Finite Group Databasebeta

← #26219942

Subgroups — #26219942

Order 12032.

1726 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×#15689669C211yesmaximal, characteristic
2C2×#15689125C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689122C211yesmaximal, characteristic, complemented
5C2×M16×Dic47C211yesmaximal, characteristic, complemented
6C2×#15688992C211yesmaximal, characteristic, complemented
7#20244203C212yesmaximal, complemented
8#20244201C212yesmaximal, complemented
9#20244192C212yesmaximal, complemented
10#20244188C212yesmaximal, complemented
11C2×C47×#417C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689669—24no—
15#15689669C418yescomplemented
16C2×#13094022C2211yescharacteristic
17C2×#13094020C2211yescharacteristic
18C2×#13094013C2211yescharacteristic
19C2×#13094011C2211yescharacteristic
20C2×#13093996—22no—
21C2×C4×C22⋊D47—22no—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4—22no—
24C22×C4×Dic47C2211yescharacteristic
25#15689125—24no—
26#15689124—24no—
27#15689122—24no—
28C2×#13093949C2212yes—
29C2×#13093948C2212yes—
30C2×#13093947C2212yes—
31C2×#13093946C2212yes—
32C2×#13093945C2212yes—
33C2×#13093944C2212yes—
34M16×Dic47—24no—
35C2×#13093934C2212yes—
36C2×C8×Dic47C2212yes—
37#15688992—22no—
38#15688992C2212yes—
39#15688992C2212yes—
40C22×#41×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#417—24no—
44C2×#103×C47C2212yes—
45C2×#102×C47C2212yes—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350C2×C414yes—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47—21no—
55C22×C22⋊D47—22no—
56C23×Dic47—21no—
57C23×Dic47D412yes—
58C22×C4×D47—21no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329D414yescomplemented
62C22×C47⋊C8C2312yes—
63#13094022—22no—
64#13094022C2×C414yescomplemented
65#13094020—22no—
66#13094020C2×C414yescomplemented
67#13094013—22no—
68#13094013C2×C414yescomplemented
69#13094011—22no—
70#13094011C2×C414yescomplemented
71#13093996—216no—
72C4×C22⋊D47—216no—
73C2×C4⋊Dic47—22no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C2×Dic47⋊C4—24no—
79C2×Dic47⋊C4—24no—
80C2×Dic47⋊C4—48no—
81C2×C22⋊Dic47—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×C4×Dic47C2×C414yes—
84C2×C4×Dic47D412yes—
85C2×C4×Dic47D414yes—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093949—24no—
89#13093948—24no—
90#13093947—24no—
91#13093946—24no—
92#13093945—24no—
93#13093944—24no—
94#13093934—24no—
95C8×Dic47—24no—
96C2×#41×C47—22no—
97C2×#41×C47—22no—
98C2×#41×C47—22no—
99C2×#41×C47—22no—
100C2×#41×C47—22no—
101C2×#41×C47C2×C414yes—
102C22×Q8×C47C2311yescharacteristic
103C22×D4×C47—21no—
104C22×D4×C47C2311yescharacteristic
105C23×C4×C47—21no—
106C23×C4×C47—21no—
107C23×C4×C47C2311yescharacteristic
108C2×M16×C47—22no—
109C2×M16×C47—22no—
110C2×M16×C47D414yescomplemented
111C22×C8×C47C2312yes—
112C2×C4⋊C4×C47—22no—
113C2×C4⋊C4×C47—22no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2312yes—
116#103×C47—24no—
117#102×C47—24no—
118C24×C47—21no—
119C24×C47C2×D411yescharacteristic
120C23×D47—21no—
121#41×C47—28no—
122#41×C47—28no—
123#41×C47—44no—
124#41×C47—44no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C22×C412yes—
131C2×D4×C47—24no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—42no—
138C2×D4×C47—42no—
139C2×D4×C47C22×C412yes—
140C22×C4×C47—21no—
141C22×C4×C47—24no—
142C22×C4×C47—22no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—44no—
146C22×C4×C47—42no—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D412yes—
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C2×C22⋊D47—24no—
156C2×C22⋊D47—28no—
157C2×C22⋊D47—42no—
158C2×C22⋊D47—42no—
159C2×C22⋊D47—44no—
160C2×C22⋊D47—44no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—21no—
164C22×Dic47—22no—
165C22×Dic47—44no—
166C22×Dic47—42no—
167C22×Dic47C22⋊C418yes—
168C22×Dic47C2×D412yes—
169C22×Dic47C2×D412yes—
170#10625350—28no—
171#10625350—28no—
172#10625350—44no—
173#10625350—44no—
174C2×D188—21no—
175C2×D188—21no—
176C2×D188—21no—
177C2×D188—21no—
178C2×D188—21no—
179C2×D188C22×C412yes—
180C2×C4×D47—24no—
181C2×C4×D47—42no—
182C2×C4×D47—42no—
183C2×Dic94—21no—
184C2×Dic94—21no—
185C2×Dic94—21no—
186C2×Dic94—21no—
187C2×Dic94—21no—
188C2×Dic94C22×C412yes—
189M16×C47—28no—
190C2×C8×C47—22no—
191C2×C8×C47—22no—
192C2×C8×C47C2×D414yes—
193C4⋊C4×C47—22no—
194C4⋊C4×C47—22no—
195C4⋊C4×C47—42no—
196C22⋊Dic47—28no—
197(C2×D47)⋊C4—28no—
198C4⋊Dic47—22no—
199C4⋊Dic47—22no—
200C4⋊Dic47—42no—
201Dic47⋊C4—28no—
202Dic47⋊C4—48no—
203Dic47⋊C4—48no—
204C4×Dic47C22⋊C418yes—
205C4×Dic47C22×C412yes—
206C4×Dic47C22×C412yes—
207C4×Dic47C2×D412yes—
208C4×Dic47C2×D412yes—
209#10625329—28no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8C2×D414yes—
213C23×C47—22no—
214C23×C47—22no—
215C23×C47—22no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—42no—
219C23×C47—42no—
220C23×C47C2×C22⋊C412yes—
221C23×C47#12412yes—
222C23×C47#12712yes—
223C23×C47C22×D411yescharacteristic
224C22×D47—22no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—42no—
228C22×D47—42no—
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—41no—
232Q8×C47—41no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47—48no—
236D4×C47—41no—
237D4×C47—41no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—48no—
242C2×C4×C47—42no—
243C2×C4×C47—44no—
244C2×C4×C47—84no—
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C4×D414yes—
247C2×C4×C47C4×D414yes—
248C2×C4×C47#12114yes—
249C2×C4×C47#12114yes—
250C2×C4×C47#12412yes—
251C2×C4×C47#12712yes—
252C2×C4×C47C23×C411yescharacteristic
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C22⋊D47—48no—
259C22⋊D47—416no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—216no—
263C2×Dic47—22no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—42no—
267C2×Dic47—44no—
268C2×Dic47—84no—
269C2×Dic47C2×C22⋊C414yes—
270C2×Dic47C2×C22⋊C414yes—
271C2×Dic47AGL(1,ℤ/8)14yes—
272C2×Dic47#13714yes—
273D188—22no—
274D188—22no—
275D188—41no—
276D188—41no—
277C4×D47—48no—
278Dic94—22no—
279Dic94—22no—
280Dic94—41no—
281Dic94—41no—
282C8×C47—24no—
283C47⋊C8—24no—
284#21200—471nomaximal, Sylow
285C22×C47—24no—
286C22×C47—24no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—44no—
290C22×C47—42no—
291C22×C47—82no—
292C22×C47—81no—
293C22×C47—81no—
294C22×C47#39014yes—
295C22×C47C22×C22⋊C411yescharacteristic
296C22×C47C2×#12411yescharacteristic
297C22×C47C2×#12711yescharacteristic
298C22×C47C2×AGL(1,ℤ/8)12yes—
299C22×C47C2×#13712yes—
300C2×D47—44no—
301C2×D47—42no—
302C2×D47—82no—
303C2×D47—81no—
304C2×D47—81no—
305C4×C47—42no—
306C4×C47—82no—
307C4×C47#39014yes—
308C4×C47C2×C4×D411yescharacteristic
309C4×C47C2×C4×D411yescharacteristic
310C4×C47C2×#12111yescharacteristic
311C4×C47C2×#12111yescharacteristic
312Dic47—24no—
313Dic47—42no—
314Dic47—82no—
315Dic47#41718yescomplemented
316C2×#532—471no—
317C2×#417—471no—
318C2×#417—471no—
319C2×#417—471no—
320C2×#417D4711yescharacteristic, complemented
321C2×C4×M16—471no—
322C2×#384—471no—
323#1959—472no—
324#1959—472no—
325#1957—472no—
326#1956—472no—
327C2×C47—24no—
328C2×C47—82no—
329C2×C47C2×#39011yescharacteristic
330C2×C47C2×#41712yes—
331C2×C47#314112yes—
332C2×C47#315111yescharacteristic
333C2×C47#315211yescharacteristic
334D47—82no—
335C22×#41—471no—
336C22×#41C2×D4711yescharacteristic
337C22×M16—471no—
338C22×M16C2×D4711yescharacteristic
339#532—478no—
340#532—944no—
341C2×C4⋊Q8—471no—
342C2×#127—471no—
343C2×#124—471no—
344C2×#124—471no—
345C2×#121—942no—
346C2×C4×D4—942no—
347C22×C4⋊C4—471no—
348C22×C4⋊C4—942no—
349C22×C4⋊C4C2×D4711yescharacteristic
350C22×C42—471no—
351#417—24no—
352#417—944no—
353#417—944no—
354#417—944no—
355C2×#103—472no—
356C2×#103—472no—
357C2×#103—472no—
358C2×#103C2×D4712yes—
359C2×#102—472no—
360C2×#102—472no—
361C2×#102—472no—
362C2×#102C2×D4712yes—
363C4×M16—944no—
364C2×#97—472no—
365C2×C4×C8—472no—
366#384—472no—
367#384—472no—
368#384—942no—
369C47#2120011yescharacteristic, complemented, Sylow
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—474no—
376C2×#41—942no—
377C2×#41—942no—
378C2×#41—942no—
379C2×#41—942no—
380C2×#41—942no—
381C2×#41C4×D4714yes—
382C22×Q8—471no—
383C22×Q8C22×D4711yescharacteristic
384C22×D4—21no—
385C22×D4—471no—
386C22×D4—942no—
387C22×D4—941no—
388C22×D4C22×D4711yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—472no—
392C23×C4—941no—
393C23×C4—941no—
394C23×C4C22×D4711yescharacteristic
395C2×M16—22no—
396C2×M16—22no—
397C2×M16—474no—
398C2×M16—942no—
399C2×M16—942no—
400C2×M16C22⋊D4714yescomplemented
401C22×C8—472no—
402C22×C8C22×D4712yes—
403C4⋊Q8—474no—
404C4⋊Q8—942no—
405#127—474no—
406#127—942no—
407#124—474no—
408#124—474no—
409#124—942no—
410#124—942no—
411#121—9416no—
412C4×D4—9416no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—472no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—942no—
419C2×C4⋊C4—942no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—944no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—1888no—
424C2×C4⋊C4C22×D4712yes—
425C2×C22⋊C4—942no—
426C2×C22⋊C4—942no—
427C2×C42—474no—
428C2×C42—472no—
429C2×C42—474no—
430C2×C42—471no—
431C2×C42—471no—
432#103—24no—
433#103—944no—
434#103—944no—
435#103—944no—
436#102—24no—
437#102—944no—
438#102—944no—
439#102—944no—
440#97—944no—
441C4×C8—944no—
442C24—21no—
443C24—941no—
444C24D4×D4711yescharacteristic
445#41—28no—
446#41—28no—
447#41—44no—
448#41—44no—
449#41—948no—
450#41—948no—
451#41—1884no—
452#41—1884no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—472no—
459C2×Q8—941no—
460C2×Q8—941no—
461C2×Q8—941no—
462C2×Q8—941no—
463C2×Q8—941no—
464C2×Q8C2×C4×D4712yes—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—472no—
474C2×D4—948no—
475C2×D4—944no—
476C2×D4—941no—
477C2×D4—941no—
478C2×D4—941no—
479C2×D4—941no—
480C2×D4—941no—
481C2×D4—1884no—
482C2×D4—1884no—
483C2×D4—1882no—
484C2×D4—1882no—
485C2×D4C2×C4×D4712yes—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—472no—
494C22×C4—478no—
495C22×C4—472no—
496C22×C4—944no—
497C22×C4—944no—
498C22×C4—941no—
499C22×C4—944no—
500C22×C4—942no—
501C22×C4—1884no—
502C22×C4—1882no—
503C22×C4—1882no—
504C22×C4—1882no—
505C22×C4C2×C4×D4712yes—
506C22×C4#1062535012yes—
507C22×C4D4×D4711yescharacteristic
508C22×C4D4×D4712yes—
509C22×C4D4×D4712yes—
510C22×C4#1062535212yes—
511C22×C4C2×C22⋊D4712yes—
512C22×C4C23×D4711yescharacteristic
513M16—28no—
514M16—948no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—474no—
518C2×C8—942no—
519C2×C8—942no—
520C2×C8C2×C22⋊D4714yes—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—42no—
524C4⋊C4—948no—
525C4⋊C4—942no—
526C4⋊C4—942no—
527C4⋊C4—1888no—
528C4⋊C4—1888no—
529C4⋊C4—1882no—
530C22⋊C4—948no—
531C22⋊C4—948no—
532C42—478no—
533C42—472no—
534C42—472no—
535C42—472no—
536C42—472no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—942no—
545C23—1881no—
546C23—1882no—
547C23—1882no—
548C23—1881no—
549C23C22⋊C4×D4712yes—
550C23#1309401012yes—
551C23#1309402012yes—
552C23C2×D4×D4711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—942no—
558Q8—942no—
559Q8—1881no—
560Q8—1881no—
561D4—22no—
562D4—22no—
563D4—48no—
564D4—41no—
565D4—41no—
566D4—942no—
567D4—942no—
568D4—18816no—
569D4—1888no—
570D4—1881no—
571D4—1881no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—474no—
580C2×C4—474no—
581C2×C4—474no—
582C2×C4—474no—
583C2×C4—9416no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4—3764no—
588C2×C4—1884no—
589C2×C4—1882no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1888no—
593C2×C4C22⋊C4×D4712yes—
594C2×C4(C22⋊D47)⋊C414yes—
595C2×C4C4×C22⋊D4714yes—
596C2×C4#1309399614yes—
597C2×C4#1309399714yes—
598C2×C4#1309401012yes—
599C2×C4#1309402012yes—
600C2×C4C22×C4×D4711yescharacteristic
601C2×C4C2×D4×D4711yescharacteristic
602C2×C4C2×D4×D4711yescharacteristic
603C2×C4C22×C22⋊D4711yescharacteristic
604C2×C4C2×#1062535211yescharacteristic
605C2×C4C2×#1062535011yescharacteristic
606C8—24no—
607C8—944no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—42no—
613C22—44no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—3762no—
618C22—3761no—
619C22—3761no—
620C22—1882no—
621C22—1884no—
622C22#1568902314yes—
623C22C2×C22⋊C4×D4711yescharacteristic
624C22C2×#1309401011yescharacteristic
625C22C2×#1309402011yescharacteristic
626C22AGL(1,ℤ/8)×D4712yes—
627C22#137×D4712yes—
628C4—42no—
629C4—82no—
630C4—478no—
631C4—944no—
632C4—3762no—
633C4—1882no—
634C4#1568902314yes—
635C4C2×(C22⋊D47)⋊C411yescharacteristic
636C4C2×C4×C22⋊D4711yescharacteristic
637C4C2×#1309399611yescharacteristic
638C4C2×#1309399711yescharacteristic
639C2—24no—
640C2—82no—
641C2—3762no—
642C2C2×#1568902311yescharacteristic
643C2#417×D4712yes—
644C2#2025377912yes—
645C2#2025383611yescharacteristic
646C2#2025384311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47C22×D4
prime radicalC2×#15689669C2
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#21200
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#15689125C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689122C2semidirect
5. C2×M16×Dic47C2semidirect
6. C2×#15688992C2semidirect
7. #20244203C2semidirect
8. #20244201C2semidirect
9. #20244192C2semidirect
10. #20244188C2semidirect
11. C2×C47×#417C2semidirect
15. #15689669C4semidirect
61. C2×#10625329D4semidirect
64. #13094022C2×C4semidirect
66. #13094020C2×C4semidirect
68. #13094013C2×C4semidirect
70. #13094011C2×C4semidirect
110. C2×M16×C47D4semidirect
315. Dic47#417semidirect
320. C2×#417D47semidirect
369. C47#21200semidirect
400. C2×M16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896692
C2×#156891252
C2×#156891242
C2×#156891222
C2×M16×Dic472
C2×#156889922
#202442032
#202442012
#202441922
#202441882
C2×C47×#4172
#2120047