Finite Group Databasebeta

← #26231747

Subgroups — #26231747

Order 12032.

1542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C47⋊C8C211yesmaximal, characteristic, complemented
2C2×#15689119C212yesmaximal, complemented
3#20246337C211yesmaximal, characteristic, complemented
4#20244316C211yesmaximal, characteristic, complemented
5#20244195C211yesmaximal, characteristic, complemented
6#20243817C212yesmaximal, complemented
7#20243711C212yesmaximal, complemented
8#20243393C211yesmaximal, characteristic, complemented
9#20243328C212yesmaximal, complemented
10C4×#13093948C211yesmaximal, characteristic, complemented
11C2×C47×#550C211yesmaximal, characteristic
12C23×C47⋊C8C2212yes—
13C2×#13093983C2212yes—
14D4×C47⋊C8—24no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16#15689119—28no—
17C2×#13093948C2211yescharacteristic
18C2×#13093948C2212yes—
19C2×#13093948C2211yescharacteristic
20C2×#13093945C2212yes—
21C2×#13093945C2212yes—
22C2×#13093945C2212yes—
23C2×#13093945C2212yes—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C4×C47⋊C8C2211yescharacteristic
26#15689026—21no—
27#15689016—22no—
28#15688993C2211yescharacteristic
29#15688983C2212yes—
30C4⋊C4×Dic47—21no—
31C4×C4⋊Dic47C2211yescharacteristic
32#15688885C2211yescharacteristic
33#15688885C2211yescharacteristic
34#15688885C2212yes—
35C22×#41×C47C2212yes—
36C47×#550—24no—
37C47×#550C418yescomplemented
38C2×#127×C47—21no—
39C2×#127×C47C2211yescharacteristic
40C2×#121×C47—22no—
41C2×#121×C47C2212yes—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47—21no—
44C2×C4×D4×C47C2211yescharacteristic
45C22×C47⋊C8—24no—
46C22×C47⋊C8—24no—
47C22×C47⋊C8—42no—
48C22×C47⋊C8C2311yescharacteristic
49C22×C47⋊C8C2312yes—
50C22×C47⋊C8C2311yescharacteristic
51#13093983—24no—
52#13093983—42no—
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47C2311yescharacteristic
56C2×C4⋊Dic47C2312yes—
57C2×C4⋊Dic47C2311yescharacteristic
58C2×C22⋊Dic47—22no—
59C2×C22⋊Dic47—22no—
60C2×C4×Dic47—21no—
61C2×C4×Dic47—22no—
62C4⋊(C47⋊C8)—22no—
63#13093948—24no—
64#13093948—22no—
65#13093948D414yescomplemented
66#13093945—24no—
67#13093945—24no—
68#13093945—42no—
69#13093945—42no—
70C22⋊(C47⋊C8)—24no—
71C4×C47⋊C8—22no—
72C47⋊#95—21no—
73C47⋊#95—22no—
74C2×#41×C47—24no—
75C2×#41×C47—24no—
76C2×#41×C47—24no—
77C2×#41×C47—42no—
78C2×#41×C47—42no—
79C2×#41×C47C2×C418yes—
80C22×Q8×C47C2311yescharacteristic
81C22×D4×C47—22no—
82C22×D4×C47—22no—
83C22×D4×C47—21no—
84C22×D4×C47—21no—
85C22×D4×C47D411yescharacteristic
86C22×D4×C47C2312yes—
87C23×C4×C47—22no—
88C23×C4×C47C2312yes—
89#127×C47—28no—
90#127×C47—22no—
91#127×C47C2×C414yes—
92#121×C47—24no—
93#121×C47—216no—
94#121×C47C2×C418yes—
95C4×Q8×C47—22no—
96C4×Q8×C47C2×C414yes—
97C4×D4×C47—28no—
98C4×D4×C47C2×C414yes—
99C4×D4×C47D414yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47—22no—
104C2×C22⋊C4×C47D412yes—
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107C24×C47—22no—
108C24×C47C2×D412yes—
109#41×C47—216no—
110#41×C47—48no—
111#41×C47—48no—
112#41×C47—48no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C22×C412yes—
117C2×Q8×C47C2×D412yes—
118C2×Q8×C47#4112yes—
119C2×D4×C47—28no—
120C2×D4×C47—28no—
121C2×D4×C47—24no—
122C2×D4×C47—24no—
123C2×D4×C47—22no—
124C2×D4×C47—22no—
125C2×D4×C47—22no—
126C2×D4×C47—22no—
127C2×D4×C47—22no—
128C2×D4×C47—22no—
129C2×D4×C47—22no—
130C2×D4×C47—44no—
131C2×D4×C47—44no—
132C2×D4×C47—42no—
133C2×D4×C47—42no—
134C2×D4×C47—81no—
135C2×D4×C47—81no—
136C2×D4×C47—82no—
137C2×D4×C47—82no—
138C2×D4×C47C22⋊C414yes—
139C2×D4×C47C22×C414yes—
140C22×C4×C47—21no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—21no—
146C22×C4×C47—28no—
147C22×C4×C47—44no—
148C22×C4×C47—44no—
149C22×C4×C47C22×C414yes—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D412yes—
152C22×C4×C47C2×D414yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158C4⋊C4×C47—24no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47C22⋊C414yes—
161C4⋊C4×C47C22×C412yes—
162C22⋊C4×C47—28no—
163C22⋊C4×C47C22⋊C418yes—
164C42×C47—24no—
165C42×C47C22×C412yes—
166C42×C47C2×D412yes—
167C22⋊Dic47—44no—
168C22⋊Dic47—82no—
169C4⋊Dic47—21no—
170C4⋊Dic47—22no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—22no—
174C4⋊Dic47—42no—
175C4⋊Dic47—81no—
176C4⋊Dic47C2×D412yes—
177C4×Dic47—44no—
178C4×Dic47—42no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—48no—
188C2×C47⋊C8C2×D412yes—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—24no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47C2×C22⋊C414yes—
203C23×C47C22×D411yescharacteristic
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47—22no—
207Q8×C47C4×D414yes—
208D4×C47—24no—
209D4×C47—24no—
210D4×C47—28no—
211D4×C47—416no—
212D4×C47—42no—
213D4×C47—42no—
214D4×C47—48no—
215D4×C47—84no—
216D4×C47—88no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—416no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47C2×C22⋊C412yes—
233C2×C4×C47C2×C22⋊C414yes—
234C2×C4×C47C2×C22⋊C412yes—
235C2×C4×C47#13512yes—
236C2×C4×C47#13512yes—
237C2×C4×C47C23×C411yescharacteristic
238C2×C4×C47C22×D411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47H(5,𝔽2)11yescharacteristic
242C2×C4×C47H(5,𝔽2)11yescharacteristic
243C2×Dic47—22no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—42no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—41no—
252C47⋊C8—22no—
253C47⋊C8—42no—
254C47⋊C8—41no—
255#22418—471nomaximal, Sylow
256C22×C47—24no—
257C22×C47—24no—
258C22×C47—24no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—44no—
264C22×C47—44no—
265C22×C47—44no—
266C22×C47—82no—
267C22×C47—82no—
268C22×C47—82no—
269C22×C47—82no—
270C22×C47C22×C22⋊C411yescharacteristic
271C22×C47D4211yescharacteristic
272C22×C47#54611yescharacteristic
273C22×C47C2×#13511yescharacteristic
274C22×C47C2×#13511yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C4×C47—24no—
278C4×C47—24no—
279C4×C47—22no—
280C4×C47—42no—
281C4×C47#41614yescomplemented
282C4×C47C2×C4×D411yescharacteristic
283C4×C47#51811yescharacteristic
284C4×C47D4211yescharacteristic
285C4×C47#54611yescharacteristic
286Dic47—42no—
287Dic47—82no—
288Dic47—81no—
289C2×#550D4711yescharacteristic
290C2×C8×D4—471no—
291C2×#416—472no—
292#2354—471no—
293#1982—471no—
294#1957—471no—
295#1893—472no—
296#1875—472no—
297#1803—471no—
298#1788—472no—
299C4×#103—471no—
300C2×C47—44no—
301C2×C47—84no—
302C2×C47D4×C22⋊C411yescharacteristic
303C2×C47C2×#41611yescharacteristic
304C2×C47#289711yescharacteristic
305C2×C47#328411yescharacteristic
306C2×C47#328711yescharacteristic
307C2×C47#330411yescharacteristic
308C2×C47#330511yescharacteristic
309C22×#41C2×D4712yes—
310C23×C8—472no—
311#550—24no—
312#550Dic4718yescomplemented
313C2×#127—21no—
314C2×#127C2×D4711yescharacteristic
315C2×#121—22no—
316C2×#121C2×D4712yes—
317C2×C4×Q8C2×D4711yescharacteristic
318C2×C4×D4—21no—
319C2×C4×D4C2×D4711yescharacteristic
320C2×#117—472no—
321C8×D4—944no—
322C2×C4⋊C8—471no—
323#416—948no—
324C2×#103—471no—
325C2×#103—472no—
326C2×#103—471no—
327C2×#102—472no—
328C2×#102—472no—
329C2×#102—472no—
330C2×#102—472no—
331C2×C22⋊C8—472no—
332C2×C4×C8—471no—
333#390—941no—
334#388—942no—
335#384—471no—
336#380—472no—
337C4×C4⋊C4—471no—
338C4×C4⋊C4—941no—
339#340—472no—
340#340—471no—
341#340—471no—
342C47#2241811yescharacteristic, complemented, Sylow
343C2×#41—24no—
344C2×#41—24no—
345C2×#41—24no—
346C2×#41—42no—
347C2×#41—42no—
348C2×#41C2×Dic4718yes—
349C22×Q8C22×D4711yescharacteristic
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4C22⋊D4711yescharacteristic
355C22×D4C22×D4712yes—
356C23×C4—22no—
357C23×C4C22×D4712yes—
358C22×C8—471no—
359C22×C8—472no—
360C22×C8—471no—
361C22×C8—944no—
362C22×C8—944no—
363C22×C8—1882no—
364#127—28no—
365#127—22no—
366#127C2×Dic4714yes—
367#121—24no—
368#121—216no—
369#121C2×Dic4718yes—
370C4×Q8—22no—
371C4×Q8C2×Dic4714yes—
372C4×D4—28no—
373C4×D4C2×Dic4714yes—
374C4×D4C22⋊D4714yes—
375#117—944no—
376#117—1882no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—472no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4C22⋊D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C22⋊C4—22no—
386C2×C22⋊C4—942no—
387C2×C22⋊C4—942no—
388C2×C22⋊C4C22⋊D4712yes—
389C2×C42—21no—
390C2×C42—942no—
391C2×C42—941no—
392C2×C42C22×D4711yescharacteristic
393C4⋊C8—942no—
394#103—474no—
395#103—944no—
396#103—942no—
397#102—944no—
398#102—944no—
399#102—1882no—
400#102—1882no—
401C22⋊C8—944no—
402C4×C8—942no—
403#95—941no—
404#95—942no—
405C24—22no—
406C24C2×C22⋊D4712yes—
407#41—216no—
408#41—48no—
409#41—48no—
410#41—48no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8D4×D4712yes—
415C2×Q8#1062535212yes—
416C2×Q8C22×Dic4712yes—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—81no—
433C2×D4—81no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4C22⋊Dic4714yes—
437C2×D4C22×Dic4714yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—28no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—942no—
450C22×C4C22×Dic4714yes—
451C22×C4C2×C22⋊D4712yes—
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4714yes—
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C23×D4711yescharacteristic
456C2×C8—472no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—942no—
460C2×C8—941no—
461C2×C8—942no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—942no—
465C2×C8—1888no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—472no—
469C4⋊C4—941no—
470C4⋊C4—942no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—942no—
474C4⋊C4—3761no—
475C4⋊C4—1882no—
476C4⋊C4C22⋊Dic4714yes—
477C4⋊C4C22×Dic4712yes—
478C22⋊C4—28no—
479C22⋊C4—3762no—
480C22⋊C4—1884no—
481C22⋊C4C22⋊Dic4718yes—
482C42—24no—
483C42—1884no—
484C42—1882no—
485C42C22×Dic4712yes—
486C42C2×C22⋊D4712yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C2×C22⋊Dic4714yes—
501C23C22×C22⋊D4711yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8D4×Dic4714yes—
506D4—24no—
507D4—24no—
508D4—28no—
509D4—416no—
510D4—42no—
511D4—42no—
512D4—48no—
513D4—84no—
514D4—88no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—416no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—942no—
531C2×C4—1882no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1882no—
535C2×C4—1881no—
536C2×C4—1882no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4C2×C22⋊Dic4712yes—
540C2×C4C2×C22⋊Dic4714yes—
541C2×C4C2×C22⋊Dic4712yes—
542C2×C4#1309404412yes—
543C2×C4#1309404412yes—
544C2×C4C23×Dic4711yescharacteristic
545C2×C4C2×D4×D4711yescharacteristic
546C2×C4C22×C22⋊D4711yescharacteristic
547C2×C4C2×#1062535211yescharacteristic
548C2×C4#1309406911yescharacteristic
549C2×C4#1309406911yescharacteristic
550C8—942no—
551C8—1882no—
552C8—1881no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22C22×C22⋊Dic4711yescharacteristic
568C22D4×C22⋊D4711yescharacteristic
569C22#1568976711yescharacteristic
570C22C2×#1309404411yescharacteristic
571C22C2×#1309404411yescharacteristic
572C22#1568992611yescharacteristic
573C22#1568993411yescharacteristic
574C4—22no—
575C4—24no—
576C4—24no—
577C4—42no—
578C4—3761no—
579C4—3762no—
580C4—1882no—
581C4#1568911914yescomplemented
582C4C2×D4×Dic4711yescharacteristic
583C4#1568959811yescharacteristic
584C4D4×C22⋊D4711yescharacteristic
585C4#1568976711yescharacteristic
586C2—44no—
587C2—84no—
588C2D4×C22⋊Dic4711yescharacteristic
589C2C2×#1568911911yescharacteristic
590C2#2025191411yescharacteristic
591C2#2025511811yescharacteristic
592C2#2025514411yescharacteristic
593C2#2025532711yescharacteristic
594C2#2025533811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#550C2
socleC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenterC2×#550D47
hypocenterC47#22418
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×D4×C47⋊C8C2semidirect
2. C2×#15689119C2semidirect
3. #20246337C2semidirect
4. #20244316C2semidirect
5. #20244195C2semidirect
6. #20243817C2semidirect
7. #20243711C2semidirect
8. #20243393C2semidirect
9. #20243328C2semidirect
10. C4×#13093948C2semidirect
37. C47×#550C4semidirect
65. #13093948D4semidirect
281. C4×C47#416semidirect
312. #550Dic47semidirect
342. C47#22418semidirect
581. C4#15689119semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C47⋊C82
C2×#156891192
#202463372
#202443162
#202441952
#202438172
#202437112
#202433932
#202433282
C4×#130939482
C2×C47×#5502
#2241847