Finite Group Databasebeta

← #26144735

Subgroups — #26144735

Order 12032.

1658 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal, complemented
2#20241637C212yesmaximal, complemented
3#20241633C212yesmaximal, complemented
4#20241629C212yesmaximal
5#20241625C212yesmaximal, complemented
6#20241620C212yesmaximal, complemented
7#20241618C211yesmaximal, characteristic, complemented
8#20241615C211yesmaximal, characteristic, complemented
9C2×C47×#390C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C4×Dic47C2212yes—
16#15689016—28no—
17C2×C47⋊#95C2212yes—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2211yescharacteristic
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C23×D4×C47C2211yescharacteristic
28C22×C4⋊C4×C47C2211yescharacteristic
29C22×C22⋊C4×C47C2212yes—
30C22×C22⋊C4×C47C2212yes—
31C22×C42×C47C2211yescharacteristic
32C47×#390C418yescomplemented
33C23×Dic47C2312yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C2×C4⋊Dic47—24no—
38C2×C4⋊Dic47—24no—
39C2×C4⋊Dic47—42no—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47—24no—
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—42no—
45C2×C22⋊Dic47—42no—
46C2×C22⋊Dic47—42no—
47C2×C22⋊Dic47—42no—
48C2×C22⋊Dic47—42no—
49C2×C22⋊Dic47—42no—
50C2×C22⋊Dic47—42no—
51C2×C22⋊Dic47—42no—
52C2×C4×Dic47—24no—
53C2×C4×Dic47—42no—
54C2×C4×Dic47—42no—
55C47⋊#95—24no—
56C47⋊#95—24no—
57C47⋊#95—22no—
58C47⋊#95—24no—
59C47⋊#95—42no—
60C47⋊#95—41no—
61C47⋊#95—42no—
62C47⋊#95—42no—
63C47⋊#95—41no—
64C47⋊#95D414yescomplemented
65C25×C47C2312yes—
66C22×D4×C47—22no—
67C22×D4×C47—22no—
68C22×D4×C47—42no—
69C22×D4×C47—42no—
70C22×D4×C47C2×C414yes—
71C23×C4×C47C2311yescharacteristic
72C23×C4×C47C2312yes—
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2311yescharacteristic
75C2×C4⋊C4×C47—42no—
76C2×C4⋊C4×C47C2×C414yes—
77C2×C22⋊C4×C47—24no—
78C2×C22⋊C4×C47—24no—
79C2×C22⋊C4×C47—42no—
80C2×C22⋊C4×C47—42no—
81C2×C22⋊C4×C47C2×C418yes—
82C2×C22⋊C4×C47C2×C418yes—
83C2×C42×C47—42no—
84C2×C42×C47C2×C414yes—
85C24×C47—22no—
86C24×C47—22no—
87C24×C47—22no—
88C24×C47—22no—
89C24×C47—22no—
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×C47C22×C414yes—
100C24×C47C2411yescharacteristic
101C2×D4×C47—28no—
102C2×D4×C47—44no—
103C2×D4×C47—44no—
104C2×D4×C47—48no—
105C2×D4×C47—48no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
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—21no—
119C22×C4×C47—21no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C414yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47#4112yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
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×Dic47C2×D414yes—
155C22×Dic47#4114yes—
156C4⋊C4×C47—44no—
157C22⋊C4×C47—216no—
158C22⋊C4×C47—48no—
159C42×C47—44no—
160C22⋊Dic47—48no—
161C22⋊Dic47—48no—
162C22⋊Dic47—48no—
163C22⋊Dic47—48no—
164C4⋊Dic47—48no—
165C4×Dic47—48no—
166C23×C47—24no—
167C23×C47—24no—
168C23×C47—24no—
169C23×C47—24no—
170C23×C47—24no—
171C23×C47—24no—
172C23×C47—24no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—42no—
177C23×C47—42no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47C23×C411yescharacteristic
202C23×C47C22×D411yescharacteristic
203C23×C47C2×#4112yes—
204C23×C47C2×#4111yescharacteristic
205C23×C47C2×#4111yescharacteristic
206C23×C47C2×#4111yescharacteristic
207C23×C47H(5,𝔽2)12yes—
208C23×C47H(5,𝔽2)12yes—
209C23×C47H(5,𝔽2)12yes—
210C23×C47H(5,𝔽2)11yescharacteristic
211C23×C472−1+411yescharacteristic
212D4×C47—416no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47#11714yes—
234C2×C4×C47C4×D414yes—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
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—
263#14444—471nomaximal, Sylow
264C22×C47—44no—
265C22×C47—44no—
266C22×C47—44no—
267C22×C47—44no—
268C22×C47—44no—
269C22×C47—44no—
270C22×C47—44no—
271C22×C47—82no—
272C22×C47—82no—
273C22×C47—82no—
274C22×C47—82no—
275C22×C47—82no—
276C22×C47—82no—
277C22×C47—82no—
278C22×C47—82no—
279C22×C47C2×#11711yescharacteristic
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C4×#4111yescharacteristic
282C22×C47#51811yescharacteristic
283C22×C47#51811yescharacteristic
284C22×C47#51811yescharacteristic
285C22×C47#52011yescharacteristic
286C22×C47#53512yes—
287C22×C47#53812yes—
288C22×C47#53812yes—
289C22×C47#53812yes—
290C22×C47#54012yes—
291C22×C47#54212yes—
292C22×C47#54612yes—
293C22×C47#54711yescharacteristic
294C22×C47#55011yescharacteristic
295C22×C47#55312yes—
296C22×C47#55312yes—
297C22×C47#55511yescharacteristic
298C22×C47#55611yescharacteristic
299C22×C47#55912yes—
300C22×C47#56012yes—
301C22×C47#56212yes—
302C4×C47—24no—
303C4×C47—24no—
304C4×C47—44no—
305C4×C47—42no—
306Dic47—44no—
307Dic47—44no—
308Dic47—82no—
309Dic47—82no—
310C2×#390D4711yescharacteristic
311C2×#388—472no—
312#1440—472no—
313#1439—472no—
314#1438—472no—
315#1437—472no—
316#1436—472no—
317C4⋊#95—471no—
318#1434—471no—
319C2×C47—84no—
320C2×C47#231211yescharacteristic
321C2×C47#232811yescharacteristic
322C2×C47#233711yescharacteristic
323C2×C47#235411yescharacteristic
324C2×C47#235711yescharacteristic
325C2×C47#236411yescharacteristic
326C2×C47#237711yescharacteristic
327C2×C47#271012yes—
328C2×C47#271412yes—
329C2×C47#280012yes—
330C2×C47#280312yes—
331C23×D4C2×D4711yescharacteristic
332C22×C4⋊C4—472no—
333C22×C4⋊C4C2×D4711yescharacteristic
334C22×C22⋊C4—472no—
335C22×C22⋊C4—472no—
336C22×C22⋊C4—472no—
337C22×C22⋊C4—472no—
338C22×C22⋊C4C2×D4712yes—
339C22×C22⋊C4C2×D4712yes—
340C22×C42—472no—
341C22×C42C2×D4711yescharacteristic
342#390Dic4718yescomplemented
343#388—948no—
344C2×#95—471no—
345C2×#95—472no—
346C2×#95—471no—
347C2×#95—472no—
348C2×#95—472no—
349C2×#95—472no—
350C2×#95—471no—
351C2×#95—471no—
352C2×#95—472no—
353C2×#95—472no—
354C47#1444411yescharacteristic, complemented, Sylow
355C25C22×D4712yes—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—42no—
359C22×D4—42no—
360C22×D4C2×Dic4714yes—
361C23×C4—472no—
362C23×C4—472no—
363C23×C4—472no—
364C23×C4—472no—
365C23×C4C22×D4712yes—
366C23×C4C22×D4711yescharacteristic
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4711yescharacteristic
369C2×C4⋊C4—42no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—1882no—
373C2×C4⋊C4C2×Dic4714yes—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—944no—
379C2×C22⋊C4—944no—
380C2×C22⋊C4—944no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—1882no—
383C2×C22⋊C4—1882no—
384C2×C22⋊C4—1882no—
385C2×C22⋊C4—1882no—
386C2×C22⋊C4—1882no—
387C2×C22⋊C4—1882no—
388C2×C22⋊C4—1882no—
389C2×C22⋊C4—1882no—
390C2×C22⋊C4C2×Dic4718yes—
391C2×C22⋊C4C2×Dic4718yes—
392C2×C42—42no—
393C2×C42—944no—
394C2×C42—1882no—
395C2×C42—1882no—
396C2×C42C2×Dic4714yes—
397#95—474no—
398#95—944no—
399#95—944no—
400#95—944no—
401#95—942no—
402#95—1882no—
403#95—1882no—
404#95—1882no—
405#95—1881no—
406#95—1881no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24C22×Dic4714yes—
422C24C23×D4711yescharacteristic
423C2×D4—28no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—48no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
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#1062535012yes—
471C22×C4#1062535012yes—
472C22×C4D4×D4712yes—
473C22×C4#1062535212yes—
474C22×C4C22×Dic4712yes—
475C22×C4C22×Dic4714yes—
476C22×C4C22×Dic4712yes—
477C22×C4C22×Dic4712yes—
478C4⋊C4—44no—
479C4⋊C4—1888no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—1888no—
483C22⋊C4—1888no—
484C22⋊C4—1888no—
485C22⋊C4—1888no—
486C42—44no—
487C42—1888no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C23×Dic4711yescharacteristic
524C23C2×D4×D4711yescharacteristic
525C23#41×D4712yes—
526C23C2×#1062535211yescharacteristic
527C23C2×#1062535011yescharacteristic
528C23C2×#1062535011yescharacteristic
529C23#1309406912yes—
530C23#1309406912yes—
531C23#1309406912yes—
532C23#1309407011yescharacteristic
533C23#1309407111yescharacteristic
534D4—416no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—944no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—1882no—
562C2×C4—1882no—
563C2×C4—1882no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4#1309398314yes—
584C2×C4D4×Dic4714yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×#1309398311yescharacteristic
601C22C2×D4×Dic4711yescharacteristic
602C22#41×Dic4711yescharacteristic
603C22#1568959811yescharacteristic
604C22#1568959811yescharacteristic
605C22#1568959811yescharacteristic
606C22#1568960611yescharacteristic
607C22#1568968612yes—
608C22#1568970712yes—
609C22#1568971112yes—
610C22#1568971112yes—
611C22#1568972712yes—
612C22#1568973712yes—
613C22#1568976212yes—
614C22#1568977811yescharacteristic
615C22#1568979411yescharacteristic
616C22#1568981612yes—
617C22#1568982312yes—
618C22#1568983811yescharacteristic
619C22#1568984311yescharacteristic
620C22#1568985612yes—
621C22#1568986112yes—
622C22#1568986812yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—3762no—
628C4—3762no—
629C4—1884no—
630C4—1884no—
631C2—84no—
632C2#2024598611yescharacteristic
633C2#2024611411yescharacteristic
634C2#2024620311yescharacteristic
635C2#2024633711yescharacteristic
636C2#2024636111yescharacteristic
637C2#2024644011yescharacteristic
638C2#2024654511yescharacteristic
639C2#2025012712yes—
640C2#2025018112yes—
641C2#2025128512yes—
642C2#2025132212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#390C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC2×#390D47
hypocenterC47#14444
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#15689016C2semidirect
2. #20241637C2semidirect
3. #20241633C2semidirect
5. #20241625C2semidirect
6. #20241620C2semidirect
7. #20241618C2semidirect
8. #20241615C2semidirect
32. C47×#390C4semidirect
64. C47⋊#95D4semidirect
342. #390Dic47semidirect
354. C47#14444semidirect

Maximal subgroups

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

subgroupindex
C2×#156890162
#202416372
#202416332
#202416292
#202416252
#202416202
#202416182
#202416152
C2×C47×#3902
#1444447