Finite Group Databasebeta

← #26243854

Subgroups — #26243854

Order 12032.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689701C211yesmaximal, characteristic
2C22×#13093955C211yesmaximal, characteristic, complemented
3C2×#15689122C211yesmaximal, characteristic, complemented
4C2×#15689117C211yesmaximal, characteristic, complemented
5C2×#15689090C211yesmaximal, characteristic, complemented
6#20244584C214yesmaximal, complemented
7#20244577C214yesmaximal, complemented
8#20243109C211yesmaximal, characteristic, complemented
9C47×#2311C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13#15689701—24no—
14#15689701—24no—
15C2×#13094003C2212yes—
16C2×#13094000—21no—
17C2×#13093998C2212yes—
18C2×#13093996—22no—
19C2×#13093995—21no—
20C22×C4⋊Dic47C2211yescharacteristic
21C2×#13093955—24no—
22C2×#13093955C2214yes—
23#15689122—24no—
24#15689117—24no—
25C2×#13093947C2212yes—
26C2×#13093947C2212yes—
27C2×#13093944C2212yes—
28C2×#13093944C2212yes—
29#15689090—24no—
30C2×C22⋊(C47⋊C8)C2214yes—
31#15688885C2214yes—
32C22×#41×C47C2211yescharacteristic
33C2×#117×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C47×#391—21no—
36C47×#390—21no—
37C47×#386—22no—
38C47×#380C2214yes—
39C2×#10625350—22no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350D414yes—
45C22×Dic94C2311yescharacteristic
46C22×D188C2311yescharacteristic
47C22×C22⋊D47—21no—
48C22×C22⋊D47—22no—
49C23×Dic47—21no—
50C22×C4×D47—21no—
51C2×#10625329—22no—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C22×C47⋊C8—22no—
55C22×C47⋊C8—22no—
56C22×C47⋊C8D414yescomplemented
57C22×C47⋊C8C2312yes—
58C22×C47⋊C8C2312yes—
59#13094003—24no—
60#13094003—24no—
61#13094000—28no—
62#13093998—24no—
63#13093998—24no—
64#13093996—216no—
65#13093995—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—22no—
68C2×C4⋊Dic47D414yes—
69C2×C4⋊Dic47C2312yes—
70C2×Dic47⋊C4—22no—
71C2×C22⋊Dic47—22no—
72C2×(C2×D47)⋊C4—22no—
73#13093955—28no—
74#13093947—24no—
75#13093947—24no—
76#13093944—24no—
77#13093944—24no—
78C22⋊(C47⋊C8)—28no—
79C2×#41×C47—22no—
80C2×#41×C47—22no—
81C2×#41×C47—22no—
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C22×Q8×C47D411yescharacteristic
86C22×D4×C47D411yescharacteristic
87C22×D4×C47D412yes—
88C23×C4×C47—21no—
89C23×C4×C47C2312yes—
90C23×C4×C47C2311yescharacteristic
91#117×C47—22no—
92#117×C47—22no—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47C2312yes—
97C2×C4⋊C4×C47C2312yes—
98C2×C22⋊C4×C47—21no—
99C2×C22⋊C4×C47—24no—
100C2×C42×C47—21no—
101#95×C47—24no—
102C24×C47C2×D412yes—
103C24×C47C2×D411yescharacteristic
104C23×D47—21no—
105#41×C47—44no—
106#41×C47—44no—
107#41×C47—44no—
108#41×C47—44no—
109C2×Q8×C47—22no—
110C2×Q8×C47—22no—
111C2×Q8×C47—22no—
112C2×D4×C47—24no—
113C2×D4×C47—24no—
114C2×D4×C47—24no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—42no—
127C22×C4×C47—82no—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47#4112yes—
134C22×C4×C47#4112yes—
135C22×C4×C47#4112yes—
136C22×C4×C47C2411yescharacteristic
137C2×C22⋊D47—24no—
138C2×C22⋊D47—28no—
139C2×C22⋊D47—42no—
140C2×C22⋊D47—42no—
141C2×C22⋊D47—84no—
142C22×Dic47—22no—
143C22×Dic47—21no—
144C22×Dic47—24no—
145C22×Dic47—82no—
146#10625350—28no—
147#10625350—28no—
148#10625350—44no—
149#10625350—44no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188C2×D412yes—
156C2×C4×D47—24no—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94C2×D412yes—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—42no—
170C22⋊C4×C47—42no—
171C22⋊C4×C47—84no—
172C42×C47—42no—
173C22⋊Dic47—28no—
174(C2×D47)⋊C4—28no—
175C4⋊Dic47—24no—
176C4⋊Dic47—22no—
177C4⋊Dic47C2×D414yes—
178Dic47⋊C4—28no—
179#10625329—44no—
180C2×C47⋊C8—28no—
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—22no—
191C23×C47—22no—
192C23×C47—24no—
193C23×C47—24no—
194C23×C47#12012yes—
195C23×C47#12112yes—
196C23×C47#12312yes—
197C23×C47C22×D411yescharacteristic
198C22×D47—22no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—42no—
202C22×D47—42no—
203Q8×C47—44no—
204D4×C47—48no—
205D4×C47—44no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—44no—
217C2×C4×C47—42no—
218C2×C4×C47—84no—
219C2×C4×C47#12012yes—
220C2×C4×C47#12112yes—
221C2×C4×C47#12114yes—
222C2×C4×C47#12114yes—
223C2×C4×C47#12314yes—
224C2×C4×C47#12312yes—
225C2×C4×C47#12414yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C22⋊D47—48no—
233C22⋊D47—88no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—44no—
239C2×Dic47—84no—
240D188—22no—
241D188—22no—
242D188—41no—
243D188—41no—
244C4×D47—48no—
245Dic94—22no—
246Dic94—22no—
247Dic94—41no—
248Dic94—41no—
249C47⋊C8—24no—
250C47⋊C8—42no—
251#23489—471nomaximal, Sylow
252C22×C47—24no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—24no—
256C22×C47—44no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—44no—
261C22×C47#39414yes—
262C22×C47C2×#12011yescharacteristic
263C22×C47C2×#12111yescharacteristic
264C22×C47C2×#12311yescharacteristic
265C22×C47#57712yes—
266C22×C47#57712yes—
267C2×D47—42no—
268C2×D47—42no—
269C2×D47—42no—
270C2×D47—82no—
271C2×D47—82no—
272C4×C47—42no—
273C4×C47—82no—
274C4×C47—82no—
275C4×C47#39414yes—
276C4×C47C2×#12111yescharacteristic
277C4×C47C2×#12111yescharacteristic
278C4×C47C2×#12311yescharacteristic
279C4×C47C2×#12411yescharacteristic
280Dic47—42no—
281Dic47—44no—
282C2×#537—471no—
283C22×#106—471no—
284C2×#417—471no—
285C2×#416—471no—
286C2×#408—471no—
287#2311D4711yescharacteristic, complemented
288#2035—474no—
289#2034—474no—
290#1736—471no—
291C2×C47—24no—
292C2×C47—82no—
293C2×C47C2×#39411yescharacteristic
294C2×C47#300911yescharacteristic
295C2×C47#301011yescharacteristic
296C2×C47#306012yes—
297C2×C47#309912yes—
298D47—82no—
299C22×#41—471no—
300C22×#41C2×D4711yescharacteristic
301C22×M16—471no—
302C23×C8—471no—
303#537—944no—
304#537—944no—
305C2×C22⋊Q8—472no—
306C2×C22⋊Q8—941no—
307C2×#121—472no—
308C2×#121—941no—
309C2×#121—942no—
310C2×#117C2×D4711yescharacteristic
311C22×C4⋊C4—471no—
312C22×C4⋊C4C2×D4711yescharacteristic
313C2×#106—474no—
314C2×#106—944no—
315#417—944no—
316#416—944no—
317C2×#103—472no—
318C2×#103—472no—
319C2×#102—472no—
320C2×#102—472no—
321#408—944no—
322C2×C22⋊C8—474no—
323#391—21no—
324#390—21no—
325#386—22no—
326#380C2×D4714yes—
327#340—474no—
328C47#2348911yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—474no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C2×#41—942no—
341C22×Q8—471no—
342C22×Q8C22⋊D4711yescharacteristic
343C22×D4—471no—
344C22×D4—942no—
345C22×D4—941no—
346C22×D4C22⋊D4712yes—
347C22×D4C22⋊D4711yescharacteristic
348C23×C4—21no—
349C23×C4—941no—
350C23×C4—941no—
351C23×C4C22×D4712yes—
352C23×C4C22×D4711yescharacteristic
353C2×M16—942no—
354C2×M16—942no—
355C2×M16—942no—
356C22×C8—474no—
357C22×C8—472no—
358C22×C8—472no—
359C22×C8—942no—
360C22×C8—942no—
361C22⋊Q8—944no—
362C22⋊Q8—944no—
363C22⋊Q8—948no—
364#121—944no—
365#121—944no—
366#121—948no—
367#121—9416no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4C22×D4712yes—
379C2×C4⋊C4C22×D4712yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—942no—
384C2×C42—21no—
385#106—948no—
386#103—944no—
387#103—944no—
388#102—944no—
389#102—944no—
390C22⋊C8—948no—
391#95—24no—
392C24—941no—
393C24C2×C22⋊D4712yes—
394C24C2×C22⋊D4711yescharacteristic
395#41—44no—
396#41—44no—
397#41—44no—
398#41—44no—
399#41—948no—
400#41—948no—
401#41—1884no—
402#41—1884no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—472no—
407C2×Q8—941no—
408C2×Q8—941no—
409C2×Q8—941no—
410C2×Q8—941no—
411C2×Q8—941no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—472no—
419C2×D4—948no—
420C2×D4—944no—
421C2×D4—941no—
422C2×D4—941no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—941no—
426C2×D4—3764no—
427C2×D4—1882no—
428C2×D4—1882no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—944no—
440C22×C4—942no—
441C22×C4—941no—
442C22×C4—944no—
443C22×C4—3762no—
444C22×C4—1882no—
445C22×C4—1882no—
446C22×C4#1062535012yes—
447C22×C4#1062535012yes—
448C22×C4D4×D4712yes—
449C22×C4D4×D4712yes—
450C22×C4D4×D4712yes—
451C22×C4#1062535212yes—
452C22×C4C2×C22⋊D4712yes—
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C23×D4711yescharacteristic
455M16—1884no—
456C2×C8—474no—
457C2×C8—948no—
458C2×C8—942no—
459C2×C8—942no—
460C2×C8—942no—
461C2×C8—942no—
462C2×C8—942no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—474no—
469C4⋊C4—948no—
470C4⋊C4—944no—
471C4⋊C4—942no—
472C22⋊C4—42no—
473C22⋊C4—84no—
474C22⋊C4—948no—
475C22⋊C4—948no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—942no—
485C23—1881no—
486C23—1881no—
487C23—1882no—
488C23—1882no—
489C23#1309399212yes—
490C23#1309399612yes—
491C23#1309400712yes—
492C23C22×C22⋊D4711yescharacteristic
493Q8—44no—
494Q8—942no—
495Q8—942no—
496Q8—1881no—
497Q8—1881no—
498D4—48no—
499D4—44no—
500D4—942no—
501D4—942no—
502D4—3768no—
503D4—1888no—
504D4—1881no—
505D4—1881no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—84no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—3764no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1884no—
525C2×C4—1888no—
526C2×C4#1309399212yes—
527C2×C4#1309399612yes—
528C2×C4#1309399714yes—
529C2×C4#1309399714yes—
530C2×C4#1309400614yes—
531C2×C4#1309400712yes—
532C2×C4#1309401014yes—
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4C2×D4×D4711yescharacteristic
536C2×C4C2×#1062535211yescharacteristic
537C2×C4C2×#1062535011yescharacteristic
538C2×C4C2×#1062535011yescharacteristic
539C8—944no—
540C8—1882no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—3762no—
551C22—3762no—
552C22—1882no—
553C22—1882no—
554C22—1882no—
555C22#1568903814yes—
556C22C2×#1309399211yescharacteristic
557C22C2×#1309399611yescharacteristic
558C22C2×#1309400711yescharacteristic
559C22#1568993112yes—
560C22#1568993412yes—
561C4—42no—
562C4—82no—
563C4—82no—
564C4—1882no—
565C4—1884no—
566C4#1568903814yes—
567C4C2×#1309399711yescharacteristic
568C4C2×#1309399711yescharacteristic
569C4C2×#1309400611yescharacteristic
570C4C2×#1309401011yescharacteristic
571C2—24no—
572C2—82no—
573C2—3762no—
574C2C2×#1568903811yescharacteristic
575C2#2025266311yescharacteristic
576C2#2025267211yescharacteristic
577C2#2025304712yes—
578C2#2025337712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2311C2
socleC23×C47C22×D4
prime radicalC2×#15689701C2
hypercenter#2311D47
hypocenterC47#23489
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C22×#13093955C2semidirect
3. C2×#15689122C2semidirect
4. C2×#15689117C2semidirect
5. C2×#15689090C2semidirect
6. #20244584C2semidirect
7. #20244577C2semidirect
8. #20243109C2semidirect
9. C47×#2311C2semidirect
56. C22×C47⋊C8D4semidirect
287. #2311D47semidirect
328. C47#23489semidirect

Maximal subgroups

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

subgroupindex
C2×#156897012
C22×#130939552
C2×#156891222
C2×#156891172
C2×#156890902
#202445842
#202445772
#202431092
C47×#23112
#2348947