Finite Group Databasebeta

← #26144678

Subgroups — #26144678

Order 12032.

2018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689003C212yesmaximal
2C2×#15688998C212yesmaximal, complemented
3#20241641C212yesmaximal, complemented
4#20241633C212yesmaximal, complemented
5#20241633C212yesmaximal, complemented
6#20241618C211yesmaximal, characteristic, complemented
7#20241615C211yesmaximal, characteristic, complemented
8#20241610C212yesmaximal, complemented
9C2×C47×#390C211yesmaximal, characteristic
10C24×Dic47C2212yes—
11C22×C4⋊Dic47C2212yes—
12C22×C4⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C22⋊Dic47C2212yes—
16#15689003—28no—
17#15688998—28no—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C23×D4×C47C2211yescharacteristic
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2212yes—
31C22×C22⋊C4×C47C2212yes—
32C22×C42×C47C2211yescharacteristic
33C47×#390C418yescomplemented
34C23×Dic47—24no—
35C23×Dic47—24no—
36C23×Dic47—24no—
37C23×Dic47—42no—
38C23×Dic47—42no—
39C23×Dic47—42no—
40C23×Dic47—42no—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C23×Dic47C2312yes—
44C23×Dic47C2312yes—
45C2×C4⋊Dic47—24no—
46C2×C4⋊Dic47—24no—
47C2×C4⋊Dic47—24no—
48C2×C4⋊Dic47—42no—
49C2×C4⋊Dic47—42no—
50C2×C4⋊Dic47—42no—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47—24no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—42no—
57C2×C22⋊Dic47—42no—
58C2×C22⋊Dic47—42no—
59C2×C22⋊Dic47—42no—
60C47⋊#95—24no—
61C47⋊#95—24no—
62C47⋊#95—22no—
63C47⋊#95—24no—
64C47⋊#95—41no—
65C47⋊#95—42no—
66C47⋊#95—42no—
67C47⋊#95—41no—
68C47⋊#95—42no—
69C47⋊#95D414yescomplemented
70C25×C47C2312yes—
71C22×D4×C47—42no—
72C22×D4×C47—42no—
73C22×D4×C47C2×C414yes—
74C22×D4×C47D414yes—
75C22×D4×C47Q814yes—
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2312yes—
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C2×C4⋊C4×C47—42no—
81C2×C4⋊C4×C47C2×C414yescomplemented
82C2×C22⋊C4×C47—24no—
83C2×C22⋊C4×C47—24no—
84C2×C22⋊C4×C47—42no—
85C2×C22⋊C4×C47—42no—
86C2×C22⋊C4×C47C2×C418yescomplemented
87C2×C22⋊C4×C47C2×C418yescomplemented
88C2×C42×C47—42no—
89C2×C42×C47C2×C414yescomplemented
90C24×C47—22no—
91C24×C47—22no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—22no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47C22×C414yes—
103C24×C47C2×D414yes—
104C24×C47C2×Q814yes—
105C24×C47C2411yescharacteristic
106C2×D4×C47—44no—
107C2×D4×C47—44no—
108C2×D4×C47—48no—
109C2×D4×C47—48no—
110C2×D4×C47C4⋊C4116yescomplemented
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×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—21no—
124C22×C4×C47—21no—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C22×C414yes—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×Q812yes—
132C22×C4×C47#4112yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
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—48no—
160C22×Dic47—48no—
161C22×Dic47—48no—
162C22×Dic47—48no—
163C22×Dic47—48no—
164C22×Dic47—48no—
165C22×Dic47—48no—
166C22×Dic47C2×D414yes—
167C22×Dic47#4114yes—
168C4⋊C4×C47—44no—
169C22⋊C4×C47—216no—
170C22⋊C4×C47—48no—
171C42×C47—44no—
172C22⋊Dic47—48no—
173C22⋊Dic47—48no—
174C22⋊Dic47—48no—
175C4⋊Dic47—48no—
176C4⋊Dic47—48no—
177C23×C47—24no—
178C23×C47—24no—
179C23×C47—24no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—24no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—24no—
189C23×C47—24no—
190C23×C47—24no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47—24no—
194C23×C47—24no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47C2×C4⋊C418yes—
212C23×C47C23×C411yescharacteristic
213C23×C47C22×D411yescharacteristic
214C23×C47C22×D411yescharacteristic
215C23×C47C22×Q811yescharacteristic
216C23×C47C2×#4112yes—
217C23×C47C2×#4111yescharacteristic
218C23×C47H(5,𝔽2)12yes—
219C23×C47H(5,𝔽2)12yes—
220C23×C47H(5,𝔽2)12yes—
221C23×C47H(5,𝔽2)11yescharacteristic
222C23×C472−1+411yescharacteristic
223D4×C47—416no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47C2×C4⋊C414yes—
245C2×C4×C47C4×D414yes—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—432no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
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—
275#14438—471nomaximal, Sylow
276C22×C47—28no—
277C22×C47—28no—
278C22×C47—28no—
279C22×C47—44no—
280C22×C47—44no—
281C22×C47—44no—
282C22×C47—44no—
283C22×C47—44no—
284C22×C47—44no—
285C22×C47—44no—
286C22×C47—44no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47—44no—
290C22×C47—44no—
291C22×C47C22×C4⋊C411yescharacteristic
292C22×C47C2×C4×D411yescharacteristic
293C22×C47C4×#4111yescharacteristic
294C22×C47#51811yescharacteristic
295C22×C47#51811yescharacteristic
296C22×C47#51811yescharacteristic
297C22×C47#52011yescharacteristic
298C22×C47#53412yes—
299C22×C47#53512yes—
300C22×C47#53812yes—
301C22×C47#54012yes—
302C22×C47#54212yes—
303C22×C47#54312yes—
304C22×C47D4211yescharacteristic
305C22×C47#54612yes—
306C22×C47#54612yes—
307C22×C47#54711yescharacteristic
308C22×C47D4×Q811yescharacteristic
309C22×C47#55312yes—
310C22×C47#55411yescharacteristic
311C22×C47#55412yes—
312C22×C47#56012yes—
313C22×C47#56212yes—
314C4×C47—24no—
315C4×C47—24no—
316C4×C47—44no—
317C4×C47—42no—
318Dic47—44no—
319Dic47—44no—
320Dic47—82no—
321Dic47—82no—
322C2×#390D4711yescharacteristic
323C2×#386—472no—
324C2×#385—472no—
325#1441—472no—
326#1439—472no—
327#1439—472no—
328C4⋊#95—471no—
329#1434—471no—
330C22⋊#95—472no—
331C2×C47—48no—
332C2×C47#231511yescharacteristic
333C2×C47#232811yescharacteristic
334C2×C47#233711yescharacteristic
335C2×C47D4×C4⋊C411yescharacteristic
336C2×C47#234911yescharacteristic
337C2×C47#236411yescharacteristic
338C2×C47#237711yescharacteristic
339C2×C47#267812yes—
340C2×C47#269212yes—
341C2×C47#270112yes—
342C2×C47#278012yes—
343C23×D4C2×D4711yescharacteristic
344C24×C4—472no—
345C22×C4⋊C4—472no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4C2×D4711yescharacteristic
348C22×C22⋊C4—472no—
349C22×C22⋊C4—472no—
350C22×C22⋊C4—472no—
351C22×C22⋊C4C2×D4712yes—
352C22×C22⋊C4C2×D4712yes—
353C22×C42C2×D4711yescharacteristic
354#390Dic4718yescomplemented
355#386—948no—
356#385—948no—
357C2×#95—471no—
358C2×#95—472no—
359C2×#95—471no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—471no—
363C2×#95—472no—
364C2×#95—472no—
365C2×#95—471no—
366C2×#95—472no—
367C47#1443811yescharacteristic, complemented, Sylow
368C25C22×D4712yes—
369C22×D4—42no—
370C22×D4—42no—
371C22×D4Dic9414yes—
372C22×D4D18814yes—
373C22×D4C2×Dic4714yes—
374C23×C4—472no—
375C23×C4—472no—
376C23×C4—472no—
377C23×C4—472no—
378C23×C4—944no—
379C23×C4—944no—
380C23×C4—944no—
381C23×C4—1882no—
382C23×C4—1882no—
383C23×C4—1882no—
384C23×C4—1882no—
385C23×C4C22×D4712yes—
386C23×C4C22×D4711yescharacteristic
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389C2×C4⋊C4—42no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4—1882no—
394C2×C4⋊C4—1882no—
395C2×C4⋊C4—1882no—
396C2×C4⋊C4C2×Dic4714yescomplemented
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—944no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—1882no—
407C2×C22⋊C4—1882no—
408C2×C22⋊C4—1882no—
409C2×C22⋊C4—1882no—
410C2×C22⋊C4C2×Dic4718yescomplemented
411C2×C22⋊C4C2×Dic4718yescomplemented
412C2×C42—42no—
413C2×C42C2×Dic4714yescomplemented
414#95—474no—
415#95—944no—
416#95—944no—
417#95—944no—
418#95—942no—
419#95—1882no—
420#95—1882no—
421#95—1882no—
422#95—1881no—
423#95—1881no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24C2×Dic9414yes—
437C24C2×D18814yes—
438C24C22×Dic4714yes—
439C24C23×D4711yescharacteristic
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4C4⋊Dic47116yescomplemented
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—474no—
460C22×C4—474no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
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—1888no—
488C22×C4—1888no—
489C22×C4—1888no—
490C22×C4—1888no—
491C22×C4—1888no—
492C22×C4—1888no—
493C22×C4—1888no—
494C22×C4C2×Dic9412yes—
495C22×C4C2×D18812yes—
496C22×C4D4×D4712yes—
497C22×C4#1062535212yes—
498C22×C4C22×Dic4712yes—
499C22×C4C22×Dic4714yes—
500C22×C4C22×Dic4712yes—
501C22×C4C22×Dic4712yes—
502C4⋊C4—44no—
503C4⋊C4—1888no—
504C4⋊C4—1888no—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—1888no—
508C22⋊C4—1888no—
509C22⋊C4—1888no—
510C42—44no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4⋊Dic4718yes—
546C23C23×Dic4711yescharacteristic
547C23C2×D4×D4711yescharacteristic
548C23C22×D18811yescharacteristic
549C23C22×Dic9411yescharacteristic
550C23#41×D4712yes—
551C23C2×#1062535211yescharacteristic
552C23#1309406912yes—
553C23#1309406912yes—
554C23#1309406912yes—
555C23#1309407011yescharacteristic
556C23#1309407111yescharacteristic
557D4—416no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—944no—
582C2×C4—944no—
583C2×C4—944no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—18832no—
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×C4C2×C4⋊Dic4714yes—
608C2×C4D4×Dic4714yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C22×C4⋊Dic4711yescharacteristic
625C22C2×D4×Dic4711yescharacteristic
626C22#41×Dic4711yescharacteristic
627C22#1568959811yescharacteristic
628C22#1568959811yescharacteristic
629C22#1568959811yescharacteristic
630C22#1568960611yescharacteristic
631C22#1568968512yes—
632C22#1568968612yes—
633C22#1568970712yes—
634C22#1568972712yes—
635C22#1568973712yes—
636C22#1568974912yes—
637C22D4×D18811yescharacteristic
638C22#1568976212yes—
639C22#1568976812yes—
640C22#1568977711yescharacteristic
641C22D4×Dic9411yescharacteristic
642C22#1568981612yes—
643C22#1568983011yescharacteristic
644C22#1568983212yes—
645C22#1568986112yes—
646C22#1568986812yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—3762no—
652C4—3762no—
653C4—1884no—
654C4—1884no—
655C2—48no—
656C2#2024600211yescharacteristic
657C2#2024611411yescharacteristic
658C2#2024620311yescharacteristic
659C2D4×C4⋊Dic4711yescharacteristic
660C2#2024629811yescharacteristic
661C2#2024644011yescharacteristic
662C2#2024654511yescharacteristic
663C2#2024970812yes—
664C2#2024988112yes—
665C2#2024999912yes—
666C2#2025103712yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#390C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC2×#390D47
hypocenterC47#14438
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#15688998C2semidirect
3. #20241641C2semidirect
4. #20241633C2semidirect
5. #20241633C2semidirect
6. #20241618C2semidirect
7. #20241615C2semidirect
8. #20241610C2semidirect
33. C47×#390C4semidirect
69. C47⋊#95D4semidirect
81. C2×C4⋊C4×C47C2×C4semidirect
86. C2×C22⋊C4×C47C2×C4semidirect
87. C2×C22⋊C4×C47C2×C4semidirect
89. C2×C42×C47C2×C4semidirect
110. C2×D4×C47C4⋊C4semidirect
354. #390Dic47semidirect
367. C47#14438semidirect
396. C2×C4⋊C4C2×Dic47semidirect
410. C2×C22⋊C4C2×Dic47semidirect
411. C2×C22⋊C4C2×Dic47semidirect
413. C2×C42C2×Dic47semidirect
444. C2×D4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890032
C2×#156889982
#202416412
#202416332
#202416182
#202416152
#202416102
C2×C47×#3902
#1443847