Finite Group Databasebeta

← #26244280

Subgroups — #26244280

Order 12032.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094049C211yesmaximal, characteristic
2C2×#15689147C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689127C211yesmaximal, characteristic, complemented
5C2×#15689025C211yesmaximal, characteristic, complemented
6#20244703C214yesmaximal, complemented
7#20244698C214yesmaximal, complemented
8#20243109C211yesmaximal, characteristic, complemented
9C47×#2297C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C2×#13094049—24no—
12C2×#13094049—24no—
13C2×#13094049—24no—
14C22×#10625336C2212yes—
15C22×#10625334C2212yes—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C22×(C2×D47)⋊C4—22no—
20#15689147—24no—
21C2×C4⋊(C47⋊C8)C2214yescomplemented
22#15689127—24no—
23#15689124—24no—
24C2×#13093948C2212yescomplemented
25C2×#13093945C2212yescomplemented
26C2×#13093944C2214yes—
27#15689025—21no—
28#15689025—21no—
29#15689025C2214yes—
30#15688885C2214yescomplemented
31C22×#41×C47C2211yescharacteristic
32C2×#117×C47C2211yescharacteristic
33C22×C42×C47C2211yescharacteristic
34C47×#391C2212yescomplemented
35C47×#390C2212yescomplemented
36C47×#386—22no—
37C4×C22⋊C4×C47—21no—
38C4×C22⋊C4×C47—21no—
39C24×D47—21no—
40C22×D188—22no—
41C22×D188—22no—
42C22×D188—22no—
43C22×D188—22no—
44C22×D188—21no—
45C22×D188—22no—
46C22×D188D414yes—
47C22×D188C2312yes—
48C23×Dic47—21no—
49#13094049—232no—
50C2×#10625336—24no—
51C2×#10625336—24no—
52C2×#10625336—24no—
53C2×#10625334—24no—
54C2×#10625334—24no—
55C2×#10625334—24no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329—22no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329D414yes—
62C22×C47⋊C8C2312yes—
63C22×C47⋊C8C2312yes—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2312yes—
68C2×(C2×D47)⋊C4—24no—
69C2×(C2×D47)⋊C4—24no—
70C2×(C2×D47)⋊C4—48no—
71C4⋊(C47⋊C8)—28no—
72#13093948—24no—
73#13093945—24no—
74#13093944—28no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47—22no—
78C2×#41×C47—22no—
79C2×#41×C47—22no—
80C2×#41×C47D414yes—
81C22×Q8×C47C2311yescharacteristic
82C22×D4×C47—21no—
83C22×D4×C47C2311yescharacteristic
84C23×C4×C47—21no—
85C23×C4×C47D412yes—
86C23×C4×C47C2311yescharacteristic
87#117×C47—22no—
88#117×C47—22no—
89C2×C4⋊C4×C47C2312yes—
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47—22no—
95C2×C42×C47—21no—
96C2×C42×C47D411yescharacteristic
97C2×C42×C47D411yescharacteristic
98C2×C42×C47C2312yes—
99#95×C47—22no—
100#95×C47—22no—
101C24×C47—21no—
102C24×C47C2×D411yescharacteristic
103C23×D47—24no—
104C23×D47—22no—
105C23×D47—44no—
106C23×D47—44no—
107C23×D47—42no—
108C23×D47—42no—
109#41×C47—28no—
110#41×C47—28no—
111#41×C47—44no—
112#41×C47—44no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×Q8×C47—21no—
118C2×Q8×C47C2×D412yes—
119C2×D4×C47—24no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—42no—
126C2×D4×C47—42no—
127C2×D4×C47C2×D412yes—
128C22×C4×C47—24no—
129C22×C4×C47—21no—
130C22×C4×C47—24no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47#4112yes—
143C22×C4×C47#4112yes—
144C22×C4×C47#4112yes—
145C22×C4×C47C2411yescharacteristic
146C22×Dic47—22no—
147C22×Dic47—44no—
148C22×Dic47—42no—
149C2×D188—28no—
150C2×D188—28no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—24no—
156C2×D188—22no—
157C2×D188—42no—
158C2×D188—42no—
159C2×D188—44no—
160C2×D188—44no—
161C2×D188C2×D414yes—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C22⋊C4×C47—44no—
165C22⋊C4×C47—42no—
166C22⋊C4×C47—44no—
167C42×C47—22no—
168C42×C47—22no—
169C42×C47—22no—
170C42×C47—22no—
171C42×C47—42no—
172#10625336—216no—
173#10625334—216no—
174(C2×D47)⋊C4—48no—
175(C2×D47)⋊C4—48no—
176C4⋊Dic47—22no—
177C4⋊Dic47—22no—
178C4⋊Dic47—42no—
179#10625329—28no—
180#10625329—44no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8C2×D414yes—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47#12112yes—
195C23×C47#12112yes—
196C23×C47#12112yes—
197C23×C47C22×D411yescharacteristic
198C22×D47—24no—
199C22×D47—416no—
200C22×D47—42no—
201C22×D47—42no—
202C22×D47—44no—
203C22×D47—44no—
204C22×D47—48no—
205C22×D47—84no—
206C22×D47—84no—
207C22×D47—88no—
208Q8×C47—22no—
209Q8×C47—22no—
210Q8×C47—41no—
211Q8×C47—41no—
212D4×C47—22no—
213D4×C47—22no—
214D4×C47—48no—
215D4×C47—41no—
216D4×C47—41no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—48no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—48no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47#12014yes—
231C2×C4×C47#12112yes—
232C2×C4×C47#12112yes—
233C2×C4×C47#12112yes—
234C2×C4×C47#12314yes—
235C2×C4×C47#12314yes—
236C2×C4×C47#12414yes—
237C2×C4×C47AGL(1,ℤ/8)14yescomplemented
238C2×C4×C47C22×D411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×Dic47—42no—
245C2×Dic47—44no—
246C2×Dic47—84no—
247D188—24no—
248D188—24no—
249D188—42no—
250D188—42no—
251D188—48no—
252C47⋊C8—24no—
253C47⋊C8—42no—
254#23546—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—24no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—82no—
263C22×C47—82no—
264C22×C47#39414yes—
265C22×C47C2×#12111yescharacteristic
266C22×C47C2×#12111yescharacteristic
267C22×C47C2×#12111yescharacteristic
268C22×C47C2×AGL(1,ℤ/8)12yes—
269C22×C47#57512yes—
270C2×D47—44no—
271C2×D47—48no—
272C2×D47—816no—
273C2×D47—82no—
274C2×D47—82no—
275C2×D47—84no—
276C4×C47—24no—
277C4×C47—42no—
278C4×C47—42no—
279C4×C47—82no—
280C4×C47#39414yes—
281C4×C47C2×#12011yescharacteristic
282C4×C47C2×#12311yescharacteristic
283C4×C47C2×#12311yescharacteristic
284C4×C47C2×#12411yescharacteristic
285Dic47—82no—
286C22×AGL(1,ℤ/8)—471no—
287C2×#423—471no—
288C2×#417—471no—
289C2×#418—471no—
290#2297D4711yescharacteristic, complemented
291C2×#390—471no—
292#2055—474no—
293#2054—474no—
294#1736—471no—
295C2×C47—24no—
296C2×C47—82no—
297C2×C47C2×#39411yescharacteristic
298C2×C47#303612yes—
299C2×C47#303811yescharacteristic
300C2×C47#304011yescharacteristic
301C2×C47#305512yes—
302D47—84no—
303C22×#41C2×D4711yescharacteristic
304C23×D4—471no—
305C2×AGL(1,ℤ/8)—944no—
306C2×AGL(1,ℤ/8)—944no—
307C2×AGL(1,ℤ/8)—944no—
308C22×SD16—472no—
309C22×D8—472no—
310C22×M16—471no—
311C22×M16—471no—
312C2×#117C2×D4711yescharacteristic
313C22×C4⋊C4—471no—
314C22×C22⋊C4—942no—
315C22×C42C2×D4711yescharacteristic
316#423—944no—
317C2×C4⋊C8—474no—
318#418—944no—
319#417—944no—
320C2×#103—472no—
321C2×#102—472no—
322C2×#102—474no—
323#391C2×D4712yescomplemented
324#390—474no—
325#390—941no—
326#390—941no—
327#390C2×D4712yescomplemented
328#386—22no—
329C4×C22⋊C4—21no—
330C4×C22⋊C4—21no—
331#340—474no—
332C47#2354611yescharacteristic, complemented, Sylow
333C25—941no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41D18814yes—
340C22×Q8C22×D4711yescharacteristic
341C22×D4—21no—
342C22×D4—474no—
343C22×D4—472no—
344C22×D4—942no—
345C22×D4—942no—
346C22×D4—942no—
347C22×D4—942no—
348C22×D4—942no—
349C22×D4—941no—
350C22×D4C22×D4711yescharacteristic
351C23×C4—21no—
352C23×C4—941no—
353C23×C4C22⋊D4712yes—
354C23×C4C22×D4711yescharacteristic
355AGL(1,ℤ/8)—9432no—
356C2×SD16—944no—
357C2×SD16—944no—
358C2×SD16—944no—
359C2×D8—944no—
360C2×D8—944no—
361C2×D8—944no—
362C2×M16—474no—
363C2×M16—942no—
364C2×M16—942no—
365C2×M16—942no—
366C2×M16—942no—
367C2×M16—942no—
368C22×C8—472no—
369C22×C8—472no—
370#117—22no—
371#117—22no—
372C2×C4⋊C4—472no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22×D4712yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—944no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—1888no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42C22⋊D4711yescharacteristic
387C2×C42C22⋊D4711yescharacteristic
388C2×C42C22×D4712yes—
389C4⋊C8—948no—
390#103—944no—
391#102—948no—
392#102—944no—
393#95—22no—
394#95—22no—
395C24—21no—
396C24—944no—
397C24—942no—
398C24—1884no—
399C24—1884no—
400C24—1882no—
401C24—1882no—
402C24C2×C22⋊D4711yescharacteristic
403#41—28no—
404#41—28no—
405#41—44no—
406#41—44no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×D18812yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—474no—
422C2×D4—948no—
423C2×D4—944no—
424C2×D4—948no—
425C2×D4—942no—
426C2×D4—942no—
427C2×D4—942no—
428C2×D4—942no—
429C2×D4—942no—
430C2×D4—1882no—
431C2×D4—1882no—
432C2×D4—1884no—
433C2×D4—1884no—
434C2×D4C2×D18812yes—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—942no—
444C22×C4—1884no—
445C22×C4—1882no—
446C22×C4C2×D18812yes—
447C22×C4#1062535012yes—
448C22×C4D4×D4712yes—
449C22×C4D4×D4712yes—
450C22×C4#1062535212yes—
451C22×C4#1062535212yes—
452C22×C4C2×C22⋊D4712yes—
453C22×C4C2×C22⋊D4712yes—
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C23×D4711yescharacteristic
456SD16—9416no—
457D8—9416no—
458M16—948no—
459M16—1884no—
460C2×C8—474no—
461C2×C8—942no—
462C2×C8—942no—
463C2×C8—942no—
464C2×C8—942no—
465C2×C8—942no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—942no—
469C4⋊C4—942no—
470C4⋊C4—1882no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—1888no—
475C22⋊C4—1888no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—42no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—944no—
489C23—3768no—
490C23—3764no—
491C23—3764no—
492C23—18816no—
493C23—1888no—
494C23—1882no—
495C23—1882no—
496C23—1884no—
497C23—1884no—
498C23#1309399512yes—
499C23#1309399612yes—
500C23#1309399812yes—
501C23C22×C22⋊D4711yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511D4—944no—
512D4—944no—
513D4—1888no—
514D4—1882no—
515D4—1882no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—3764no—
530C2×C4—1882no—
531C2×C4—1884no—
532C2×C4#1309399414yes—
533C2×C4#1309399512yes—
534C2×C4#1309399612yes—
535C2×C4#1309399812yes—
536C2×C4#1309400614yes—
537C2×C4#1309400814yes—
538C2×C4#1309401014yes—
539C2×C4#1309404914yescomplemented
540C2×C4C2×D4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C22×D18811yescharacteristic
543C2×C4C2×#1062535211yescharacteristic
544C2×C4C2×#1062535211yescharacteristic
545C2×C4C2×#1062535011yescharacteristic
546C8—944no—
547C8—1882no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22—37616no—
558C22—3764no—
559C22—3762no—
560C22—3762no—
561C22—1884no—
562C22—1888no—
563C22#1568903914yes—
564C22C2×#1309399511yescharacteristic
565C22C2×#1309399611yescharacteristic
566C22C2×#1309399811yescharacteristic
567C22C2×#1309404912yes—
568C22#1568992112yes—
569C4—24no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—3762no—
574C4#1568903914yes—
575C4C2×#1309399411yescharacteristic
576C4C2×#1309400611yescharacteristic
577C4C2×#1309400811yescharacteristic
578C4C2×#1309401011yescharacteristic
579C2—24no—
580C2—82no—
581C2—3764no—
582C2C2×#1568903911yescharacteristic
583C2#2025284812yes—
584C2#2025286811yescharacteristic
585C2#2025288211yescharacteristic
586C2#2025300912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2297C2
socleC23×C47C22×D4
prime radicalC22×#13094049C2
hypercenter#2297D47
hypocenterC47#23546
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C2×#15689147C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689127C2semidirect
5. C2×#15689025C2semidirect
6. #20244703C2semidirect
7. #20244698C2semidirect
8. #20243109C2semidirect
9. C47×#2297C2semidirect
21. C2×C4⋊(C47⋊C8)C22semidirect
24. C2×#13093948C22semidirect
25. C2×#13093945C22semidirect
30. #15688885C22semidirect
34. C47×#391C22semidirect
35. C47×#390C22semidirect
237. C2×C4×C47AGL(1,ℤ/8)semidirect
290. #2297D47semidirect
323. #391C2×D47semidirect
327. #390C2×D47semidirect
332. C47#23546semidirect
539. C2×C4#13094049semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C2×#156891472
C2×#156891242
C2×#156891272
C2×#156890252
#202447032
#202446982
#202431092
C47×#22972
#2354647