Finite Group Databasebeta

← #26221136

Subgroups — #26221136

Order 12032.

958 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689175C211yesmaximal, characteristic
2#20245781C211yesmaximal, characteristic
3C4×C4⋊C4×D47C211yesmaximal, characteristic
4#20244158C211yesmaximal, characteristic, complemented
5#20244156C211yesmaximal, characteristic, complemented
6#20244154C211yesmaximal, characteristic, complemented
7#20243582C211yesmaximal, characteristic
8#20243579C211yesmaximal, characteristic
9#20243577C211yesmaximal, characteristic
10#20243574C211yesmaximal, characteristic
11#107×Dic47C211yesmaximal, characteristic, complemented
12#20243280C211yesmaximal, characteristic, complemented
13#106×Dic47C211yesmaximal, characteristic, complemented
14#20243276C211yesmaximal, characteristic, complemented
15C47×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D47C2211yescharacteristic
17C2×C4⋊C4×D47C2211yescharacteristic
18C2×C42×D47—21no—
19C2×C42×D47C2211yescharacteristic
20#15689175C418yescomplemented
21C2×#13093958C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×(C2×D47)⋊C8C2211yescharacteristic
29C2×(C2×D47)⋊C8C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32#15688990—21no—
33#15688989C2211yescharacteristic
34#15688985—21no—
35#15688980C2211yescharacteristic
36#15688980C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4×Dic47⋊C4—21no—
42C4×C4⋊Dic47C2211yescharacteristic
43#15688974—21no—
44#15688970—21no—
45#15688968—21no—
46#95×D47—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×C4⋊C8×C47C2211yescharacteristic
56C47×#382C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D411yescharacteristic
64C22×C4×D47C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C4⋊C4×D47—22no—
74C4⋊C4×D47—22no—
75C4⋊C4×D47—22no—
76C4⋊C4×D47—22no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×C4×Dic47—21no—
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
87C42×D47—44no—
88C42×D47C2×C414yes—
89C42×D47Q814yes—
90#13093958—22no—
91#13093957—22no—
92#13093957—22no—
93#13093956—22no—
94#13093955—22no—
95#13093955—22no—
96C4⋊(C47⋊C8)C2×C414yescomplemented
97(C2×D47)⋊C8C2×C414yes—
98(C2×D47)⋊C8C2×C414yes—
99C8×Dic47C2×C414yescomplemented
100C8×Dic47C2×C414yescomplemented
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C42×C47—21no—
117C2×C42×C47C2311yescharacteristic
118#107×C47—22no—
119#106×C47—22no—
120C4⋊C8×C47C2×C414yescomplemented
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124C23×D47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C2×C4×D47—22no—
140C2×C4×D47—22no—
141C2×C4×D47—21no—
142C2×C4×D47—21no—
143C2×C4×D47—21no—
144C2×C4×D47—21no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—24no—
148C2×C4×D47—24no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47—42no—
153C2×C4×D47C4⋊C414yes—
154C2×C4×D47C4⋊C414yes—
155C2×C4×D47C22×C412yes—
156C2×C4×D47C2×Q812yes—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47C22×C412yes—
162C2×C8×C47C22×C412yes—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47#4112yes—
170C4⋊C4×C47#4112yes—
171C42×C47—42no—
172C42×C47C22×C412yes—
173C42×C47C2×Q812yes—
174(C2×D47)⋊C4—42no—
175(C2×D47)⋊C4—42no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47#4112yes—
183C4⋊Dic47#4112yes—
184Dic47⋊C4—24no—
185Dic47⋊C4—24no—
186C4×Dic47—42no—
187C4×Dic47—42no—
188C4×Dic47—42no—
189C4×Dic47—42no—
190C4×Dic47—42no—
191C4×Dic47C4⋊C414yes—
192C4×Dic47C4⋊C414yes—
193C4×Dic47C22×C412yes—
194C4×Dic47C2×Q812yes—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8C22×C412yes—
200C2×C47⋊C8C22×C412yes—
201C23×C47C22×D411yescharacteristic
202C22×D47—21no—
203C22×D47—21no—
204C22×D47—21no—
205C22×D47—21no—
206C22×D47C2×C4⋊C412yes—
207C22×D47#13512yes—
208C22×D47#13512yes—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
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×C47C2×C4⋊C412yes—
226C2×C4×C47C2×C4⋊C412yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47C2×C4⋊C412yes—
250C2×Dic47C2×C4⋊C412yes—
251C2×Dic47C2×C4⋊C412yes—
252C2×Dic47AGL(1,ℤ/8)12yes—
253C2×Dic47#13712yes—
254C4×D47—28no—
255C4×D47—22no—
256C4×D47—22no—
257C4×D47—22no—
258C4×D47—22no—
259C4×D47—28no—
260C4×D47—48no—
261C4×D47—48no—
262C8×C47—22no—
263C8×C47—22no—
264C47⋊C8—22no—
265C47⋊C8—22no—
266#21361—471nomaximal, Sylow
267C22×C47C22×C4⋊C411yescharacteristic
268C22×C47#52511yescharacteristic
269C22×C47#52511yescharacteristic
270C22×C47C2×#13511yescharacteristic
271C22×C47C2×#13511yescharacteristic
272C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C47C2×#13711yescharacteristic
274C2×D47—22no—
275C2×D47—22no—
276C2×D47—22no—
277C2×D47—22no—
278C2×D47—22no—
279C2×D47—22no—
280C2×D47#42714yes—
281C4×C47—22no—
282C4×C47—22no—
283C4×C47—42no—
284C4×C47—42no—
285C4×C47—81no—
286C4×C47—81no—
287C4×C47C4×#4111yescharacteristic
288C4×C47C4×#4111yescharacteristic
289C4×C47#53311yescharacteristic
290C4×C47#53311yescharacteristic
291Dic47—22no—
292Dic47—22no—
293Dic47—42no—
294Dic47—42no—
295Dic47—81no—
296Dic47—81no—
297Dic47#42814yes—
298C2×#431—471no—
299#2283—471no—
300C2×C4×C4⋊C4—471no—
301#1949—471no—
302#1949—471no—
303#1949—471no—
304#1949D4711yescharacteristic, complemented
305#1853—471no—
306#1853—471no—
307#1852—471no—
308#1852—471no—
309C4×#107—471no—
310C4×#107—471no—
311C4×#106—471no—
312C4×#106—471no—
313C2×C47#229611yescharacteristic
314C2×C47C2×#42711yescharacteristic
315C2×C47C2×#42811yescharacteristic
316C2×C47#310511yescharacteristic
317C2×C47#310511yescharacteristic
318C2×C47#310811yescharacteristic
319C2×C47#310811yescharacteristic
320D47—24no—
321C22×C4⋊C4—471no—
322C22×C4⋊C4—471no—
323C22×C42—471no—
324C22×C42—941no—
325#431—478no—
326C2×#107—471no—
327C2×#107—471no—
328C2×#107—471no—
329C2×#107C2×D4711yescharacteristic
330C2×#106—471no—
331C2×#106—471no—
332C2×#106—471no—
333C2×#106C2×D4711yescharacteristic
334C2×C4⋊C8—471no—
335C2×C4⋊C8C2×D4711yescharacteristic
336C2×C22⋊C8—471no—
337C2×C22⋊C8—471no—
338C2×C4×C8—471no—
339C2×C4×C8—471no—
340#383—941no—
341#382—471no—
342#382C2×D4711yescharacteristic
343#381—941no—
344#380—471no—
345#380—471no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4—471no—
351C4×C4⋊C4—941no—
352C4×C4⋊C4—941no—
353C4×C4⋊C4C2×D4711yescharacteristic
354C4×C22⋊C4—941no—
355#376—941no—
356C2×#95—941no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340C2×D4711yescharacteristic
364#340C2×D4711yescharacteristic
365C47#2136111yescharacteristic, complemented, Sylow
366C23×C4—471no—
367C23×C4—471no—
368C23×C4—471no—
369C23×C4—941no—
370C23×C4—941no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—942no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C4⋊C4C22×D4711yescharacteristic
389C2×C22⋊C4—941no—
390C2×C22⋊C4—941no—
391C2×C42—21no—
392C2×C42—471no—
393C2×C42—471no—
394C2×C42—471no—
395C2×C42—474no—
396C2×C42—474no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42—941no—
400C2×C42—941no—
401C2×C42—941no—
402C2×C42—1884no—
403C2×C42C22×D4711yescharacteristic
404#107—22no—
405#107—942no—
406#107—942no—
407#107—942no—
408#106—22no—
409#106—942no—
410#106—942no—
411#106—942no—
412C4⋊C8—474no—
413C4⋊C8C4×D4714yescomplemented
414C22⋊C8—474no—
415C22⋊C8—474no—
416C4×C8—474no—
417C4×C8—474no—
418#95—21no—
419#95—21no—
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—
429#95—941no—
430C24—471no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—474no—
439C22×C4—474no—
440C22×C4—472no—
441C22×C4—472no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—941no—
449C22×C4—941no—
450C22×C4—941no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—1882no—
457C22×C4—1882no—
458C22×C4—1882no—
459C22×C4—1882no—
460C22×C4D4×D4711yescharacteristic
461C22×C4D4×D4711yescharacteristic
462C22×C4C23×D4711yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—472no—
468C2×C8—472no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8C2×C4×D4712yes—
474C2×C8C2×C4×D4712yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—472no—
482C4⋊C4—472no—
483C4⋊C4—944no—
484C4⋊C4—944no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4#1062535212yes—
492C4⋊C4#1062535412yes—
493C22⋊C4—1882no—
494C22⋊C4—1882no—
495C42—42no—
496C42—474no—
497C42—474no—
498C42—472no—
499C42—472no—
500C42—1882no—
501C42—1882no—
502C42—1882no—
503C42—1882no—
504C42—1882no—
505C42C2×C4×D4712yes—
506C42Q8×D4712yes—
507C23—472no—
508C23—472no—
509C23—472no—
510C23—941no—
511C23—941no—
512C23—941no—
513C23—941no—
514C23C2×D4×D4711yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—472no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—948no—
537C2×C4—948no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—1888no—
549C2×C4—1888no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4C4⋊C4×D4712yes—
561C2×C4C4⋊C4×D4712yes—
562C2×C4C22×C4×D4711yescharacteristic
563C2×C4C2×Q8×D4711yescharacteristic
564C2×C4#41×D4711yescharacteristic
565C2×C4#41×D4711yescharacteristic
566C2×C4C2×#1062535211yescharacteristic
567C2×C4C2×#1062535411yescharacteristic
568C8—22no—
569C8—22no—
570C8—942no—
571C8—942no—
572C22—474no—
573C22—942no—
574C22—942no—
575C22—942no—
576C22—942no—
577C22—942no—
578C22—942no—
579C22C2×C4⋊C4×D4711yescharacteristic
580C22#1568963311yescharacteristic
581C22#1568963611yescharacteristic
582C22#135×D4711yescharacteristic
583C22#135×D4711yescharacteristic
584C22AGL(1,ℤ/8)×D4711yescharacteristic
585C22#137×D4711yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—474no—
593C4—942no—
594C4—942no—
595C4—3761no—
596C4—3761no—
597C4—1882no—
598C4—1882no—
599C4C4×#1062535211yescharacteristic
600C4C4×#1062535411yescharacteristic
601C4#1568967811yescharacteristic
602C4#1568967911yescharacteristic
603C2—944no—
604C2#2024586911yescharacteristic
605C2#427×D4711yescharacteristic
606C2#428×D4711yescharacteristic
607C2#2025343411yescharacteristic
608C2#2025344011yescharacteristic
609C2#2025346711yescharacteristic
610C2#2025347311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1949C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1949D47
hypocenterC47#21361
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #20244158C2semidirect
5. #20244156C2semidirect
6. #20244154C2semidirect
11. #107×Dic47C2semidirect
12. #20243280C2semidirect
13. #106×Dic47C2semidirect
14. #20243276C2semidirect
15. C47×#1949C2semidirect
20. #15689175C4semidirect
96. C4⋊(C47⋊C8)C2×C4semidirect
99. C8×Dic47C2×C4semidirect
100. C8×Dic47C2×C4semidirect
120. C4⋊C8×C47C2×C4semidirect
304. #1949D47semidirect
365. C47#21361semidirect
413. C4⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891752
#202457812
C4×C4⋊C4×D472
#202441582
#202441562
#202441542
#202435822
#202435792
#202435772
#202435742
#107×Dic472
#202432802
#106×Dic472
#202432762
C47×#19492
#2136147