Finite Group Databasebeta

← #26138201

Subgroups — #26138201

Order 12032.

1918 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688984C211yesmaximal, characteristic
2C2×#15688984C211yesmaximal, characteristic
3C2×#15688981C211yesmaximal, characteristic
4C22×C47⋊#95C211yesmaximal, characteristic
5#20241648C212yesmaximal, complemented
6#20241643C212yesmaximal, complemented
7#20241621C212yesmaximal, complemented
8#20241615C212yesmaximal, complemented
9#20241611C211yesmaximal, characteristic
10#20241610C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C24×Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×C22⋊Dic47C2211yescharacteristic
16C22×C22⋊Dic47C2211yescharacteristic
17#15688984—24no—
18#15688984—24no—
19#15688981—24no—
20C2×C47⋊#95C418yescomplemented
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2211yescharacteristic
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2211yescharacteristic
33C2×C47⋊#95C2211yescharacteristic
34C24×C4×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C22×C42×C47C2212yes—
38C4×C22⋊C4×C47—24no—
39C2×#95×C47C2212yes—
40C23×Dic47—22no—
41C23×Dic47—22no—
42C23×Dic47—22no—
43C23×Dic47—22no—
44C23×Dic47—22no—
45C23×Dic47—22no—
46C23×Dic47—22no—
47C23×Dic47—22no—
48C23×Dic47—24no—
49C23×Dic47—42no—
50C23×Dic47C2×C414yes—
51C23×Dic47C2×C414yes—
52C23×Dic47C2312yes—
53C23×Dic47C2312yes—
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2311yescharacteristic
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4—24no—
60C2×Dic47⋊C4—42no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—22no—
65C2×C22⋊Dic47—22no—
66C2×C22⋊Dic47—22no—
67C47⋊#95—24no—
68C47⋊#95—24no—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—42no—
75C47⋊#95—42no—
76C47⋊#95C2×C414yes—
77C47⋊#95C2×C414yes—
78C47⋊#95C2×C414yes—
79C47⋊#95C2×C414yes—
80C25×C47C2311yescharacteristic
81C23×C4×C47—22no—
82C23×C4×C47—22no—
83C23×C4×C47—24no—
84C23×C4×C47—42no—
85C23×C4×C47C2×C414yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2312yes—
88C23×C4×C47C2312yes—
89C23×C4×C47C2311yescharacteristic
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—22no—
94C2×C22⋊C4×C47D414yes—
95C2×C22⋊C4×C47D414yes—
96C2×C42×C47—24no—
97C2×C42×C47—24no—
98C2×C42×C47—42no—
99#95×C47—24no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47C22×C412yes—
105C24×C47C2×D412yes—
106C24×C47C2×D412yes—
107C24×C47#4112yes—
108C24×C47#4112yes—
109C24×C47#4112yes—
110C24×C47#4112yes—
111C24×C47C2411yescharacteristic
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—28no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
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×C47—28no—
133C22×C4×C47—44no—
134C22×C4×C47—48no—
135C22×C4×C47—48no—
136C22×C4×C47C22×C412yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D414yes—
139C22×C4×C47C2×D414yes—
140C22×C4×C47#4114yes—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—28no—
145C22×Dic47—28no—
146C22×Dic47—28no—
147C22×Dic47—28no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—22no—
164C22×Dic47—28no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—22no—
172C22×Dic47—22no—
173C22×Dic47—22no—
174C22×Dic47—22no—
175C22×Dic47—22no—
176C22×Dic47—22no—
177C22×Dic47—22no—
178C22×Dic47—22no—
179C22×Dic47—44no—
180C22×Dic47—48no—
181C22×Dic47—48no—
182C22×Dic47—44no—
183C22×Dic47—44no—
184C22×Dic47—44no—
185C22×Dic47—44no—
186C22×Dic47C22×C412yes—
187C22×Dic47C22×C412yes—
188C22×Dic47C22×C412yes—
189C22×Dic47C22×C412yes—
190C22×Dic47#4114yes—
191C22⋊C4×C47—28no—
192C22⋊C4×C47—28no—
193C42×C47—48no—
194C22⋊Dic47—44no—
195C22⋊Dic47—44no—
196Dic47⋊C4—48no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—24no—
210C23×C47—24no—
211C23×C47—24no—
212C23×C47—24no—
213C23×C47#11714yes—
214C23×C47C4×D414yes—
215C23×C47C4×D414yes—
216C23×C47#12114yes—
217C23×C47#12314yes—
218C23×C47#12414yes—
219C23×C47#12614yes—
220C23×C47C23×C411yescharacteristic
221C23×C47C22×D411yescharacteristic
222C23×C47C22×D411yescharacteristic
223C23×C47C2×#4112yes—
224C23×C47C2×#4112yes—
225C23×C47C2×#4111yescharacteristic
226C23×C47C2×#4111yescharacteristic
227C23×C47C2×#4111yescharacteristic
228C23×C47C2×#4111yescharacteristic
229C23×C47H(5,𝔽2)12yes—
230C23×C472−1+412yes—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—416no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
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#11714yes—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—22no—
269C2×Dic47—416no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—416no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47—42no—
285C2×Dic47—42no—
286C2×Dic47—42no—
287C2×Dic47—42no—
288C2×Dic47—42no—
289C2×Dic47—42no—
290C2×Dic47—42no—
291C2×Dic47—42no—
292C2×Dic47—42no—
293C2×Dic47—42no—
294C2×Dic47—42no—
295C2×Dic47#11714yes—
296#13874—471nomaximal, Sylow
297C22×C47—24no—
298C22×C47—24no—
299C22×C47—24no—
300C22×C47—24no—
301C22×C47—24no—
302C22×C47—28no—
303C22×C47—28no—
304C22×C47—28no—
305C22×C47—28no—
306C22×C47—24no—
307C22×C47#38818yes—
308C22×C47C2×#11711yescharacteristic
309C22×C47C2×#11711yescharacteristic
310C22×C47C2×#11711yescharacteristic
311C22×C47C2×C4×D411yescharacteristic
312C22×C47C2×C4×D411yescharacteristic
313C22×C47#51811yescharacteristic
314C22×C47#51911yescharacteristic
315C22×C47C2×#12111yescharacteristic
316C22×C47C2×#12311yescharacteristic
317C22×C47C2×#12411yescharacteristic
318C22×C47C2×#12611yescharacteristic
319C22×C47#52912yes—
320C22×C47#52912yes—
321C22×C47#53912yes—
322C22×C47#53912yes—
323C22×C47#54612yes—
324C22×C47#54612yes—
325C22×C47#54712yes—
326C22×C47#54712yes—
327C22×C47#55112yes—
328C22×C47#55112yes—
329C22×C47#55212yes—
330C22×C47#55212yes—
331C4×C47—24no—
332C4×C47—42no—
333C4×C47—44no—
334C4×C47—44no—
335Dic47—24no—
336Dic47—42no—
337Dic47—42no—
338Dic47—42no—
339Dic47—82no—
340Dic47—82no—
341C2×#381—471no—
342C2×#381—471no—
343C2×#380—471no—
344C2×C4×C22⋊C4D4711yescharacteristic
345C22×#95—471no—
346#1443—472no—
347#1442—472no—
348#1436—472no—
349#1434—472no—
350C22⋊#95—471no—
351C22⋊#95—471no—
352C2×C47—28no—
353C2×C47#228311yescharacteristic
354C2×C47C2×#38811yescharacteristic
355C2×C47#231311yescharacteristic
356C2×C47#234211yescharacteristic
357C2×C47#234211yescharacteristic
358C2×C47#234411yescharacteristic
359C2×C47#234411yescharacteristic
360C2×C47#241912yes—
361C2×C47#242512yes—
362C2×C47#248812yes—
363C2×C47#249312yes—
364C24×C4—471no—
365C24×C4—471no—
366C24×C4C2×D4711yescharacteristic
367C22×C4⋊C4—472no—
368C22×C22⋊C4—471no—
369C22×C22⋊C4—471no—
370C22×C22⋊C4C2×D4711yescharacteristic
371C22×C22⋊C4C2×D4711yescharacteristic
372C22×C42C2×D4712yes—
373#381—944no—
374#381—944no—
375#380—944no—
376C4×C22⋊C4—24no—
377C2×#95—478no—
378C2×#95—471no—
379C2×#95—471no—
380C2×#95—471no—
381C2×#95—471no—
382C2×#95—472no—
383C2×#95—472no—
384C2×#95—472no—
385C2×#95—472no—
386C2×#95—472no—
387C2×#95—472no—
388C2×#95—472no—
389C2×#95—472no—
390C2×#95—472no—
391C2×#95C2×D4712yes—
392C47#1387411yescharacteristic, complemented, Sylow
393C25C22×D4711yescharacteristic
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—24no—
397C23×C4—42no—
398C23×C4—471no—
399C23×C4—474no—
400C23×C4—471no—
401C23×C4—471no—
402C23×C4—471no—
403C23×C4—474no—
404C23×C4—472no—
405C23×C4—472no—
406C23×C4—944no—
407C23×C4—942no—
408C23×C4—942no—
409C23×C4—942no—
410C23×C4—942no—
411C23×C4—942no—
412C23×C4—942no—
413C23×C4—942no—
414C23×C4—942no—
415C23×C4—1882no—
416C23×C4C4×D4714yes—
417C23×C4C22×D4711yescharacteristic
418C23×C4C22×D4712yes—
419C23×C4C22×D4712yes—
420C23×C4C22×D4711yescharacteristic
421C2×C4⋊C4—944no—
422C2×C4⋊C4—944no—
423C2×C4⋊C4—1882no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—942no—
429C2×C22⋊C4—942no—
430C2×C22⋊C4—942no—
431C2×C22⋊C4—942no—
432C2×C22⋊C4—942no—
433C2×C22⋊C4—942no—
434C2×C22⋊C4D18814yes—
435C2×C22⋊C4C22⋊D4714yes—
436C2×C42—24no—
437C2×C42—24no—
438C2×C42—42no—
439#95—24no—
440#95—474no—
441#95—474no—
442#95—474no—
443#95—474no—
444#95—944no—
445#95—944no—
446#95—944no—
447#95—944no—
448#95—944no—
449#95—944no—
450#95—944no—
451#95—1882no—
452#95—1882no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×C4×D4712yes—
458C24C2×D18812yes—
459C24#1062535012yes—
460C24#1062535012yes—
461C24#1062535012yes—
462C24#1062535012yes—
463C24C2×C22⋊D4712yes—
464C24C23×D4711yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—28no—
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—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—28no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—472no—
490C22×C4—474no—
491C22×C4—472no—
492C22×C4—472no—
493C22×C4—472no—
494C22×C4—948no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—942no—
498C22×C4—948no—
499C22×C4—948no—
500C22×C4—948no—
501C22×C4—948no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4—941no—
506C22×C4—941no—
507C22×C4—942no—
508C22×C4—941no—
509C22×C4—941no—
510C22×C4—941no—
511C22×C4—941no—
512C22×C4—941no—
513C22×C4—941no—
514C22×C4—941no—
515C22×C4—941no—
516C22×C4—941no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—942no—
521C22×C4—942no—
522C22×C4—942no—
523C22×C4—942no—
524C22×C4—942no—
525C22×C4—942no—
526C22×C4—942no—
527C22×C4—942no—
528C22×C4—942no—
529C22×C4—942no—
530C22×C4—942no—
531C22×C4—942no—
532C22×C4—1884no—
533C22×C4—1888no—
534C22×C4—1888no—
535C22×C4—1884no—
536C22×C4—1884no—
537C22×C4—1884no—
538C22×C4—1884no—
539C22×C4C2×C4×D4712yes—
540C22×C4C2×C4×D4712yes—
541C22×C4C2×D18814yes—
542C22×C4#1062535214yes—
543C22×C4C2×C22⋊D4714yes—
544C4⋊C4—1888no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—1884no—
548C22⋊C4—1884no—
549C42—48no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23#1309398114yes—
567C23C4×C22⋊D4714yes—
568C23C4×D18814yes—
569C23#1309399814yes—
570C23#1309400714yes—
571C23#1309401214yes—
572C23#1309401814yes—
573C23C22×C4×D4711yescharacteristic
574C23C22×C22⋊D4711yescharacteristic
575C23C22×D18811yescharacteristic
576C23#41×D4712yes—
577C23C2×#1062535212yes—
578C23C2×#1062535011yescharacteristic
579C23C2×#1062535011yescharacteristic
580C23C2×#1062535011yescharacteristic
581C23C2×#1062535011yescharacteristic
582C23#1309406912yes—
583C23#1309407112yes—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—474no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—942no—
612C2×C4—942no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—942no—
616C2×C4—942no—
617C2×C4—942no—
618C2×C4—942no—
619C2×C4—942no—
620C2×C4—942no—
621C2×C4—942no—
622C2×C4—18816no—
623C2×C4—1882no—
624C2×C4—1882no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4—1882no—
629C2×C4—1882no—
630C2×C4—1882no—
631C2×C4—1882no—
632C2×C4—1882no—
633C2×C4—1882no—
634C2×C4—1882no—
635C2×C4—1882no—
636C2×C4—1882no—
637C2×C4—1882no—
638C2×C4—18816no—
639C2×C4—1882no—
640C2×C4—1881no—
641C2×C4—1881no—
642C2×C4—1881no—
643C2×C4—1881no—
644C2×C4—1881no—
645C2×C4—1881no—
646C2×C4—1881no—
647C2×C4—1881no—
648C2×C4#1309398014yes—
649C22—24no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22—28no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22#1568901418yes—
660C22#117×D4711yescharacteristic
661C22C2×#1309398011yescharacteristic
662C22C2×#1309398111yescharacteristic
663C22C2×C4×C22⋊D4711yescharacteristic
664C22C2×C4×D18811yescharacteristic
665C22#1568959611yescharacteristic
666C22#1568960011yescharacteristic
667C22C2×#1309399811yescharacteristic
668C22C2×#1309400711yescharacteristic
669C22C2×#1309401211yescharacteristic
670C22C2×#1309401811yescharacteristic
671C22#1568965312yes—
672C22#1568965312yes—
673C22#1568971512yes—
674C22#1568971512yes—
675C22#1568976712yes—
676C22#1568976812yes—
677C22#1568977212yes—
678C22#1568977712yes—
679C22#1568979912yes—
680C22#1568980212yes—
681C22#1568980612yes—
682C22#1568981312yes—
683C4—24no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—944no—
688C4—3762no—
689C4—3762no—
690C4—1882no—
691C4—1882no—
692C4—1882no—
693C2—28no—
694C2#2024578211yescharacteristic
695C2C2×#1568901411yescharacteristic
696C2#2024598811yescharacteristic
697C2#2024623311yescharacteristic
698C2#2024624011yescharacteristic
699C2#2024625511yescharacteristic
700C2#2024625811yescharacteristic
701C2#2024685912yes—
702C2#2024693312yes—
703C2#2024770812yes—
704C2#2024776712yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#13874
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. #20241648C2semidirect
6. #20241643C2semidirect
7. #20241621C2semidirect
8. #20241615C2semidirect
20. C2×C47⋊#95C4semidirect
392. C47#13874semidirect

Maximal subgroups

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

subgroupindex
C2×#156889842
C2×#156889812
C22×C47⋊#952
#202416482
#202416432
#202416212
#202416152
#202416112
#202416102
C2×C4×C22⋊C4×C472
#1387447