Finite Group Databasebeta

← #26137647

Subgroups — #26137647

Order 12032.

2194 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689003C212yesmaximal
2C2×C22⋊C4×Dic47C212yesmaximal, complemented
3#20241625C212yesmaximal, complemented
4#20241625C212yesmaximal, complemented
5#20241620C212yesmaximal, complemented
6#20241615C211yesmaximal, characteristic, complemented
7#20241610C212yesmaximal, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C2×C47×#390C211yesmaximal, characteristic
10C24×Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C4×Dic47C2211yescharacteristic
15C22×C4×Dic47C2212yes—
16C22×C4×Dic47C2211yescharacteristic
17#15689003C4116yescomplemented
18C22⋊C4×Dic47—28no—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
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—42no—
37C23×Dic47—42no—
38C23×Dic47—42no—
39C23×Dic47—42no—
40C23×Dic47C2×C418yescomplemented
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C23×Dic47C2312yes—
44C23×Dic47C2312yes—
45C2×C22⋊Dic47—24no—
46C2×C22⋊Dic47—24no—
47C2×C22⋊Dic47—42no—
48C2×C22⋊Dic47—42no—
49C2×C22⋊Dic47—42no—
50C2×C22⋊Dic47—42no—
51C2×C22⋊Dic47C2×C418yes—
52C2×C22⋊Dic47C2×C418yes—
53C2×C22⋊Dic47C2×C418yescomplemented
54C2×C4×Dic47—22no—
55C2×C4×Dic47—24no—
56C2×C4×Dic47—41no—
57C2×C4×Dic47—41no—
58C2×C4×Dic47—42no—
59C2×C4×Dic47—42no—
60C2×C4×Dic47—41no—
61C2×C4×Dic47—41no—
62C2×C4×Dic47D414yescomplemented
63C47⋊#95—24no—
64C47⋊#95—24no—
65C47⋊#95—24no—
66C47⋊#95—42no—
67C47⋊#95—41no—
68C47⋊#95—42no—
69C47⋊#95—41no—
70C47⋊#95C2×C418yescomplemented
71C47⋊#95C2×C418yescomplemented
72C25×C47C2312yes—
73C22×D4×C47—41no—
74C22×D4×C47—41no—
75C22×D4×C47—42no—
76C22×D4×C47C2×C418yes—
77C22×D4×C47C2×C414yes—
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2312yes—
80C23×C4×C47C2311yescharacteristic
81C23×C4×C47C2311yescharacteristic
82C2×C4⋊C4×C47—42no—
83C2×C4⋊C4×C47C2×C414yescomplemented
84C2×C22⋊C4×C47—24no—
85C2×C22⋊C4×C47—24no—
86C2×C22⋊C4×C47—42no—
87C2×C22⋊C4×C47—42no—
88C2×C22⋊C4×C47C2×C418yescomplemented
89C2×C22⋊C4×C47C2×C418yescomplemented
90C2×C42×C47—41no—
91C2×C42×C47—41no—
92C2×C42×C47C2×C414yescomplemented
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×C47—22no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47C22×C414yes—
106C24×C47C22×C414yes—
107C24×C47C22×C414yes—
108C24×C47C2411yescharacteristic
109C2×D4×C47—44no—
110C2×D4×C47—44no—
111C2×D4×C47—48no—
112C2×D4×C47—48no—
113C2×D4×C47C42116yescomplemented
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C414yes—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47#4112yes—
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—22no—
160C22×Dic47—216no—
161C22×Dic47—48no—
162C22×Dic47—48no—
163C22×Dic47—48no—
164C22×Dic47—48no—
165C22×Dic47—48no—
166C22×Dic47—48no—
167C22×Dic47C22×C414yes—
168C22×Dic47C22×C414yes—
169C22×Dic47C22×C414yes—
170C22×Dic47C22×C414yes—
171C22×Dic47C2×D414yes—
172C22×Dic47#4114yes—
173C4⋊C4×C47—44no—
174C22⋊C4×C47—216no—
175C22⋊C4×C47—48no—
176C42×C47—44no—
177C22⋊Dic47—216no—
178C22⋊Dic47—48no—
179C22⋊Dic47—48no—
180C4×Dic47—44no—
181C4×Dic47—48no—
182C4×Dic47—44no—
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—24no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—24no—
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×C47—42no—
212C23×C47—42no—
213C23×C47—42no—
214C23×C47—42no—
215C23×C47—42no—
216C23×C47—42no—
217C23×C47C2×C4218yes—
218C23×C47C23×C412yes—
219C23×C47C23×C411yescharacteristic
220C23×C47C22×D411yescharacteristic
221C23×C47C2×#4112yes—
222C23×C47C2×#4111yescharacteristic
223C23×C47H(5,𝔽2)11yescharacteristic
224C23×C47H(5,𝔽2)12yes—
225C23×C47H(5,𝔽2)11yescharacteristic
226C23×C47H(5,𝔽2)12yes—
227C23×C47H(5,𝔽2)11yescharacteristic
228C23×C472−1+411yescharacteristic
229D4×C47—416no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47C2×C4214yes—
253C2×C4×C47C4×D414yes—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—432no—
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—
282C2×Dic47C4×D418yes—
283#13791—471nomaximal, Sylow
284C22×C47—28no—
285C22×C47—28no—
286C22×C47—28no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47—44no—
290C22×C47—44no—
291C22×C47—44no—
292C22×C47—44no—
293C22×C47—44no—
294C22×C47—44no—
295C22×C47—44no—
296C22×C47—44no—
297C22×C47—44no—
298C22×C47—44no—
299C22×C47C22×C4211yescharacteristic
300C22×C47C2×C4×D412yes—
301C22×C47C2×C4×D411yescharacteristic
302C22×C47C4×#4112yes—
303C22×C47C4×#4111yescharacteristic
304C22×C47#51812yes—
305C22×C47#51812yes—
306C22×C47#51812yes—
307C22×C47#51811yescharacteristic
308C22×C47#51811yescharacteristic
309C22×C47#51811yescharacteristic
310C22×C47#52012yes—
311C22×C47#52011yescharacteristic
312C22×C47#53411yescharacteristic
313C22×C47#53512yes—
314C22×C47#53711yescharacteristic
315C22×C47#53812yes—
316C22×C47#53812yes—
317C22×C47#53811yescharacteristic
318C22×C47#53911yescharacteristic
319C22×C47#54012yes—
320C22×C47#54012yes—
321C22×C47#54212yes—
322C4×C47—24no—
323C4×C47—24no—
324C4×C47—44no—
325C4×C47—42no—
326Dic47—28no—
327Dic47—44no—
328Dic47—44no—
329Dic47—44no—
330C2×#390D4711yescharacteristic
331C2×#386—472no—
332C2×C4×C22⋊C4—472no—
333#1437—472no—
334#1437—472no—
335#1436—472no—
336#1434—471no—
337C22⋊#95—472no—
338C4×#95—471no—
339C2×C47—48no—
340C2×C47C42×D411yescharacteristic
341C2×C47#227311yescharacteristic
342C2×C47#227511yescharacteristic
343C2×C47#232112yes—
344C2×C47#232412yes—
345C2×C47#232511yescharacteristic
346C2×C47#232811yescharacteristic
347C2×C47#233312yes—
348C2×C47#233311yescharacteristic
349C2×C47#233512yes—
350C2×C47#233711yescharacteristic
351C23×D4C2×D4711yescharacteristic
352C24×C4—472no—
353C22×C4⋊C4C2×D4711yescharacteristic
354C22×C22⋊C4—472no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4—472no—
357C22×C22⋊C4C2×D4712yes—
358C22×C22⋊C4C2×D4712yes—
359C22×C42—471no—
360C22×C42—471no—
361C22×C42—472no—
362C22×C42C2×D4711yescharacteristic
363#390Dic4718yescomplemented
364#386—4716no—
365C4×C22⋊C4—948no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95—471no—
372C2×#95—472no—
373C2×#95—471no—
374C2×#95—472no—
375C47#1379111yescharacteristic, complemented, Sylow
376C25C22×D4712yes—
377C22×D4—41no—
378C22×D4—41no—
379C22×D4—42no—
380C22×D4C4×D4718yes—
381C22×D4C2×Dic4714yes—
382C23×C4—472no—
383C23×C4—472no—
384C23×C4—472no—
385C23×C4—472no—
386C23×C4—478no—
387C23×C4—944no—
388C23×C4—944no—
389C23×C4—1882no—
390C23×C4—1882no—
391C23×C4—1882no—
392C23×C4—1882no—
393C23×C4C22×D4712yes—
394C23×C4C22×D4711yescharacteristic
395C23×C4C22×D4711yescharacteristic
396C23×C4C22×D4711yescharacteristic
397C2×C4⋊C4—42no—
398C2×C4⋊C4C2×Dic4714yescomplemented
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—478no—
404C2×C22⋊C4—478no—
405C2×C22⋊C4—478no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—944no—
408C2×C22⋊C4—1882no—
409C2×C22⋊C4—1882no—
410C2×C22⋊C4—1882no—
411C2×C22⋊C4—1882no—
412C2×C22⋊C4C2×Dic4718yescomplemented
413C2×C22⋊C4C2×Dic4718yescomplemented
414C2×C42—41no—
415C2×C42—41no—
416C2×C42—474no—
417C2×C42—944no—
418C2×C42—942no—
419C2×C42—1881no—
420C2×C42—1881no—
421C2×C42—1881no—
422C2×C42—1881no—
423C2×C42—1882no—
424C2×C42—1882no—
425C2×C42C2×Dic4714yescomplemented
426#95—478no—
427#95—478no—
428#95—944no—
429#95—944no—
430#95—944no—
431#95—1881no—
432#95—1881no—
433#95—1882no—
434#95—1882no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24C2×C4×D4714yes—
448C24C2×C4×D4714yes—
449C24C22×Dic4714yes—
450C24C23×D4711yescharacteristic
451C2×D4—48no—
452C2×D4—48no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4C4×Dic47116yescomplemented
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—474no—
474C22×C4—474no—
475C22×C4—474no—
476C22×C4—474no—
477C22×C4—474no—
478C22×C4—474no—
479C22×C4—9416no—
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—942no—
501C22×C4—942no—
502C22×C4—1888no—
503C22×C4—1888no—
504C22×C4—1888no—
505C22×C4—1888no—
506C22×C4—1888no—
507C22×C4—1888no—
508C22×C4C2×C4×D4714yes—
509C22×C4D4×D4712yes—
510C22×C4#1062535212yes—
511C22×C4C22×Dic4712yes—
512C22×C4C22×Dic4714yes—
513C22×C4C22×Dic4712yes—
514C22×C4C22×Dic4712yes—
515C4⋊C4—44no—
516C22⋊C4—216no—
517C22⋊C4—48no—
518C22⋊C4—9416no—
519C22⋊C4—1888no—
520C22⋊C4—1888no—
521C42—44no—
522C42—1884no—
523C42—1884no—
524C42—1888no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C2×C4×Dic4718yes—
560C23C22×C4×D4712yes—
561C23C23×Dic4711yescharacteristic
562C23C2×D4×D4711yescharacteristic
563C23#41×D4712yes—
564C23C2×#1062535211yescharacteristic
565C23#1309406912yes—
566C23#1309406912yes—
567C23#1309406911yescharacteristic
568C23#1309406911yescharacteristic
569C23#1309407011yescharacteristic
570C23#1309407111yescharacteristic
571D4—416no—
572C2×C4—24no—
573C2×C4—24no—
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—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—478no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—944no—
606C2×C4—944no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—18832no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4—1882no—
619C2×C4—1882no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1882no—
623C2×C4C2×C4×Dic4714yes—
624C2×C4D4×Dic4714yes—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22C22×C4×Dic4711yescharacteristic
641C22C4×D4×D4712yes—
642C22C2×D4×Dic4711yescharacteristic
643C22C4×#1062535212yes—
644C22#41×Dic4711yescharacteristic
645C22#1568959512yes—
646C22#1568959612yes—
647C22#1568959612yes—
648C22#1568959811yescharacteristic
649C22#1568959811yescharacteristic
650C22#1568959811yescharacteristic
651C22#1568960212yes—
652C22#1568960611yescharacteristic
653C22#1568968111yescharacteristic
654C22#1568968612yes—
655C22#1568970011yescharacteristic
656C22#1568970512yes—
657C22#1568970712yes—
658C22#1568971011yescharacteristic
659C22#1568971811yescharacteristic
660C22#1568972312yes—
661C22#1568972712yes—
662C22#1568973712yes—
663C4—24no—
664C4—24no—
665C4—44no—
666C4—42no—
667C4—948no—
668C4—1884no—
669C4—1884no—
670C4—1884no—
671C2—48no—
672C2C4×D4×Dic4711yescharacteristic
673C2#2024574111yescharacteristic
674C2#2024574911yescharacteristic
675C2#2024604712yes—
676C2#2024607412yes—
677C2#2024609311yescharacteristic
678C2#2024611411yescharacteristic
679C2#2024614712yes—
680C2#2024615511yescharacteristic
681C2#2024617512yes—
682C2#2024620311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC2×C47×#390C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC2×#390D47
hypocenterC47#13791
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×Dic47C2semidirect
3. #20241625C2semidirect
4. #20241625C2semidirect
5. #20241620C2semidirect
6. #20241615C2semidirect
7. #20241610C2semidirect
8. C4×C47⋊#95C2semidirect
17. #15689003C4semidirect
33. C47×#390C4semidirect
40. C23×Dic47C2×C4semidirect
53. C2×C22⋊Dic47C2×C4semidirect
62. C2×C4×Dic47D4semidirect
70. C47⋊#95C2×C4semidirect
71. C47⋊#95C2×C4semidirect
83. C2×C4⋊C4×C47C2×C4semidirect
88. C2×C22⋊C4×C47C2×C4semidirect
89. C2×C22⋊C4×C47C2×C4semidirect
92. C2×C42×C47C2×C4semidirect
113. C2×D4×C47C42semidirect
363. #390Dic47semidirect
375. C47#13791semidirect
398. C2×C4⋊C4C2×Dic47semidirect
412. C2×C22⋊C4C2×Dic47semidirect
413. C2×C22⋊C4C2×Dic47semidirect
425. C2×C42C2×Dic47semidirect
455. 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×C22⋊C4×Dic472
#202416252
#202416202
#202416152
#202416102
C4×C47⋊#952
C2×C47×#3902
#1379147