Finite Group Databasebeta

← #26147607

Subgroups — #26147607

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×#13094008C211yesmaximal, characteristic
2C23×C4×Dic47C211yesmaximal, characteristic, complemented
3C2×#15689002C211yesmaximal, characteristic
4C2×#15689000C211yesmaximal, characteristic
5C2×#15688984C211yesmaximal, characteristic, complemented
6#20241624C214yesmaximal, complemented
7#20241622C214yesmaximal, complemented
8#20241610C211yesmaximal, characteristic, complemented
9C47×C22⋊#95C211yesmaximal, characteristic, complemented
10C23×C22⋊D47C2211yescharacteristic
11C24×Dic47C2211yescharacteristic
12C24×Dic47C2211yescharacteristic
13C2×#13094008—24no—
14C2×#13094008—24no—
15C2×#13094008C418yescomplemented
16C22×C4⋊Dic47C2212yes—
17C22×C22⋊Dic47C2211yescharacteristic
18C22×(C2×D47)⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2212yes—
21C22×C4×Dic47—24no—
22C22×C4×Dic47—24no—
23C22×C4×Dic47—24no—
24C22×C4×Dic47C2214yes—
25#15689002—24no—
26#15689000—24no—
27#15688984—24no—
28C2×C47⋊#95C2214yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2214yes—
32C24×C4×C47C2211yescharacteristic
33C22×C22⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47C2211yescharacteristic
35C2×#95×C47C2214yes—
36C24×D47C2311yescharacteristic
37C22×C22⋊D47—22no—
38C22×C22⋊D47—22no—
39C22×C22⋊D47—22no—
40C22×C22⋊D47—22no—
41C22×C22⋊D47—22no—
42C22×C22⋊D47—22no—
43C22×C22⋊D47C2×C414yes—
44C23×Dic47—22no—
45C23×Dic47—22no—
46C23×Dic47—22no—
47C23×Dic47—22no—
48C23×Dic47—22no—
49C23×Dic47—22no—
50C23×Dic47—28no—
51C23×Dic47C2×C414yes—
52C23×Dic47D414yes—
53C23×Dic47D414yes—
54C23×Dic47D414yes—
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2312yes—
58C23×Dic47C2312yes—
59#13094008—232no—
60C2×C4⋊Dic47—24no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47C2×C418yescomplemented
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—22no—
65C2×C22⋊Dic47—22no—
66C2×(C2×D47)⋊C4—24no—
67C2×(C2×D47)⋊C4—24no—
68C2×(C2×D47)⋊C4—24no—
69C2×(C2×D47)⋊C4—28no—
70C2×(C2×D47)⋊C4—24no—
71C2×(C2×D47)⋊C4—28no—
72C2×(C2×D47)⋊C4C2×C418yescomplemented
73C2×C4×Dic47—232no—
74C2×C4×Dic47—28no—
75C2×C4×Dic47—28no—
76C2×C4×Dic47—28no—
77C47⋊#95—24no—
78C47⋊#95—28no—
79C47⋊#95—24no—
80C47⋊#95—28no—
81C25×C47C2311yescharacteristic
82C23×C4×C47—22no—
83C23×C4×C47—22no—
84C23×C4×C47—28no—
85C23×C4×C47D414yes—
86C23×C4×C47C2312yes—
87C23×C4×C47C2312yes—
88C23×C4×C47C2312yes—
89C2×C22⋊C4×C47—22no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—22no—
94C2×C22⋊C4×C47C2×C414yes—
95#95×C47—28no—
96C24×C47—24no—
97C24×C47—24no—
98C24×C47C22×C412yes—
99C24×C47C2×D412yes—
100C24×C47C2×D412yes—
101C24×C47C2×D412yes—
102C24×C47C2×D412yes—
103C24×C47#4112yes—
104C24×C47#4112yes—
105C24×C47C2411yescharacteristic
106C23×D47—21no—
107C23×D47—21no—
108C23×D47—21no—
109C23×D47—21no—
110C23×D47—24no—
111C23×D47—24no—
112C23×D47—21no—
113C23×D47—21no—
114C23×D47C22×C412yes—
115C22×C4×C47—216no—
116C22×C4×C47—216no—
117C22×C4×C47—28no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—216no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—24no—
125C22×C4×C47—24no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—24no—
131C22×C4×C47C22×C414yes—
132C22×C4×C47C2×D414yes—
133C22×C4×C47#4118yes—
134C2×C22⋊D47—28no—
135C2×C22⋊D47—28no—
136C2×C22⋊D47—28no—
137C2×C22⋊D47—44no—
138C2×C22⋊D47—44no—
139C2×C22⋊D47—44no—
140C2×C22⋊D47—44no—
141C22×Dic47—216no—
142C22×Dic47—216no—
143C22×Dic47—28no—
144C22×Dic47—28no—
145C22×Dic47—28no—
146C22×Dic47—28no—
147C22×Dic47—28no—
148C22×Dic47—28no—
149C22×Dic47—28no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—24no—
157C22×Dic47—216no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47—22no—
169C22×Dic47C22⋊C4116yescomplemented
170C22×Dic47C22×C412yes—
171C22×Dic47C22×C414yes—
172C22×Dic47C2×D414yes—
173C22×Dic47C2×D414yes—
174C22×Dic47C2×D414yes—
175C22×Dic47#4118yes—
176C22⋊C4×C47—28no—
177C22⋊C4×C47—44no—
178C22⋊Dic47—44no—
179(C2×D47)⋊C4—216no—
180(C2×D47)⋊C4—48no—
181(C2×D47)⋊C4—48no—
182C4⋊Dic47—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×C47C2×C22⋊C414yes—
195C23×C47C4×D414yes—
196C23×C47C4×D414yes—
197C23×C47#12014yes—
198C23×C47#12114yes—
199C23×C47#12114yes—
200C23×C47#12314yes—
201C23×C47C23×C411yescharacteristic
202C23×C47C22×D411yescharacteristic
203C23×C47C22×D411yescharacteristic
204C23×C47C22×D411yescharacteristic
205C23×C47C22×D411yescharacteristic
206C23×C47C2×#4114yes—
207C23×C47C2×#4114yes—
208C23×C47C2×#4111yescharacteristic
209C23×C47C2×#4111yescharacteristic
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—28no—
214C22×D47—41no—
215C22×D47—41no—
216C22×D47—41no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—41no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—42no—
223C22×D47—42no—
224C22×D47—44no—
225C22×D47—44no—
226C22×D47—44no—
227C2×C4×C47—232no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—28no—
231C2×C4×C47—28no—
232C2×C4×C47—24no—
233C2×C4×C47—28no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—44no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—44no—
241C2×C4×C47—44no—
242C2×C4×C47#12318yescomplemented
243C22⋊D47—416no—
244C2×Dic47—232no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—28no—
254C2×Dic47—28no—
255C2×Dic47—232no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
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#38618yes—
280C22×C47C22×C22⋊C411yescharacteristic
281C22×C47C2×C4×D411yescharacteristic
282C22×C47C2×C4×D411yescharacteristic
283C22×C47C4×#4114yes—
284C22×C47C2×#12011yescharacteristic
285C22×C47C2×#12111yescharacteristic
286C22×C47C2×#12111yescharacteristic
287C22×C47C2×#12312yes—
288C22×C47C2×#12312yes—
289C22×C47C2×#12311yescharacteristic
290C22×C47#52512yes—
291C22×C47#52512yes—
292C22×C47#52512yes—
293C22×C47#52512yes—
294C22×C47#52914yes—
295C22×C47#52914yes—
296C22×C47#53214yes—
297C2×D47—42no—
298C2×D47—44no—
299C2×D47—44no—
300C2×D47—44no—
301C2×D47—81no—
302C2×D47—81no—
303C2×D47—82no—
304C2×D47—82no—
305C2×D47—82no—
306C4×C47—28no—
307C4×C47—44no—
308C4×C47—82no—
309Dic47—28no—
310Dic47—28no—
311Dic47—42no—
312Dic47—82no—
313C22×#123—471no—
314C23×C42—471no—
315C2×#386—471no—
316C2×#386—471no—
317C2×#381—471no—
318#1437—474no—
319#1436—474no—
320C22⋊#95—471no—
321C22⋊#95D4711yescharacteristic, complemented
322C2×C47—28no—
323C2×C47C2×#38611yescharacteristic
324C2×C47#229712yes—
325C2×C47C4×#12312yes—
326C2×C47C4×#12312yes—
327C2×C47#284911yescharacteristic
328C2×C47#284911yescharacteristic
329C2×C47#285312yes—
330C2×C47#285412yes—
331C2×C47#285412yes—
332D47—82no—
333C23×D4—471no—
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—472no—
342C22×C22⋊C4—471no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4C2×D4711yescharacteristic
346C22×C22⋊C4C2×D4711yescharacteristic
347C22×C42—474no—
348C22×C42—944no—
349C22×C42—944no—
350C22×C42—944no—
351#386—944no—
352#386—944no—
353#381—944no—
354C2×#95—474no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95—474no—
358C2×#95C2×D4714yes—
359C47#1470611yescharacteristic, complemented, Sylow
360C25—471no—
361C25C22×D4711yescharacteristic
362C22×D4—474no—
363C22×D4—942no—
364C22×D4—942no—
365C22×D4—942no—
366C22×D4—942no—
367C22×D4—942no—
368C22×D4—942no—
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—471no—
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×C4D18814yes—
388C23×C4C22×D4712yes—
389C23×C4C22×D4712yes—
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—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—478no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—948no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—942no—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—944no—
409C2×C22⋊C4—948no—
410C2×C22⋊C4C4×D4714yes—
411C2×C42—948no—
412C2×C42—948no—
413C2×C42—9432no—
414C2×C42—948no—
415#95—28no—
416#95—948no—
417#95—948no—
418#95—944no—
419#95—944no—
420C24—24no—
421C24—24no—
422C24—472no—
423C24—941no—
424C24—941no—
425C24—941no—
426C24—941no—
427C24—944no—
428C24—944no—
429C24—941no—
430C24—941no—
431C24C2×C4×D4712yes—
432C24C2×D18812yes—
433C24#1062535012yes—
434C24D4×D4712yes—
435C24D4×D4712yes—
436C24D4×D4712yes—
437C24#1062535412yes—
438C24C23×D4711yescharacteristic
439C2×D4—948no—
440C2×D4—948no—
441C2×D4—948no—
442C2×D4—1884no—
443C2×D4—1884no—
444C2×D4—1884no—
445C2×D4—1884no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—4716no—
463C22×C4—474no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—474no—
467C22×C4—478no—
468C22×C4—472no—
469C22×C4—9416no—
470C22×C4—9416no—
471C22×C4—9416no—
472C22×C4—948no—
473C22×C4—948no—
474C22×C4—948no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—948no—
478C22×C4—948no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—944no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—944no—
495C22×C4—944no—
496C22×C4—942no—
497C22×C4C2×C4×D4714yes—
498C22×C4C2×D18814yes—
499C22×C4#1062535218yes—
500C4⋊C4—9416no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—9416no—
504C22⋊C4—1888no—
505C22⋊C4—1884no—
506C22⋊C4—1888no—
507C42—9432no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—942no—
519C23—942no—
520C23—942no—
521C23—948no—
522C23—1881no—
523C23—1881no—
524C23—1881no—
525C23—1882no—
526C23—1882no—
527C23—1881no—
528C23—1882no—
529C23—1882no—
530C23—1882no—
531C23—1882no—
532C23—1884no—
533C23—1884no—
534C23—1884no—
535C23C22⋊C4×D4714yes—
536C23#1309398614yes—
537C23C4×D18814yes—
538C23#1309399414yes—
539C23#1309399714yes—
540C23#1309399914yes—
541C23#1309400914yes—
542C23C22×C4×D4711yescharacteristic
543C23C2×D4×D4711yescharacteristic
544C23C2×D4×D4711yescharacteristic
545C23C2×D4×D4711yescharacteristic
546C23C22×D18811yescharacteristic
547C23#41×D4714yes—
548C23C2×#1062535214yes—
549C23C2×#1062535011yescharacteristic
550C23C2×#1062535411yescharacteristic
551D4—18816no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—478no—
568C2×C4—478no—
569C2×C4—9432no—
570C2×C4—9432no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—948no—
580C2×C4—948no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1884no—
588C2×C4—1884no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4#1309400818yescomplemented
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—3761no—
603C22—3761no—
604C22—3762no—
605C22—3762no—
606C22—3762no—
607C22—1882no—
608C22—1884no—
609C22—1884no—
610C22—1884no—
611C22#1568900118yes—
612C22C2×C22⋊C4×D4711yescharacteristic
613C22C2×#1309398611yescharacteristic
614C22C2×C4×D18811yescharacteristic
615C22C4×#1062535214yes—
616C22C2×#1309399411yescharacteristic
617C22C2×#1309399711yescharacteristic
618C22C2×#1309399911yescharacteristic
619C22#123×D4712yes—
620C22C2×#1309400812yes—
621C22C2×#1309400911yescharacteristic
622C22#1568962912yes—
623C22#1568962912yes—
624C22#1568963112yes—
625C22#1568963512yes—
626C22#1568965314yes—
627C22#1568965914yes—
628C22#1568966814yes—
629C4—28no—
630C4—44no—
631C4—82no—
632C4—948no—
633C4—948no—
634C4—3762no—
635C4—1882no—
636C2—28no—
637C2—3762no—
638C2C2×#1568900111yescharacteristic
639C2#2024587312yes—
640C2#2024590712yes—
641C2C4×#1309400812yes—
642C2#2025168111yescharacteristic
643C2#2025168211yescharacteristic
644C2#2025171312yes—
645C2#2025171612yes—
646C2#2025172012yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C22⋊#95C2
socle, cosocleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC22⋊#95D47
hypocenterC47#14706
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic47C2semidirect
5. C2×#15688984C2semidirect
6. #20241624C2semidirect
7. #20241622C2semidirect
8. #20241610C2semidirect
9. C47×C22⋊#95C2semidirect
15. C2×#13094008C4semidirect
62. C2×C4⋊Dic47C2×C4semidirect
72. C2×(C2×D47)⋊C4C2×C4semidirect
169. C22×Dic47C22⋊C4semidirect
242. C2×C4×C47#123semidirect
269. C2×Dic47#123semidirect
321. C22⋊#95D47semidirect
359. C47#14706semidirect
593. C2×C4#13094008semidirect

Maximal subgroups

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

subgroupindex
C22×#130940082
C23×C4×Dic472
C2×#156890022
C2×#156890002
C2×#156889842
#202416242
#202416222
#202416102
C47×C22⋊#952
#1470647