Finite Group Databasebeta

← #26246287

Subgroups — #26246287

Order 12032.

796 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic47C211yesmaximal, characteristic
2#20245790C211yesmaximal, characteristic
3#20244949C211yesmaximal, characteristic, complemented
4#20244929C211yesmaximal, characteristic, complemented
5#20244867C211yesmaximal, characteristic, complemented
6#20244824C211yesmaximal, characteristic
7#20244821C211yesmaximal, characteristic
8#20244799C211yesmaximal, characteristic, complemented
9#20244581C211yesmaximal, characteristic
10#106×Dic47C211yesmaximal, characteristic, complemented
11#20243276C211yesmaximal, characteristic, complemented
12#103×Dic47C211yesmaximal, characteristic, complemented
13#103×Dic47C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic47C211yesmaximal, characteristic
15C47×#2035C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×#13093980—21no—
18C22×C4×Dic47C2211yescharacteristic
19C2×#13093956C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C8×Dic47C2211yescharacteristic
26C2×C8×Dic47C2211yescharacteristic
27#15689030C2211yescharacteristic
28#15689022C2211yescharacteristic
29#15689009—21no—
30#15688998—21no—
31#15688992C2211yescharacteristic
32#15688992C2211yescharacteristic
33#15688987—21no—
34#15688983C2211yescharacteristic
35#15688983C2211yescharacteristic
36#15688981—21no—
37C4⋊C4×Dic47—21no—
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41#15688974—21no—
42C22⋊C4×Dic47—21no—
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C2×C22⋊Q8×C47C2211yescharacteristic
53C2×#106×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C23×Dic47D411yescharacteristic
60C23×Dic47D411yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63Q8×Dic47—22no—
64Q8×Dic47—22no—
65#13093980—44no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—22no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093956—22no—
89#13093955—22no—
90#13093955—22no—
91#13093948—22no—
92#13093948—22no—
93C22⋊(C47⋊C8)—22no—
94C8×Dic47—22no—
95C8×Dic47—22no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C22×Q8×C47C2311yescharacteristic
105C23×C4×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C22⋊Q8×C47—22no—
109C22⋊Q8×C47—22no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47—21no—
116#106×C47—22no—
117#103×C47—22no—
118#103×C47—22no—
119C22⋊C8×C47—22no—
120#95×C47—21no—
121C24×C47C2×D411yescharacteristic
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47#4112yes—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—41no—
149C22×Dic47SD1614yes—
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47#4112yes—
160C4⋊C4×C47—24no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47#4112yes—
167C22⋊C4×C47—24no—
168C22⋊C4×C47—42no—
169C22⋊Dic47—42no—
170C22⋊Dic47—42no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—42no—
180C4⋊Dic47—42no—
181Dic47⋊C4—42no—
182Dic47⋊C4—42no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—22no—
188C4×Dic47—22no—
189C4×Dic47—42no—
190C4×Dic47—42no—
191C4×Dic47—42no—
192C4×Dic47—42no—
193C4×Dic47#4112yes—
194C4×Dic47#4112yes—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8#4112yes—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47C2×SD1612yes—
208C23×C47C22×D411yescharacteristic
209Q8×C47—22no—
210Q8×C47—22no—
211Q8×C47—41no—
212Q8×C47—41no—
213C2×C4×C47—21no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—28no—
245C2×Dic47—44no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47#12312yes—
259C2×Dic47#12312yes—
260C2×Dic47C2×SD1612yes—
261C2×Dic47C2×SD1612yes—
262C2×Dic47#13512yes—
263C2×Dic47#13712yes—
264C8×C47—22no—
265C8×C47—41no—
266C47⋊C8—22no—
267C47⋊C8—41no—
268#23756—471nomaximal, Sylow
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47C2×#12311yescharacteristic
277C22×C47#52511yescharacteristic
278C22×C47#53211yescharacteristic
279C22×C47C22×SD1611yescharacteristic
280C22×C47C2×#13511yescharacteristic
281C22×C47C2×#13711yescharacteristic
282C22×C47#57511yescharacteristic
283C4×C47—42no—
284C4×C47—41no—
285C4×C47—42no—
286C4×C47—81no—
287C4×C47—81no—
288C4×C47#52911yescharacteristic
289C4×C47#52911yescharacteristic
290C4×C47#52911yescharacteristic
291C4×C47#52911yescharacteristic
292Dic47—22no—
293Dic47—22no—
294Dic47—22no—
295Dic47—81no—
296Dic47—81no—
297Dic47—81no—
298Dic47—81no—
299Dic47#48314yes—
300C4×C22⋊Q8—471no—
301#2285—471no—
302#2099—471no—
303#2096—471no—
304#2084—471no—
305#2077—471no—
306#2077—471no—
307#2072—471no—
308#2035—471no—
309#2035D4711yescharacteristic
310C4×#106—471no—
311C4×#106—471no—
312C4×#103—471no—
313C4×#103—471no—
314C4×C22⋊C8—471no—
315C2×C47—42no—
316C2×C47#239511yescharacteristic
317C2×C47C2×#48311yescharacteristic
318C2×C47#309111yescharacteristic
319C2×C47#310311yescharacteristic
320C2×C47#310711yescharacteristic
321C2×C47#316711yescharacteristic
322C2×C47#317211yescharacteristic
323C2×C22⋊Q8C2×D4711yescharacteristic
324C2×C4×Q8—471no—
325C2×#117—941no—
326C22×C42—471no—
327C2×#106—471no—
328C2×#106—471no—
329C2×#106—471no—
330C2×#106C2×D4711yescharacteristic
331C2×#103—471no—
332C2×#103—471no—
333C2×#103C2×D4711yescharacteristic
334C2×#103C2×D4711yescharacteristic
335C2×C22⋊C8—471no—
336C2×C22⋊C8C2×D4711yescharacteristic
337C2×C4×C8—471no—
338C2×C4×C8—471no—
339#391—471no—
340#389—471no—
341#387—941no—
342#385—941no—
343#384—471no—
344#384—471no—
345#382—941no—
346#380—471no—
347#380—471no—
348#380—941no—
349#380C2×D4711yescharacteristic
350C4×C4⋊C4—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4—941no—
354C4×C4⋊C4—941no—
355C4×C22⋊C4—941no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340C2×D4711yescharacteristic
364#336—471no—
365#336—471no—
366C47#2375611yescharacteristic, complemented, Sylow
367C22×Q8C22×D4711yescharacteristic
368C23×C4—471no—
369C23×C4—471no—
370C23×C4C22×D4711yescharacteristic
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C4×Q8—942no—
378C4×Q8—942no—
379#117—1884no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C4⋊C4C22×D4711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—942no—
403C2×C42—942no—
404C2×C42—941no—
405C2×C42—941no—
406C2×C42—941no—
407C2×C42—941no—
408#106—22no—
409#106—942no—
410#106—942no—
411#106—942no—
412#103—22no—
413#103—22no—
414#103—942no—
415#103—942no—
416C22⋊C8—22no—
417C22⋊C8—942no—
418C4×C8—942no—
419C4×C8—942no—
420#95—21no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429C24D4×D4711yescharacteristic
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8#1062535212yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—471no—
447C22×C4—474no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—1881no—
460C22×C4D4×D4711yescharacteristic
461C22×C4C23×D4711yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—472no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8#1062535212yes—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—1882no—
489C4⋊C4—1882no—
490C4⋊C4—1882no—
491C4⋊C4—1882no—
492C4⋊C4—1882no—
493C4⋊C4—1882no—
494C4⋊C4#1062535212yes—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—1882no—
498C22⋊C4—1882no—
499C42—472no—
500C42—472no—
501C42—942no—
502C42—942no—
503C42—941no—
504C42—941no—
505C42—1882no—
506C42—1882no—
507C42—1882no—
508C42—1882no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23SD16×D4712yes—
516C23C2×D4×D4711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—472no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—472no—
546C2×C4—948no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—1884no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4#41×D4711yescharacteristic
567C2×C4#41×D4711yescharacteristic
568C2×C4#41×D4711yescharacteristic
569C2×C4C2×#1062535211yescharacteristic
570C2×C4C2×#1062535211yescharacteristic
571C2×C4C2×#1062535211yescharacteristic
572C8—22no—
573C8—41no—
574C8—942no—
575C8—1881no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22#123×D4711yescharacteristic
584C22#1568962911yescharacteristic
585C22#1568966811yescharacteristic
586C22C2×SD16×D4711yescharacteristic
587C22#135×D4711yescharacteristic
588C22#137×D4711yescharacteristic
589C22#1568992211yescharacteristic
590C4—42no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—474no—
596C4—942no—
597C4—942no—
598C4—942no—
599C4—3761no—
600C4—3761no—
601C4—3761no—
602C4—3761no—
603C4#1568965011yescharacteristic
604C4#1568965211yescharacteristic
605C4#1568965411yescharacteristic
606C4#1568965411yescharacteristic
607C2—42no—
608C2#2024664411yescharacteristic
609C2#483×D4711yescharacteristic
610C2#2025330611yescharacteristic
611C2#2025341411yescharacteristic
612C2#2025345211yescharacteristic
613C2#2025394211yescharacteristic
614C2#2025398911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2035C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2035D47
hypocenterC47#23756
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #20244949C2semidirect
4. #20244929C2semidirect
5. #20244867C2semidirect
8. #20244799C2semidirect
10. #106×Dic47C2semidirect
11. #20243276C2semidirect
12. #103×Dic47C2semidirect
13. #103×Dic47C2semidirect
366. C47#23756semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic472
#202457902
#202449492
#202449292
#202448672
#202448242
#202448212
#202447992
#202445812
#106×Dic472
#202432762
#103×Dic472
C22⋊C8×Dic472
C47×#20352
#2375647