Finite Group Databasebeta

← #26215777

Subgroups — #26215777

Order 12032.

1510 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2C2×#15689145C211yesmaximal, characteristic, complemented
3C2×#15689123C212yesmaximal
4#382×D47C211yesmaximal, characteristic, complemented
5#20243403C211yesmaximal, characteristic, complemented
6#20243401C212yesmaximal, complemented
7#20243399C211yesmaximal, characteristic, complemented
8#20243397C212yesmaximal, complemented
9#20243395C211yesmaximal, characteristic, complemented
10#20243106C212yesmaximal, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C22×#10625324C2212yes—
14C4×#10625354—24no—
15C4×#10625354C418yescomplemented
16C2×Q8×Dic47C2211yescharacteristic
17C2×C4×D188—21no—
18C2×C4×D188C2211yescharacteristic
19C2×#13093986—21no—
20C2×#13093986C2211yescharacteristic
21C2×#13093982—21no—
22C2×#13093982C2211yescharacteristic
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689145—24no—
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×Dic47⋊C8C2212yes—
29#15689123—28no—
30C2×#13093948C2212yes—
31C2×#13093946C2212yes—
32C2×#13093944C2212yes—
33#15688989C2211yescharacteristic
34#15688988C2212yes—
35#15688987—21no—
36#15688987—21no—
37#95×D47—21no—
38#95×D47—21no—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4⋊C8×C47C2211yescharacteristic
44C2×#103×C47C2212yes—
45C47×#382C2211yescharacteristic
46C47×#340C2212yes—
47C2×#10625354—22no—
48C2×#10625354—22no—
49C2×#10625354—22no—
50C2×#10625354C2×C414yes—
51C2×#10625354D414yes—
52C2×#10625354Q814yes—
53C22×D188—21no—
54C22×D188C2311yescharacteristic
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47—21no—
58C22×C4×D47—22no—
59C22×C4×D47D411yescharacteristic
60C22×C4×D47D411yescharacteristic
61C22×C4×D47C2311yescharacteristic
62C2×#10625324—24no—
63C2×#10625324—24no—
64C2×#10625324—24no—
65C22×C47⋊C8C2312yes—
66Q8×Dic47—22no—
67Q8×Dic47C2×C414yescomplemented
68C4×D188—28no—
69C4×D188—22no—
70C4×D188C2×C414yescomplemented
71#13093986—28no—
72#13093986—22no—
73#13093986C2×C414yescomplemented
74#13093982—22no—
75#13093982—28no—
76#13093982C2×C414yes—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47C2312yes—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—22no—
81C4⋊C4×D47—24no—
82C4⋊C4×D47—24no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C42×D47—28no—
92C42×D47C2×C414yes—
93C42×D47D414yes—
94C4⋊(C47⋊C8)—22no—
95Dic47⋊C8—24no—
96#13093948—24no—
97#13093946—24no—
98#13093944—24no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—22no—
102C47⋊#95—22no—
103C22×Q8×C47C2311yescharacteristic
104C22×C8×C47C2312yes—
105C4×Q8×C47—22no—
106C4×Q8×C47C2×C414yescomplemented
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2312yes—
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112C4⋊C8×C47—22no—
113#103×C47—24no—
114#95×C47—21no—
115#95×C47—21no—
116C23×D47—21no—
117C23×D47C2×D411yescharacteristic
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47C22×C412yes—
122C2×Q8×C47C2×D412yes—
123C2×Q8×C47C2×Q812yes—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2411yescharacteristic
130C22×Dic47—21no—
131C22×Dic47—22no—
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C22×Dic47C2×D411yescharacteristic
135#10625354—28no—
136#10625354—28no—
137#10625354—28no—
138#10625354C4⋊C4116yes—
139C2×D188—24no—
140C2×D188—24no—
141C2×D188—24no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188C22×C412yes—
146C2×D188C2×D412yes—
147C2×D188C2×Q812yes—
148C2×C4×D47—24no—
149C2×C4×D47—24no—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—24no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×C4×D47—44no—
156C2×C4×D47—44no—
157C2×C4×D47—44no—
158C2×C4×D47—44no—
159C2×C4×D47—44no—
160C2×C4×D47C22⋊C414yes—
161C2×C4×D47C22⋊C414yes—
162C2×C4×D47C22×C412yes—
163C2×C4×D47C2×D412yes—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×Q812yes—
166C2×C4×D47#4112yes—
167C2×C4×D47#4112yes—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47C22×C412yes—
176C42×C47—24no—
177C42×C47C22×C412yes—
178C42×C47C2×D412yes—
179(C2×D47)⋊C4—24no—
180(C2×D47)⋊C4—24no—
181(C2×D47)⋊C4—24no—
182C4⋊Dic47—21no—
183C4⋊Dic47—24no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47C22×C412yes—
187Dic47⋊C4—44no—
188C4×Dic47—24no—
189C4×Dic47—24no—
190C4×Dic47C22⋊C414yes—
191C4×Dic47C22⋊C414yes—
192C4×Dic47C22×C412yes—
193C4×Dic47C2×D412yes—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197#10625324—48no—
198C23×C47C22×D411yescharacteristic
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47C2×C22⋊C412yes—
207C22×D47#12112yes—
208C22×D47C22⋊Q812yes—
209Q8×C47—22no—
210Q8×C47—22no—
211Q8×C47—22no—
212Q8×C47C2×C4⋊C414yes—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47C2×C22⋊C412yes—
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47C2×C22⋊C412yes—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×C22⋊C412yes—
241C2×Dic47#12112yes—
242C2×Dic47C22⋊Q812yes—
243D188—216no—
244D188—22no—
245D188—22no—
246D188—22no—
247D188C2×C4⋊C414yes—
248C4×D47—216no—
249C4×D47—44no—
250C4×D47—44no—
251C4×D47—48no—
252C4×D47—48no—
253C4×D47—88no—
254C4×D47C2×C4⋊C414yes—
255C4×D47#11714yes—
256C4×D47#12114yes—
257C4×D47C22⋊Q814yes—
258C8×C47—42no—
259C47⋊C8—42no—
260#20649—471nomaximal, Sylow
261C22×C47C22×C22⋊C411yescharacteristic
262C22×C47C2×#12111yescharacteristic
263C22×C47C2×C22⋊Q811yescharacteristic
264C22×C47#57512yes—
265C22×C47#57512yes—
266C2×D47—24no—
267C2×D47—22no—
268C2×D47—22no—
269C2×D47—24no—
270C2×D47—24no—
271C2×D47—44no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—44no—
275C2×D47#38014yes—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—24no—
279C4×C47—42no—
280C4×C47—82no—
281C4×C47C22×C4⋊C411yescharacteristic
282C4×C47C2×#11711yescharacteristic
283C4×C47C2×#12111yescharacteristic
284C4×C47C2×C22⋊Q811yescharacteristic
285Dic47—22no—
286Dic47—22no—
287Dic47—42no—
288Dic47—82no—
289Dic47#38014yes—
290C2×C4×#41—471no—
291C2×#423—471no—
292C2×#417—472no—
293C2×#382—471no—
294#1805—472no—
295#1805—471no—
296#1805D4711yescharacteristic, complemented
297#1804—471no—
298#1804—471no—
299#1804—472no—
300#1736—472no—
301C2×C47C2×#38011yescharacteristic
302C2×C47#289611yescharacteristic
303C2×C47#289611yescharacteristic
304C2×C47#304012yes—
305C2×C47#308012yes—
306D47—24no—
307D47—44no—
308C22×#41—471no—
309C22×M16—472no—
310C4×#41—478no—
311C4×#41—944no—
312C2×C4×Q8—471no—
313C2×C4×Q8C2×D4711yescharacteristic
314C2×C4×D4—471no—
315C2×C4×D4—471no—
316C2×C4×D4—941no—
317C2×C4×D4—941no—
318C2×#117—471no—
319C2×#117—941no—
320C22×C4⋊C4—472no—
321C22×C42—471no—
322C22×C42—941no—
323#423—944no—
324C2×C4⋊C8—472no—
325C2×C4⋊C8—471no—
326C2×C4⋊C8C2×D4711yescharacteristic
327#417—948no—
328C2×#103—472no—
329C2×#103C2×D4712yes—
330C2×#102—472no—
331C2×#102—472no—
332#382—472no—
333#382—471no—
334#382—941no—
335#382—941no—
336#382C2×D4711yescharacteristic
337C2×#95—941no—
338C2×#95—941no—
339#340—472no—
340#340—472no—
341#340—472no—
342#340C2×D4712yes—
343C47#2064911yescharacteristic, complemented, Sylow
344C2×#41—474no—
345C2×#41—474no—
346C2×#41—474no—
347C2×#41—942no—
348C2×#41—942no—
349C2×#41—942no—
350C22×Q8C22×D4711yescharacteristic
351C22×D4—471no—
352C22×D4—941no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4—471no—
356C23×C4—941no—
357C23×C4—941no—
358C23×C4—942no—
359C23×C4—941no—
360C2×M16—944no—
361C2×M16—944no—
362C2×M16—944no—
363C22×C8—472no—
364C22×C8C22×D4712yes—
365C4×Q8—22no—
366C4×Q8—474no—
367C4×Q8—942no—
368C4×Q8C4×D4714yescomplemented
369C4×D4—474no—
370C4×D4—474no—
371C4×D4—948no—
372C4×D4—942no—
373C4×D4—948no—
374C4×D4—942no—
375#117—474no—
376#117—948no—
377#117—942no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—944no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4C22×D4712yes—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C22⋊C4—941no—
388C2×C22⋊C4—941no—
389C2×C22⋊C4—941no—
390C2×C42—21no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—471no—
394C2×C42—474no—
395C2×C42—474no—
396C2×C42—948no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42C22×D4711yescharacteristic
400C4⋊C8—22no—
401C4⋊C8—944no—
402C4⋊C8—942no—
403#103—24no—
404#103—944no—
405#102—944no—
406#102—944no—
407#95—21no—
408#95—21no—
409#95—942no—
410#95—942no—
411#95—941no—
412#95—941no—
413C24—471no—
414C24—941no—
415#41—4716no—
416#41—948no—
417#41—948no—
418#41—948no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8C2×C4×D4712yes—
423C2×Q8D4×D4712yes—
424C2×Q8Q8×D4712yes—
425C2×D4—472no—
426C2×D4—472no—
427C2×D4—472no—
428C2×D4—944no—
429C2×D4—944no—
430C2×D4—944no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—471no—
438C22×C4—472no—
439C22×C4—472no—
440C22×C4—472no—
441C22×C4—472no—
442C22×C4—472no—
443C22×C4—472no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—941no—
453C22×C4—944no—
454C22×C4—942no—
455C22×C4—1882no—
456C22×C4—1882no—
457C22×C4—1884no—
458C22×C4—1884no—
459C22×C4—1884no—
460C22×C4—1884no—
461C22×C4—1884no—
462C22×C4D4×D4711yescharacteristic
463C22×C4D4×D4711yescharacteristic
464C22×C4C23×D4711yescharacteristic
465M16—1888no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—942no—
470C2×C8—942no—
471C2×C8—942no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—472no—
477C4⋊C4—944no—
478C4⋊C4—941no—
479C4⋊C4—942no—
480C4⋊C4—942no—
481C4⋊C4—1884no—
482C4⋊C4C2×C4×D4712yes—
483C22⋊C4—944no—
484C22⋊C4—944no—
485C22⋊C4—944no—
486C42—24no—
487C42—474no—
488C42—474no—
489C42—472no—
490C42—472no—
491C42—944no—
492C42—944no—
493C42C2×C4×D4712yes—
494C42D4×D4712yes—
495C23—472no—
496C23—472no—
497C23—472no—
498C23—942no—
499C23—942no—
500C23—942no—
501C23—942no—
502C23—942no—
503C23—1882no—
504C23—1882no—
505C23C2×D4×D4711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8C4⋊C4×D4714yes—
510D4—474no—
511D4—9416no—
512D4—942no—
513D4—942no—
514D4—942no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—472no—
526C2×C4—472no—
527C2×C4—472no—
528C2×C4—474no—
529C2×C4—474no—
530C2×C4—474no—
531C2×C4—474no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—9416no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—3768no—
539C2×C4—1884no—
540C2×C4—1884no—
541C2×C4—1888no—
542C2×C4—1888no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C2×C4C22⋊C4×D4712yes—
548C2×C4C22⋊C4×D4712yes—
549C2×C4C22×C4×D4711yescharacteristic
550C2×C4C2×D4×D4711yescharacteristic
551C2×C4C2×D4×D4711yescharacteristic
552C2×C4C2×Q8×D4711yescharacteristic
553C2×C4#41×D4711yescharacteristic
554C2×C4#41×D4711yescharacteristic
555C8—42no—
556C8—1882no—
557C22—474no—
558C22—942no—
559C22—944no—
560C22—944no—
561C22—942no—
562C22—944no—
563C22—1882no—
564C22—1884no—
565C22—1884no—
566C22—1882no—
567C22C2×C22⋊C4×D4711yescharacteristic
568C22#121×D4711yescharacteristic
569C22C22⋊Q8×D4711yescharacteristic
570C22#1568992012yes—
571C22#1568992212yes—
572C4—22no—
573C4—24no—
574C4—22no—
575C4—42no—
576C4—82no—
577C4—474no—
578C4—942no—
579C4—942no—
580C4—3762no—
581C4—1882no—
582C4C2×C4⋊C4×D4711yescharacteristic
583C4#117×D4711yescharacteristic
584C4#121×D4711yescharacteristic
585C4C22⋊Q8×D4711yescharacteristic
586C2—944no—
587C2—1884no—
588C2#380×D4711yescharacteristic
589C2#2025190311yescharacteristic
590C2#2025190611yescharacteristic
591C2#2025287912yes—
592C2#2025320712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47#20649
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#15689145C2semidirect
4. #382×D47C2semidirect
5. #20243403C2semidirect
6. #20243401C2semidirect
7. #20243399C2semidirect
8. #20243397C2semidirect
9. #20243395C2semidirect
10. #20243106C2semidirect
11. C47×#1805C2semidirect
15. C4×#10625354C4semidirect
67. Q8×Dic47C2×C4semidirect
70. C4×D188C2×C4semidirect
73. #13093986C2×C4semidirect
106. C4×Q8×C47C2×C4semidirect
296. #1805D47semidirect
343. C47#20649semidirect
368. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×#156891452
C2×#156891232
#382×D472
#202434032
#202434012
#202433992
#202433972
#202433952
#202431062
C47×#18052
#2064947