Finite Group Databasebeta

← #26138604

Subgroups — #26138604

Order 12032.

1730 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×#15689016C212yesmaximal
2C2×#15688989C211yesmaximal, characteristic, complemented
3C2×#15688983C211yesmaximal, characteristic
4#20241653C212yesmaximal, complemented
5#20241633C212yesmaximal
6#20241622C212yesmaximal, complemented
7#20241618C211yesmaximal, characteristic, complemented
8#20241615C211yesmaximal, characteristic, complemented
9#20241611C211yesmaximal, characteristic
10C4×C47⋊#95C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
12C22×C4⋊Dic47C2212yes—
13C22×C4⋊Dic47C2212yes—
14C22×C4⋊Dic47C2212yes—
15C22×C22⋊Dic47C2212yes—
16C22×C22⋊Dic47C2212yes—
17C22×C4×Dic47C2212yes—
18#15689016—28no—
19#15688989—24no—
20#15688983—24no—
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
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×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C2×C4⋊Dic47—24no—
44C2×C4⋊Dic47—24no—
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×C4⋊Dic47—42no—
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×C4×Dic47—24no—
59C2×C4×Dic47—24no—
60C2×C4×Dic47—42no—
61C47⋊#95—22no—
62C47⋊#95—22no—
63C47⋊#95—24no—
64C47⋊#95—24no—
65C47⋊#95—24no—
66C47⋊#95—22no—
67C47⋊#95—24no—
68C47⋊#95—24no—
69C47⋊#95—24no—
70C47⋊#95D414yes—
71C25×C47C2311yescharacteristic
72C23×C4×C47—22no—
73C23×C4×C47—24no—
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×C414yes—
92C2×C42×C47C2×C414yes—
93#95×C47C2×C414yes—
94#95×C47C2×C414yes—
95C24×C47—21no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—21no—
99C24×C47—22no—
100C24×C47—22no—
101C24×C47—21no—
102C24×C47—21no—
103C24×C47C22×C412yes—
104C24×C47C2×D412yes—
105C24×C47C2×Q812yes—
106C24×C47C2411yescharacteristic
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—28no—
126C22×C4×C47—28no—
127C22×C4×C47—44no—
128C22×C4×C47—48no—
129C22×C4×C47—48no—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×Q812yes—
140C22×C4×C47C2×Q812yes—
141C22×C4×C47#4114yes—
142C22×C4×C47#4114yes—
143C22×C4×C47#4112yes—
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—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×Dic47C2×D414yes—
171C22×Dic47#4114yes—
172C22⋊C4×C47—28no—
173C22⋊C4×C47C4⋊C4116yescomplemented
174C42×C47—28no—
175C42×C47—44no—
176C22⋊Dic47—48no—
177C22⋊Dic47—48no—
178C4⋊Dic47—48no—
179C4⋊Dic47—48no—
180C4⋊Dic47—48no—
181C4×Dic47—48no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—24no—
189C23×C47—24no—
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—24no—
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×C47C2×C4⋊C414yes—
205C23×C47C23×C411yescharacteristic
206C23×C47C22×D411yescharacteristic
207C23×C47C22×D411yescharacteristic
208C23×C47C22×Q811yescharacteristic
209C23×C47C2×#4112yes—
210C23×C47C2×#4111yescharacteristic
211C23×C47C2×#4112yes—
212C23×C47C2×#4112yes—
213C23×C47H(5,𝔽2)12yes—
214C23×C47H(5,𝔽2)11yescharacteristic
215C23×C472−1+411yescharacteristic
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
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—24no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—416no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47C2×C4⋊C414yes—
245C2×C4×C47C2×C4⋊C414yes—
246C2×C4×C47#11714yes—
247C2×C4×C47#11714yes—
248C2×C4×C47C4×D414yes—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
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—
275C2×Dic47—42no—
276C2×Dic47—42no—
277#13915—471nomaximal, Sylow
278C22×C47—24no—
279C22×C47—24no—
280C22×C47—24no—
281C22×C47—42no—
282C22×C47—44no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—44no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47C22×C4⋊C411yescharacteristic
290C22×C47C2×#11711yescharacteristic
291C22×C47C2×#11711yescharacteristic
292C22×C47C2×C4×D411yescharacteristic
293C22×C47C4×#4111yescharacteristic
294C22×C47#51811yescharacteristic
295C22×C47#52011yescharacteristic
296C22×C47#52912yes—
297C22×C47#53212yes—
298C22×C47#53212yes—
299C22×C47#53312yes—
300C22×C47#53812yes—
301C22×C47#53912yes—
302C22×C47D4211yescharacteristic
303C22×C47#54612yes—
304C22×C47#54612yes—
305C22×C47#54711yescharacteristic
306C22×C47D4×Q811yescharacteristic
307C22×C47#55112yes—
308C22×C47#55212yes—
309C22×C47#55411yescharacteristic
310C22×C47#55412yes—
311C22×C47#55512yes—
312C4×C47—24no—
313C4×C47—24no—
314C4×C47—24no—
315C4×C47—24no—
316C4×C47—24no—
317C4×C47—42no—
318Dic47—44no—
319Dic47—44no—
320Dic47—82no—
321Dic47—82no—
322C2×#388—472no—
323C2×#382—471no—
324C2×#380—471no—
325C2×C4×C22⋊C4D4711yescharacteristic
326#1444—472no—
327#1439—472no—
328#1436—472no—
329C4⋊#95—471no—
330#1434—471no—
331C22⋊#95—471no—
332C4×#95—471no—
333C2×C47—44no—
334C2×C47#228511yescharacteristic
335C2×C47#229311yescharacteristic
336C2×C47#231811yescharacteristic
337C2×C47#234211yescharacteristic
338C2×C47#234611yescharacteristic
339C2×C47D4×C4⋊C411yescharacteristic
340C2×C47#234911yescharacteristic
341C2×C47#255312yes—
342C2×C47#255512yes—
343C2×C47#255612yes—
344C2×C47#257412yes—
345C24×C4C2×D4711yescharacteristic
346C22×C4⋊C4—472no—
347C22×C4⋊C4—472no—
348C22×C4⋊C4—472no—
349C22×C22⋊C4—472no—
350C22×C22⋊C4—472no—
351C22×C22⋊C4C2×D4711yescharacteristic
352C22×C22⋊C4C2×D4711yescharacteristic
353C22×C42—472no—
354C22×C42C2×D4711yescharacteristic
355C22×C42C2×D4711yescharacteristic
356#388—948no—
357#382—944no—
358#380—944no—
359C4×C22⋊C4Dic4718yescomplemented
360C2×#95—471no—
361C2×#95—471no—
362C2×#95—472no—
363C2×#95—471no—
364C2×#95—472no—
365C2×#95—471no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95C2×D4711yescharacteristic
371C2×#95C2×D4711yescharacteristic
372C47#1391511yescharacteristic, complemented, Sylow
373C25C22×D4711yescharacteristic
374C23×C4—24no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—42no—
378C23×C4—472no—
379C23×C4—472no—
380C23×C4—472no—
381C23×C4—472no—
382C23×C4C2×Dic4714yes—
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4711yescharacteristic
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389C2×C4⋊C4—944no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4—944no—
394C2×C4⋊C4—1882no—
395C2×C4⋊C4—1882no—
396C2×C4⋊C4—1882no—
397C2×C4⋊C4—1882no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—1882no—
404C2×C22⋊C4—1882no—
405C2×C22⋊C4Dic9414yes—
406C2×C22⋊C4D18814yes—
407C2×C22⋊C4C2×Dic4714yes—
408C2×C22⋊C4C2×Dic4714yes—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—42no—
412C2×C42—944no—
413C2×C42—944no—
414C2×C42—1882no—
415C2×C42C2×Dic4714yes—
416C2×C42C2×Dic4714yes—
417#95—474no—
418#95—944no—
419#95—944no—
420#95—944no—
421#95—944no—
422#95—944no—
423#95—944no—
424#95—942no—
425#95—942no—
426#95—942no—
427#95C2×Dic4714yes—
428#95C2×Dic4714yes—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24C2×Dic9412yes—
438C24C2×D18812yes—
439C24C22×Dic4712yes—
440C24C23×D4711yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—474no—
465C22×C4—474no—
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—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4C2×Dic9412yes—
493C22×C4C2×Dic9412yes—
494C22×C4C2×D18812yes—
495C22×C4C2×D18812yes—
496C22×C4#1062535014yes—
497C22×C4#1062535014yes—
498C22×C4D4×D4712yes—
499C22×C4#1062535212yes—
500C22×C4C22×Dic4712yes—
501C22×C4C22×Dic4712yes—
502C22×C4C22×Dic4712yes—
503C22×C4C22×Dic4712yes—
504C22×C4C22×Dic4712yes—
505C22×C4C22×Dic4712yes—
506C4⋊C4—1888no—
507C4⋊C4—1888no—
508C4⋊C4—1888no—
509C22⋊C4—28no—
510C22⋊C4—1888no—
511C22⋊C4—1888no—
512C22⋊C4C4⋊Dic47116yescomplemented
513C42—28no—
514C42—44no—
515C42—1888no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C2×C4⋊Dic4714yes—
539C23C23×Dic4711yescharacteristic
540C23C2×D4×D4711yescharacteristic
541C23C22×D18811yescharacteristic
542C23C22×Dic9411yescharacteristic
543C23#41×D4712yes—
544C23C2×#1062535211yescharacteristic
545C23C2×#1062535012yes—
546C23C2×#1062535012yes—
547C23#1309406912yes—
548C23#1309407011yescharacteristic
549C23#1309407111yescharacteristic
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—416no—
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—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4C2×C4⋊Dic4714yes—
607C2×C4C2×C4⋊Dic4714yes—
608C2×C4#1309398314yes—
609C2×C4#1309398314yes—
610C2×C4D4×Dic4714yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22C22×C4⋊Dic4711yescharacteristic
623C22C2×#1309398311yescharacteristic
624C22C2×#1309398311yescharacteristic
625C22C2×D4×Dic4711yescharacteristic
626C22#41×Dic4711yescharacteristic
627C22#1568959811yescharacteristic
628C22#1568960611yescharacteristic
629C22#1568965512yes—
630C22#1568967012yes—
631C22#1568967312yes—
632C22#1568968012yes—
633C22#1568970612yes—
634C22#1568971512yes—
635C22D4×D18811yescharacteristic
636C22#1568976312yes—
637C22#1568976812yes—
638C22#1568977711yescharacteristic
639C22D4×Dic9411yescharacteristic
640C22#1568979912yes—
641C22#1568981212yes—
642C22#1568983011yescharacteristic
643C22#1568983212yes—
644C22#1568983612yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—3762no—
652C4—3762no—
653C4—1884no—
654C4—1884no—
655C2—44no—
656C2#2024579511yescharacteristic
657C2#2024585311yescharacteristic
658C2#2024602111yescharacteristic
659C2#2024623711yescharacteristic
660C2#2024627311yescharacteristic
661C2D4×C4⋊Dic4711yescharacteristic
662C2#2024629811yescharacteristic
663C2#2024842712yes—
664C2#2024845712yes—
665C2#2024847212yes—
666C2#2024866212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#13915
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. #20241653C2semidirect
6. #20241622C2semidirect
7. #20241618C2semidirect
8. #20241615C2semidirect
10. C4×C47⋊#95C2semidirect
36. C4×C22⋊C4×C47C4semidirect
173. C22⋊C4×C47C4⋊C4semidirect
359. C4×C22⋊C4Dic47semidirect
372. C47#13915semidirect
512. C22⋊C4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890162
C2×#156889892
C2×#156889832
#202416532
#202416332
#202416222
#202416182
#202416152
#202416112
C4×C47⋊#952
C2×C4×C22⋊C4×C472
#1391547