Finite Group Databasebeta

← #26138433

Subgroups — #26138433

Order 12032.

1962 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688998C212yesmaximal
2C2×#15688989C211yesmaximal, characteristic, complemented
3C22×C47⋊#95C211yesmaximal, characteristic
4#20241645C212yesmaximal, complemented
5#20241645C212yesmaximal, complemented
6#20241618C211yesmaximal, characteristic, complemented
7#20241615C211yesmaximal, characteristic, complemented
8#20241615C211yesmaximal, characteristic, complemented
9#20241611C211yesmaximal, characteristic
10#20241610C212yesmaximal
11C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
12C24×Dic47C2212yes—
13C22×C4⋊Dic47C2212yes—
14C22×C4⋊Dic47C2212yes—
15C22×C22⋊Dic47C2212yes—
16#15688998—28no—
17#15688989—24no—
18C2×C47⋊#95—24no—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2211yescharacteristic
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2211yescharacteristic
31C24×C4×C47C2211yescharacteristic
32C22×C22⋊C4×C47C2211yescharacteristic
33C22×C22⋊C4×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C22×C42×C47C2211yescharacteristic
36C4×C22⋊C4×C47C418yescomplemented
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C23×Dic47—24no—
40C23×Dic47—24no—
41C23×Dic47—24no—
42C23×Dic47—24no—
43C23×Dic47—24no—
44C23×Dic47—42no—
45C23×Dic47—42no—
46C23×Dic47C2312yes—
47C23×Dic47C2312yes—
48C23×Dic47C2312yes—
49C23×Dic47C2312yes—
50C2×C4⋊Dic47—24no—
51C2×C4⋊Dic47—24no—
52C2×C4⋊Dic47—24no—
53C2×C4⋊Dic47—24no—
54C2×C4⋊Dic47—42no—
55C2×C4⋊Dic47—42no—
56C2×C22⋊Dic47—24no—
57C2×C22⋊Dic47—24no—
58C2×C22⋊Dic47—24no—
59C47⋊#95—22no—
60C47⋊#95—22no—
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—22no—
64C47⋊#95—24no—
65C47⋊#95—24no—
66C47⋊#95—24no—
67C47⋊#95—42no—
68C47⋊#95—42no—
69C47⋊#95—42no—
70C47⋊#95D414yes—
71C25×C47C2311yescharacteristic
72C23×C4×C47—24no—
73C23×C4×C47—22no—
74C23×C4×C47—22no—
75C23×C4×C47—42no—
76C23×C4×C47C2×C414yes—
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2311yescharacteristic
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C2×C22⋊C4×C47—24no—
84C2×C22⋊C4×C47C2×C414yes—
85C2×C22⋊C4×C47C2×C414yes—
86C2×C22⋊C4×C47D414yes—
87C2×C22⋊C4×C47Q814yes—
88C2×C42×C47—22no—
89C2×C42×C47—22no—
90C2×C42×C47—42no—
91C2×C42×C47C2×C414yescomplemented
92C2×C42×C47C2×C414yescomplemented
93#95×C47C2×C414yescomplemented
94#95×C47C2×C414yescomplemented
95C24×C47—21no—
96C24×C47—21no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—21no—
100C24×C47—21no—
101C24×C47C22×C412yes—
102C24×C47C2×D412yes—
103C24×C47C2×Q812yes—
104C24×C47#4114yes—
105C24×C47#4114yes—
106C24×C47C2411yescharacteristic
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—28no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—28no—
129C22×C4×C47—44no—
130C22×C4×C47—48no—
131C22×C4×C47—48no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C22×C412yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×Q812yes—
142C22×C4×C47C2×Q812yes—
143C22×C4×C47#4112yes—
144C22×Dic47—216no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—48no—
172C22×Dic47—48no—
173C22×Dic47—48no—
174C22×Dic47—48no—
175C22×Dic47—48no—
176C22×Dic47—48no—
177C22×Dic47C2×D414yes—
178C22×Dic47#4114yes—
179C22⋊C4×C47—28no—
180C22⋊C4×C47C4⋊C4116yes—
181C42×C47—28no—
182C42×C47—44no—
183C22⋊Dic47—48no—
184C4⋊Dic47—48no—
185C4⋊Dic47—48no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—24no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47—24no—
194C23×C47—22no—
195C23×C47—24no—
196C23×C47—24no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—24no—
200C23×C47—24no—
201C23×C47—24no—
202C23×C47—22no—
203C23×C47—24no—
204C23×C47C2×C4⋊C414yes—
205C23×C47#11714yes—
206C23×C47#11714yes—
207C23×C47#12418yes—
208C23×C47#12518yes—
209C23×C47C23×C411yescharacteristic
210C23×C47C22×D411yescharacteristic
211C23×C47C22×D411yescharacteristic
212C23×C47C22×Q811yescharacteristic
213C23×C47C2×#4112yes—
214C23×C47C2×#4111yescharacteristic
215C23×C47C2×#4112yes—
216C23×C47C2×#4112yes—
217C23×C47H(5,𝔽2)12yes—
218C23×C47H(5,𝔽2)11yescharacteristic
219C23×C472−1+411yescharacteristic
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—416no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47C2×C4⋊C414yes—
251C2×C4×C47C2×C4⋊C414yes—
252C2×C4×C47C4×D414yes—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—432no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282#13896—471nomaximal, Sylow
283C22×C47—24no—
284C22×C47—24no—
285C22×C47—24no—
286C22×C47—24no—
287C22×C47—24no—
288C22×C47—28no—
289C22×C47—28no—
290C22×C47—28no—
291C22×C47—28no—
292C22×C47—24no—
293C22×C47#38218yescomplemented
294C22×C47C22×C4⋊C411yescharacteristic
295C22×C47C2×#11711yescharacteristic
296C22×C47C2×#11711yescharacteristic
297C22×C47C2×C4×D411yescharacteristic
298C22×C47C4×#4111yescharacteristic
299C22×C47#51811yescharacteristic
300C22×C47#52011yescharacteristic
301C22×C47#52512yes—
302C22×C47C2×#12412yes—
303C22×C47C2×#12512yes—
304C22×C47#52912yes—
305C22×C47#53812yes—
306C22×C47#53912yes—
307C22×C47D4211yescharacteristic
308C22×C47#54612yes—
309C22×C47#54612yes—
310C22×C47#54711yescharacteristic
311C22×C47D4×Q811yescharacteristic
312C22×C47#55112yes—
313C22×C47#55112yes—
314C22×C47#55212yes—
315C22×C47#55411yescharacteristic
316C22×C47#55412yes—
317C4×C47—24no—
318C4×C47—24no—
319C4×C47—24no—
320C4×C47—42no—
321C4×C47—42no—
322C4×C47—42no—
323Dic47—44no—
324Dic47—44no—
325Dic47—82no—
326Dic47—82no—
327C2×#385—472no—
328C2×#382—471no—
329C2×C4×C22⋊C4D4711yescharacteristic
330C22×#95—471no—
331#1442—472no—
332#1442—472no—
333C4⋊#95—471no—
334#1434—471no—
335#1434—471no—
336C22⋊#95—471no—
337C22⋊#95—472no—
338C2×C47—28no—
339C2×C47C2×#38211yescharacteristic
340C2×C47#229011yescharacteristic
341C2×C47#231811yescharacteristic
342C2×C47#234211yescharacteristic
343C2×C47#234311yescharacteristic
344C2×C47D4×C4⋊C411yescharacteristic
345C2×C47#234911yescharacteristic
346C2×C47#247212yes—
347C2×C47#247312yes—
348C2×C47#248812yes—
349C2×C47#250512yes—
350C24×C4—472no—
351C24×C4C2×D4711yescharacteristic
352C22×C4⋊C4—472no—
353C22×C4⋊C4—472no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4C2×D4711yescharacteristic
356C22×C22⋊C4C2×D4711yescharacteristic
357C22×C42C2×D4711yescharacteristic
358C22×C42C2×D4711yescharacteristic
359#385—948no—
360#382—944no—
361C4×C22⋊C4Dic4718yescomplemented
362C2×#95—471no—
363C2×#95—471no—
364C2×#95—472no—
365C2×#95—471no—
366C2×#95—472no—
367C2×#95—471no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95—472no—
374C2×#95—944no—
375C2×#95C2×D4711yescharacteristic
376C2×#95C2×D4711yescharacteristic
377C47#1389611yescharacteristic, complemented, Sylow
378C25C22×D4711yescharacteristic
379C23×C4—24no—
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—42no—
383C23×C4—472no—
384C23×C4—472no—
385C23×C4—472no—
386C23×C4—472no—
387C23×C4—944no—
388C23×C4—944no—
389C23×C4—944no—
390C23×C4—944no—
391C23×C4—944no—
392C23×C4—1882no—
393C23×C4—1882no—
394C23×C4C2×Dic4714yes—
395C23×C4C22×D4711yescharacteristic
396C23×C4C22×D4711yescharacteristic
397C23×C4C22×D4711yescharacteristic
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401C2×C4⋊C4—944no—
402C2×C4⋊C4—944no—
403C2×C4⋊C4—944no—
404C2×C4⋊C4—944no—
405C2×C4⋊C4—1882no—
406C2×C4⋊C4—1882no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—944no—
409C2×C22⋊C4—944no—
410C2×C22⋊C4—944no—
411C2×C22⋊C4Dic9414yes—
412C2×C22⋊C4D18814yes—
413C2×C22⋊C4C2×Dic4714yes—
414C2×C22⋊C4C2×Dic4714yes—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42—42no—
418C2×C42C2×Dic4714yescomplemented
419C2×C42C2×Dic4714yescomplemented
420#95—474no—
421#95—944no—
422#95—944no—
423#95—944no—
424#95—944no—
425#95—944no—
426#95—942no—
427#95—942no—
428#95—942no—
429#95—1882no—
430#95—1882no—
431#95—1882no—
432#95C2×Dic4714yescomplemented
433#95C2×Dic4714yescomplemented
434C24—21no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—21no—
440C24C2×Dic9412yes—
441C24C2×D18812yes—
442C24#1062535014yes—
443C24#1062535014yes—
444C24C22×Dic4712yes—
445C24C23×D4711yescharacteristic
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—28no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—28no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—474no—
472C22×C4—474no—
473C22×C4—9416no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—1888no—
501C22×C4—1888no—
502C22×C4—1888no—
503C22×C4—1888no—
504C22×C4—1888no—
505C22×C4—1888no—
506C22×C4C2×Dic9412yes—
507C22×C4C2×Dic9412yes—
508C22×C4C2×D18812yes—
509C22×C4C2×D18812yes—
510C22×C4D4×D4712yes—
511C22×C4#1062535212yes—
512C22×C4C22×Dic4712yes—
513C22×C4C22×Dic4712yes—
514C22×C4C22×Dic4712yes—
515C22×C4C22×Dic4712yes—
516C22×C4C22×Dic4712yes—
517C22×C4C22×Dic4712yes—
518C4⋊C4—1888no—
519C4⋊C4—1888no—
520C22⋊C4—28no—
521C22⋊C4—1888no—
522C22⋊C4C4⋊Dic47116yes—
523C42—28no—
524C42—44no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23C2×C4⋊Dic4714yes—
544C23#1309398314yes—
545C23#1309398314yes—
546C23#1309401218yes—
547C23#1309401618yes—
548C23C23×Dic4711yescharacteristic
549C23C2×D4×D4711yescharacteristic
550C23C22×D18811yescharacteristic
551C23C22×Dic9411yescharacteristic
552C23#41×D4712yes—
553C23C2×#1062535211yescharacteristic
554C23C2×#1062535012yes—
555C23C2×#1062535012yes—
556C23#1309406912yes—
557C23#1309407011yescharacteristic
558C23#1309407111yescharacteristic
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—944no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—944no—
593C2×C4—944no—
594C2×C4—944no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—18832no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4C2×C4⋊Dic4714yes—
619C2×C4C2×C4⋊Dic4714yes—
620C2×C4D4×Dic4714yes—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22#1568898918yescomplemented
632C22C22×C4⋊Dic4711yescharacteristic
633C22C2×#1309398311yescharacteristic
634C22C2×#1309398311yescharacteristic
635C22C2×D4×Dic4711yescharacteristic
636C22#41×Dic4711yescharacteristic
637C22#1568959811yescharacteristic
638C22#1568960611yescharacteristic
639C22#1568963212yes—
640C22C2×#1309401212yes—
641C22C2×#1309401612yes—
642C22#1568965312yes—
643C22#1568970612yes—
644C22#1568971512yes—
645C22D4×D18811yescharacteristic
646C22#1568976312yes—
647C22#1568976812yes—
648C22#1568977711yescharacteristic
649C22D4×Dic9411yescharacteristic
650C22#1568979712yes—
651C22#1568979912yes—
652C22#1568980612yes—
653C22#1568983011yescharacteristic
654C22#1568983212yes—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—3762no—
662C4—3762no—
663C4—1884no—
664C4—1884no—
665C2—28no—
666C2C2×#1568898911yescharacteristic
667C2#2024583311yescharacteristic
668C2#2024602111yescharacteristic
669C2#2024623711yescharacteristic
670C2#2024624811yescharacteristic
671C2D4×C4⋊Dic4711yescharacteristic
672C2#2024629811yescharacteristic
673C2#2024751412yes—
674C2#2024752912yes—
675C2#2024770812yes—
676C2#2024788312yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#13896
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15688989C2semidirect
4. #20241645C2semidirect
5. #20241645C2semidirect
6. #20241618C2semidirect
7. #20241615C2semidirect
8. #20241615C2semidirect
36. C4×C22⋊C4×C47C4semidirect
91. C2×C42×C47C2×C4semidirect
92. C2×C42×C47C2×C4semidirect
93. #95×C47C2×C4semidirect
94. #95×C47C2×C4semidirect
293. C22×C47#382semidirect
361. C4×C22⋊C4Dic47semidirect
377. C47#13896semidirect
418. C2×C42C2×Dic47semidirect
419. C2×C42C2×Dic47semidirect
432. #95C2×Dic47semidirect
433. #95C2×Dic47semidirect
631. C22#15688989semidirect

Maximal subgroups

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

subgroupindex
C2×#156889982
C2×#156889892
C22×C47⋊#952
#202416452
#202416182
#202416152
#202416112
#202416102
C2×C4×C22⋊C4×C472
#1389647