Finite Group Databasebeta

← #26142569

Subgroups — #26142569

Order 12032.

1470 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689048C211yesmaximal, characteristic
2C2×#15689016C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
4#20241657C211yesmaximal, characteristic, complemented
5#20241647C211yesmaximal, characteristic, complemented
6#20241625C212yesmaximal
7#20241625C212yesmaximal
8#20241622C212yesmaximal, complemented
9#20241612C212yesmaximal, complemented
10#95×Dic47C211yesmaximal, characteristic, complemented
11C47×#1439C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×C22⋊Dic47C2211yescharacteristic
15C22×C22⋊Dic47C2211yescharacteristic
16C22×C22⋊Dic47C2212yes—
17C22×C4×Dic47C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19C22×C4×Dic47C2212yes—
20#15689048C418yescomplemented
21#15689016—24no—
22C4⋊C4×Dic47—24no—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47C2212yes—
35C22×C22⋊C4×C47C2211yescharacteristic
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C23×Dic47C2311yescharacteristic
39C23×Dic47C2311yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—24no—
48C2×C4⋊Dic47C2×C414yes—
49C2×C22⋊Dic47—22no—
50C2×C22⋊Dic47—22no—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—42no—
54C2×C22⋊Dic47—42no—
55C2×C22⋊Dic47C2×C418yes—
56C2×C4×Dic47—24no—
57C2×C4×Dic47—24no—
58C2×C4×Dic47—24no—
59C2×C4×Dic47—24no—
60C2×C4×Dic47—42no—
61C2×C4×Dic47—42no—
62C2×C4×Dic47—42no—
63C2×C4×Dic47D414yes—
64C47⋊#95—22no—
65C47⋊#95—24no—
66C47⋊#95—24no—
67C47⋊#95—22no—
68C47⋊#95—24no—
69C47⋊#95—42no—
70C47⋊#95—42no—
71C47⋊#95—42no—
72C47⋊#95C2×C414yes—
73C47⋊#95C2×C418yes—
74C25×C47C2311yescharacteristic
75C23×C4×C47C2311yescharacteristic
76C23×C4×C47C2312yes—
77C23×C4×C47C2312yes—
78C23×C4×C47C2311yescharacteristic
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—24no—
81C2×C22⋊C4×C47—22no—
82C2×C22⋊C4×C47—22no—
83C2×C22⋊C4×C47—24no—
84C2×C22⋊C4×C47—24no—
85C2×C22⋊C4×C47—42no—
86C2×C22⋊C4×C47C2×C414yes—
87#95×C47—24no—
88#95×C47—22no—
89C24×C47—21no—
90C24×C47—21no—
91C24×C47—22no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—21no—
96C24×C47—21no—
97C24×C47—21no—
98C24×C47—21no—
99C24×C47C22×C412yes—
100C24×C47C2411yescharacteristic
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—21no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
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—22no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C2×D412yes—
122C22×C4×C47#4114yes—
123C22×C4×C47#4114yes—
124C22×C4×C47#4112yes—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—21no—
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—21no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47C22×C414yes—
154C22×Dic47C22×C414yes—
155C22×Dic47C2×D412yes—
156C22×Dic47C2×D412yes—
157C22×Dic47#4114yes—
158C22×Dic47#4114yes—
159C4⋊C4×C47—28no—
160C22⋊C4×C47—28no—
161C22⋊C4×C47—48no—
162C22⋊Dic47—48no—
163C22⋊Dic47—44no—
164C22⋊Dic47—44no—
165C4⋊Dic47—28no—
166C4⋊Dic47—44no—
167C4×Dic47—28no—
168C4×Dic47—48no—
169C4×Dic47—48no—
170C23×C47—22no—
171C23×C47—24no—
172C23×C47—24no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—41no—
176C23×C47—41no—
177C23×C47—41no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—41no—
183C23×C47—41no—
184C23×C47—41no—
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×C47C23×C411yescharacteristic
194C23×C47C22×D411yescharacteristic
195C23×C47C2×#4112yes—
196C23×C47C2×#4112yes—
197C23×C47C2×#4111yescharacteristic
198C23×C47C2×#4112yes—
199C23×C47C2×#4112yes—
200C23×C47H(5,𝔽2)11yescharacteristic
201C23×C47H(5,𝔽2)11yescharacteristic
202C23×C47H(5,𝔽2)11yescharacteristic
203C23×C472−1+411yescharacteristic
204C2×C4×C47—24no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—24no—
208C2×C4×C47—22no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—41no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47C4×D414yes—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—22no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
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—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47#12314yes—
261C2×Dic47#12414yes—
262#14249—471nomaximal, Sylow
263C22×C47—42no—
264C22×C47—44no—
265C22×C47—44no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—81no—
269C22×C47—81no—
270C22×C47—81no—
271C22×C47—81no—
272C22×C47—82no—
273C22×C47—82no—
274C22×C47C2×C4×D411yescharacteristic
275C22×C47C4×#4112yes—
276C22×C47C4×#4112yes—
277C22×C47#51811yescharacteristic
278C22×C47#52011yescharacteristic
279C22×C47C2×#12311yescharacteristic
280C22×C47#52511yescharacteristic
281C22×C47C2×#12411yescharacteristic
282C22×C47#52912yes—
283C22×C47#52912yes—
284C22×C47#53211yescharacteristic
285C22×C47#53511yescharacteristic
286C22×C47#53711yescharacteristic
287C22×C47#54011yescharacteristic
288C22×C47#54211yescharacteristic
289C22×C47#55112yes—
290C22×C47#55112yes—
291C22×C47#55212yes—
292C22×C47#55312yes—
293C22×C47#55312yes—
294C22×C47#55312yes—
295C22×C47#55612yes—
296C22×C47#55912yes—
297C4×C47—24no—
298C4×C47—44no—
299C4×C47—44no—
300C4×C47—42no—
301Dic47—24no—
302Dic47—24no—
303Dic47—44no—
304Dic47—44no—
305Dic47—81no—
306Dic47—81no—
307C2×#397—471no—
308C2×#388—471no—
309C2×C4×C4⋊C4—471no—
310#1445—471no—
311#1442—471no—
312#1439D4711yescharacteristic
313#1437—472no—
314#1437—472no—
315#1436—472no—
316#1434—472no—
317C4×#95—471no—
318C2×C47—82no—
319C2×C47C4×#12311yescharacteristic
320C2×C47C4×#12411yescharacteristic
321C2×C47#232411yescharacteristic
322C2×C47#237312yes—
323C2×C47#237412yes—
324C2×C47#249811yescharacteristic
325C2×C47#250411yescharacteristic
326C2×C47#251612yes—
327C2×C47#252512yes—
328C2×C47#259211yescharacteristic
329C2×C47#260311yescharacteristic
330C22×C4⋊C4—471no—
331C22×C4⋊C4—471no—
332C22×C4⋊C4C2×D4711yescharacteristic
333C22×C22⋊C4—472no—
334C22×C22⋊C4—471no—
335C22×C22⋊C4—471no—
336C22×C22⋊C4C2×D4712yes—
337C22×C22⋊C4C2×D4711yescharacteristic
338C22×C42—471no—
339C22×C42—472no—
340C22×C42—472no—
341#397—478no—
342#388—944no—
343C4×C4⋊C4—944no—
344C2×#95—471no—
345C2×#95—472no—
346C2×#95—472no—
347C2×#95—471no—
348C2×#95—472no—
349C2×#95—472no—
350C2×#95—471no—
351C2×#95—472no—
352C2×#95—472no—
353C2×#95—472no—
354C2×#95C2×D4711yescharacteristic
355C2×#95C2×D4712yes—
356C47#1424911yescharacteristic, complemented, Sylow
357C25C22×D4711yescharacteristic
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—472no—
361C23×C4—472no—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4C22×D4711yescharacteristic
365C23×C4C22×D4712yes—
366C23×C4C22×D4712yes—
367C23×C4C22×D4711yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4—944no—
374C2×C4⋊C4—942no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—944no—
383C2×C22⋊C4—942no—
384C2×C22⋊C4—942no—
385C2×C22⋊C4—1882no—
386C2×C22⋊C4—1882no—
387C2×C22⋊C4C4×D4714yes—
388C2×C42—474no—
389C2×C42—944no—
390C2×C42—944no—
391C2×C42—944no—
392C2×C42—944no—
393C2×C42—1882no—
394C2×C42—1882no—
395C2×C42—1882no—
396#95—24no—
397#95—22no—
398#95—474no—
399#95—478no—
400#95—944no—
401#95—944no—
402#95—942no—
403#95—942no—
404#95—944no—
405#95—1882no—
406#95—1882no—
407#95—1882no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—21no—
412C24—22no—
413C24—22no—
414C24—21no—
415C24—21no—
416C24—21no—
417C24—21no—
418C24C2×C4×D4712yes—
419C24C23×D4711yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—472no—
439C22×C4—472no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—941no—
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—941no—
463C22×C4—941no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4C2×C4×D4712yes—
473C22×C4C2×C4×D4712yes—
474C22×C4D4×D4712yes—
475C22×C4#1062535214yes—
476C22×C4#1062535214yes—
477C22×C4#1062535212yes—
478C4⋊C4—28no—
479C4⋊C4—948no—
480C4⋊C4—1884no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—1884no—
484C22⋊C4—1884no—
485C22⋊C4—1888no—
486C42—948no—
487C42—1888no—
488C42—1888no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C22×C4×D4711yescharacteristic
513C23C2×D4×D4711yescharacteristic
514C23#41×D4712yes—
515C23#41×D4712yes—
516C23C2×#1062535211yescharacteristic
517C23C2×#1062535212yes—
518C23C2×#1062535212yes—
519C23#1309406911yescharacteristic
520C23#1309407011yescharacteristic
521C23#1309407011yescharacteristic
522C23#1309407211yescharacteristic
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—474no—
547C2×C4—474no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—944no—
552C2×C4—942no—
553C2×C4—942no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—1882no—
563C2×C4—1882no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
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(C22⋊D47)⋊C414yes—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—82no—
592C22C2×(C22⋊D47)⋊C411yescharacteristic
593C22C4×#1062535212yes—
594C22C4×#1062535212yes—
595C22#1568959711yescharacteristic
596C22#1568960411yescharacteristic
597C22#123×D4711yescharacteristic
598C22#1568962911yescharacteristic
599C22#124×D4711yescharacteristic
600C22#1568965412yes—
601C22#1568965412yes—
602C22#1568967111yescharacteristic
603C22#1568968911yescharacteristic
604C22#1568970311yescharacteristic
605C22#1568972611yescharacteristic
606C22#1568974611yescharacteristic
607C22#1568979612yes—
608C22#1568980412yes—
609C22#1568980712yes—
610C22#1568981812yes—
611C22#1568982412yes—
612C22#1568982512yes—
613C22#1568984412yes—
614C22#1568985512yes—
615C4—24no—
616C4—42no—
617C4—44no—
618C4—44no—
619C4—944no—
620C4—944no—
621C4—3761no—
622C4—3761no—
623C4—1884no—
624C4—1884no—
625C2—82no—
626C2#2024591311yescharacteristic
627C2#2024593511yescharacteristic
628C2#2024607711yescharacteristic
629C2#2024650312yes—
630C2#2024651812yes—
631C2#2024782511yescharacteristic
632C2#2024787511yescharacteristic
633C2#2024802312yes—
634C2#2024814612yes—
635C2#2024887011yescharacteristic
636C2#2024900811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1439C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1439D47
hypocenterC47#14249
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×Dic47C2semidirect
4. #20241657C2semidirect
5. #20241647C2semidirect
8. #20241622C2semidirect
9. #20241612C2semidirect
10. #95×Dic47C2semidirect
20. #15689048C4semidirect
356. C47#14249semidirect

Maximal subgroups

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

subgroupindex
C2×#156890482
C2×#156890162
C2×C4⋊C4×Dic472
#202416572
#202416472
#202416252
#202416222
#202416122
#95×Dic472
C47×#14392
#1424947