Finite Group Databasebeta

← #26137643

Subgroups — #26137643

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×#15689025C211yesmaximal, characteristic
2C2×#15689001C212yesmaximal
3C2×#15688970C212yesmaximal, complemented
4#20241624C212yesmaximal, complemented
5#20241624C212yesmaximal, complemented
6#20241622C212yesmaximal, complemented
7(C2×D47)⋊#95C212yesmaximal, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C47×#1434C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C23×C4×D47C2212yes—
12C22×C4⋊Dic47C2211yescharacteristic
13C22×(C2×D47)⋊C4C2212yes—
14C22×(C2×D47)⋊C4C2212yes—
15C22×(C2×D47)⋊C4C2212yes—
16C22×(C2×D47)⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19C22×C4×Dic47C2212yes—
20C22×C4×Dic47C2211yescharacteristic
21#15689025C418yescomplemented
22#15689001C4116yescomplemented
23#15688970—28no—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C22×C42×C47C2211yescharacteristic
30C2×#95×C47C2212yes—
31C2×#95×C47C2211yescharacteristic
32C2×#95×C47C2211yescharacteristic
33C2×#95×C47C2212yes—
34C24×D47C2312yes—
35C22×D188—41no—
36C22×D188—41no—
37C22×D188—42no—
38C22×D188C2×C418yes—
39C22×D188C2×C414yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C22×C4×D47—24no—
44C22×C4×D47—24no—
45C22×C4×D47—42no—
46C22×C4×D47—42no—
47C22×C4×D47—42no—
48C22×C4×D47—42no—
49C22×C4×D47C2×C418yescomplemented
50C2×C4⋊Dic47—42no—
51C2×C4⋊Dic47C2×C414yescomplemented
52C2×(C2×D47)⋊C4—24no—
53C2×(C2×D47)⋊C4—24no—
54C2×(C2×D47)⋊C4—24no—
55C2×(C2×D47)⋊C4—24no—
56C2×(C2×D47)⋊C4—42no—
57C2×(C2×D47)⋊C4—42no—
58C2×(C2×D47)⋊C4—42no—
59C2×(C2×D47)⋊C4—42no—
60C2×(C2×D47)⋊C4—42no—
61C2×(C2×D47)⋊C4—42no—
62C2×(C2×D47)⋊C4C2×C418yes—
63C2×(C2×D47)⋊C4C2×C418yes—
64C2×(C2×D47)⋊C4C2×C418yescomplemented
65C2×(C2×D47)⋊C4C2×C418yescomplemented
66C2×(C2×D47)⋊C4C2×C418yescomplemented
67C2×C4×Dic47—22no—
68C2×C4×Dic47—24no—
69C2×C4×Dic47—41no—
70C2×C4×Dic47—41no—
71C2×C4×Dic47—42no—
72C2×C4×Dic47—42no—
73C2×C4×Dic47—41no—
74C2×C4×Dic47—41no—
75C2×C4×Dic47D414yescomplemented
76C47⋊#95—24no—
77C47⋊#95—24no—
78C47⋊#95—42no—
79C47⋊#95—42no—
80C47⋊#95C2×C418yescomplemented
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2312yes—
83C23×C4×C47C2312yes—
84C23×C4×C47C2311yescharacteristic
85C23×C4×C47C2311yescharacteristic
86C2×C42×C47—41no—
87C2×C42×C47—41no—
88C2×C42×C47C2×C414yescomplemented
89#95×C47—24no—
90#95×C47—41no—
91#95×C47—41no—
92#95×C47C2×C418yescomplemented
93C24×C47C2411yescharacteristic
94C23×D47—22no—
95C23×D47—22no—
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—22no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47C22×C414yes—
107C23×D47C22×C414yes—
108C23×D47C22×C414yes—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C414yes—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C22×C414yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47#4112yes—
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×Dic47C22×C414yes—
157C22×Dic47C22×C414yes—
158C22×Dic47C22×C414yes—
159C22×Dic47C2×D414yes—
160C22×Dic47#4114yes—
161C2×D188—44no—
162C2×D188—44no—
163C2×D188—48no—
164C2×D188—48no—
165C2×D188C42116yescomplemented
166C2×C4×D47—216no—
167C2×C4×D47—48no—
168C2×C4×D47—48no—
169C2×C4×D47—48no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C2×C4×D47—48no—
173C42×C47—44no—
174(C2×D47)⋊C4—216no—
175(C2×D47)⋊C4—216no—
176(C2×D47)⋊C4—48no—
177(C2×D47)⋊C4—48no—
178(C2×D47)⋊C4—48no—
179C4⋊Dic47—44no—
180C4×Dic47—44no—
181C4×Dic47—48no—
182C4×Dic47—44no—
183C23×C47C23×C412yes—
184C23×C47C23×C411yescharacteristic
185C23×C47C22×D411yescharacteristic
186C23×C47C2×#4112yes—
187C23×C47C2×#4111yescharacteristic
188C23×C47H(5,𝔽2)11yescharacteristic
189C23×C47H(5,𝔽2)11yescharacteristic
190C23×C47H(5,𝔽2)12yes—
191C23×C47H(5,𝔽2)11yescharacteristic
192C23×C47H(5,𝔽2)12yes—
193C23×C472−1+411yescharacteristic
194C22×D47—24no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—24no—
202C22×D47—24no—
203C22×D47—24no—
204C22×D47—24no—
205C22×D47—24no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—42no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—42no—
223C22×D47—42no—
224C22×D47—42no—
225C22×D47—42no—
226C22×D47—42no—
227C22×D47—42no—
228C22×D47C2×C4218yes—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
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—24no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—22no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47—42no—
258C2×C4×C47C2×C4214yes—
259C2×C4×C47C4×D414yes—
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—24no—
270C2×Dic47—24no—
271C2×Dic47—24no—
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×Dic47C4×D418yes—
281D188—416no—
282C4×D47—432no—
283#13791—471nomaximal, Sylow
284C22×C47C22×C4211yescharacteristic
285C22×C47C2×C4×D412yes—
286C22×C47C2×C4×D411yescharacteristic
287C22×C47C4×#4111yescharacteristic
288C22×C47C4×#4112yes—
289C22×C47#51812yes—
290C22×C47#51812yes—
291C22×C47#51811yescharacteristic
292C22×C47#51812yes—
293C22×C47#51811yescharacteristic
294C22×C47#51811yescharacteristic
295C22×C47#52012yes—
296C22×C47#52011yescharacteristic
297C22×C47#53411yescharacteristic
298C22×C47#53512yes—
299C22×C47#53711yescharacteristic
300C22×C47#53812yes—
301C22×C47#53812yes—
302C22×C47#53811yescharacteristic
303C22×C47#53911yescharacteristic
304C22×C47#54012yes—
305C22×C47#54012yes—
306C22×C47#54212yes—
307C2×D47—28no—
308C2×D47—28no—
309C2×D47—28no—
310C2×D47—44no—
311C2×D47—44no—
312C2×D47—44no—
313C2×D47—44no—
314C2×D47—44no—
315C2×D47—44no—
316C2×D47—44no—
317C2×D47—44no—
318C2×D47—44no—
319C2×D47—44no—
320C2×D47—44no—
321C2×D47—44no—
322C4×C47—24no—
323C4×C47—24no—
324C4×C47—44no—
325C4×C47—44no—
326C4×C47—42no—
327Dic47—28no—
328Dic47—44no—
329Dic47—44no—
330C2×#390—471no—
331C2×#386—472no—
332C2×C4×C22⋊C4—472no—
333#1437—472no—
334#1437—472no—
335#1436—472no—
336#1434D4711yescharacteristic, complemented
337C22⋊#95—472no—
338C4×#95—471no—
339C2×C47C42×D411yescharacteristic
340C2×C47#227311yescharacteristic
341C2×C47#227511yescharacteristic
342C2×C47#232112yes—
343C2×C47#232412yes—
344C2×C47#232511yescharacteristic
345C2×C47#232811yescharacteristic
346C2×C47#233311yescharacteristic
347C2×C47#233312yes—
348C2×C47#233512yes—
349C2×C47#233711yescharacteristic
350D47—48no—
351C23×D4—471no—
352C24×C4—472no—
353C22×C4⋊C4—471no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4—472no—
357C22×C22⋊C4—472no—
358C22×C22⋊C4—472no—
359C22×C42—471no—
360C22×C42—471no—
361C22×C42—472no—
362C22×C42C2×D4711yescharacteristic
363#390—478no—
364#386—4716no—
365C4×C22⋊C4—948no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95C2×D4712yes—
372C2×#95C2×D4711yescharacteristic
373C2×#95C2×D4711yescharacteristic
374C2×#95C2×D4712yes—
375C47#1379111yescharacteristic, complemented, Sylow
376C25—472no—
377C22×D4—478no—
378C22×D4—474no—
379C22×D4—1881no—
380C22×D4—1881no—
381C22×D4—1882no—
382C23×C4—472no—
383C23×C4—472no—
384C23×C4—472no—
385C23×C4—478no—
386C23×C4—944no—
387C23×C4—944no—
388C23×C4—1882no—
389C23×C4—1882no—
390C23×C4—1882no—
391C23×C4—1882no—
392C23×C4C22×D4711yescharacteristic
393C23×C4C22×D4712yes—
394C23×C4C22×D4712yes—
395C23×C4C22×D4711yescharacteristic
396C23×C4C22×D4711yescharacteristic
397C2×C4⋊C4—474no—
398C2×C4⋊C4—1882no—
399C2×C22⋊C4—478no—
400C2×C22⋊C4—478no—
401C2×C22⋊C4—478no—
402C2×C22⋊C4—478no—
403C2×C22⋊C4—478no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—944no—
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⋊C4—1882no—
413C2×C22⋊C4—1882no—
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×C42C4×D4714yescomplemented
426#95—24no—
427#95—41no—
428#95—41no—
429#95—478no—
430#95—944no—
431#95—944no—
432#95—1882no—
433#95—1882no—
434#95C4×D4718yescomplemented
435C24—474no—
436C24—474no—
437C24—474no—
438C24—942no—
439C24—942no—
440C24—942no—
441C24—942no—
442C24—942no—
443C24—942no—
444C24—942no—
445C24—942no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24—942no—
450C24C23×D4711yescharacteristic
451C2×D4—4716no—
452C2×D4—1888no—
453C2×D4—1888no—
454C2×D4—1884no—
455C2×D4—1884no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—474no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—474no—
483C22×C4—474no—
484C22×C4—9416no—
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—1888no—
502C22×C4—1888no—
503C22×C4—1888no—
504C22×C4—1888no—
505C22×C4—1888no—
506C22×C4—1888no—
507C22×C4C2×C4×D4714yes—
508C22×C4C2×C4×D4714yes—
509C22×C4C2×C4×D4712yes—
510C22×C4C2×C4×D4714yes—
511C22×C4C2×C4×D4712yes—
512C22×C4C2×C4×D4712yes—
513C22×C4D4×D4712yes—
514C22×C4#1062535412yes—
515C4⋊C4—1884no—
516C22⋊C4—9416no—
517C22⋊C4—9416no—
518C22⋊C4—1888no—
519C22⋊C4—1888no—
520C22⋊C4—1888no—
521C42—44no—
522C42—1884no—
523C42—1884no—
524C42—1888no—
525C23—478no—
526C23—944no—
527C23—944no—
528C23—944no—
529C23—944no—
530C23—944no—
531C23—944no—
532C23—944no—
533C23—944no—
534C23—944no—
535C23—944no—
536C23—944no—
537C23—944no—
538C23—944no—
539C23—944no—
540C23—944no—
541C23—944no—
542C23—944no—
543C23—944no—
544C23—1882no—
545C23—1882no—
546C23—1882no—
547C23—1882no—
548C23—1882no—
549C23—1882no—
550C23—1882no—
551C23—1882no—
552C23—1882no—
553C23—1882no—
554C23—1882no—
555C23—1882no—
556C23—1882no—
557C23—1882no—
558C23—1882no—
559C23—1882no—
560C23C22×C4×D4712yes—
561C23C22×C4×D4711yescharacteristic
562C23C2×D4×D4711yescharacteristic
563C23#41×D4712yes—
564C23C2×#1062535411yescharacteristic
565C23#1309406911yescharacteristic
566C23#1309406911yescharacteristic
567C23#1309407012yes—
568C23#1309407012yes—
569C23#1309407011yescharacteristic
570C23#1309407111yescharacteristic
571D4—18816no—
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—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—478no—
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—944no—
611C2×C4—944no—
612C2×C4—944no—
613C2×C4—944no—
614C2×C4—18832no—
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×C4C42×D4714yes—
624C2×C4#1309398614yes—
625C22—948no—
626C22—948no—
627C22—948no—
628C22—1884no—
629C22—1884no—
630C22—1884no—
631C22—1884no—
632C22—1884no—
633C22—1884no—
634C22—1884no—
635C22—1884no—
636C22—1884no—
637C22—1884no—
638C22—1884no—
639C22—1884no—
640C22C2×C42×D4711yescharacteristic
641C22C4×D4×D4712yes—
642C22C2×#1309398611yescharacteristic
643C22#1568959111yescharacteristic
644C22C4×#1062535412yes—
645C22#1568959512yes—
646C22#1568959512yes—
647C22#1568959611yescharacteristic
648C22#1568959712yes—
649C22#1568959711yescharacteristic
650C22#1568959711yescharacteristic
651C22#1568960312yes—
652C22#1568960311yescharacteristic
653C22#1568968111yescharacteristic
654C22#1568969112yes—
655C22#1568970011yescharacteristic
656C22#1568970912yes—
657C22#1568971212yes—
658C22#1568971311yescharacteristic
659C22#1568972211yescharacteristic
660C22#1568972512yes—
661C22#1568972912yes—
662C22#1568973912yes—
663C4—24no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—948no—
669C4—1884no—
670C4—1884no—
671C2—1888no—
672C2C4×#1309398611yescharacteristic
673C2#2024574011yescharacteristic
674C2#2024575011yescharacteristic
675C2#2024604112yes—
676C2#2024607612yes—
677C2#2024608811yescharacteristic
678C2#2024611111yescharacteristic
679C2#2024615011yescharacteristic
680C2#2024615212yes—
681C2#2024617712yes—
682C2#2024619811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC47×#1434C2
socle, cosocleC24×C47C24
prime radicalC23×D188C22
hypercenter#1434D47
hypocenterC47#13791
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. C2×#15688970C2semidirect
4. #20241624C2semidirect
5. #20241624C2semidirect
6. #20241622C2semidirect
7. (C2×D47)⋊#95C2semidirect
8. C4×C47⋊#95C2semidirect
9. C47×#1434C2semidirect
21. #15689025C4semidirect
22. #15689001C4semidirect
49. C22×C4×D47C2×C4semidirect
51. C2×C4⋊Dic47C2×C4semidirect
64. C2×(C2×D47)⋊C4C2×C4semidirect
65. C2×(C2×D47)⋊C4C2×C4semidirect
66. C2×(C2×D47)⋊C4C2×C4semidirect
75. C2×C4×Dic47D4semidirect
80. C47⋊#95C2×C4semidirect
88. C2×C42×C47C2×C4semidirect
92. #95×C47C2×C4semidirect
165. C2×D188C42semidirect
336. #1434D47semidirect
375. C47#13791semidirect
425. C2×C42C4×D47semidirect
434. #95C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890252
C2×#156890012
C2×#156889702
#202416242
#202416222
(C2×D47)⋊#952
C4×C47⋊#952
C47×#14342
#1379147