Finite Group Databasebeta

← #26217461

Subgroups — #26217461

Order 12032.

1782 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689889C211yesmaximal, characteristic, complemented
2#20256717C211yesmaximal, characteristic
3C2×#15689140C211yesmaximal, characteristic, complemented
4#20245878C211yesmaximal, characteristic, complemented
5#20243687C218yesmaximal, complemented
6#20243099C211yesmaximal, characteristic, complemented
7C2×#15688883C211yesmaximal, characteristic, complemented
8C22×#104×C47C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10#15689889—24no—
11#15689889—24no—
12C2×#13094032—22no—
13C2×#13094030—22no—
14C22×#10625329C2211yescharacteristic
15C2×M16×D47C2214yescomplemented
16#15689701C414yescomplemented
17#15689697C412yescomplemented
18#15689694—21no—
19#15689690C412yescomplemented
20#15689687—21no—
21#15689682—22no—
22#15689669—22no—
23#15689634—22no—
24#15689630C2214yes—
25C2×#13093983C2211yescharacteristic
26C2×#13093983C2211yescharacteristic
27#15689140—24no—
28C2×#13093951C2214yes—
29#15689029—21no—
30#15689025—21no—
31#15689002—22no—
32C4×(C2×D47)⋊C4C2214yescomplemented
33#15688884C2214yescomplemented
34#15688883C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C2×#104×C47—24no—
39C2×#104×C47C2214yes—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C412yes—
45C2×#10625350C2×C412yes—
46C2×#10625350D412yes—
47C2×#10625350D412yes—
48C22×Dic94D411yescharacteristic
49C22×D188D411yescharacteristic
50C22×C22⋊D47D412yes—
51C22×C4×D47C2312yes—
52#13094032—216no—
53#13094030—216no—
54C2×#10625329—21no—
55C2×#10625329—22no—
56C2×#10625329D414yescomplemented
57C2×#10625329C2312yes—
58C2×#10625324—24no—
59M16×D47—28no—
60M16×D47—28no—
61C22×C47⋊C8—21no—
62C2×C8×D47—24no—
63#13094022—22no—
64#13094020—22no—
65#13094013—22no—
66#13094011—22no—
67#13094007—24no—
68#13094005—24no—
69#13094003—24no—
70#13094000C2×C418yescomplemented
71#13093998—24no—
72#13093996—28no—
73#13093995C2×C418yescomplemented
74#13093993—24no—
75#13093992—24no—
76C4×C22⋊D47—28no—
77#13093983—21no—
78#13093983—21no—
79#13093983—22no—
80#13093983—22no—
81#13093983C2×C412yes—
82#13093983C2312yes—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47C2×C412yes—
85C2×C22⋊Dic47—21no—
86C2×C22⋊Dic47—21no—
87C2×(C2×D47)⋊C4—24no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47C2312yes—
90#13093951—28no—
91C47⋊#95—24no—
92C2×#41×C47—21no—
93C2×#41×C47—21no—
94C2×#41×C47—22no—
95C2×#41×C47—22no—
96C2×#41×C47—22no—
97C2×#41×C47—22no—
98C2×#41×C47C2×C412yes—
99C2×#41×C47C2312yes—
100C22×Q8×C47C2×C411yescharacteristic
101C22×D4×C47—21no—
102C22×D4×C47C2×C411yescharacteristic
103C23×C4×C47—21no—
104C23×C4×C47—21no—
105C23×C4×C47C2311yescharacteristic
106C2×M16×C47—21no—
107C2×M16×C47—22no—
108C2×M16×C47D414yescomplemented
109C2×M16×C47C2312yes—
110C22×C8×C47—21no—
111C2×C42×C47—22no—
112C2×C42×C47—22no—
113C2×C42×C47—21no—
114C2×C42×C47C2312yes—
115#104×C47—28no—
116C24×C47—21no—
117C24×C47C2×D411yescharacteristic
118C23×D47C2×D412yes—
119#41×C47—24no—
120#41×C47—24no—
121#41×C47—22no—
122#41×C47—22no—
123#41×C47—22no—
124#41×C47—22no—
125#41×C47—42no—
126#41×C47—42no—
127#41×C47—42no—
128#41×C47—42no—
129C2×Q8×C47—21no—
130C2×Q8×C47—22no—
131C2×Q8×C47—22no—
132C2×Q8×C47C22×C412yes—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—21no—
136C2×D4×C47—22no—
137C2×D4×C47—22no—
138C2×D4×C47—42no—
139C2×D4×C47—42no—
140C2×D4×C47C22×C412yes—
141C22×C4×C47—21no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—44no—
146C22×C4×C47—42no—
147C22×C4×C47—42no—
148C22×C4×C47—42no—
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C22×C411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2×D412yes—
156C22×C4×C47#4112yes—
157C22×C4×C47#4112yes—
158C22×C4×C47C2411yescharacteristic
159C2×C22⋊D47—24no—
160C2×C22⋊D47—24no—
161C2×C22⋊D47—28no—
162C2×C22⋊D47C22⋊C414yes—
163C2×C22⋊D47C22⋊C414yes—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47C2×D412yes—
167#10625350—24no—
168#10625350—24no—
169#10625350—24no—
170#10625350—24no—
171#10625350—24no—
172#10625350—24no—
173#10625350—24no—
174#10625350—24no—
175C2×D188—22no—
176C2×D188—22no—
177C2×D188C22⋊C412yes—
178C2×D188C22⋊C412yes—
179C2×C4×D47—22no—
180C2×C4×D47—22no—
181C2×C4×D47—22no—
182C2×C4×D47—22no—
183C2×C4×D47C22×C414yes—
184C2×C4×D47C2×D414yes—
185C2×Dic94—22no—
186C2×Dic94—22no—
187C2×Dic94C22⋊C412yes—
188C2×Dic94C22⋊C412yes—
189M16×C47—24no—
190M16×C47—22no—
191M16×C47C2×D414yes—
192C2×C8×C47—24no—
193C2×C8×C47—22no—
194C42×C47—22no—
195C42×C47—22no—
196C42×C47—42no—
197C22⋊Dic47—22no—
198C22⋊Dic47—22no—
199C22⋊Dic47—28no—
200C22⋊Dic47—42no—
201(C2×D47)⋊C4—28no—
202(C2×D47)⋊C4—48no—
203C4⋊Dic47—22no—
204C4⋊Dic47—42no—
205C4⋊Dic47C22×C414yes—
206Dic47⋊C4—28no—
207C4×Dic47—22no—
208C4×Dic47—22no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211#10625329—24no—
212#10625329—22no—
213#10625329C2×D414yes—
214C2×C47⋊C8—24no—
215C2×C47⋊C8—22no—
216#10625324—216no—
217C8×D47—216no—
218C23×C47—22no—
219C23×C47—42no—
220C23×C47—42no—
221C23×C47—82no—
222C23×C47C2×C22⋊C411yescharacteristic
223C23×C47C2×C22⋊C411yescharacteristic
224C23×C47#12112yes—
225C23×C47#12112yes—
226C23×C47C22×D411yescharacteristic
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—44no—
230C22×D47C2×C22⋊C414yes—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—42no—
234D4×C47—22no—
235D4×C47—22no—
236D4×C47—42no—
237D4×C47—42no—
238D4×C47—44no—
239D4×C47—42no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—42no—
243C2×C4×C47—48no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—44no—
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C4×D412yes—
252C2×C4×C47C4×D412yes—
253C2×C4×C47C4×D412yes—
254C2×C4×C47C4×D412yes—
255C2×C4×C47#12012yes—
256C2×C4×C47#12012yes—
257C2×C4×C47#12112yes—
258C2×C4×C47#12112yes—
259C2×C4×C47#12312yes—
260C2×C4×C47#12312yes—
261C2×C4×C47C23×C411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C22⋊D47—28no—
268C22⋊D47—28no—
269C22⋊D47—88no—
270C2×Dic47—22no—
271C2×Dic47—22no—
272C2×Dic47—24no—
273C2×Dic47—44no—
274C2×Dic47—42no—
275C2×Dic47—82no—
276C2×Dic47C2×C22⋊C414yes—
277D188—24no—
278D188—24no—
279C4×D47—24no—
280C4×D47—24no—
281C4×D47—24no—
282C4×D47—24no—
283Dic94—24no—
284Dic94—24no—
285C8×C47—24no—
286C47⋊C8—24no—
287#20903—471nomaximal, Sylow
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—44no—
294C22×C47—42no—
295C22×C47—84no—
296C22×C47#38612yes—
297C22×C47#38612yes—
298C22×C47C22×C22⋊C411yescharacteristic
299C22×C47C2×#12111yescharacteristic
300C22×C47C2×#12111yescharacteristic
301C2×D47—28no—
302C2×D47—24no—
303C2×D47—82no—
304C2×D47—82no—
305C4×C47—44no—
306C4×C47—42no—
307C4×C47#38612yes—
308C4×C47#38612yes—
309C4×C47C2×C4×D411yescharacteristic
310C4×C47C2×C4×D411yescharacteristic
311C4×C47C2×#12011yescharacteristic
312C4×C47C2×#12311yescharacteristic
313Dic47—24no—
314Dic47—44no—
315Dic47—82no—
316C2×#568—471no—
317#3449—471no—
318C2×#421—471no—
319C22×#104D4711yescharacteristic, complemented
320#2297—471no—
321#1871—478no—
322#1733—471no—
323C2×#339—471no—
324C2×C47—21no—
325C2×C47—21no—
326C2×C47—22no—
327C2×C47—22no—
328C2×C47—82no—
329C2×C47C2×#38611yescharacteristic
330C2×C47#290112yes—
331D47—44no—
332C22×#41—471no—
333C22×#41C2×D4711yescharacteristic
334#568—944no—
335#568—944no—
336C2×#131—942no—
337C2×#131—942no—
338C22×M16—471no—
339C22×M16—474no—
340C22×M16C2×D4711yescharacteristic
341#537—474no—
342#536—472no—
343#536—941no—
344#535—472no—
345#535—941no—
346#534—942no—
347#532—942no—
348#525—474no—
349#525—942no—
350C2×#117—471no—
351C2×#117—471no—
352C22×C42C2×D4711yescharacteristic
353#421—944no—
354C2×#104—24no—
355C2×#104—474no—
356C2×#104C2×D4714yes—
357#391—941no—
358#390—941no—
359#386—942no—
360C4×C22⋊C4—474no—
361#339—474no—
362#339—474no—
363C47#2090311yescharacteristic, complemented, Sylow
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—472no—
371C2×#41—472no—
372C2×#41—472no—
373C2×#41—472no—
374C2×#41—942no—
375C2×#41—942no—
376C2×#41—942no—
377C2×#41—942no—
378C2×#41C4×D4712yes—
379C2×#41C22×D4712yes—
380C22×Q8—471no—
381C22×Q8C4×D4711yescharacteristic
382C22×D4—21no—
383C22×D4—472no—
384C22×D4—471no—
385C22×D4C4×D4711yescharacteristic
386C23×C4—21no—
387C23×C4—21no—
388C23×C4—472no—
389C23×C4C22×D4711yescharacteristic
390#131—9416no—
391#131—9416no—
392C2×M16—21no—
393C2×M16—22no—
394C2×M16—474no—
395C2×M16—472no—
396C2×M16—944no—
397C2×M16—948no—
398C2×M16—941no—
399C2×M16—942no—
400C2×M16—948no—
401C2×M16C22⋊D4714yescomplemented
402C2×M16C22×D4712yes—
403C22×C8—21no—
404C22×C8—944no—
405C22×C8—941no—
406C4⋊Q8—942no—
407#127—942no—
408#124—942no—
409#124—942no—
410#123—944no—
411#123—944no—
412C22⋊Q8—478no—
413C22⋊Q8—944no—
414#121—478no—
415#121—944no—
416#121—948no—
417#120—944no—
418#120—944no—
419C4×D4—948no—
420#117—472no—
421#117—472no—
422#117—941no—
423#117—941no—
424#117—942no—
425#117—942no—
426C2×C4⋊C4—472no—
427C2×C4⋊C4—941no—
428C2×C22⋊C4—941no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—941no—
431C2×C42—22no—
432C2×C42—22no—
433C2×C42—21no—
434C2×C42—472no—
435C2×C42—941no—
436C2×C42C22×D4712yes—
437#104—28no—
438#104—948no—
439#95—944no—
440C24—21no—
441C24—472no—
442C24D4×D4711yescharacteristic
443#41—24no—
444#41—24no—
445#41—22no—
446#41—22no—
447#41—22no—
448#41—22no—
449#41—42no—
450#41—42no—
451#41—42no—
452#41—42no—
453#41—944no—
454#41—944no—
455#41—944no—
456#41—944no—
457#41—944no—
458#41—944no—
459#41—944no—
460#41—944no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—472no—
465C2×Q8—472no—
466C2×Q8—942no—
467C2×Q8—942no—
468C2×Q8C2×C4×D4712yes—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—474no—
477C2×D4—474no—
478C2×D4—472no—
479C2×D4—472no—
480C2×D4—944no—
481C2×D4—944no—
482C2×D4—948no—
483C2×D4—942no—
484C2×D4—942no—
485C2×D4C2×C4×D4712yes—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—472no—
495C22×C4—474no—
496C22×C4—474no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4C2×C4×D4711yescharacteristic
504C22×C4C2×C4×D4711yescharacteristic
505C22×C4#1062535012yes—
506C22×C4D4×D4711yescharacteristic
507C22×C4D4×D4711yescharacteristic
508C22×C4D4×D4711yescharacteristic
509C22×C4D4×D4712yes—
510C22×C4#1062535212yes—
511C22×C4C2×C22⋊D4712yes—
512C22×C4C23×D4711yescharacteristic
513M16—24no—
514M16—22no—
515M16—474no—
516M16—9416no—
517M16—944no—
518M16—942no—
519M16C2×C22⋊D4714yes—
520C2×C8—24no—
521C2×C8—22no—
522C2×C8—9416no—
523C2×C8—944no—
524C2×C8—942no—
525C4⋊C4—474no—
526C4⋊C4—948no—
527C4⋊C4—942no—
528C4⋊C4—1882no—
529C22⋊C4—948no—
530C22⋊C4—948no—
531C22⋊C4—942no—
532C22⋊C4—942no—
533C22⋊C4—1888no—
534C22⋊C4—1882no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—942no—
539C42—942no—
540C42—942no—
541C42—942no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—82no—
546C23—474no—
547C23—942no—
548C23—942no—
549C23—1884no—
550C23C22⋊C4×D4711yescharacteristic
551C23C22⋊C4×D4711yescharacteristic
552C23#1309399512yes—
553C23#1309399712yes—
554C23C2×D4×D4711yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—42no—
558Q8—944no—
559Q8—944no—
560D4—22no—
561D4—22no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—948no—
567D4—948no—
568D4—944no—
569D4—944no—
570D4—3768no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—474no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—3762no—
589C2×C4—1884no—
590C2×C4—1882no—
591C2×C4C22⋊C4×D4711yescharacteristic
592C2×C4C22⋊C4×D4711yescharacteristic
593C2×C4(C22⋊D47)⋊C412yes—
594C2×C4(C22⋊D47)⋊C412yes—
595C2×C4C4×C22⋊D4712yes—
596C2×C4C4×C22⋊D4712yes—
597C2×C4#1309399212yes—
598C2×C4#1309399212yes—
599C2×C4#1309399512yes—
600C2×C4#1309399712yes—
601C2×C4#1309400612yes—
602C2×C4#1309400612yes—
603C2×C4C22×C4×D4711yescharacteristic
604C2×C4C2×D4×D4711yescharacteristic
605C2×C4C2×D4×D4711yescharacteristic
606C2×C4C22×C22⋊D4711yescharacteristic
607C2×C4C2×#1062535211yescharacteristic
608C2×C4C2×#1062535011yescharacteristic
609C8—24no—
610C8—944no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—84no—
619C22—948no—
620C22—944no—
621C22—3762no—
622C22—3762no—
623C22#1568899912yes—
624C22#1568899912yes—
625C22C2×C22⋊C4×D4711yescharacteristic
626C22C2×#1309399511yescharacteristic
627C22C2×#1309399711yescharacteristic
628C4—44no—
629C4—42no—
630C4—944no—
631C4—3762no—
632C4—1884no—
633C4#1568899912yes—
634C4#1568899912yes—
635C4C2×(C22⋊D47)⋊C411yescharacteristic
636C4C2×C4×C22⋊D4711yescharacteristic
637C4C2×#1309399211yescharacteristic
638C4C2×#1309400611yescharacteristic
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—82no—
644C2—1884no—
645C2C2×#1568899911yescharacteristic
646C2#2025193212yes—
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#104×C47C2
socleC22×C47C22×C22⋊C4
prime radical#20256717C2
cosocleC22×C4×C47C24
hypercenterC22×#104D47
hypocenterC47#20903
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
1. C2×#15689889C2semidirect
3. C2×#15689140C2semidirect
4. #20245878C2semidirect
5. #20243687C2semidirect
6. #20243099C2semidirect
7. C2×#15688883C2semidirect
8. C22×#104×C47C2semidirect
15. C2×M16×D47C22semidirect
16. #15689701C4semidirect
17. #15689697C4semidirect
19. #15689690C4semidirect
32. C4×(C2×D47)⋊C4C22semidirect
33. #15688884C22semidirect
34. #15688883C22semidirect
56. C2×#10625329D4semidirect
70. #13094000C2×C4semidirect
73. #13093995C2×C4semidirect
108. C2×M16×C47D4semidirect
319. C22×#104D47semidirect
363. C47#20903semidirect
401. C2×M16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156898892
#202567172
C2×#156891402
#202458782
#202436872
#202430992
C2×#156888832
C22×#104×C472
#2090347