Finite Group Databasebeta

← #26217554

Subgroups — #26217554

Order 12032.

1842 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689704C211yesmaximal, characteristic
2C2×#15689603C211yesmaximal, characteristic, complemented
3C2×#15689125C211yesmaximal, characteristic, complemented
4C2×#15689117C211yesmaximal, characteristic, complemented
5C2×#15689091C211yesmaximal, characteristic, complemented
6#20243709C212yesmaximal, complemented
7#20243704C212yesmaximal, complemented
8#20243699C212yesmaximal, complemented
9#20243694C212yesmaximal, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#1736C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625350C2211yescharacteristic
14C23×C47⋊C8C2211yescharacteristic
15#15689704—24no—
16#15689704C418yescomplemented
17C2×#13094004—21no—
18C2×#13094004C2212yes—
19C2×#13093999—21no—
20C2×#13093999C2212yes—
21C2×#13093998—22no—
22#15689603—24no—
23#15689603—24no—
24C2×C4×Dic94—21no—
25C2×C4×D188—21no—
26C2×(C22⋊D47)⋊C4—22no—
27C2×#13093982C2212yes—
28C22×C4⋊Dic47C2211yescharacteristic
29C2×C4⋊C4×D47C2212yes—
30#15689125—24no—
31#15689117—24no—
32C2×#13093949C2212yes—
33C2×#13093947C2211yescharacteristic
34C2×#13093947C2211yescharacteristic
35C2×#13093946C2212yes—
36C2×#13093944C2211yescharacteristic
37C2×#13093944C2211yescharacteristic
38#15689091—24no—
39C2×(C2×D47)⋊C8C2212yes—
40C2×(C2×D47)⋊C8C2212yes—
41#15688885C2212yes—
42#15688885C2212yes—
43C22×M16×C47C2211yescharacteristic
44C2×#117×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C47×#340C2212yes—
47C47×#340C2212yes—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350—22no—
52C2×#10625350—22no—
53C2×#10625350—22no—
54C2×#10625350—22no—
55C2×#10625350—22no—
56C2×#10625350—22no—
57C2×#10625350C2×C414yes—
58C2×#10625350C2×C414yescomplemented
59C2×#10625350D414yes—
60C22×Dic94D411yescharacteristic
61C22×Dic94C2311yescharacteristic
62C22×D188—22no—
63C22×D188D411yescharacteristic
64C22×D188C2311yescharacteristic
65C22×C22⋊D47—21no—
66C22×C22⋊D47D412yes—
67C23×Dic47—21no—
68C22×C4×D47—21no—
69C22×C4×D47—22no—
70C22×C4×D47C2312yes—
71C22×C47⋊C8—22no—
72C22×C47⋊C8—22no—
73C22×C47⋊C8—22no—
74C22×C47⋊C8C2311yescharacteristic
75C22×C47⋊C8C2311yescharacteristic
76#13094004—24no—
77#13094004—28no—
78#13094004C2×C418yescomplemented
79#13093999—28no—
80#13093999—24no—
81#13093999C2×C418yescomplemented
82#13093998—216no—
83C4×Dic94—28no—
84C4×D188—28no—
85(C22⋊D47)⋊C4—216no—
86#13093982—24no—
87#13093982—24no—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—22no—
90C2×C4⋊Dic47—22no—
91C2×C4⋊Dic47D414yescomplemented
92C2×C4⋊Dic47C2312yes—
93C2×Dic47⋊C4—22no—
94C4⋊C4×D47—24no—
95C4⋊C4×D47—24no—
96C2×(C2×D47)⋊C4—22no—
97C2×(C2×D47)⋊C4—22no—
98C2×(C2×D47)⋊C4—22no—
99C2×C4×Dic47—22no—
100#13093949—24no—
101#13093947—22no—
102#13093947—22no—
103#13093946—24no—
104#13093944—22no—
105#13093944—22no—
106(C2×D47)⋊C8—24no—
107(C2×D47)⋊C8—24no—
108C23×C4×C47—21no—
109C23×C4×C47C2311yescharacteristic
110C2×M16×C47—22no—
111C2×M16×C47—22no—
112C2×M16×C47—22no—
113C22×C8×C47C2312yes—
114#117×C47—22no—
115#117×C47D414yescomplemented
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—22no—
118C2×C4⋊C4×C47C2×C414yescomplemented
119C2×C4⋊C4×C47C2312yes—
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C22⋊C4×C47—21no—
123C2×C42×C47—21no—
124C24×C47C2×D411yescharacteristic
125C23×D47—21no—
126C23×D47C2×D412yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—22no—
130C22×C4×C47—24no—
131C22×C4×C47—81no—
132C22×C4×C47—81no—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47#4112yes—
139C22×C4×C47#4112yes—
140C22×C4×C47C2411yescharacteristic
141C2×C22⋊D47—24no—
142C2×C22⋊D47—24no—
143C2×C22⋊D47—24no—
144C2×C22⋊D47—42no—
145C2×C22⋊D47—42no—
146C2×C22⋊D47C22⋊C418yes—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47—24no—
150C22×Dic47—81no—
151C22×Dic47—81no—
152C22×Dic47C2×D412yes—
153#10625350—28no—
154#10625350—28no—
155#10625350—28no—
156#10625350—28no—
157#10625350—28no—
158#10625350—28no—
159#10625350—44no—
160#10625350—44no—
161C2×D188—22no—
162C2×D188—22no—
163C2×D188—21no—
164C2×D188—21no—
165C2×D188—21no—
166C2×D188—21no—
167C2×D188—21no—
168C2×D188—28no—
169C2×D188—82no—
170C2×D188—82no—
171C2×D188C22⋊C414yes—
172C2×D188C22×C412yes—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×C4×D47—22no—
176C2×C4×D47—22no—
177C2×C4×D47—24no—
178C2×C4×D47—28no—
179C2×C4×D47—42no—
180C2×C4×D47—42no—
181C2×C4×D47—82no—
182C2×C4×D47—82no—
183C2×C4×D47C22×C414yes—
184C2×C4×D47C2×D414yes—
185C2×Dic94—22no—
186C2×Dic94—22no—
187C2×Dic94—21no—
188C2×Dic94—21no—
189C2×Dic94—21no—
190C2×Dic94—21no—
191C2×Dic94—21no—
192C2×Dic94C22⋊C414yes—
193C2×Dic94C22×C412yes—
194M16×C47—44no—
195C2×C8×C47—22no—
196C2×C8×C47—22no—
197C2×C8×C47—22no—
198C4⋊C4×C47—24no—
199C4⋊C4×C47—22no—
200C4⋊C4×C47—21no—
201C4⋊C4×C47—21no—
202C4⋊C4×C47C22×C414yes—
203C4⋊C4×C47C2×D412yes—
204C4⋊C4×C47C2×D412yes—
205C22⋊C4×C47—24no—
206C42×C47—24no—
207(C2×D47)⋊C4—28no—
208(C2×D47)⋊C4—28no—
209(C2×D47)⋊C4—28no—
210C4⋊Dic47—24no—
211C4⋊Dic47—22no—
212C4⋊Dic47—28no—
213C4⋊Dic47C2×D414yes—
214Dic47⋊C4—28no—
215C4×Dic47—28no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—44no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—21no—
226C23×C47—21no—
227C23×C47C2×C22⋊C412yes—
228C23×C47#12012yes—
229C23×C47#12312yes—
230C23×C47C22×D411yescharacteristic
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—22no—
234C22×D47—22no—
235C22×D47—22no—
236C22×D47—22no—
237C22×D47—22no—
238C22×D47—41no—
239C22×D47—41no—
240C22×D47—41no—
241C22×D47—41no—
242C22×D47—41no—
243C22×D47—41no—
244C22×D47C2×C22⋊C414yes—
245C2×C4×C47—24no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—41no—
249C2×C4×C47—42no—
250C2×C4×C47—41no—
251C2×C4×C47—42no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—84no—
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C4×D414yes—
257C2×C4×C47C4×D414yes—
258C2×C4×C47#12012yes—
259C2×C4×C47#12114yes—
260C2×C4×C47#12114yes—
261C2×C4×C47#12312yes—
262C2×C4×C47C23×C411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C22⋊D47—216no—
269C22⋊D47—48no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—24no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—84no—
279C2×Dic47C2×C22⋊C414yes—
280D188—22no—
281D188—22no—
282D188—28no—
283D188—41no—
284D188—41no—
285D188—88no—
286C4×D47—24no—
287C4×D47—24no—
288C4×D47—24no—
289C4×D47—24no—
290C4×D47—48no—
291C4×D47—88no—
292Dic94—28no—
293Dic94—22no—
294Dic94—22no—
295Dic94—41no—
296Dic94—41no—
297C8×C47—42no—
298C47⋊C8—41no—
299C47⋊C8—41no—
300#20915—471nomaximal, Sylow
301C22×C47—22no—
302C22×C47—22no—
303C22×C47—22no—
304C22×C47—22no—
305C22×C47—22no—
306C22×C47—22no—
307C22×C47#38614yes—
308C22×C47C22×C22⋊C411yescharacteristic
309C22×C47C2×#12011yescharacteristic
310C22×C47C2×#12311yescharacteristic
311C22×C47#57611yescharacteristic
312C22×C47#57711yescharacteristic
313C22×C47#57711yescharacteristic
314C22×C47#57811yescharacteristic
315C2×D47—24no—
316C2×D47—24no—
317C2×D47—24no—
318C2×D47—42no—
319C2×D47—42no—
320C2×D47—42no—
321C2×D47—42no—
322C2×D47—42no—
323C2×D47—42no—
324C2×D47—42no—
325C2×D47—81no—
326C2×D47—81no—
327C2×D47—81no—
328C2×D47—81no—
329C4×C47—44no—
330C4×C47—42no—
331C4×C47—42no—
332C4×C47#38614yes—
333C4×C47C2×C4×D411yescharacteristic
334C4×C47C2×C4×D411yescharacteristic
335C4×C47C2×#12111yescharacteristic
336C4×C47C2×#12111yescharacteristic
337Dic47—24no—
338Dic47—44no—
339Dic47—42no—
340C2×#537—471no—
341C2×#520—471no—
342C2×#417—471no—
343C2×#416—471no—
344C2×#408—471no—
345#1875—472no—
346#1874—472no—
347#1873—472no—
348#1872—472no—
349#1736D4711yescharacteristic, complemented
350C2×#340—471no—
351C2×C47—24no—
352C2×C47C2×#38611yescharacteristic
353C2×C47#289711yescharacteristic
354C2×C47#289811yescharacteristic
355C2×C47#301211yescharacteristic
356C2×C47#301311yescharacteristic
357C2×C47#309811yescharacteristic
358C2×C47#309911yescharacteristic
359D47—44no—
360D47—82no—
361C22×#41—471no—
362C22×#41—471no—
363C22×M16C2×D4711yescharacteristic
364C23×C8—471no—
365#537—478no—
366#537—944no—
367C2×C22⋊Q8—472no—
368C2×C22⋊Q8—941no—
369C2×#121—472no—
370C2×#121—941no—
371C2×#121—942no—
372#520—944no—
373#520—944no—
374C2×C4×Q8—941no—
375C2×C4×D4—941no—
376C2×C4×D4—942no—
377C2×#117—472no—
378C2×#117C2×D4711yescharacteristic
379C22×C4⋊C4—471no—
380C22×C4⋊C4—472no—
381C22×C4⋊C4C2×D4711yescharacteristic
382#417—944no—
383#416—944no—
384C2×#103—472no—
385C2×#103—471no—
386C2×#103—471no—
387C2×#102—472no—
388C2×#102—471no—
389C2×#102—471no—
390#408—944no—
391C2×C22⋊C8—472no—
392C2×C22⋊C8—472no—
393#340—472no—
394#340—472no—
395#340C2×D4712yes—
396#340C2×D4712yes—
397C47#2091511yescharacteristic, complemented, Sylow
398C2×#41—474no—
399C2×#41—474no—
400C2×#41—474no—
401C2×#41—942no—
402C2×#41—942no—
403C2×#41—942no—
404C2×#41—942no—
405C2×#41—942no—
406C2×#41—942no—
407C2×#41—942no—
408C2×#41—942no—
409C2×#41—942no—
410C22×Q8—471no—
411C22×Q8—471no—
412C22×D4—472no—
413C22×D4—471no—
414C22×D4—471no—
415C22×D4—942no—
416C22×D4—941no—
417C23×C4—21no—
418C23×C4—472no—
419C23×C4—942no—
420C23×C4—941no—
421C23×C4—941no—
422C23×C4C22×D4711yescharacteristic
423C2×M16—22no—
424C2×M16—22no—
425C2×M16—22no—
426C22×C8—471no—
427C22×C8—471no—
428C22×C8—942no—
429C22×C8—942no—
430C22×C8—942no—
431C22×C8C22×D4712yes—
432C22⋊Q8—478no—
433C22⋊Q8—944no—
434C22⋊Q8—948no—
435#121—478no—
436#121—944no—
437#121—948no—
438#121—9416no—
439C4×Q8—948no—
440C4×D4—948no—
441C4×D4—9416no—
442#117—22no—
443#117—944no—
444#117—944no—
445#117D18814yescomplemented
446C2×C4⋊C4—21no—
447C2×C4⋊C4—22no—
448C2×C4⋊C4—474no—
449C2×C4⋊C4—472no—
450C2×C4⋊C4—944no—
451C2×C4⋊C4—942no—
452C2×C4⋊C4—942no—
453C2×C4⋊C4—942no—
454C2×C4⋊C4—941no—
455C2×C4⋊C4—944no—
456C2×C4⋊C4C4×D4714yescomplemented
457C2×C4⋊C4C22×D4712yes—
458C2×C4⋊C4C22×D4711yescharacteristic
459C2×C4⋊C4C22×D4711yescharacteristic
460C2×C22⋊C4—21no—
461C2×C22⋊C4—942no—
462C2×C22⋊C4—942no—
463C2×C22⋊C4—942no—
464C2×C42—21no—
465C2×C42—942no—
466#103—944no—
467#103—942no—
468#103—942no—
469#102—944no—
470#102—942no—
471#102—942no—
472C22⋊C8—944no—
473C22⋊C8—944no—
474C24—472no—
475C24—941no—
476C24D4×D4711yescharacteristic
477#41—948no—
478#41—948no—
479#41—948no—
480#41—948no—
481#41—948no—
482#41—948no—
483#41—1884no—
484#41—1884no—
485C2×Q8—474no—
486C2×Q8—472no—
487C2×Q8—942no—
488C2×Q8—942no—
489C2×Q8—941no—
490C2×Q8—941no—
491C2×Q8—941no—
492C2×Q8—941no—
493C2×Q8—941no—
494C2×D4—478no—
495C2×D4—474no—
496C2×D4—472no—
497C2×D4—948no—
498C2×D4—944no—
499C2×D4—944no—
500C2×D4—944no—
501C2×D4—942no—
502C2×D4—942no—
503C2×D4—941no—
504C2×D4—941no—
505C2×D4—941no—
506C2×D4—941no—
507C2×D4—941no—
508C2×D4—3762no—
509C2×D4—3762no—
510C2×D4—1882no—
511C2×D4—1882no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—24no—
516C22×C4—81no—
517C22×C4—81no—
518C22×C4—472no—
519C22×C4—474no—
520C22×C4—474no—
521C22×C4—948no—
522C22×C4—942no—
523C22×C4—942no—
524C22×C4—942no—
525C22×C4—942no—
526C22×C4—944no—
527C22×C4—942no—
528C22×C4—941no—
529C22×C4—944no—
530C22×C4—3762no—
531C22×C4—3762no—
532C22×C4—3761no—
533C22×C4—3761no—
534C22×C4—1882no—
535C22×C4—1882no—
536C22×C4C2×C4×D4712yes—
537C22×C4C2×D18812yes—
538C22×C4#1062535012yes—
539C22×C4D4×D4711yescharacteristic
540C22×C4D4×D4712yes—
541C22×C4D4×D4712yes—
542C22×C4#1062535412yes—
543C22×C4C23×D4711yescharacteristic
544M16—44no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—941no—
549C2×C8—941no—
550C2×C8—941no—
551C2×C8—941no—
552C2×C8—941no—
553C2×C8—941no—
554C2×C8—1884no—
555C4⋊C4—24no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—474no—
560C4⋊C4—948no—
561C4⋊C4—948no—
562C4⋊C4—944no—
563C4⋊C4—942no—
564C4⋊C4C2×C4×D4714yes—
565C4⋊C4C2×D18812yes—
566C4⋊C4C2×D18812yes—
567C22⋊C4—24no—
568C22⋊C4—948no—
569C22⋊C4—948no—
570C22⋊C4—948no—
571C42—24no—
572C42—948no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—474no—
578C23—942no—
579C23—942no—
580C23—942no—
581C23—942no—
582C23—942no—
583C23—942no—
584C23—942no—
585C23—1881no—
586C23—1881no—
587C23—1881no—
588C23—1881no—
589C23—1881no—
590C23—1881no—
591C23C22⋊C4×D4712yes—
592C23#1309399412yes—
593C23#1309400912yes—
594C23C2×D4×D4711yescharacteristic
595Q8—948no—
596Q8—942no—
597Q8—942no—
598Q8—1881no—
599Q8—1881no—
600D4—9416no—
601D4—948no—
602D4—942no—
603D4—942no—
604D4—3768no—
605D4—1888no—
606D4—1881no—
607D4—1881no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—84no—
618C2×C4—474no—
619C2×C4—942no—
620C2×C4—942no—
621C2×C4—944no—
622C2×C4—944no—
623C2×C4—944no—
624C2×C4—944no—
625C2×C4—942no—
626C2×C4—944no—
627C2×C4—3768no—
628C2×C4—3764no—
629C2×C4—1881no—
630C2×C4—1881no—
631C2×C4—1882no—
632C2×C4—1882no—
633C2×C4—1888no—
634C2×C4C22⋊C4×D4712yes—
635C2×C4#1309398614yes—
636C2×C4C4×D18814yes—
637C2×C4#1309399412yes—
638C2×C4#1309399714yes—
639C2×C4#1309399914yes—
640C2×C4#1309400912yes—
641C2×C4C22×C4×D4711yescharacteristic
642C2×C4C2×D4×D4711yescharacteristic
643C2×C4C2×D4×D4711yescharacteristic
644C2×C4C22×D18811yescharacteristic
645C2×C4C2×#1062535011yescharacteristic
646C2×C4C2×#1062535411yescharacteristic
647C8—42no—
648C8—1881no—
649C8—1881no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—944no—
657C22—944no—
658C22—944no—
659C22—3761no—
660C22—3761no—
661C22—3761no—
662C22—3761no—
663C22—1882no—
664C22—1882no—
665C22—1882no—
666C22—1882no—
667C22—1882no—
668C22—1882no—
669C22—1882no—
670C22#1568900114yes—
671C22C2×C22⋊C4×D4711yescharacteristic
672C22C2×#1309399411yescharacteristic
673C22C2×#1309400911yescharacteristic
674C22#1568992711yescharacteristic
675C22#1568993211yescharacteristic
676C22#1568993311yescharacteristic
677C22#1568993911yescharacteristic
678C4—44no—
679C4—42no—
680C4—42no—
681C4—944no—
682C4—1884no—
683C4—1882no—
684C4#1568900114yes—
685C4C2×#1309398611yescharacteristic
686C4C2×C4×D18811yescharacteristic
687C4C2×#1309399711yescharacteristic
688C4C2×#1309399911yescharacteristic
689C2—24no—
690C2—3762no—
691C2—1884no—
692C2C2×#1568900111yescharacteristic
693C2#2025191311yescharacteristic
694C2#2025192011yescharacteristic
695C2#2025269411yescharacteristic
696C2#2025270511yescharacteristic
697C2#2025337011yescharacteristic
698C2#2025338111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC2×#15689704C2
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20915
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#15689603C2semidirect
3. C2×#15689125C2semidirect
4. C2×#15689117C2semidirect
5. C2×#15689091C2semidirect
6. #20243709C2semidirect
7. #20243704C2semidirect
8. #20243699C2semidirect
9. #20243694C2semidirect
10. C2×#15688885C2semidirect
11. C47×#1736C2semidirect
16. #15689704C4semidirect
58. C2×#10625350C2×C4semidirect
78. #13094004C2×C4semidirect
81. #13093999C2×C4semidirect
91. C2×C4⋊Dic47D4semidirect
115. #117×C47D4semidirect
118. C2×C4⋊C4×C47C2×C4semidirect
349. #1736D47semidirect
397. C47#20915semidirect
445. #117D188semidirect
456. C2×C4⋊C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897042
C2×#156896032
C2×#156891252
C2×#156891172
C2×#156890912
#202437092
#202437042
#202436992
#202436942
C2×#156888852
C47×#17362
#2091547