Finite Group Databasebeta

← #26224756

Subgroups — #26224756

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689504C211yesmaximal, characteristic
2C2×#15689233C211yesmaximal, characteristic
3C2×#15689228C211yesmaximal, characteristic, complemented
4C2×#15689225C211yesmaximal, characteristic, complemented
5C4×#13094013C211yesmaximal, characteristic, complemented
6C4×#13094011C211yesmaximal, characteristic, complemented
7#20244271C211yesmaximal, characteristic, complemented
8#20244267C211yesmaximal, characteristic, complemented
9#20244261C211yesmaximal, characteristic
10#20244256C211yesmaximal, characteristic, complemented
11#20244198C211yesmaximal, characteristic, complemented
12#20244196C211yesmaximal, characteristic, complemented
13#20243551C211yesmaximal, characteristic, complemented
14#97×Dic47C211yesmaximal, characteristic, complemented
15C2×C47×#440C211yesmaximal, characteristic, complemented
16C22×#10625337C2211yescharacteristic
17C22×#10625325C2211yescharacteristic
18C22×#10625334C2211yescharacteristic
19C2×#13094013C2211yescharacteristic
20C2×#13094011C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23#15689504C418yescomplemented
24#15689233—24no—
25#15689228—24no—
26#15689225—24no—
27C2×#13093958C2211yescharacteristic
28C2×#13093955C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093949C2211yescharacteristic
31C2×#13093948C2211yescharacteristic
32C2×#13093947C2211yescharacteristic
33C2×#13093946C2211yescharacteristic
34C2×#13093945C2211yescharacteristic
35C2×#13093944C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×#13093934C2211yescharacteristic
38C2×#13093934C2211yescharacteristic
39C2×C8×Dic47C2211yescharacteristic
40C2×C8×Dic47C2211yescharacteristic
41#15689030C2211yescharacteristic
42#15689028C2211yescharacteristic
43#15689026C2211yescharacteristic
44#15689024C2211yescharacteristic
45#15689017—22no—
46#15689013—22no—
47#15688988C2211yescharacteristic
48#15688988C2211yescharacteristic
49C4×(C2×D47)⋊C4—22no—
50C4×C22⋊Dic47—22no—
51C42×Dic47C2211yescharacteristic
52C22×SD16×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C2×C4×D4×C47C2211yescharacteristic
55C47×#440—24no—
56C2×#107×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yescomplemented
66C2×#10625325—22no—
67C2×#10625325—22no—
68C2×#10625325C2×C414yescomplemented
69C2×#10625334—22no—
70C2×#10625334—22no—
71C2×#10625334C2×C414yes—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74#13094013—22no—
75#13094013C2×C414yescomplemented
76#13094011—22no—
77#13094011C2×C414yescomplemented
78C4×Dic94—22no—
79C4×Dic94—22no—
80C4×D188—22no—
81C4×D188—22no—
82C2×C4⋊Dic47C2311yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×Dic47⋊C4—21no—
85C2×C22⋊Dic47—22no—
86C2×C22⋊Dic47—22no—
87C2×(C2×D47)⋊C4—22no—
88C2×(C2×D47)⋊C4—21no—
89C2×(C2×D47)⋊C4—22no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47D412yes—
92C2×C4×Dic47D412yes—
93C2×C4×Dic47C2311yescharacteristic
94C2×C4×Dic47C2311yescharacteristic
95#13093958—22no—
96#13093955—22no—
97#13093955—22no—
98#13093949—22no—
99#13093948—22no—
100#13093947—22no—
101#13093946—22no—
102#13093945—22no—
103#13093944—22no—
104#13093935D414yes—
105#13093934—22no—
106#13093934—22no—
107C8×Dic47—22no—
108C8×Dic47C2×C414yescomplemented
109C47⋊#95—22no—
110C47⋊#95—22no—
111C47⋊#95—22no—
112C47⋊#95—22no—
113C22×Q8×C47C2311yescharacteristic
114C22×D4×C47C2311yescharacteristic
115C23×C4×C47—21no—
116C2×SD16×C47—22no—
117C2×SD16×C47—22no—
118C2×SD16×C47C2×C414yescomplemented
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C4×Q8×C47—22no—
122C4×Q8×C47—22no—
123C4×D4×C47—22no—
124C4×D4×C47—22no—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C22⋊C4×C47—21no—
129C2×C42×C47—21no—
130C2×C42×C47C2311yescharacteristic
131#107×C47—22no—
132#103×C47—22no—
133#102×C47—22no—
134#97×C47D414yes—
135C24×C47—21no—
136C23×D47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47C22×C412yes—
143C2×D4×C47—21no—
144C2×D4×C47—21no—
145C2×D4×C47—21no—
146C2×D4×C47—21no—
147C2×D4×C47—21no—
148C2×D4×C47C22×C412yes—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—44no—
153C22×C4×C47—42no—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47C2×D412yes—
161C22×Dic47C2×D412yes—
162C2×D188—21no—
163C2×D188—21no—
164C2×D188—21no—
165C2×D188—21no—
166C2×D188—21no—
167C2×D188C22×C412yes—
168C2×C4×D47—44no—
169C2×C4×D47—42no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94C22×C412yes—
176SD16×C47—28no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47C22×C412yes—
180C2×C8×C47C2×D412yes—
181C2×C8×C47C2×D412yes—
182C2×C8×C47#4112yes—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47—42no—
188C22⋊C4×C47—42no—
189C42×C47—42no—
190C42×C47C2×D412yes—
191C42×C47C2×D412yes—
192C22⋊Dic47—28no—
193C22⋊Dic47—44no—
194#10625337—28no—
195#10625334—28no—
196(C2×D47)⋊C4—28no—
197(C2×D47)⋊C4—44no—
198(C2×D47)⋊C4—42no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203Dic47⋊C4—42no—
204C4×Dic47—21no—
205C4×Dic47—21no—
206C4×Dic47—42no—
207C4×Dic47—42no—
208C4×Dic47—42no—
209C4×Dic47C22×C412yes—
210C4×Dic47#4112yes—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8C22×C412yes—
214C2×C47⋊C8C2×D412yes—
215C2×C47⋊C8C2×D412yes—
216C2×C47⋊C8#4112yes—
217#10625325—28no—
218C23×C47—22no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—42no—
222C23×C47—42no—
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—41no—
236D4×C47—41no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—84no—
247C2×C4×C47#12712yes—
248C2×C4×C47#12712yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47C4×D414yes—
268D188—22no—
269D188—22no—
270D188—41no—
271D188—41no—
272C4×D47—84no—
273Dic94—22no—
274Dic94—22no—
275Dic94—41no—
276Dic94—41no—
277C8×C47—22no—
278C8×C47C4×D414yes—
279C47⋊C8—22no—
280C47⋊C8C4×D414yes—
281#21693—471nomaximal, Sylow
282C22×C47—42no—
283C22×C47—44no—
284C22×C47—81no—
285C22×C47—81no—
286C22×C47—82no—
287C22×C47C2×C4×D411yescharacteristic
288C22×C47C2×#12711yescharacteristic
289C22×C47#53211yescharacteristic
290C22×C47#57512yes—
291C22×C47#57512yes—
292C2×D47—42no—
293C2×D47—44no—
294C2×D47—81no—
295C2×D47—81no—
296C2×D47—82no—
297C4×C47—22no—
298C4×C47—22no—
299C4×C47—42no—
300C4×C47—81no—
301C4×C47—81no—
302C4×C47C2×C4×D411yescharacteristic
303C4×C47C2×C4×D411yescharacteristic
304C4×C47#53211yescharacteristic
305C4×C47#53211yescharacteristic
306Dic47—24no—
307Dic47—42no—
308Dic47—42no—
309Dic47—81no—
310Dic47—81no—
311C2×#495—471no—
312C2×#442—471no—
313C2×#441—471no—
314C2×#440—471no—
315C2×#440D4711yescharacteristic, complemented
316C4×#124—471no—
317C4×#124—471no—
318#1973—471no—
319#1973—471no—
320#1972—471no—
321#1971—471no—
322#1958—471no—
323#1958—471no—
324#1846—471no—
325C4×#97—471no—
326C2×C47—82no—
327C2×C47C4×#12711yescharacteristic
328C2×C47#294312yes—
329C2×C47#315311yescharacteristic
330C2×C47#315311yescharacteristic
331C2×C47#318312yes—
332D47—82no—
333C22×Q16—471no—
334C22×SD16—471no—
335C22×SD16C2×D4711yescharacteristic
336C22×D8—471no—
337C2×#124—471no—
338C2×#124—471no—
339C2×C4×Q8—471no—
340C2×C4×Q8C2×D4711yescharacteristic
341C2×C4×D4—471no—
342C2×C4×D4C2×D4711yescharacteristic
343#495—478no—
344#442—944no—
345#441—944no—
346#440—24no—
347#440—944no—
348C2×#107—471no—
349C2×#107C2×D4711yescharacteristic
350C2×#106—471no—
351C2×#106—471no—
352C2×#103—471no—
353C2×#103—471no—
354C2×#103—471no—
355C2×#103C2×D4711yescharacteristic
356C2×#102—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#102C2×D4711yescharacteristic
360C2×#97—471no—
361C2×#97—471no—
362C2×#97—471no—
363C2×#97C2×D4711yescharacteristic
364C2×C4×C8—471no—
365C2×C4×C8—471no—
366#391—471no—
367#391—471no—
368#390—471no—
369#390—471no—
370#388—942no—
371#388—942no—
372#382—471no—
373#382—471no—
374C4×C22⋊C4—942no—
375C4×C22⋊C4—942no—
376C43—471no—
377C47#2169311yescharacteristic, complemented, Sylow
378C22×Q8—471no—
379C22×Q8C22×D4711yescharacteristic
380C22×D4—471no—
381C22×D4C22×D4711yescharacteristic
382C23×C4—21no—
383C23×C4—941no—
384C2×Q16—474no—
385C2×Q16—942no—
386C2×Q16—942no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16—474no—
390C2×SD16—942no—
391C2×SD16—942no—
392C2×SD16C4×D4714yescomplemented
393C2×D8—474no—
394C2×D8—942no—
395C2×D8—942no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8C22×D4711yescharacteristic
399C22×C8C22×D4711yescharacteristic
400#124—474no—
401#124—474no—
402#124—942no—
403#124—942no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—942no—
407C4×Q8—942no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—942no—
411C4×D4—942no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—942no—
420C2×C22⋊C4—942no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4—942no—
423C2×C22⋊C4—941no—
424C2×C42—21no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42—472no—
428C2×C42—472no—
429C2×C42—941no—
430C2×C42C22×D4711yescharacteristic
431#107—22no—
432#107—942no—
433#106—942no—
434#106—942no—
435#103—22no—
436#103—942no—
437#103—942no—
438#103—942no—
439#102—22no—
440#102—942no—
441#102—942no—
442#102—942no—
443#97—474no—
444#97—942no—
445#97—942no—
446#97C22⋊D4714yes—
447C4×C8—474no—
448C4×C8—942no—
449#95—942no—
450#95—942no—
451#95—942no—
452#95—942no—
453C24—21no—
454C24—941no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—472no—
461C2×Q8—941no—
462C2×Q8—941no—
463C2×Q8—941no—
464C2×Q8—941no—
465C2×Q8—941no—
466C2×Q8C2×C4×D4712yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—472no—
473C2×D4—941no—
474C2×D4—941no—
475C2×D4—941no—
476C2×D4—941no—
477C2×D4—941no—
478C2×D4C2×C4×D4712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—472no—
485C22×C4—472no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—1884no—
490C22×C4—1882no—
491C22×C4D4×D4711yescharacteristic
492C22×C4D4×D4711yescharacteristic
493C22×C4C23×D4711yescharacteristic
494Q16—948no—
495SD16—28no—
496SD16—948no—
497D8—948no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—472no—
501C2×C8—472no—
502C2×C8—472no—
503C2×C8—472no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8C2×C4×D4712yes—
507C2×C8#1062535012yes—
508C2×C8C2×C22⋊D4712yes—
509C2×C8C2×C22⋊D4712yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—1882no—
520C22⋊C4—42no—
521C22⋊C4—948no—
522C22⋊C4—948no—
523C22⋊C4—1884no—
524C22⋊C4—1884no—
525C22⋊C4—1882no—
526C42—42no—
527C42—472no—
528C42—472no—
529C42—941no—
530C42—941no—
531C42—1882no—
532C42—1882no—
533C42—1882no—
534C42D4×D4712yes—
535C42C2×C22⋊D4712yes—
536C23—22no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—942no—
542C23—1881no—
543C23—1882no—
544C23—1882no—
545C23—1881no—
546C23C2×D4×D4711yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—942no—
552Q8—942no—
553Q8—1881no—
554Q8—1881no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—942no—
560D4—942no—
561D4—1881no—
562D4—1881no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—84no—
573C2×C4—474no—
574C2×C4—942no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—3764no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4#1309402012yes—
588C2×C4#1309402012yes—
589C2×C4C22×C4×D4711yescharacteristic
590C2×C4C2×D4×D4711yescharacteristic
591C2×C4C22×C22⋊D4711yescharacteristic
592C2×C4#41×D4711yescharacteristic
593C2×C4#41×D4711yescharacteristic
594C2×C4C2×#1062535011yescharacteristic
595C8—22no—
596C8—474no—
597C8—942no—
598C8C4×C22⋊D4714yes—
599C22—44no—
600C22—42no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22—3761no—
605C22—3761no—
606C22—3762no—
607C22—1884no—
608C22—1882no—
609C22C4×D4×D4711yescharacteristic
610C22C2×#1309402011yescharacteristic
611C22#1568967011yescharacteristic
612C22#1568991912yes—
613C22#1568992212yes—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—944no—
620C4—3761no—
621C4—3761no—
622C4—1882no—
623C4—1882no—
624C4C4×D4×D4711yescharacteristic
625C4C2×C4×C22⋊D4711yescharacteristic
626C4#1568966911yescharacteristic
627C4#1568967011yescharacteristic
628C2—82no—
629C2—3762no—
630C2C4×#1309402011yescharacteristic
631C2#2025218312yes—
632C2#2025385011yescharacteristic
633C2#2025385511yescharacteristic
634C2#2025411312yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#440C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
cosocleC22×C4×C47C24
hypercenterC2×#440D47
hypocenterC47#21693
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. C2×#15689228C2semidirect
4. C2×#15689225C2semidirect
5. C4×#13094013C2semidirect
6. C4×#13094011C2semidirect
7. #20244271C2semidirect
8. #20244267C2semidirect
10. #20244256C2semidirect
11. #20244198C2semidirect
12. #20244196C2semidirect
13. #20243551C2semidirect
14. #97×Dic47C2semidirect
15. C2×C47×#440C2semidirect
23. #15689504C4semidirect
65. C2×#10625337C2×C4semidirect
68. C2×#10625325C2×C4semidirect
75. #13094013C2×C4semidirect
77. #13094011C2×C4semidirect
108. C8×Dic47C2×C4semidirect
118. C2×SD16×C47C2×C4semidirect
315. C2×#440D47semidirect
377. C47#21693semidirect
392. C2×SD16C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895042
C2×#156892332
C2×#156892282
C2×#156892252
C4×#130940132
C4×#130940112
#202442712
#202442672
#202442612
#202442562
#202441982
#202441962
#202435512
#97×Dic472
C2×C47×#4402
#2169347