yes—
162C22×C4×C47C22×C412yes—
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47C2×D412yes—
166C22×C4×C47C2×D412yes—
167C22×C4×C47C2×D412yes—
168C22×C4×C47C2411yescharacteristic
169C22×Dic47—21no—
170C22×Dic47—21no—
171C22×Dic47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47C22⋊C411yescharacteristic
186C4⋊C4×C47C22⋊C412yes—
187C4⋊C4×C47C22⋊C411yescharacteristic
188C4⋊C4×C47C22×C411yescharacteristic
189C4⋊C4×C47C22×C412yes—
190C4⋊C4×C47C22×C411yescharacteristic
191C22⋊C4×C47—22no—
192C22⋊C4×C47—22no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—21no—
196C22⋊C4×C47—22no—
197C22⋊C4×C47—21no—
198C22⋊C4×C47—21no—
199C22⋊C4×C47—22no—
200C22⋊C4×C47—21no—
201C22⋊C4×C47—21no—
202C22⋊C4×C47C22⋊C412yes—
203C22⋊C4×C47C22⋊C412yes—
204C22⋊C4×C47C22⋊C412yes—
205C22⋊C4×C47C22⋊C412yes—
206C42×C47—22no—
207C42×C47—22no—
208C42×C47—21no—
209C42×C47C22×C411yescharacteristic
210C42×C47C22×C411yescharacteristic
211C42×C47C2×D411yescharacteristic
212C42×C47C2×D411yescharacteristic
213C22⋊Dic47—82no—
214C22⋊Dic47—82no—
215C4⋊Dic47—42no—
216C4⋊Dic47—82no—
217C4×Dic47—42no—
218C4×Dic47—81no—
219#10625329—82no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—22no—
223C2×C47⋊C8—82no—
224C23×C47—22no—
225C23×C47—22no—
226C23×C47—21no—
227C23×C47—42no—
228C23×C47—42no—
229C23×C47C2×C22⋊C412yes—
230C23×C47C2×C22⋊C412yes—
231C23×C47C22×D411yescharacteristic
232Q8×C47—41no—
233Q8×C47—42no—
234Q8×C47—42no—
235Q8×C47—82no—
236D4×C47—22no—
237D4×C47—22no—
238D4×C47—42no—
239D4×C47—41no—
240D4×C47—44no—
241D4×C47—41no—
242D4×C47—41no—
243D4×C47—82no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—21no—
250C2×C4×C47—22no—
251C2×C4×C47—21no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—41no—
256C2×C4×C47—42no—
257C2×C4×C47—42no—
258C2×C4×C47—44no—
259C2×C4×C47—42no—
260C2×C4×C47—82no—
261C2×C4×C47—82no—
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C2×C22⋊C412yes—
264C2×C4×C47C2×C22⋊C411yescharacteristic
265C2×C4×C47C2×C22⋊C411yescharacteristic
266C2×C4×C47C2×C22⋊C412yes—
267C2×C4×C47C2×C22⋊C411yescharacteristic
268C2×C4×C47C23×C411yescharacteristic
269C2×C4×C47C22×D411yescharacteristic
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—42no—
273C2×Dic47—41no—
274C2×Dic47—42no—
275C2×Dic47—41no—
276C2×Dic47—42no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C47⋊C8—41no—
280C47⋊C8—42no—
281C47⋊C8—41no—
282#17600—471nomaximal, Sylow
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47—41no—
294C22×C47—82no—
295C22×C47—82no—
296C22×C47C22×C22⋊C411yescharacteristic
297C22×C47#57711yescharacteristic
298C22×C47#57811yescharacteristic
299C4×C47—21no—
300C4×C47—21no—
301C4×C47—22no—
302C4×C47—22no—
303C4×C47—21no—
304C4×C47—21no—
305C4×C47—21no—
306C4×C47—42no—
307C4×C47—41no—
308C4×C47—41no—
309C4×C47—41no—
310C4×C47—42no—
311C4×C47#42112yes—
312Dic47—81no—
313Dic47—82no—
314Dic47—81no—
315#3520D4711yescharacteristic
316#2974—471no—
317#2319—471no—
318#1520—472no—
319#1516—472no—
320#1513—472no—
321#1511—472no—
322#1509—471no—
323#1506—471no—
324#1503—471no—
325#1500—471no—
326C2×C47—21no—
327C2×C47—21no—
328C2×C47—42no—
329C2×C47—42no—
330C2×C47—81no—
331C2×C47#288311yescharacteristic
332C2×C47#289811yescharacteristic
333C2×C47C2×#42111yescharacteristic
334#556Dic4711yescharacteristic, complemented
335#554—21no—
336#552Dic4712yescomplemented
337#551—22no—
338D4×Q8Dic4711yescharacteristic, complemented
339#548Dic4712yescomplemented
340#547Dic4711yescharacteristic, complemented
341#547Dic4711yescharacteristic, complemented
342#546—21no—
343#542—21no—
344#539—22no—
345#533C2×D4711yescharacteristic
346#529C2×D4711yescharacteristic
347#525C2×D4712yes—
348C2×C22⋊Q8C2×D4712yes—
349#520—21no—
350C2×C4×D4C2×D4711yescharacteristic
351#436—941no—
352#435—942no—
353C8×D4—941no—
354#432—471no—
355#431—471no—
356#407—472no—
357C2×C22⋊C8—472no—
358#390—941no—
359#389—941no—
360#388—942no—
361#381—472no—
362#380—472no—
363C4×C4⋊C4—471no—
364#376—471no—
365#328—471no—
366#328—471no—
367#328—472no—
368#328—472no—
369#328—472no—
370#328—472no—
371#328—472no—
372#327—471no—
373#327—471no—
374#327—472no—
375C47#1760011yescharacteristic, complemented, Sylow
376C2×#41—21no—
377C2×#41C2×Dic4712yes—
378C22×Q8C2×Dic4712yes—
379C22×D4C22⋊D4711yescharacteristic
380C23×C4C22×D4712yes—
381C2×M16—942no—
382C22×C8—942no—
383C4⋊Q8—21no—
384C4⋊Q8C2×Dic4711yescharacteristic
385C4⋊Q8C2×Dic4711yescharacteristic
386#126—22no—
387#126—22no—
388#126—21no—
389#125—22no—
390#125—21no—
391#125C2×Dic4711yescharacteristic
392#124—21no—
393#124—22no—
394#124C2×Dic4711yescharacteristic
395#123—22no—
396#123—22no—
397#123—21no—
398#123—22no—
399#123—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C2×Dic4712yes—
407C22⋊Q8C2×Dic4712yes—
408C22⋊Q8C2×Dic4712yes—
409C22⋊Q8C22×D4711yescharacteristic
410C22⋊Q8C22×D4712yes—
411#121—22no—
412#121—21no—
413#121C2×Dic4712yes—
414#121C22×D4711yescharacteristic
415#120—22no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8C2×Dic4711yescharacteristic
419C4×Q8C2×Dic4712yes—
420C4×D4—21no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—21no—
424C4×D4—22no—
425C4×D4C2×Dic4711yescharacteristic
426C4×D4C2×Dic4711yescharacteristic
427C4×D4C2×Dic4711yescharacteristic
428C4×D4C2×Dic4712yes—
429C4×D4C22⋊D4711yescharacteristic
430C4×D4C22⋊D4712yes—
431C4×D4C22⋊D4711yescharacteristic
432#117—21no—
433#117—22no—
434#117—21no—
435#117C2×Dic4712yes—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—942no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4C2×Dic4712yes—
441C2×C4⋊C4C22⋊D4711yescharacteristic
442C2×C22⋊C4—22no—
443C2×C22⋊C4—942no—
444C2×C22⋊C4—942no—
445C2×C22⋊C4C22⋊D4712yes—
446C2×C42—941no—
447C2×C42—941no—
448C2×C42C22×D4711yescharacteristic
449C4⋊C8—941no—
450C4⋊C8—941no—
451C22⋊C8—471no—
452C22⋊C8—472no—
453C22⋊C8—471no—
454C22⋊C8—942no—
455C22⋊C8—942no—
456#97—941no—
457C4×C8—941no—
458#95—471no—
459#95—471no—
460#95—472no—
461#95—941no—
462#95—942no—
463C24C2×C22⋊D4712yes—
464#41—44no—
465#41—82no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—82no—
469C2×Q8C22×Dic4711yescharacteristic
470C2×Q8C22×Dic4712yes—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—42no—
475C2×D4C22⋊Dic4711yescharacteristic
476C2×D4C22⋊Dic4711yescharacteristic
477C2×D4C22⋊Dic4712yes—
478C2×D4C22×Dic4711yescharacteristic
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—82no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—942no—
487C22×C4C22×Dic4712yes—
488C22×C4C22×Dic4712yes—
489C22×C4C2×C22⋊D4711yescharacteristic
490C22×C4C2×C22⋊D4711yescharacteristic
491C22×C4C2×C22⋊D4712yes—
492C22×C4C2×C22⋊D4712yes—
493C22×C4C2×C22⋊D4712yes—
494C22×C4C23×D4711yescharacteristic
495M16—3762no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—942no—
499C2×C8—3762no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—3762no—
514C4⋊C4—1882no—
515C4⋊C4C22⋊Dic4711yescharacteristic
516C4⋊C4C22⋊Dic4712yes—
517C4⋊C4C22⋊Dic4711yescharacteristic
518C4⋊C4C22×Dic4711yescharacteristic
519C4⋊C4C22×Dic4712yes—
520C4⋊C4C22×Dic4711yescharacteristic
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—3762no—
533C22⋊C4—3762no—
534C22⋊C4C22⋊Dic4712yes—
535C22⋊C4C22⋊Dic4712yes—
536C22⋊C4C22⋊Dic4712yes—
537C22⋊C4C22⋊Dic4712yes—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—3761no—
542C42—1882no—
543C42C22×Dic4711yescharacteristic
544C42C22×Dic4711yescharacteristic
545C42C2×C22⋊D4711yescharacteristic
546C42C2×C22⋊D4711yescharacteristic
547C23—22no—
548C23—22no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23C2×C22⋊Dic4712yes—
553C23C2×C22⋊Dic4712yes—
554C23C22×C22⋊D4711yescharacteristic
555Q8—41no—
556Q8—42no—
557Q8—42no—
558Q8—82no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—41no—
563D4—44no—
564D4—41no—
565D4—41no—
566D4—82no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1882no—
589C2×C4—1881no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4C2×C22⋊Dic4711yescharacteristic
595C2×C4C2×C22⋊Dic4712yes—
596C2×C4C2×C22⋊Dic4711yescharacteristic
597C2×C4C2×C22⋊Dic4711yescharacteristic
598C2×C4C2×C22⋊Dic4712yes—
599C2×C4C2×C22⋊Dic4711yescharacteristic
600C2×C4C23×Dic4711yescharacteristic
601C2×C4C22×C22⋊D4711yescharacteristic
602C8—1881no—
603C8—1882no—
604C8—1881no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—82no—
617C22—82no—
618C22C22×C22⋊Dic4711yescharacteristic
619C22#1568993111yescharacteristic
620C22#1568993811yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—3761no—
634C4—3762no—
635C4—3761no—
636C4#1568914012yes—
637C2—21no—
638C2—21no—
639C2—42no—
640C2—42no—
641C2—81no—
642C2#2025184911yescharacteristic
643C2#2025192111yescharacteristic
644C2C2×#1568914011yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3520C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenter#3520D47
hypocenterC47#17600
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3520

Complements

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

NH ≅ G/Ntype
1. #20252423C2semidirect
2. #20246031C2semidirect
3. #20241960C2semidirect
4. #20241944C2semidirect
5. #20241931C2semidirect
6. #20241921C2semidirect
7. #20241913C2semidirect
8. #20241900C2semidirect
9. #20241887C2semidirect
10. #20241874C2semidirect
36. C47×#556C4semidirect
38. C47×#552C4semidirect
40. D4×Q8×C47C4semidirect
41. C47×#548C4semidirect
42. C47×#547C4semidirect
43. C47×#547C4semidirect
334. #556Dic47semidirect
336. #552Dic47semidirect
338. D4×Q8Dic47semidirect
339. #548Dic47semidirect
340. #547Dic47semidirect
341. #547Dic47semidirect
375. C47#17600semidirect

Maximal subgroups

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

subgroupindex
#202524232
#202460312
#202419602
#202419442
#202419312
#202419212
#202419132
#202419002
#202418872
#202418742
C47×#35202
#1760047