Finite Group Databasebeta

← #26244962

Subgroups — #26244962

Order 12032.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689163C211yesmaximal, characteristic, complemented
2C2×#15689150C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic
4C2×#15689119C211yesmaximal, characteristic
5#20246008C211yesmaximal, characteristic, complemented
6#20244783C212yesmaximal, complemented
7#20244776C212yesmaximal, complemented
8#20244769C212yesmaximal, complemented
9#20244762C212yesmaximal, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#2311C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14C2×#13093983C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16#15689163—24no—
17C2×#13093957C2212yes—
18C2×#13093955C2212yes—
19#15689150—24no—
20C2×C4⋊(C47⋊C8)C2212yes—
21C2×C4⋊(C47⋊C8)C2212yes—
22#15689124—24no—
23#15689119—24no—
24C2×#13093948C2211yescharacteristic
25C2×#13093948C2212yes—
26C2×#13093948C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093945C2212yes—
29C2×#13093945C2211yescharacteristic
30#15689021C2212yes—
31#15689021C2212yes—
32#15688995—22no—
33#15688992—21no—
34#15688988—21no—
35#15688885C2212yes—
36#15688885C2212yes—
37C22×#41×C47C2211yescharacteristic
38C2×#117×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2211yescharacteristic
40C47×#391C2212yes—
41C47×#390C2212yes—
42C47×#386—22no—
43C47×#380—21no—
44C47×#380—21no—
45C23×Dic47—21no—
46C2×#10625329—22no—
47C2×#10625329—22no—
48C2×#10625329D414yes—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8C2311yescharacteristic
53C22×C47⋊C8C2311yescharacteristic
54C22×C47⋊C8C2312yes—
55#13093983—22no—
56#13093983D414yes—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47C2312yes—
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2311yescharacteristic
63C2×Dic47⋊C4—22no—
64C2×Dic47⋊C4—22no—
65C2×C22⋊Dic47—21no—
66C2×C4×Dic47—21no—
67#13093957—24no—
68#13093955—24no—
69C4⋊(C47⋊C8)—24no—
70C4⋊(C47⋊C8)—24no—
71#13093948—22no—
72#13093948—22no—
73#13093948—24no—
74#13093945—22no—
75#13093945—22no—
76#13093945—24no—
77C47⋊#95—22no—
78C47⋊#95—22no—
79C2×#41×C47—22no—
80C2×#41×C47—22no—
81C2×#41×C47—22no—
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47Q814yes—
85C22×Q8×C47C2311yescharacteristic
86C22×D4×C47—21no—
87C22×D4×C47C2311yescharacteristic
88C23×C4×C47—21no—
89C23×C4×C47—21no—
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×C47D411yescharacteristic
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C4⋊C4×C47C2312yes—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47D412yes—
102C2×C42×C47C2312yes—
103#95×C47—22no—
104#95×C47—22no—
105C24×C47—21no—
106C24×C47C2×D411yescharacteristic
107#41×C47—28no—
108#41×C47—28no—
109#41×C47—44no—
110#41×C47—44no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C2×Q812yes—
117C2×D4×C47—24no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—42no—
124C2×D4×C47—42no—
125C2×D4×C47C2×Q812yes—
126C22×C4×C47—21no—
127C22×C4×C47—24no—
128C22×C4×C47—22no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—81no—
133C22×C4×C47—81no—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×Q812yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47#4112yes—
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—22no—
147C22×Dic47—42no—
148C22×Dic47—81no—
149C22×Dic47—81no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—42no—
155C22⋊C4×C47—24no—
156C22⋊C4×C47—82no—
157C22⋊C4×C47—82no—
158C42×C47—22no—
159C42×C47—22no—
160C22⋊Dic47—24no—
161C4⋊Dic47—21no—
162C4⋊Dic47—21no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—42no—
166C4⋊Dic47C2×D412yes—
167C4⋊Dic47C2×D412yes—
168Dic47⋊C4—82no—
169Dic47⋊C4—82no—
170C4×Dic47—24no—
171#10625329—28no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—44no—
181C2×C47⋊C8C2×D414yes—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—22no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47#12112yes—
194C23×C47#12112yes—
195C23×C47C22⋊Q812yes—
196C23×C47C22×D411yescharacteristic
197Q8×C47—22no—
198Q8×C47—22no—
199Q8×C47—41no—
200Q8×C47—41no—
201D4×C47—22no—
202D4×C47—22no—
203D4×C47—48no—
204D4×C47—41no—
205D4×C47—41no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—48no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—84no—
220C2×C4×C47#12112yes—
221C2×C4×C47#12112yes—
222C2×C4×C47#12114yes—
223C2×C4×C47C22⋊Q814yes—
224C2×C4×C47C22⋊Q814yes—
225C2×C4×C47C22⋊Q812yes—
226C2×C4×C47#12614yes—
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×Q811yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—84no—
243C47⋊C8—24no—
244C47⋊C8—41no—
245C47⋊C8—41no—
246#23616—471nomaximal, Sylow
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—81no—
257C22×C47—81no—
258C22×C47—81no—
259C22×C47—81no—
260C22×C47#39614yes—
261C22×C47C2×#12111yescharacteristic
262C22×C47C2×#12111yescharacteristic
263C22×C47C2×C22⋊Q811yescharacteristic
264C22×C47#57611yescharacteristic
265C22×C47#57711yescharacteristic
266C22×C47#57711yescharacteristic
267C22×C47#57811yescharacteristic
268C4×C47—42no—
269C4×C47—42no—
270C4×C47—42no—
271C4×C47—82no—
272C4×C47#39614yes—
273C4×C47C2×#12111yescharacteristic
274C4×C47C2×C22⋊Q811yescharacteristic
275C4×C47C2×C22⋊Q811yescharacteristic
276C4×C47C2×#12611yescharacteristic
277Dic47—42no—
278Dic47—42no—
279Dic47—82no—
280C2×#428—471no—
281C2×#424—471no—
282C2×#417—471no—
283C2×#416—471no—
284#2317—471no—
285#2311D4711yescharacteristic
286#2068—472no—
287#2067—472no—
288#2066—472no—
289#2065—472no—
290C2×#340—471no—
291C2×C47—24no—
292C2×C47—82no—
293C2×C47C2×#39611yescharacteristic
294C2×C47#304511yescharacteristic
295C2×C47#304611yescharacteristic
296C2×C47#306211yescharacteristic
297C2×C47#306311yescharacteristic
298C2×C47#308311yescharacteristic
299C2×C47#308411yescharacteristic
300C22×#41C2×D4711yescharacteristic
301C22×M16—471no—
302C23×C8—471no—
303C2×#117—471no—
304C2×#117C2×D4711yescharacteristic
305C22×C4⋊C4—471no—
306C22×C4⋊C4C2×D4711yescharacteristic
307#428—944no—
308C2×#107—472no—
309C2×#106—472no—
310#424—944no—
311C2×C4⋊C8—472no—
312C2×C4⋊C8—472no—
313#417—944no—
314#416—944no—
315C2×#103—471no—
316C2×#103—472no—
317C2×#103—471no—
318C2×#102—471no—
319C2×#102—472no—
320C2×#102—471no—
321#391C2×D4712yes—
322#390C2×D4712yes—
323#389—472no—
324#389—472no—
325#386—22no—
326#385—942no—
327#384—941no—
328#382—941no—
329#380—21no—
330#380—21no—
331#340—472no—
332#340—472no—
333C47#2361611yescharacteristic, complemented, Sylow
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41Dic9414yes—
340C22×Q8C22×D4711yescharacteristic
341C22×D4—21no—
342C22×D4C22×D4711yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—941no—
346C23×C4C22×D4711yescharacteristic
347C2×M16—474no—
348C2×M16—942no—
349C2×M16—942no—
350C22×C8—472no—
351C22×C8—471no—
352C22×C8—471no—
353C22×C8—942no—
354C22×C8—942no—
355C22×C8—942no—
356#117—22no—
357#117—22no—
358#117—474no—
359#117—942no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—472no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—942no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22⋊D4711yescharacteristic
372C2×C4⋊C4C22⋊D4711yescharacteristic
373C2×C4⋊C4C22×D4712yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4C22⋊D4712yes—
378C2×C42—941no—
379C2×C42C22×D4712yes—
380#107—944no—
381#106—944no—
382C4⋊C8—944no—
383C4⋊C8—944no—
384#103—942no—
385#103—942no—
386#103—944no—
387#102—942no—
388#102—942no—
389#102—944no—
390#95—22no—
391#95—22no—
392#95—942no—
393#95—942no—
394C24—21no—
395C24C2×C22⋊D4711yescharacteristic
396#41—28no—
397#41—28no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8C2×Dic9412yes—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×Dic9412yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—942no—
426C22×C4—3761no—
427C22×C4—3761no—
428C22×C4—1882no—
429C22×C4C2×Dic9412yes—
430C22×C4#1062535012yes—
431C22×C4D4×D4712yes—
432C22×C4D4×D4712yes—
433C22×C4#1062535212yes—
434C22×C4#1062535212yes—
435C22×C4C2×C22⋊D4712yes—
436C22×C4C2×C22⋊D4712yes—
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C23×D4711yescharacteristic
439M16—948no—
440C2×C8—474no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—1884no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—472no—
456C4⋊C4—472no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—942no—
460C4⋊C4—942no—
461C4⋊C4—3762no—
462C4⋊C4—3762no—
463C4⋊C4—1882no—
464C22⋊C4—24no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—944no—
468C42—22no—
469C42—22no—
470C42—944no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23#1309399512yes—
483C23#1309399612yes—
484C23#1309400312yes—
485C23C22×C22⋊D4711yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490D4—22no—
491D4—22no—
492D4—48no—
493D4—41no—
494D4—41no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—84no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—3764no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4#1309399512yes—
520C2×C4#1309399612yes—
521C2×C4#1309399714yes—
522C2×C4#1309400114yes—
523C2×C4#1309400114yes—
524C2×C4#1309400312yes—
525C2×C4#1309401714yes—
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4C2×D4×D4711yescharacteristic
528C2×C4C22×Dic9411yescharacteristic
529C2×C4C2×#1062535211yescharacteristic
530C2×C4C2×#1062535211yescharacteristic
531C2×C4C2×#1062535011yescharacteristic
532C8—944no—
533C8—1881no—
534C8—1881no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#1568904614yes—
549C22C2×#1309399511yescharacteristic
550C22C2×#1309399611yescharacteristic
551C22C2×#1309400311yescharacteristic
552C22#1568992611yescharacteristic
553C22#1568993111yescharacteristic
554C22#1568993411yescharacteristic
555C22#1568993811yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—3762no—
561C4—1882no—
562C4—1882no—
563C4#1568904614yes—
564C4C2×#1309399711yescharacteristic
565C4C2×#1309400111yescharacteristic
566C4C2×#1309400111yescharacteristic
567C4C2×#1309401711yescharacteristic
568C2—24no—
569C2—82no—
570C2C2×#1568904611yescharacteristic
571C2#2025292311yescharacteristic
572C2#2025293411yescharacteristic
573C2#2025307011yescharacteristic
574C2#2025308111yescharacteristic
575C2#2025323911yescharacteristic
576C2#2025325011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2311C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
hypercenter#2311D47
hypocenterC47#23616
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C2×#15689163C2semidirect
2. C2×#15689150C2semidirect
5. #20246008C2semidirect
6. #20244783C2semidirect
7. #20244776C2semidirect
8. #20244769C2semidirect
9. #20244762C2semidirect
10. C2×#15688885C2semidirect
333. C47#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#156891632
C2×#156891502
C2×#156891242
C2×#156891192
#202460082
#202447832
#202447762
#202447692
#202447622
C2×#156888852
C47×#23112
#2361647