Finite Group Databasebeta

← #26147604

Subgroups — #26147604

Order 12032.

2842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094005C211yesmaximal, characteristic
2C23×C4×Dic47C211yesmaximal, characteristic, complemented
3C2×#15689003C211yesmaximal, characteristic
4C2×#15688984C211yesmaximal, characteristic, complemented
5#20241625C214yesmaximal, complemented
6#20241619C214yesmaximal, complemented
7#20241611C211yesmaximal, characteristic, complemented
8#20241610C211yesmaximal, characteristic, complemented
9C2×C47×#386C211yesmaximal, characteristic
10C24×Dic47C2211yescharacteristic
11C24×Dic47C2211yescharacteristic
12C2×#13094005—24no—
13C2×#13094005—24no—
14C2×#13094005C418yescomplemented
15C22×Dic47⋊C4C2212yes—
16C22×C22⋊Dic47C2211yescharacteristic
17C22×C22⋊Dic47C2212yes—
18C22×C22⋊Dic47C2211yescharacteristic
19C22×C22⋊Dic47C2212yes—
20C22×C4×Dic47—24no—
21C22×C4×Dic47—24no—
22C22×C4×Dic47—24no—
23C22×C4×Dic47C2214yes—
24#15689003—24no—
25#15688984—24no—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2214yes—
29C2×C47⋊#95C2214yes—
30C23×D4×C47C2211yescharacteristic
31C24×C4×C47C2211yescharacteristic
32C22×C22⋊C4×C47C2211yescharacteristic
33C22×C22⋊C4×C47C2212yes—
34C47×#386—24no—
35C2×#95×C47C2212yes—
36C23×Dic47—22no—
37C23×Dic47—22no—
38C23×Dic47—22no—
39C23×Dic47—22no—
40C23×Dic47—22no—
41C23×Dic47—22no—
42C23×Dic47—28no—
43C23×Dic47C2×C414yes—
44C23×Dic47D414yes—
45C23×Dic47D414yes—
46C23×Dic47D414yes—
47C23×Dic47C2312yes—
48C23×Dic47C2312yes—
49C23×Dic47C2312yes—
50C23×Dic47C2312yes—
51#13094005—232no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4C2×C418yescomplemented
55C2×C22⋊Dic47—22no—
56C2×C22⋊Dic47—22no—
57C2×C22⋊Dic47—24no—
58C2×C22⋊Dic47—24no—
59C2×C22⋊Dic47—22no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—24no—
63C2×C22⋊Dic47—28no—
64C2×C22⋊Dic47C2×C414yes—
65C2×C22⋊Dic47C2×C418yescomplemented
66C2×C4×Dic47—232no—
67C2×C4×Dic47—28no—
68C2×C4×Dic47—28no—
69C2×C4×Dic47—28no—
70C47⋊#95—28no—
71C47⋊#95—28no—
72C47⋊#95—24no—
73C47⋊#95—28no—
74C25×C47C2311yescharacteristic
75C25×C47C2311yescharacteristic
76C22×D4×C47—22no—
77C22×D4×C47—22no—
78C22×D4×C47—22no—
79C22×D4×C47—22no—
80C22×D4×C47—22no—
81C22×D4×C47—22no—
82C22×D4×C47C2×C414yes—
83C23×C4×C47—22no—
84C23×C4×C47—22no—
85C23×C4×C47—28no—
86C23×C4×C47D414yes—
87C23×C4×C47C2312yes—
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2312yes—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—24no—
94C2×C22⋊C4×C47—28no—
95#95×C47—24no—
96C24×C47—24no—
97C24×C47—24no—
98C24×C47—21no—
99C24×C47—21no—
100C24×C47—21no—
101C24×C47—21no—
102C24×C47—24no—
103C24×C47—21no—
104C24×C47—21no—
105C24×C47—24no—
106C24×C47C22×C412yes—
107C24×C47C22×C412yes—
108C24×C47C2×D412yes—
109C24×C47C2×D412yes—
110C24×C47C2×D412yes—
111C24×C47C2×D412yes—
112C24×C47#4112yes—
113C24×C47#4112yes—
114C24×C47C2411yescharacteristic
115C2×D4×C47—28no—
116C2×D4×C47—28no—
117C2×D4×C47—28no—
118C2×D4×C47—44no—
119C2×D4×C47—44no—
120C2×D4×C47—44no—
121C2×D4×C47—44no—
122C22×C4×C47—216no—
123C22×C4×C47—216no—
124C22×C4×C47—28no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—24no—
128C22×C4×C47—216no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D414yes—
142C22×C4×C47#4118yes—
143C22×Dic47—216no—
144C22×Dic47—216no—
145C22×Dic47—28no—
146C22×Dic47—28no—
147C22×Dic47—28no—
148C22×Dic47—28no—
149C22×Dic47—28no—
150C22×Dic47—28no—
151C22×Dic47—28no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—24no—
159C22×Dic47—216no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—24no—
167C22×Dic47—22no—
168C22×Dic47—24no—
169C22×Dic47C22⋊C4116yescomplemented
170C22×Dic47C22×C414yes—
171C22×Dic47C22×C414yes—
172C22×Dic47C2×D414yes—
173C22×Dic47C2×D414yes—
174C22×Dic47C2×D414yes—
175C22×Dic47#4118yes—
176C22⋊C4×C47—44no—
177C22⋊C4×C47—48no—
178C22⋊Dic47—28no—
179C22⋊Dic47—216no—
180C22⋊Dic47—44no—
181C22⋊Dic47—48no—
182Dic47⋊C4—216no—
183C4×Dic47—232no—
184C23×C47—28no—
185C23×C47—24no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—24no—
189C23×C47—24no—
190C23×C47—24no—
191C23×C47—28no—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—28no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—41no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—44no—
209C23×C47—44no—
210C23×C47—44no—
211C23×C47C2×C22⋊C414yes—
212C23×C47C4×D414yes—
213C23×C47C4×D414yes—
214C23×C47#12014yes—
215C23×C47#12114yes—
216C23×C47#12114yes—
217C23×C47#12314yes—
218C23×C47C23×C411yescharacteristic
219C23×C47C22×D411yescharacteristic
220C23×C47C22×D411yescharacteristic
221C23×C47C22×D411yescharacteristic
222C23×C47C22×D411yescharacteristic
223C23×C47C2×#4114yes—
224C23×C47C2×#4111yescharacteristic
225C23×C47C2×#4114yes—
226C23×C47C2×#4111yescharacteristic
227D4×C47—416no—
228C2×C4×C47—232no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—28no—
232C2×C4×C47—28no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—44no—
244C2×C4×C47—44no—
245C2×C4×C47#12318yescomplemented
246C2×Dic47—232no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—28no—
256C2×Dic47—28no—
257C2×Dic47—232no—
258C2×Dic47—24no—
259C2×Dic47—28no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—44no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—44no—
267C2×Dic47—44no—
268C2×Dic47C2×C22⋊C418yes—
269C2×Dic47#12318yescomplemented
270#14706—471nomaximal, Sylow
271C22×C47—24no—
272C22×C47—24no—
273C22×C47—28no—
274C22×C47—28no—
275C22×C47—28no—
276C22×C47—28no—
277C22×C47—28no—
278C22×C47—28no—
279C22×C47—42no—
280C22×C47—44no—
281C22×C47—44no—
282C22×C47—44no—
283C22×C47—81no—
284C22×C47—81no—
285C22×C47—82no—
286C22×C47—82no—
287C22×C47—82no—
288C22×C47#38618yes—
289C22×C47C22×C22⋊C411yescharacteristic
290C22×C47C2×C4×D411yescharacteristic
291C22×C47C2×C4×D411yescharacteristic
292C22×C47C4×#4114yes—
293C22×C47C2×#12011yescharacteristic
294C22×C47C2×#12111yescharacteristic
295C22×C47C2×#12111yescharacteristic
296C22×C47C2×#12312yes—
297C22×C47C2×#12312yes—
298C22×C47C2×#12311yescharacteristic
299C22×C47#52512yes—
300C22×C47#52512yes—
301C22×C47#52512yes—
302C22×C47#52512yes—
303C22×C47#52914yes—
304C22×C47#52914yes—
305C22×C47#53214yes—
306C4×C47—28no—
307C4×C47—42no—
308C4×C47—82no—
309Dic47—28no—
310Dic47—28no—
311Dic47—44no—
312Dic47—82no—
313C22×#123—471no—
314C23×C42—471no—
315C2×#386—471no—
316C2×#386D4711yescharacteristic
317C2×#381—471no—
318#1437—474no—
319#1436—474no—
320C22⋊#95—471no—
321C22⋊#95—471no—
322C2×C47—28no—
323C2×C47—82no—
324C2×C47C2×#38611yescharacteristic
325C2×C47#229712yes—
326C2×C47C4×#12312yes—
327C2×C47C4×#12312yes—
328C2×C47#284911yescharacteristic
329C2×C47#284911yescharacteristic
330C2×C47#285312yes—
331C2×C47#285412yes—
332C2×C47#285412yes—
333C23×D4C2×D4711yescharacteristic
334C24×C4—471no—
335C24×C4—471no—
336C24×C4C2×D4711yescharacteristic
337C2×#123—478no—
338C2×#123—944no—
339C2×#123—944no—
340C22×C4⋊C4—472no—
341C22×C22⋊C4—471no—
342C22×C22⋊C4—472no—
343C22×C22⋊C4—471no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4C2×D4711yescharacteristic
346C22×C22⋊C4C2×D4712yes—
347C22×C42—474no—
348C22×C42—944no—
349C22×C42—944no—
350C22×C42—944no—
351#386—24no—
352#386—944no—
353#381—944no—
354C2×#95—474no—
355C2×#95—474no—
356C2×#95—472no—
357C2×#95—474no—
358C2×#95C2×D4712yes—
359C47#1470611yescharacteristic, complemented, Sylow
360C25C22×D4711yescharacteristic
361C25C22×D4711yescharacteristic
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4C4×D4714yes—
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—28no—
372C23×C4—474no—
373C23×C4—474no—
374C23×C4—474no—
375C23×C4—472no—
376C23×C4—472no—
377C23×C4—474no—
378C23×C4—472no—
379C23×C4—472no—
380C23×C4—948no—
381C23×C4—942no—
382C23×C4—942no—
383C23×C4—942no—
384C23×C4—942no—
385C23×C4—942no—
386C23×C4—942no—
387C23×C4C22⋊D4714yes—
388C23×C4C22×D4712yes—
389C23×C4C22×D4711yescharacteristic
390C23×C4C22×D4712yes—
391#123—9432no—
392C2×C4⋊C4—478no—
393C2×C4⋊C4—944no—
394C2×C4⋊C4—944no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—28no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—474no—
401C2×C22⋊C4—478no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—944no—
404C2×C22⋊C4—942no—
405C2×C22⋊C4—942no—
406C2×C22⋊C4—948no—
407C2×C22⋊C4—944no—
408C2×C22⋊C4—942no—
409C2×C22⋊C4—942no—
410C2×C22⋊C4—942no—
411C2×C42—948no—
412C2×C42—948no—
413C2×C42—9432no—
414C2×C42—948no—
415#95—24no—
416#95—948no—
417#95—948no—
418#95—944no—
419#95—948no—
420C24—24no—
421C24—24no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—21no—
426C24—24no—
427C24—24no—
428C24—21no—
429C24—21no—
430C24C2×C4×D4712yes—
431C24C2×C4×D4712yes—
432C24#1062535012yes—
433C24D4×D4712yes—
434C24D4×D4712yes—
435C24D4×D4712yes—
436C24#1062535212yes—
437C24C2×C22⋊D4712yes—
438C24C23×D4711yescharacteristic
439C2×D4—28no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—4716no—
465C22×C4—474no—
466C22×C4—474no—
467C22×C4—474no—
468C22×C4—474no—
469C22×C4—478no—
470C22×C4—474no—
471C22×C4—9416no—
472C22×C4—9416no—
473C22×C4—9416no—
474C22×C4—948no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—948no—
478C22×C4—948no—
479C22×C4—948no—
480C22×C4—948no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—944no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—944no—
495C22×C4—944no—
496C22×C4—942no—
497C22×C4C2×C4×D4712yes—
498C22×C4#1062535218yes—
499C22×C4C2×C22⋊D4714yes—
500C4⋊C4—9416no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—9416no—
504C22⋊C4—948no—
505C22⋊C4—1888no—
506C22⋊C4—1884no—
507C42—9432no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23C22⋊C4×D4714yes—
536C23(C22⋊D47)⋊C414yes—
537C23C4×C22⋊D4714yes—
538C23#1309399214yes—
539C23#1309399514yes—
540C23#1309399714yes—
541C23#1309400614yes—
542C23C22×C4×D4711yescharacteristic
543C23C2×D4×D4711yescharacteristic
544C23C2×D4×D4711yescharacteristic
545C23C2×D4×D4711yescharacteristic
546C23C22×C22⋊D4711yescharacteristic
547C23#41×D4714yes—
548C23C2×#1062535211yescharacteristic
549C23C2×#1062535214yes—
550C23C2×#1062535011yescharacteristic
551D4—416no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—478no—
570C2×C4—478no—
571C2×C4—9432no—
572C2×C4—9432no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—948no—
582C2×C4—948no—
583C2×C4—944no—
584C2×C4—948no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1884no—
589C2×C4—1884no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1884no—
593C2×C4#1309400518yescomplemented
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—81no—
607C22—81no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22#1568899918yes—
612C22C2×C22⋊C4×D4711yescharacteristic
613C22C2×(C22⋊D47)⋊C411yescharacteristic
614C22C2×C4×C22⋊D4711yescharacteristic
615C22C4×#1062535214yes—
616C22C2×#1309399211yescharacteristic
617C22C2×#1309399511yescharacteristic
618C22C2×#1309399711yescharacteristic
619C22#123×D4712yes—
620C22C2×#1309400512yes—
621C22C2×#1309400611yescharacteristic
622C22#1568962912yes—
623C22#1568962912yes—
624C22#1568963012yes—
625C22#1568963112yes—
626C22#1568965314yes—
627C22#1568965414yes—
628C22#1568966814yes—
629C4—28no—
630C4—42no—
631C4—82no—
632C4—948no—
633C4—948no—
634C4—3762no—
635C4—1884no—
636C2—28no—
637C2—82no—
638C2C2×#1568899911yescharacteristic
639C2#2024587312yes—
640C2C4×#1309400512yes—
641C2#2024591312yes—
642C2#2025167711yescharacteristic
643C2#2025167811yescharacteristic
644C2#2025170912yes—
645C2#2025171412yes—
646C2#2025171612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#386C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC2×#386D47
hypocenterC47#14706
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic47C2semidirect
4. C2×#15688984C2semidirect
5. #20241625C2semidirect
6. #20241619C2semidirect
7. #20241611C2semidirect
8. #20241610C2semidirect
14. C2×#13094005C4semidirect
54. C2×Dic47⋊C4C2×C4semidirect
65. C2×C22⋊Dic47C2×C4semidirect
169. C22×Dic47C22⋊C4semidirect
245. C2×C4×C47#123semidirect
269. C2×Dic47#123semidirect
359. C47#14706semidirect
593. C2×C4#13094005semidirect

Maximal subgroups

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

subgroupindex
C22×#130940052
C23×C4×Dic472
C2×#156890032
C2×#156889842
#202416252
#202416192
#202416112
#202416102
C2×C47×#3862
#1470647