Finite Group Databasebeta

← #26222671

Subgroups — #26222671

Order 12032.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689325C211yesmaximal, characteristic
2C2×#15689221C211yesmaximal, characteristic
3C2×C4×#10625336C211yesmaximal, characteristic
4#20245750C211yesmaximal, characteristic
5#20244312C211yesmaximal, characteristic, complemented
6#20244154C211yesmaximal, characteristic, complemented
7#20244026C211yesmaximal, characteristic, complemented
8#20244019C211yesmaximal, characteristic, complemented
9#20243725C211yesmaximal, characteristic
10#20243725C211yesmaximal, characteristic
11#20243395C211yesmaximal, characteristic
12#20243245C211yesmaximal, characteristic, complemented
13#20243218C211yesmaximal, characteristic, complemented
14C4×#13093948C211yesmaximal, characteristic, complemented
15C47×#2327C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C22×#10625336C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19C2×C4×D188C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21#15689325C418yescomplemented
22#15689221—24no—
23C4×#10625336—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093948C2211yescharacteristic
31C2×#13093948C2211yescharacteristic
32C2×#13093948C2211yescharacteristic
33C2×#13093948C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×#13093944C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41#15688970—21no—
42#15688970—21no—
43#15688970—21no—
44#95×D47—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#391C2211yescharacteristic
55C47×#391C2211yescharacteristic
56C47×#387—21no—
57C47×#387—21no—
58C47×#387—21no—
59C47×#387—21no—
60C47×#382C2211yescharacteristic
61C47×#376C2211yescharacteristic
62C22×D188C2311yescharacteristic
63C22×C4×D47—21no—
64C22×C4×D47—21no—
65C22×C4×D47—21no—
66C2×#10625336—22no—
67C2×#10625336—22no—
68C2×#10625336—22no—
69C2×#10625336—22no—
70C2×#10625336C2×C414yes—
71C2×#10625336C2×C414yes—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75C22×C47⋊C8C2311yescharacteristic
76C4×D188—22no—
77C4×D188—22no—
78C4×D188—22no—
79C4×D188—22no—
80C4×D188C2×C414yes—
81C4×D188D414yes—
82C2×C4⋊Dic47C2311yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×C4⋊Dic47C2311yescharacteristic
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91#13093957—22no—
92#13093955—22no—
93C4⋊(C47⋊C8)—22no—
94C4⋊(C47⋊C8)C2×C414yescomplemented
95#13093948—22no—
96#13093948—22no—
97#13093948—22no—
98#13093948—22no—
99#13093948C2×C414yescomplemented
100#13093948C2×C414yescomplemented
101#13093944—22no—
102#13093944—22no—
103#13093935—22no—
104C4×C47⋊C8—22no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C22×Q8×C47C2311yescharacteristic
110C4⋊Q8×C47—22no—
111C4⋊Q8×C47C2×C414yescomplemented
112C4×Q8×C47—22no—
113C4×Q8×C47—22no—
114C4×Q8×C47—22no—
115C4×Q8×C47—22no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C42×C47—21no—
123C2×C42×C47—21no—
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126C2×C42×C47C2311yescharacteristic
127#95×C47—21no—
128#95×C47—21no—
129#95×C47—21no—
130#95×C47—21no—
131#95×C47—21no—
132#95×C47—21no—
133#95×C47D411yescharacteristic
134#95×C47D411yescharacteristic
135#95×C47D411yescharacteristic
136#95×C47D411yescharacteristic
137C23×D47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47—21no—
147C2×Q8×C47—21no—
148C2×Q8×C47C22×C412yes—
149C2×Q8×C47C22×C412yes—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2411yescharacteristic
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C2×D188—21no—
165C2×D188—21no—
166C2×D188—21no—
167C2×D188C22×C412yes—
168C2×D188C2×D412yes—
169C2×D188#4112yes—
170C2×C4×D47—24no—
171C2×C4×D47—24no—
172C2×C4×D47—24no—
173C2×C4×D47—81no—
174C2×C4×D47—81no—
175C2×C4×D47—81no—
176C2×C4×D47—81no—
177C2×C4×D47—81no—
178C2×C4×D47—81no—
179C4⋊C4×C47—24no—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—42no—
186C4⋊C4×C47—42no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—42no—
190C42×C47—42no—
191C42×C47C22×C412yes—
192C42×C47C2×D412yes—
193C42×C47#4112yes—
194C42×C47#4112yes—
195#10625336—28no—
196#10625336—28no—
197(C2×D47)⋊C4—24no—
198(C2×D47)⋊C4—24no—
199(C2×D47)⋊C4—24no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47C22×C412yes—
204C4⋊Dic47C2×D412yes—
205C4⋊Dic47#4112yes—
206C4×Dic47—81no—
207C4×Dic47—81no—
208C4×Dic47—81no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8C22×C412yes—
220C2×C47⋊C8C22×C412yes—
221C23×C47C22×D411yescharacteristic
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—41no—
234Q8×C47—41no—
235Q8×C47—41no—
236Q8×C47—41no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47—21no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—41no—
263C2×C4×C47C4×D412yes—
264C2×C4×C47C4×D412yes—
265C2×C4×C47#12112yes—
266C2×C4×C47#12112yes—
267C2×C4×C47C23×C411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—22no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282D188—22no—
283D188—22no—
284D188—22no—
285D188C4×D414yes—
286C4×D47—84no—
287C4×D47—84no—
288C4×D47—84no—
289C47⋊C8—22no—
290C47⋊C8—22no—
291C47⋊C8—41no—
292C47⋊C8—41no—
293#21514—471nomaximal, Sylow
294C22×C47C2×C4×D411yescharacteristic
295C22×C47C2×#12111yescharacteristic
296C22×C47#52511yescharacteristic
297C22×C47#57711yescharacteristic
298C22×C47#57711yescharacteristic
299C22×C47#57711yescharacteristic
300C22×C47#57811yescharacteristic
301C2×D47—24no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C2×D47—42no—
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—41no—
313C4×C47—41no—
314C4×C47—42no—
315C4×C47—42no—
316C4×C47—81no—
317C4×C47—81no—
318C4×C47C2×C4×D411yescharacteristic
319C4×C47C4×#4111yescharacteristic
320C4×C47#52911yescharacteristic
321C4×C47#53211yescharacteristic
322Dic47—42no—
323Dic47—42no—
324Dic47—42no—
325C2×#461—471no—
326C2×#440—471no—
327C2×C4×SD16—471no—
328#2327D4711yescharacteristic, complemented
329#2275—471no—
330#1981—471no—
331#1949—471no—
332#1928—471no—
333#1927—471no—
334#1877—471no—
335#1877—471no—
336#1804—471no—
337#1768—471no—
338#1763—471no—
339C4×#103—471no—
340C2×C47C4×#12111yescharacteristic
341C2×C47#294711yescharacteristic
342C2×C47#294911yescharacteristic
343C2×C47#304211yescharacteristic
344C2×C47#304611yescharacteristic
345C2×C47#312011yescharacteristic
346C2×C47#312311yescharacteristic
347D47—44no—
348C22×SD16—471no—
349C22×SD16—471no—
350C2×C4⋊Q8C2×D4711yescharacteristic
351C2×C4×Q8C2×D4711yescharacteristic
352C2×C4×Q8C2×D4711yescharacteristic
353C2×C4×D4—471no—
354C2×C4×D4—471no—
355C2×C4×D4—471no—
356#461—478no—
357#440—944no—
358C4×SD16—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103—471no—
368C2×#103—471no—
369C2×#102—471no—
370C2×#102—471no—
371C2×#97—471no—
372C2×C4×C8—471no—
373#391C2×D4711yescharacteristic
374#391C2×D4711yescharacteristic
375#387—21no—
376#387—21no—
377#387—21no—
378#387—21no—
379#382C2×D4711yescharacteristic
380C4×C4⋊C4—471no—
381C4×C4⋊C4—471no—
382C4×C4⋊C4—471no—
383C4×C22⋊C4—941no—
384C4×C22⋊C4—941no—
385C4×C22⋊C4—941no—
386#376C2×D4711yescharacteristic
387C2×#95—941no—
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#336—471no—
393#336—471no—
394C47#2151411yescharacteristic, complemented, Sylow
395C22×Q8C22×D4711yescharacteristic
396C22×Q8C22×D4711yescharacteristic
397C22×D4—471no—
398C23×C4—941no—
399C23×C4—941no—
400C23×C4—941no—
401C2×SD16—474no—
402C2×SD16—474no—
403C2×SD16—942no—
404C2×SD16—942no—
405C2×SD16—942no—
406C2×SD16—942no—
407C22×C8—471no—
408C22×C8—471no—
409C22×C8—471no—
410C22×C8—471no—
411C4⋊Q8—22no—
412C4⋊Q8C4×D4714yescomplemented
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×Q8—22no—
417C4×D4—474no—
418C4×D4—474no—
419C4×D4—942no—
420C4×D4—942no—
421C4×D4—942no—
422C4×D4—942no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—471no—
430C2×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—941no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42C22×D4711yescharacteristic
441C2×C42C22×D4711yescharacteristic
442C2×C42C22×D4711yescharacteristic
443#107—942no—
444#106—942no—
445C4⋊C8—474no—
446C4⋊C8—942no—
447#103—474no—
448#103—474no—
449#103—942no—
450#103—942no—
451#103—942no—
452#103—942no—
453#102—942no—
454#102—942no—
455#97—942no—
456C4×C8—942no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95C22⋊D4711yescharacteristic
467#95C22⋊D4711yescharacteristic
468#95C22⋊D4711yescharacteristic
469#95C22⋊D4711yescharacteristic
470C24—941no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8C2×C4×D4712yes—
482C2×Q8C2×C4×D4712yes—
483C2×D4—472no—
484C2×D4—472no—
485C2×D4—472no—
486C2×D4—941no—
487C2×D4—941no—
488C2×D4—941no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—944no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—944no—
498C22×C4—944no—
499C22×C4—3761no—
500C22×C4—3761no—
501C22×C4—3761no—
502C22×C4—3761no—
503C22×C4—3761no—
504C22×C4—3761no—
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C2×C22⋊D4711yescharacteristic
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C2×C22⋊D4711yescharacteristic
510C22×C4C2×C22⋊D4711yescharacteristic
511C22×C4C23×D4711yescharacteristic
512SD16—948no—
513SD16—948no—
514C2×C8—472no—
515C2×C8—472no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8—941no—
522C2×C8—941no—
523C2×C8—941no—
524C2×C8—941no—
525C2×C8—941no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—472no—
535C4⋊C4—472no—
536C4⋊C4—472no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4—941no—
540C22⋊C4—944no—
541C22⋊C4—944no—
542C22⋊C4—944no—
543C42—21no—
544C42—21no—
545C42—42no—
546C42—42no—
547C42—3761no—
548C42—3761no—
549C42—3761no—
550C42C2×C4×D4712yes—
551C42#1062535012yes—
552C42D4×D4712yes—
553C42#1062535212yes—
554C23—942no—
555C23—942no—
556C23—942no—
557C23—1881no—
558C23—1881no—
559C23—1881no—
560C23—1881no—
561C23C22×C22⋊D4711yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570D4—474no—
571D4—942no—
572D4—942no—
573D4—942no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—3764no—
604C2×C4—3764no—
605C2×C4—3764no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4C4×C22⋊D4712yes—
613C2×C4C4×C22⋊D4712yes—
614C2×C4#1309399512yes—
615C2×C4#1309399512yes—
616C2×C4C22×C4×D4711yescharacteristic
617C2×C4C2×D4×D4711yescharacteristic
618C2×C4#41×D4711yescharacteristic
619C2×C4#41×D4711yescharacteristic
620C2×C4C2×#1062535211yescharacteristic
621C2×C4C2×#1062535011yescharacteristic
622C8—942no—
623C8—942no—
624C8—1881no—
625C8—1881no—
626C22—944no—
627C22—1882no—
628C22—1882no—
629C22—1882no—
630C22—1882no—
631C22—1882no—
632C22—1882no—
633C22C2×C4×C22⋊D4711yescharacteristic
634C22C2×#1309399511yescharacteristic
635C22#1568963011yescharacteristic
636C22#1568993111yescharacteristic
637C22#1568993111yescharacteristic
638C22#1568993111yescharacteristic
639C22#1568993811yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—1882no—
651C4—1882no—
652C4—1882no—
653C4C4×D4×D4711yescharacteristic
654C4C4×#1062535211yescharacteristic
655C4#1568965511yescharacteristic
656C4#1568967011yescharacteristic
657C2—1884no—
658C2C4×#1309399511yescharacteristic
659C2#2025223111yescharacteristic
660C2#2025224711yescharacteristic
661C2#2025289611yescharacteristic
662C2#2025293411yescharacteristic
663C2#2025359611yescharacteristic
664C2#2025363511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2327C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#2327D47
hypocenterC47#21514
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2327

Complements

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

NH ≅ G/Ntype
5. #20244312C2semidirect
6. #20244154C2semidirect
7. #20244026C2semidirect
8. #20244019C2semidirect
12. #20243245C2semidirect
13. #20243218C2semidirect
14. C4×#13093948C2semidirect
15. C47×#2327C2semidirect
21. #15689325C4semidirect
94. C4⋊(C47⋊C8)C2×C4semidirect
99. #13093948C2×C4semidirect
100. #13093948C2×C4semidirect
111. C4⋊Q8×C47C2×C4semidirect
328. #2327D47semidirect
394. C47#21514semidirect
412. C4⋊Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156893252
C2×#156892212
C2×C4×#106253362
#202457502
#202443122
#202441542
#202440262
#202440192
#202437252
#202433952
#202432452
#202432182
C4×#130939482
C47×#23272
#2151447