Finite Group Databasebeta

← #26183828

Subgroups — #26183828

Order 12032.

1046 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257420C211yesmaximal, characteristic
2#20252423C211yesmaximal, characteristic, complemented
3#20241955C212yesmaximal, complemented
4#20241942C212yesmaximal, complemented
5#20241929C212yesmaximal, complemented
6#20241916C212yesmaximal, complemented
7#20241911C211yesmaximal, characteristic, complemented
8#20241895C211yesmaximal, characteristic, complemented
9#20241882C211yesmaximal, characteristic, complemented
10#20241872C211yesmaximal, characteristic, complemented
11C47×#2319C211yesmaximal, characteristic, complemented
12#15689838C411yescharacteristic, complemented
13#15689830C411yescharacteristic, complemented
14#15689821—21no—
15#15689819C412yescomplemented
16#15689794C411yescharacteristic, complemented
17#15689769C412yescomplemented
18#15689763—22no—
19D4×D188C411yescharacteristic, complemented
20D4×C22⋊D47—21no—
21#15689724—21no—
22#15689706—22no—
23#15689673C2211yescharacteristic
24#15689657C2211yescharacteristic
25#15689634C2212yes—
26C2×#13093998C2212yes—
27#15689595—21no—
28#15689201—21no—
29#15689195—22no—
30D4×C47⋊C8—21no—
31#15689180C2211yescharacteristic, complemented
32#15689176C2211yescharacteristic, complemented
33#15689087C2212yescomplemented
34C2×C22⋊(C47⋊C8)C2212yescomplemented
35#15688859C2211yescharacteristic, complemented
36#15688859C2212yescomplemented
37#15688859C2211yescharacteristic, complemented
38#15688856C2211yescharacteristic
39#15688856C2212yes—
40#15688856C2211yescharacteristic
41#15688856C2212yescomplemented
42#15688856C2212yescomplemented
43#15688856C2212yescomplemented
44#15688856C2212yescomplemented
45C2×C4×D4×C47C2211yescharacteristic
46C47×#390—21no—
47C47×#389—21no—
48C47×#388—22no—
49C47×#381C2212yescomplemented
50C47×#380C2212yescomplemented
51C4×C4⋊C4×C47C2211yescharacteristic, complemented
52C47×#376C2211yescharacteristic, complemented
53C2×#10625350—21no—
54C2×#10625350C2×C412yes—
55C22×D188C2×C412yes—
56C22×C22⋊D47—22no—
57C2×D4×D47—21no—
58C2×#10625329—22no—
59C22×C47⋊C8—22no—
60#13094024C2×C411yescharacteristic
61#13094021C2×C411yescharacteristic
62#13094020—21no—
63#13094019—22no—
64#13094018—21no—
65#13094016C2×C411yescharacteristic
66#13094015—21no—
67#13094012—21no—
68#13094012C2×C411yescharacteristic
69#13094010—22no—
70#13094009—22no—
71#13094007—21no—
72#13094007—22no—
73#13094006—22no—
74#13094003C2×C412yes—
75#13094003C2311yescharacteristic
76#13094001—22no—
77#13093999—21no—
78#13093998—21no—
79#13093998—22no—
80#13093998—22no—
81#13093998C2×C412yes—
82#13093998C2×C412yes—
83#13093998C2×C412yes—
84#13093998C2311yescharacteristic
85#13093998C2312yes—
86#13093997—22no—
87#13093997—22no—
88#13093996—22no—
89#13093995—21no—
90#13093994—22no—
91#13093993—22no—
92#13093992—22no—
93C4×Dic94C2×C411yescharacteristic
94C4×D188—21no—
95C4×D188C2×C411yescharacteristic
96C4×D188C2×C412yes—
97D4×Dic47—21no—
98#13093986—21no—
99C4×C22⋊D47—21no—
100C4×C22⋊D47—22no—
101C4×C22⋊D47—22no—
102(C22⋊D47)⋊C4—22no—
103#13093983C2×C412yes—
104#13093982—21no—
105#13093981—21no—
106C2×C4⋊Dic47C2×C412yes—
107C2×(C2×D47)⋊C4—22no—
108C22⋊C4×D47—22no—
109C4⋊(C47⋊C8)—21no—
110C4⋊(C47⋊C8)—21no—
111C22⋊(C47⋊C8)—22no—
112C22⋊(C47⋊C8)—22no—
113C22⋊(C47⋊C8)C2311yescharacteristic
114C22⋊(C47⋊C8)C2312yes—
115C22⋊(C47⋊C8)C2311yescharacteristic
116#13093935—21no—
117C4×C47⋊C8—21no—
118C22×D4×C47D411yescharacteristic
119C23×C4×C47C2312yes—
120C4×D4×C47C2×C411yescharacteristic
121C4×D4×C47C2×C412yes—
122C4×D4×C47C2×C411yescharacteristic
123C4×D4×C47D411yescharacteristic
124C4×D4×C47D411yescharacteristic
125C4×D4×C47D412yes—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47—22no—
128C2×C4⋊C4×C47D411yescharacteristic
129C2×C22⋊C4×C47—22no—
130C2×C22⋊C4×C47—22no—
131C2×C22⋊C4×C47D412yes—
132C2×C42×C47—21no—
133C2×C42×C47—21no—
134C2×C42×C47C2311yescharacteristic
135#95×C47—21no—
136#95×C47—22no—
137#95×C47C2311yescharacteristic
138#95×C47C2311yescharacteristic
139#95×C47C2312yes—
140C24×C47C2×D412yes—
141C23×D47—22no—
142C2×D4×C47—42no—
143C2×D4×C47C22⋊C411yescharacteristic
144C2×D4×C47C22⋊C411yescharacteristic
145C2×D4×C47C22⋊C412yes—
146C22×C4×C47—21no—
147C22×C4×C47—22no—
148C22×C4×C47—21no—
149C22×C4×C47—82no—
150C22×C4×C47C22×C412yes—
151C22×C4×C47C22×C412yes—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2411yescharacteristic
158C2×C22⋊D47—22no—
159C2×C22⋊D47—22no—
160C2×C22⋊D47—21no—
161C2×C22⋊D47—22no—
162C2×C22⋊D47—22no—
163C2×C22⋊D47—22no—
164C2×C22⋊D47—82no—
165C2×C22⋊D47—82no—
166C22×Dic47—22no—
167D4×D47—44no—
168#10625350—82no—
169#10625350—81no—
170#10625350—81no—
171C2×D188—21no—
172C2×D188—21no—
173C2×D188—22no—
174C2×D188—82no—
175C2×D188C22×C411yescharacteristic
176C2×D188C22×C412yes—
177C2×C4×D47—21no—
178C2×C4×D47—22no—
179C2×C4×D47—22no—
180C2×Dic94—21no—
181C2×Dic94C22×C411yescharacteristic
182C4⋊C4×C47—42no—
183C4⋊C4×C47—82no—
184C4⋊C4×C47C22⋊C411yescharacteristic
185C4⋊C4×C47C22⋊C411yescharacteristic
186C4⋊C4×C47C22⋊C412yes—
187C22⋊C4×C47—82no—
188C22⋊C4×C47—82no—
189C22⋊C4×C47C22⋊C412yes—
190C22⋊C4×C47C22⋊C412yes—
191C22⋊C4×C47C22⋊C412yes—
192C22⋊C4×C47C22⋊C412yes—
193C42×C47—42no—
194C42×C47—81no—
195C42×C47C22×C411yescharacteristic
196C42×C47C22×C411yescharacteristic
197C42×C47C2×D411yescharacteristic
198C42×C47C2×D411yescharacteristic
199C22⋊Dic47—22no—
200C22⋊Dic47—22no—
201C22⋊Dic47—21no—
202(C2×D47)⋊C4—21no—
203(C2×D47)⋊C4—22no—
204(C2×D47)⋊C4—22no—
205(C2×D47)⋊C4—21no—
206(C2×D47)⋊C4—22no—
207(C2×D47)⋊C4—22no—
208(C2×D47)⋊C4—21no—
209(C2×D47)⋊C4—21no—
210(C2×D47)⋊C4—22no—
211C4⋊Dic47—22no—
212C4⋊Dic47—21no—
213C4⋊Dic47C22×C411yescharacteristic
214C4⋊Dic47C22×C412yes—
215C4⋊Dic47C22×C411yescharacteristic
216Dic47⋊C4—21no—
217Dic47⋊C4—21no—
218Dic47⋊C4—22no—
219Dic47⋊C4—21no—
220Dic47⋊C4—21no—
221C4×Dic47—21no—
222C4×Dic47—22no—
223#10625329—82no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—21no—
226C2×C47⋊C8—22no—
227C2×C47⋊C8—82no—
228C23×C47—22no—
229C23×C47—22no—
230C23×C47—42no—
231C23×C47—42no—
232C23×C47C2×C22⋊C412yes—
233C23×C47C2×C22⋊C412yes—
234C23×C47C22×D411yescharacteristic
235C22×D47—21no—
236C22×D47—22no—
237C22×D47—44no—
238C22×D47—82no—
239C22×D47—82no—
240D4×C47—22no—
241D4×C47—22no—
242D4×C47—42no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47—41no—
258C2×C4×C47—42no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47C2×C22⋊C412yes—
262C2×C4×C47C2×C22⋊C412yes—
263C2×C4×C47C2×C22⋊C411yescharacteristic
264C2×C4×C47C2×C22⋊C411yescharacteristic
265C2×C4×C47C2×C22⋊C411yescharacteristic
266C2×C4×C47C2×C22⋊C411yescharacteristic
267C2×C4×C47C23×C411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C22⋊D47—44no—
270C22⋊D47—42no—
271C22⋊D47—41no—
272C22⋊D47—42no—
273C22⋊D47—41no—
274C22⋊D47—82no—
275C22⋊D47—82no—
276C22⋊D47—82no—
277C2×Dic47—21no—
278C2×Dic47—21no—
279C2×Dic47—21no—
280C2×Dic47—22no—
281C2×Dic47—82no—
282D188—41no—
283D188—42no—
284D188—42no—
285D188—82no—
286D188—81no—
287C4×D47—42no—
288C4×D47—82no—
289C4×D47—82no—
290Dic94—41no—
291Dic94—81no—
292C47⋊C8—41no—
293C47⋊C8—42no—
294C47⋊C8—41no—
295#17599—471nomaximal, Sylow
296C22×C47—21no—
297C22×C47—21no—
298C22×C47—22no—
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—22no—
302C22×C47—22no—
303C22×C47—22no—
304C22×C47—82no—
305C22×C47—82no—
306C22×C47C22×C22⋊C411yescharacteristic
307C22×C47#57611yescharacteristic
308C22×C47#57711yescharacteristic
309C2×D47—42no—
310C2×D47—41no—
311C2×D47—42no—
312C2×D47—41no—
313C2×D47—41no—
314C2×D47—42no—
315C2×D47—164no—
316C4×C47—21no—
317C4×C47—21no—
318C4×C47—22no—
319C4×C47—22no—
320C4×C47—21no—
321C4×C47—21no—
322C4×C47—21no—
323C4×C47—81no—
324C4×C47—82no—
325C4×C47—81no—
326C4×C47#42112yes—
327Dic47—41no—
328Dic47—42no—
329Dic47—41no—
330Dic47—41no—
331#3519—471no—
332#2974—471no—
333#2319D4711yescharacteristic, complemented
334#1519—472no—
335#1516—472no—
336#1513—472no—
337#1510—472no—
338#1509—471no—
339#1505—471no—
340#1502—471no—
341#1500—471no—
342C2×C47—21no—
343C2×C47—21no—
344C2×C47—42no—
345C2×C47—42no—
346C2×C47#288311yescharacteristic
347C2×C47#289711yescharacteristic
348C2×C47C2×#42111yescharacteristic
349D47—82no—
350D47—81no—
351#555—471no—
352#554—471no—
353#553—472no—
354#553—941no—
355#550—471no—
356#546—472no—
357#546—942no—
358D42—471no—
359D42—941no—
360#540—941no—
361#538—942no—
362#532—471no—
363#529—471no—
364#525—472no—
365C2×#121—472no—
366#518—941no—
367C2×C4×D4C2×D4711yescharacteristic
368#436—941no—
369#435—942no—
370C8×D4—941no—
371#432—471no—
372#431—471no—
373#407—472no—
374C2×C22⋊C8—472no—
375#390—21no—
376#389—21no—
377#388—22no—
378#381C2×D4712yescomplemented
379#380C2×D4712yescomplemented
380C4×C4⋊C4C2×D4711yescharacteristic, complemented
381#376C2×D4711yescharacteristic, complemented
382#328—471no—
383#328—471no—
384#328—472no—
385#327—471no—
386#327—471no—
387#327—472no—
388#327—472no—
389#327—472no—
390#327—472no—
391#327—472no—
392C47#1759911yescharacteristic, complemented, Sylow
393C2×#41—472no—
394C2×#41—941no—
395C22×D4—472no—
396C22×D4—942no—
397C22×D4—941no—
398C22×D4C22⋊D4711yescharacteristic
399C23×C4C22×D4712yes—
400C2×M16—942no—
401C22×C8—942no—
402C4⋊Q8—471no—
403#127—471no—
404#127—941no—
405#126—942no—
406#126—941no—
407#125—471no—
408#125—941no—
409#124—471no—
410#124—942no—
411#124—941no—
412#123—942no—
413#123—941no—
414#123—942no—
415#123—942no—
416C22⋊Q8—472no—
417C22⋊Q8—471no—
418C22⋊Q8—942no—
419#121—472no—
420#121—472no—
421#121—472no—
422#121—471no—
423#121—472no—
424#121—942no—
425#121—941no—
426#121—942no—
427#121—942no—
428#121—941no—
429#121—941no—
430#121—942no—
431#121—942no—
432#120—942no—
433#120—942no—
434#120—942no—
435C4×Q8—471no—
436C4×D4—471no—
437C4×D4—472no—
438C4×D4—941no—
439C4×D4—942no—
440C4×D4—942no—
441C4×D4—941no—
442C4×D4—941no—
443C4×D4—942no—
444C4×D4—941no—
445C4×D4C4×D4711yescharacteristic
446C4×D4C4×D4712yes—
447C4×D4C4×D4711yescharacteristic
448C4×D4D18811yescharacteristic
449C4×D4D18812yes—
450C4×D4D18811yescharacteristic
451#117—472no—
452#117—941no—
453#117—941no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—22no—
456C2×C4⋊C4—472no—
457C2×C4⋊C4C22⋊D4711yescharacteristic
458C2×C22⋊C4—22no—
459C2×C22⋊C4—22no—
460C2×C22⋊C4—942no—
461C2×C22⋊C4—942no—
462C2×C22⋊C4C22⋊D4712yes—
463C2×C42—21no—
464C2×C42—21no—
465C2×C42C22×D4711yescharacteristic
466C4⋊C8—941no—
467C4⋊C8—941no—
468C22⋊C8—471no—
469C22⋊C8—472no—
470C22⋊C8—471no—
471C22⋊C8—942no—
472C22⋊C8—942no—
473#97—941no—
474C4×C8—941no—
475#95—21no—
476#95—22no—
477#95C22×D4711yescharacteristic
478#95C22×D4711yescharacteristic
479#95C22×D4712yes—
480C24—942no—
481C24C2×C22⋊D4712yes—
482#41—3761no—
483#41—3761no—
484#41—3762no—
485C2×Q8—471no—
486C2×Q8—941no—
487C2×D4—42no—
488C2×D4—471no—
489C2×D4—472no—
490C2×D4—942no—
491C2×D4—942no—
492C2×D4—942no—
493C2×D4—942no—
494C2×D4—942no—
495C2×D4—941no—
496C2×D4—941no—
497C2×D4—942no—
498C2×D4—941no—
499C2×D4—3762no—
500C2×D4—3762no—
501C2×D4—3762no—
502C2×D4—1884no—
503C2×D4(C2×D47)⋊C411yescharacteristic
504C2×D4(C2×D47)⋊C411yescharacteristic
505C2×D4(C2×D47)⋊C412yes—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—82no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—941no—
514C22×C4C2×C4×D4712yes—
515C22×C4C2×C4×D4712yes—
516C22×C4C2×D18812yes—
517C22×C4C2×D18812yes—
518C22×C4C2×C22⋊D4711yescharacteristic
519C22×C4C2×C22⋊D4711yescharacteristic
520C22×C4C2×C22⋊D4712yes—
521C22×C4C23×D4711yescharacteristic
522M16—3762no—
523C2×C8—941no—
524C2×C8—941no—
525C2×C8—942no—
526C2×C8—3762no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—471no—
530C4⋊C4—472no—
531C4⋊C4—471no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—942no—
535C4⋊C4—941no—
536C4⋊C4—942no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4(C2×D47)⋊C411yescharacteristic
540C4⋊C4(C2×D47)⋊C411yescharacteristic
541C4⋊C4(C2×D47)⋊C412yes—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—942no—
545C22⋊C4—942no—
546C22⋊C4—941no—
547C22⋊C4—942no—
548C22⋊C4—941no—
549C22⋊C4—942no—
550C22⋊C4—941no—
551C22⋊C4—942no—
552C22⋊C4—941no—
553C22⋊C4—942no—
554C22⋊C4—941no—
555C22⋊C4—942no—
556C22⋊C4(C2×D47)⋊C412yes—
557C22⋊C4(C2×D47)⋊C412yes—
558C22⋊C4(C2×D47)⋊C412yes—
559C22⋊C4(C2×D47)⋊C412yes—
560C42—42no—
561C42—81no—
562C42—942no—
563C42—941no—
564C42C2×C4×D4711yescharacteristic
565C42C2×C4×D4711yescharacteristic
566C42C2×D18811yescharacteristic
567C42C2×D18811yescharacteristic
568C23—22no—
569C23—22no—
570C23—42no—
571C23—42no—
572C23—941no—
573C23—942no—
574C23—3762no—
575C23—3762no—
576C23—1884no—
577C23C2×(C2×D47)⋊C412yes—
578C23C2×(C2×D47)⋊C412yes—
579C23C22×C22⋊D4711yescharacteristic
580Q8—3761no—
581Q8—1881no—
582D4—22no—
583D4—22no—
584D4—42no—
585D4—3762no—
586D4—3761no—
587D4—3762no—
588D4—3762no—
589D4—3762no—
590D4—1881no—
591D4—1882no—
592D4—1884no—
593D4—1881no—
594D4—1882no—
595D4—1881no—
596D4—1882no—
597D4—1882no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—941no—
617C2×C4—942no—
618C2×C4—941no—
619C2×C4—941no—
620C2×C4—3762no—
621C2×C4—3762no—
622C2×C4—3762no—
623C2×C4—1882no—
624C2×C4C2×(C2×D47)⋊C412yes—
625C2×C4C2×(C2×D47)⋊C411yescharacteristic
626C2×C4C2×(C2×D47)⋊C411yescharacteristic
627C2×C4C2×(C2×D47)⋊C411yescharacteristic
628C2×C4C2×(C2×D47)⋊C412yes—
629C2×C4C2×(C2×D47)⋊C411yescharacteristic
630C2×C4C22×C4×D4711yescharacteristic
631C2×C4C22×D18811yescharacteristic
632C8—1881no—
633C8—1882no—
634C8—1881no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—82no—
644C22—82no—
645C22—1882no—
646C22—1881no—
647C22—1881no—
648C22—1882no—
649C22—1882no—
650C22—1881no—
651C22—7524no—
652C22C22×(C2×D47)⋊C411yescharacteristic
653C22#1568992611yescharacteristic
654C22#1568993111yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—22no—
661C4—22no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—1881no—
666C4—1882no—
667C4—1881no—
668C4—1881no—
669C4#1568914112yes—
670C2—21no—
671C2—21no—
672C2—42no—
673C2—42no—
674C2—3761no—
675C2—3762no—
676C2#2025184811yescharacteristic
677C2#2025191111yescharacteristic
678C2C2×#1568914111yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2319C2
socleC22×C47C22×C22⋊C4
prime radical#20257420C2
cosocleC22×C4×C47C24
hypercenter#2319D47
hypocenterC47#17599
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #20252423C2semidirect
3. #20241955C2semidirect
4. #20241942C2semidirect
5. #20241929C2semidirect
6. #20241916C2semidirect
7. #20241911C2semidirect
8. #20241895C2semidirect
9. #20241882C2semidirect
10. #20241872C2semidirect
11. C47×#2319C2semidirect
12. #15689838C4semidirect
13. #15689830C4semidirect
15. #15689819C4semidirect
16. #15689794C4semidirect
17. #15689769C4semidirect
19. D4×D188C4semidirect
31. #15689180C22semidirect
32. #15689176C22semidirect
33. #15689087C22semidirect
34. C2×C22⋊(C47⋊C8)C22semidirect
35. #15688859C22semidirect
36. #15688859C22semidirect
37. #15688859C22semidirect
41. #15688856C22semidirect
42. #15688856C22semidirect
43. #15688856C22semidirect
44. #15688856C22semidirect
49. C47×#381C22semidirect
50. C47×#380C22semidirect
51. C4×C4⋊C4×C47C22semidirect
52. C47×#376C22semidirect
333. #2319D47semidirect
378. #381C2×D47semidirect
379. #380C2×D47semidirect
380. C4×C4⋊C4C2×D47semidirect
381. #376C2×D47semidirect
392. C47#17599semidirect

Maximal subgroups

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

subgroupindex
#202574202
#202524232
#202419552
#202419422
#202419292
#202419162
#202419112
#202418952
#202418822
#202418722
C47×#23192
#1759947