Finite Group Databasebeta

← #26219792

Subgroups — #26219792

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689162C211yesmaximal, characteristic
2C2×#15689160C211yesmaximal, characteristic
3C2×#15689148C211yesmaximal, characteristic
4#20246013C211yesmaximal, characteristic
5#20245742C211yesmaximal, characteristic
6#20244159C211yesmaximal, characteristic, complemented
7#20244154C211yesmaximal, characteristic, complemented
8#20244153C211yesmaximal, characteristic, complemented
9#20244151C211yesmaximal, characteristic, complemented
10#20244148C211yesmaximal, characteristic, complemented
11#20244147C211yesmaximal, characteristic, complemented
12#20244145C211yesmaximal, characteristic, complemented
13#20243106C211yesmaximal, characteristic
14#340×D47C211yesmaximal, characteristic
15C47×#1949C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×#13093981C2211yescharacteristic
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C4⋊C4×D47C2211yescharacteristic
23#15689162—24no—
24#15689160—24no—
25C2×#13093958C2211yescharacteristic
26C2×#13093958C2211yescharacteristic
27C2×#13093957C2211yescharacteristic
28C2×#13093955C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31#15689148C418yescomplemented
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33C2×Dic47⋊C8C2211yescharacteristic
34C2×Dic47⋊C8C2211yescharacteristic
35#15689020—21no—
36#15689020—21no—
37#15688997—21no—
38#15688997—21no—
39#15688993C2211yescharacteristic
40#15688992C2211yescharacteristic
41#15688988C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44C4×Dic47⋊C4—21no—
45C4×C4⋊Dic47C2211yescharacteristic
46#15688974—21no—
47C4×(C2×D47)⋊C4—21no—
48#15688970—21no—
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×#107×C47C2211yescharacteristic
56C2×#106×C47C2211yescharacteristic
57C2×C4⋊C8×C47C2211yescharacteristic
58C47×#382C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47C2311yescharacteristic
65C2×#10625324—22no—
66C2×#10625324—22no—
67C2×#10625324C2×C414yes—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C2×C8×D47—22no—
71C2×C8×D47—22no—
72C2×C8×D47C2×C414yes—
73#13093982—22no—
74#13093982—22no—
75#13093982—22no—
76#13093982D414yes—
77#13093981C2×C414yes—
78#13093981Q814yes—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×C4⋊Dic47C2311yescharacteristic
85C2×Dic47⋊C4—21no—
86C2×Dic47⋊C4—21no—
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4D411yescharacteristic
90C2×Dic47⋊C4D411yescharacteristic
91C4⋊C4×D47—22no—
92C4⋊C4×D47—22no—
93C4⋊C4×D47—22no—
94C4⋊C4×D47Q814yes—
95C2×(C2×D47)⋊C4—21no—
96C2×(C2×D47)⋊C4—21no—
97C2×(C2×D47)⋊C4D411yescharacteristic
98C2×(C2×D47)⋊C4D411yescharacteristic
99C2×C4×Dic47—21no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—21no—
102C2×C4×Dic47—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47C2311yescharacteristic
105#13093958—22no—
106#13093958—22no—
107#13093957—22no—
108#13093955—22no—
109#13093955—22no—
110#13093955—22no—
111C4⋊(C47⋊C8)C2×C414yescomplemented
112Dic47⋊C8C2×C414yescomplemented
113Dic47⋊C8C2×C414yescomplemented
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C42×C47—21no—
126C2×C42×C47C2311yescharacteristic
127#107×C47—22no—
128#106×C47—22no—
129C4⋊C8×C47C2×C414yescomplemented
130#95×C47—21no—
131#95×C47—21no—
132#95×C47—21no—
133C23×D47C2×D411yescharacteristic
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C2×C4×D47—24no—
149C2×C4×D47—42no—
150C2×C4×D47—81no—
151C2×C4×D47—81no—
152C2×C4×D47—81no—
153C2×C4×D47—81no—
154C2×C4×D47C22×C412yes—
155C2×C4×D47C2×D412yes—
156C2×C4×D47C2×Q812yes—
157C2×C4×D47C2×Q812yes—
158C2×C4×D47#4112yes—
159C2×C4×D47#4112yes—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47C22×C412yes—
165C2×C8×C47C22×C412yes—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47C2×D412yes—
173C4⋊C4×C47C2×Q812yes—
174C42×C47—81no—
175C42×C47C22×C412yes—
176C42×C47C2×Q812yes—
177(C2×D47)⋊C4—24no—
178(C2×D47)⋊C4—42no—
179(C2×D47)⋊C4C4⋊C414yes—
180(C2×D47)⋊C4C4⋊C414yes—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—81no—
188C4⋊Dic47—81no—
189C4⋊Dic47C2×D412yes—
190C4⋊Dic47C2×Q812yes—
191Dic47⋊C4—24no—
192Dic47⋊C4—42no—
193Dic47⋊C4—81no—
194Dic47⋊C4—81no—
195Dic47⋊C4C4⋊C414yes—
196Dic47⋊C4C4⋊C414yes—
197C4×Dic47—24no—
198C4×Dic47—42no—
199C4×Dic47—81no—
200C4×Dic47—81no—
201C4×Dic47—81no—
202C4×Dic47C22×C412yes—
203C4×Dic47C2×Q812yes—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8C22×C412yes—
209C2×C47⋊C8C22×C412yes—
210#10625324—28no—
211C8×D47—28no—
212C23×C47C22×D411yescharacteristic
213C22×D47—21no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47C2×C4⋊C412yes—
218C22×D47#12112yes—
219C22×D47C22⋊Q812yes—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
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×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47C2×C4⋊C412yes—
237C2×C4×C47C2×C4⋊C412yes—
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×Q811yescharacteristic
241C2×C4×C47C22×Q811yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—22no—
249C2×Dic47—22no—
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—41no—
259C2×Dic47—41no—
260C2×Dic47C2×C4⋊C412yes—
261C2×Dic47C2×C4⋊C412yes—
262C2×Dic47C2×C4⋊C412yes—
263C2×Dic47#12112yes—
264C2×Dic47C22⋊Q812yes—
265C4×D47—84no—
266C4×D47—84no—
267C4×D47C4×D414yes—
268C4×D47C4×Q814yes—
269C4×D47#12514yes—
270C4×D47C4⋊Q814yes—
271C8×C47—22no—
272C8×C47—22no—
273C47⋊C8—22no—
274C47⋊C8—22no—
275#21175—471nomaximal, Sylow
276C22×C47C22×C4⋊C411yescharacteristic
277C22×C47C2×#12111yescharacteristic
278C22×C47C2×C22⋊Q811yescharacteristic
279C22×C47#57611yescharacteristic
280C22×C47#57711yescharacteristic
281C22×C47#57711yescharacteristic
282C22×C47#57811yescharacteristic
283C2×D47—22no—
284C2×D47—22no—
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47#38914yes—
290C4×C47—22no—
291C4×C47—22no—
292C4×C47—42no—
293C4×C47—42no—
294C4×C47—81no—
295C4×C47—81no—
296C4×C47C2×C4×D411yescharacteristic
297C4×C47C2×C4×Q811yescharacteristic
298C4×C47C2×#12511yescharacteristic
299C4×C47C2×C4⋊Q811yescharacteristic
300Dic47—22no—
301Dic47—22no—
302Dic47—42no—
303Dic47—42no—
304Dic47—81no—
305Dic47—81no—
306Dic47#38914yes—
307C2×#428—471no—
308C2×#427—471no—
309C2×#424—471no—
310#2317—471no—
311#2274—471no—
312#1949—471no—
313#1949—471no—
314#1949—471no—
315#1949D4711yescharacteristic, complemented
316#1948—471no—
317#1948—471no—
318#1947—471no—
319#1947—471no—
320#1736—471no—
321C2×#340—471no—
322C2×C47C2×#38911yescharacteristic
323C2×C47#291211yescharacteristic
324C2×C47#291311yescharacteristic
325C2×C47#304511yescharacteristic
326C2×C47#304611yescharacteristic
327C2×C47#308111yescharacteristic
328C2×C47#308211yescharacteristic
329D47—24no—
330C22×M16—471no—
331C23×C8—471no—
332C2×#117—471no—
333C2×#117—471no—
334C2×#117—471no—
335C22×C4⋊C4—471no—
336C22×C4⋊C4—471no—
337#428—944no—
338#427—944no—
339C2×#107—471no—
340C2×#107—471no—
341C2×#107—471no—
342C2×#107C2×D4711yescharacteristic
343C2×#106—471no—
344C2×#106—471no—
345C2×#106—471no—
346C2×#106C2×D4711yescharacteristic
347#424—478no—
348C2×C4⋊C8—471no—
349C2×C4⋊C8—471no—
350C2×C4⋊C8—471no—
351C2×C4⋊C8C2×D4711yescharacteristic
352#389—941no—
353#389—941no—
354#385—941no—
355#385—941no—
356#384—471no—
357#384—471no—
358#382—471no—
359#382C2×D4711yescharacteristic
360C4×C4⋊C4—471no—
361C4×C4⋊C4—471no—
362C4×C4⋊C4—471no—
363C4×C4⋊C4—941no—
364C4×C4⋊C4—941no—
365C4×C4⋊C4C2×D4711yescharacteristic
366C4×C22⋊C4—941no—
367C4×C22⋊C4—941no—
368#340—471no—
369#340—471no—
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#340C2×D4711yescharacteristic
375#340C2×D4711yescharacteristic
376C47#2117511yescharacteristic, complemented, Sylow
377C23×C4—471no—
378C23×C4—941no—
379C23×C4—941no—
380C2×M16—474no—
381C2×M16—942no—
382C2×M16—942no—
383C22×C8—471no—
384C22×C8—471no—
385C22×C8—474no—
386C22×C8—942no—
387C22×C8—942no—
388C22×C8C22×D4711yescharacteristic
389C22×C8C22×D4711yescharacteristic
390#117—474no—
391#117—474no—
392#117—474no—
393#117—942no—
394#117—942no—
395#117—942no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—474no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—942no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4C22×D4711yescharacteristic
413C2×C4⋊C4C22×D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C22⋊C4—471no—
417C2×C22⋊C4—471no—
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C42—21no—
421C2×C42—471no—
422C2×C42—941no—
423C2×C42—941no—
424C2×C42—941no—
425C2×C42—941no—
426C2×C42—941no—
427C2×C42C22×D4711yescharacteristic
428#107—22no—
429#107—942no—
430#107—942no—
431#107—942no—
432#106—22no—
433#106—942no—
434#106—942no—
435#106—942no—
436C4⋊C8—474no—
437C4⋊C8—474no—
438C4⋊C8—474no—
439C4⋊C8C4×D4714yescomplemented
440#95—21no—
441#95—21no—
442#95—21no—
443#95—941no—
444#95—941no—
445#95—941no—
446#95—941no—
447#95—941no—
448C24—471no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—471no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—944no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—3761no—
468C22×C4—3761no—
469C22×C4—3761no—
470C22×C4—3761no—
471C22×C4—1882no—
472C22×C4D4×D4711yescharacteristic
473C22×C4D4×D4711yescharacteristic
474C22×C4C23×D4711yescharacteristic
475M16—948no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—472no—
481C2×C8—472no—
482C2×C8—948no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8C2×C4×D4712yes—
488C2×C8C2×C4×D4712yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—474no—
496C4⋊C4—474no—
497C4⋊C4—472no—
498C4⋊C4—472no—
499C4⋊C4—944no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4—941no—
505C4⋊C4—941no—
506C4⋊C4—3761no—
507C4⋊C4—3761no—
508C4⋊C4—3761no—
509C4⋊C4—3761no—
510C4⋊C4—1882no—
511C4⋊C4D4×D4712yes—
512C4⋊C4Q8×D4712yes—
513C22⋊C4—474no—
514C22⋊C4—474no—
515C22⋊C4—944no—
516C22⋊C4—1882no—
517C42—81no—
518C42—472no—
519C42—472no—
520C42—944no—
521C42—3761no—
522C42—3761no—
523C42—3761no—
524C42—1882no—
525C42C2×C4×D4712yes—
526C42Q8×D4712yes—
527C23—472no—
528C23—472no—
529C23—472no—
530C23—941no—
531C23—941no—
532C23—941no—
533C23—941no—
534C23C2×D4×D4711yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—472no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—474no—
555C2×C4—474no—
556C2×C4—474no—
557C2×C4—474no—
558C2×C4—472no—
559C2×C4—472no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—3764no—
567C2×C4—3764no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4C4⋊C4×D4712yes—
579C2×C4C4⋊C4×D4712yes—
580C2×C4C22×C4×D4711yescharacteristic
581C2×C4C2×D4×D4711yescharacteristic
582C2×C4C2×Q8×D4711yescharacteristic
583C2×C4C2×Q8×D4711yescharacteristic
584C2×C4#41×D4711yescharacteristic
585C2×C4#41×D4711yescharacteristic
586C8—22no—
587C8—22no—
588C8—942no—
589C8—942no—
590C22—474no—
591C22—942no—
592C22—942no—
593C22—942no—
594C22—942no—
595C22—942no—
596C22—942no—
597C22C2×C4⋊C4×D4711yescharacteristic
598C22#121×D4711yescharacteristic
599C22C22⋊Q8×D4711yescharacteristic
600C22#1568992511yescharacteristic
601C22#1568993211yescharacteristic
602C22#1568993311yescharacteristic
603C22#1568993711yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—474no—
611C4—942no—
612C4—942no—
613C4—3761no—
614C4—3761no—
615C4—1882no—
616C4—1882no—
617C4C4×D4×D4711yescharacteristic
618C4C4×Q8×D4711yescharacteristic
619C4#125×D4711yescharacteristic
620C4C4⋊Q8×D4711yescharacteristic
621C2—944no—
622C2#389×D4711yescharacteristic
623C2#2025198211yescharacteristic
624C2#2025198511yescharacteristic
625C2#2025293011yescharacteristic
626C2#2025293911yescharacteristic
627C2#2025321611yescharacteristic
628C2#2025322911yescharacteristic
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#21175
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
6. #20244159C2semidirect
7. #20244154C2semidirect
8. #20244153C2semidirect
9. #20244151C2semidirect
10. #20244148C2semidirect
11. #20244147C2semidirect
12. #20244145C2semidirect
15. C47×#1949C2semidirect
31. #15689148C4semidirect
111. C4⋊(C47⋊C8)C2×C4semidirect
112. Dic47⋊C8C2×C4semidirect
113. Dic47⋊C8C2×C4semidirect
129. C4⋊C8×C47C2×C4semidirect
315. #1949D47semidirect
376. C47#21175semidirect
439. C4⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891622
C2×#156891602
C2×#156891482
#202460132
#202457422
#202441592
#202441542
#202441532
#202441512
#202441482
#202441472
#202441452
#202431062
#340×D472
C47×#19492
#2117547