Finite Group Databasebeta

← #26138239

Subgroups — #26138239

Order 12032.

4046 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic47C211yesmaximal, characteristic
2C2×#15688998C212yesmaximal, complemented
3C2×#15688983C212yesmaximal, complemented
4C2×#15688979C211yesmaximal, characteristic
5#20241633C214yesmaximal, complemented
6#20241611C212yesmaximal, complemented
7#20241610C212yesmaximal, complemented
8C23×C22⋊C4×C47C211yesmaximal, characteristic
9C24×Dic47C2212yes—
10C22×C4⋊Dic47C2214yescomplemented
11C22×C22⋊Dic47—28no—
12C22×C22⋊Dic47—24no—
13C22×C22⋊Dic47C2211yescharacteristic
14C22×C22⋊Dic47C2212yes—
15C22×C22⋊Dic47C2212yes—
16C22×C22⋊Dic47C2211yescharacteristic
17C22×C22⋊Dic47C2212yes—
18C22×C22⋊Dic47C2212yes—
19#15688998—28no—
20#15688983—28no—
21#15688979—24no—
22C2×C47⋊#95C2214yescomplemented
23C2×C47⋊#95C2214yescomplemented
24C2×C47⋊#95C2214yescomplemented
25C26×C47C2211yescharacteristic
26C24×C4×C47C2212yes—
27C22×C22⋊C4×C47—28no—
28C22×C22⋊C4×C47C418yescomplemented
29C22×C22⋊C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2212yes—
31C22×C22⋊C4×C47C2211yescharacteristic
32C23×Dic47—24no—
33C23×Dic47—28no—
34C23×Dic47—24no—
35C23×Dic47—24no—
36C23×Dic47—44no—
37C23×Dic47C2312yes—
38C23×Dic47C2312yes—
39C23×Dic47C2314yes—
40C2×C4⋊Dic47—28no—
41C2×C4⋊Dic47—28no—
42C2×C4⋊Dic47—44no—
43C2×C22⋊Dic47—232no—
44C2×C22⋊Dic47—24no—
45C2×C22⋊Dic47—24no—
46C2×C22⋊Dic47—24no—
47C2×C22⋊Dic47—28no—
48C2×C22⋊Dic47—24no—
49C2×C22⋊Dic47—22no—
50C2×C22⋊Dic47—24no—
51C2×C22⋊Dic47—22no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47—24no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—44no—
57C2×C22⋊Dic47D418yes—
58C47⋊#95—28no—
59C47⋊#95—28no—
60C47⋊#95—44no—
61C25×C47—24no—
62C25×C47—24no—
63C25×C47—24no—
64C25×C47—24no—
65C25×C47—44no—
66C25×C47C2×C414yes—
67C25×C47D414yes—
68C25×C47Q814yes—
69C25×C47C2311yescharacteristic
70C25×C47C2311yescharacteristic
71C25×C47C2311yescharacteristic
72C23×C4×C47—24no—
73C23×C4×C47—24no—
74C23×C4×C47—28no—
75C23×C4×C47—44no—
76C23×C4×C47C2×C418yescomplemented
77C23×C4×C47C2312yes—
78C23×C4×C47C2312yes—
79C2×C22⋊C4×C47—24no—
80C2×C22⋊C4×C47—28no—
81C2×C22⋊C4×C47—232no—
82C2×C22⋊C4×C47C2×C414yes—
83C2×C22⋊C4×C47C2×C414yes—
84C2×C22⋊C4×C47C2×C418yescomplemented
85C2×C22⋊C4×C47D418yes—
86C24×C47—22no—
87C24×C47—22no—
88C24×C47—22no—
89C24×C47—22no—
90C24×C47—24no—
91C24×C47—24no—
92C24×C47—28no—
93C24×C47—28no—
94C24×C47—216no—
95C24×C47—28no—
96C24×C47—216no—
97C24×C47—216no—
98C24×C47—216no—
99C24×C47—28no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—24no—
105C24×C47—48no—
106C24×C47—48no—
107C24×C47—48no—
108C24×C47—416no—
109C24×C47—416no—
110C24×C47—416no—
111C24×C47—416no—
112C24×C47C4⋊C4116yescomplemented
113C24×C47C22×C412yes—
114C24×C47C22×C412yes—
115C24×C47C22×C412yes—
116C24×C47C2×D412yes—
117C24×C47C2×D414yes—
118C24×C47C2×D412yes—
119C24×C47C2×D412yes—
120C24×C47C2×D414yes—
121C24×C47C2×Q812yes—
122C24×C47C2×Q812yes—
123C24×C47C2×Q812yes—
124C24×C47#4114yes—
125C24×C47#4114yes—
126C24×C47C2411yescharacteristic
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—216no—
131C22×C4×C47—24no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—24no—
135C22×C4×C47—216no—
136C22×C4×C47—48no—
137C22×C4×C47—416no—
138C22×C4×C47—416no—
139C22×C4×C47C22×C414yes—
140C22×C4×C47C22×C414yes—
141C22×C4×C47C2×D418yes—
142C22×Dic47—216no—
143C22×Dic47—24no—
144C22×Dic47—24no—
145C22×Dic47—24no—
146C22×Dic47—24no—
147C22×Dic47—24no—
148C22×Dic47—24no—
149C22×Dic47—22no—
150C22×Dic47—24no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—416no—
160C22×Dic47—416no—
161C22×Dic47—48no—
162C22×Dic47—48no—
163C22×Dic47C2×D418yes—
164C22⋊C4×C47—28no—
165C22⋊C4×C47—216no—
166C22⋊C4×C47C22⋊C4116yescomplemented
167C22⋊Dic47—216no—
168C22⋊Dic47—216no—
169C22⋊Dic47—48no—
170C22⋊Dic47—44no—
171C22⋊Dic47—48no—
172C22⋊Dic47—44no—
173C4⋊Dic47—416no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—24no—
177C23×C47—24no—
178C23×C47—24no—
179C23×C47—22no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—22no—
184C23×C47—24no—
185C23×C47—24no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—28no—
189C23×C47—28no—
190C23×C47—28no—
191C23×C47—28no—
192C23×C47—28no—
193C23×C47—28no—
194C23×C47—28no—
195C23×C47—28no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—22no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—22no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—232no—
208C23×C47—232no—
209C23×C47—416no—
210C23×C47—432no—
211C23×C47—432no—
212C23×C47—432no—
213C23×C47—432no—
214C23×C47—464no—
215C23×C47—44no—
216C23×C47—44no—
217C23×C47—44no—
218C23×C47—44no—
219C23×C47C2×C22⋊C414yes—
220C23×C47C2×C4⋊C414yes—
221C23×C47C2×C4⋊C414yes—
222C23×C47C2×C4⋊C414yes—
223C23×C47#11714yes—
224C23×C47C4×D418yes—
225C23×C47#12014yes—
226C23×C47#12118yes—
227C23×C47C22⋊Q818yes—
228C23×C47C22⋊Q818yes—
229C23×C47#12314yes—
230C23×C47C23×C411yescharacteristic
231C23×C47C22×D412yes—
232C23×C47C22×D412yes—
233C23×C47C22×D411yescharacteristic
234C23×C47C22×Q811yescharacteristic
235C23×C47C2×#4112yes—
236C23×C47C2×#4112yes—
237C23×C47H(5,𝔽2)14yes—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—28no—
241C2×C4×C47—28no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—432no—
246C2×C4×C47—44no—
247C2×C4×C47—44no—
248C2×C4×C47C2×C22⋊C418yescomplemented
249C2×Dic47—28no—
250C2×Dic47—28no—
251C2×Dic47—28no—
252C2×Dic47—432no—
253C2×Dic47—44no—
254C2×Dic47—44no—
255C2×Dic47—44no—
256C2×Dic47—44no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—44no—
264C2×Dic47—44no—
265C2×Dic47—44no—
266C2×Dic47—44no—
267#13877—471nomaximal, Sylow
268C22×C47—24no—
269C22×C47—24no—
270C22×C47—28no—
271C22×C47—28no—
272C22×C47—28no—
273C22×C47—28no—
274C22×C47—28no—
275C22×C47—28no—
276C22×C47—28no—
277C22×C47—28no—
278C22×C47—216no—
279C22×C47—216no—
280C22×C47—24no—
281C22×C47—24no—
282C22×C47—24no—
283C22×C47—24no—
284C22×C47—24no—
285C22×C47—464no—
286C22×C47—44no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47—44no—
290C22×C47—48no—
291C22×C47#38018yescomplemented
292C22×C47#38518yescomplemented
293C22×C47C22×C22⋊C411yescharacteristic
294C22×C47C22×C4⋊C411yescharacteristic
295C22×C47C2×#11711yescharacteristic
296C22×C47C2×C4×D412yes—
297C22×C47#51812yes—
298C22×C47C2×#12011yescharacteristic
299C22×C47C2×#12112yes—
300C22×C47C2×C22⋊Q812yes—
301C22×C47C2×C22⋊Q812yes—
302C22×C47C2×#12311yescharacteristic
303C22×C47#53412yes—
304C22×C47#54312yes—
305C22×C47D4214yes—
306C22×C47#54614yes—
307C22×C47#54614yes—
308C22×C47#55114yes—
309C4×C47—28no—
310C4×C47—44no—
311Dic47—48no—
312Dic47—82no—
313Dic47—82no—
314C23×C22⋊C4—471no—
315C23×C22⋊C4D4711yescharacteristic
316C2×#385—472no—
317C2×#380—472no—
318C2×#379—471no—
319#1439—474no—
320C22⋊#95—472no—
321C22⋊#95—472no—
322C2×C47—28no—
323C2×C47—28no—
324C2×C47—44no—
325C2×C47C2×#38011yescharacteristic
326C2×C47C2×#38511yescharacteristic
327C2×C47#231511yescharacteristic
328C2×C47D4×C22⋊C412yes—
329C2×C47#234212yes—
330C2×C47#241012yes—
331C2×C47#241212yes—
332C2×C47#243714yes—
333C26C2×D4711yescharacteristic
334C24×C4—472no—
335C24×C4C2×D4712yes—
336C22×C4⋊C4—474no—
337C22×C22⋊C4—28no—
338C22×C22⋊C4—472no—
339C22×C22⋊C4—472no—
340C22×C22⋊C4—471no—
341C22×C22⋊C4—472no—
342C22×C22⋊C4—471no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—948no—
345C22×C22⋊C4—944no—
346C22×C22⋊C4Dic4718yescomplemented
347C22×C22⋊C4C2×D4711yescharacteristic
348C22×C22⋊C4C2×D4712yes—
349C22×C22⋊C4C2×D4711yescharacteristic
350#385—948no—
351#380—948no—
352#379—944no—
353C2×#95—474no—
354C2×#95—474no—
355C2×#95—474no—
356C47#1387711yescharacteristic, complemented, Sylow
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25Dic9414yes—
363C25D18814yes—
364C25C2×Dic4714yes—
365C25C22×D4711yescharacteristic
366C25C22×D4711yescharacteristic
367C25C22×D4711yescharacteristic
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—28no—
371C23×C4—44no—
372C23×C4—474no—
373C23×C4—472no—
374C23×C4—472no—
375C23×C4—948no—
376C23×C4—944no—
377C23×C4—944no—
378C23×C4—944no—
379C23×C4—1884no—
380C23×C4C2×Dic4718yescomplemented
381C23×C4C22×D4712yes—
382C23×C4C22×D4712yes—
383C2×C4⋊C4—948no—
384C2×C4⋊C4—948no—
385C2×C4⋊C4—1884no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—28no—
388C2×C22⋊C4—232no—
389C2×C22⋊C4—478no—
390C2×C22⋊C4—948no—
391C2×C22⋊C4—9432no—
392C2×C22⋊C4—944no—
393C2×C22⋊C4—944no—
394C2×C22⋊C4—944no—
395C2×C22⋊C4—944no—
396C2×C22⋊C4—942no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—1884no—
404C2×C22⋊C4C2×Dic4714yes—
405C2×C22⋊C4C2×Dic4714yes—
406C2×C22⋊C4C2×Dic4718yescomplemented
407C2×C22⋊C4C22⋊D4718yes—
408#95—948no—
409#95—948no—
410#95—1884no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—28no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4⋊Dic47116yescomplemented
438C24C2×Dic9412yes—
439C24C2×Dic9412yes—
440C24C2×Dic9412yes—
441C24C2×D18812yes—
442C24C2×D18812yes—
443C24C2×D18812yes—
444C24#1062535014yes—
445C24D4×D4714yes—
446C24#1062535214yes—
447C24C22×Dic4712yes—
448C24C22×Dic4712yes—
449C24C22×Dic4712yes—
450C24C2×C22⋊D4714yes—
451C24C23×D4711yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—216no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—478no—
465C22×C4—9416no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—944no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—944no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—944no—
481C22×C4—944no—
482C22×C4—18816no—
483C22×C4—18816no—
484C22×C4—1888no—
485C22×C4—1888no—
486C22×C4C22×Dic4714yes—
487C22×C4C22×Dic4714yes—
488C22×C4C2×C22⋊D4718yes—
489C4⋊C4—18816no—
490C22⋊C4—28no—
491C22⋊C4—216no—
492C22⋊C4—9416no—
493C22⋊C4—9416no—
494C22⋊C4—1888no—
495C22⋊C4—1884no—
496C22⋊C4—1888no—
497C22⋊C4—1884no—
498C22⋊C4C22⋊Dic47116yescomplemented
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—232no—
523C23—232no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—416no—
535C23—464no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊Dic4714yes—
545C23C2×C4⋊Dic4714yes—
546C23C2×C4⋊Dic4714yes—
547C23C2×C4⋊Dic4714yes—
548C23#1309398314yes—
549C23D4×Dic4718yes—
550C23#1309399414yes—
551C23#1309399818yes—
552C23#1309400118yes—
553C23#1309400318yes—
554C23#1309400814yes—
555C23C23×Dic4711yescharacteristic
556C23C2×D4×D4712yes—
557C23C22×C22⋊D4712yes—
558C23C22×D18811yescharacteristic
559C23C22×Dic9411yescharacteristic
560C23C2×#1062535212yes—
561C23C2×#1062535012yes—
562C23#1309406914yes—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—948no—
574C2×C4—948no—
575C2×C4—948no—
576C2×C4—1884no—
577C2×C4—1884no—
578C2×C4—1884no—
579C2×C4—1884no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—18832no—
587C2×C4—1884no—
588C2×C4—1884no—
589C2×C4—1884no—
590C2×C4—1884no—
591C2×C4C2×C22⋊Dic4718yescomplemented
592C22—24no—
593C22—216no—
594C22—216no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22#1568898318yescomplemented
616C22#1568899818yescomplemented
617C22C22×C22⋊Dic4711yescharacteristic
618C22C22×C4⋊Dic4711yescharacteristic
619C22C2×#1309398311yescharacteristic
620C22C2×D4×Dic4712yes—
621C22#1568959812yes—
622C22C2×#1309399411yescharacteristic
623C22C2×#1309399812yes—
624C22C2×#1309400112yes—
625C22C2×#1309400312yes—
626C22C2×#1309400811yescharacteristic
627C22#1568968512yes—
628C22#1568974912yes—
629C22D4×C22⋊D4714yes—
630C22#1568976314yes—
631C22#1568976714yes—
632C22#1568979914yes—
633C4—28no—
634C4—44no—
635C4—3762no—
636C4—3762no—
637C4—1888no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×#1568898311yescharacteristic
642C2C2×#1568899811yescharacteristic
643C2#2024600211yescharacteristic
644C2D4×C22⋊Dic4712yes—
645C2#2024623712yes—
646C2#2024675612yes—
647C2#2024677812yes—
648C2#2024704914yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC23×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC26×C47C22
hypercenterC23×C22⋊C4D47
hypocenterC47#13877
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15688998C2semidirect
3. C2×#15688983C2semidirect
5. #20241633C2semidirect
6. #20241611C2semidirect
7. #20241610C2semidirect
10. C22×C4⋊Dic47C22semidirect
22. C2×C47⋊#95C22semidirect
23. C2×C47⋊#95C22semidirect
24. C2×C47⋊#95C22semidirect
28. C22×C22⋊C4×C47C4semidirect
76. C23×C4×C47C2×C4semidirect
84. C2×C22⋊C4×C47C2×C4semidirect
112. C24×C47C4⋊C4semidirect
166. C22⋊C4×C47C22⋊C4semidirect
248. C2×C4×C47C2×C22⋊C4semidirect
291. C22×C47#380semidirect
292. C22×C47#385semidirect
346. C22×C22⋊C4Dic47semidirect
356. C47#13877semidirect
380. C23×C4C2×Dic47semidirect
406. C2×C22⋊C4C2×Dic47semidirect
437. C24C4⋊Dic47semidirect
498. C22⋊C4C22⋊Dic47semidirect
591. C2×C4C2×C22⋊Dic47semidirect
615. C22#15688983semidirect
616. C22#15688998semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic472
C2×#156889982
C2×#156889832
C2×#156889792
#202416332
#202416112
#202416102
C23×C22⋊C4×C472
#1387747