Finite Group Databasebeta

← #26217526

Subgroups — #26217526

Order 12032.

2278 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093999C211yesmaximal, characteristic
2C2×#117×D47C211yesmaximal, characteristic, complemented
3C2×#15689129C211yesmaximal, characteristic, complemented
4C2×#15689127C211yesmaximal, characteristic, complemented
5C2×#15689086C211yesmaximal, characteristic, complemented
6#20243709C214yesmaximal, complemented
7#20243704C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C47×#1736C211yesmaximal, characteristic, complemented
10C23×D188—21no—
11C23×D188C2211yescharacteristic
12C23×C4×D47C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C2×#13093999—24no—
15C2×#13093999—21no—
16C2×#13093999—21no—
17C2×#13093999C418yescomplemented
18C2×#13093999C2212yes—
19C2×#13093999C2212yes—
20C2×#13093983C2211yescharacteristic
21C2×#13093982C2212yescomplemented
22C2×#13093981—21no—
23C2×#13093980—21no—
24#117×D47—24no—
25#117×D47—24no—
26C2×C4⋊C4×D47C2212yescomplemented
27C22×(C2×D47)⋊C4—21no—
28C2×C22⋊C4×D47—21no—
29C2×C42×D47—21no—
30#15689129—24no—
31#15689127—24no—
32C2×#13093946C2214yescomplemented
33C2×#13093944C2214yescomplemented
34#15689086—24no—
35C2×(C2×D47)⋊C8C2212yescomplemented
36C2×(C2×D47)⋊C8C2212yescomplemented
37#15688885C2214yescomplemented
38C22×M16×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C47×#340C2214yescomplemented
42C24×D47—21no—
43C24×D47D411yescharacteristic
44C22×D188—22no—
45C22×D188—22no—
46C22×D188—22no—
47C22×D188—22no—
48C22×D188—28no—
49C22×D188—22no—
50C22×D188—22no—
51C22×D188—21no—
52C22×D188—22no—
53C22×D188—48no—
54C22×D188C2×C414yescomplemented
55C22×D188C2312yes—
56C23×Dic47D411yescharacteristic
57C22×C4×D47—22no—
58C22×C4×D47—22no—
59C22×C4×D47—22no—
60C22×C4×D47—22no—
61C22×C4×D47—22no—
62C22×C4×D47C2×C414yes—
63C22×C4×D47D411yescharacteristic
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D414yes—
66C22×C4×D47C2311yescharacteristic
67C22×C4×D47C2311yescharacteristic
68C2×#10625329—22no—
69C2×#10625329—22no—
70C2×#10625329—22no—
71C22×C47⋊C8C2312yes—
72#13093999—28no—
73#13093999—28no—
74#13093999—24no—
75#13093999—24no—
76#13093999C2×C418yes—
77#13093999C2×C418yescomplemented
78#13093983—22no—
79#13093983D414yescomplemented
80#13093982—24no—
81#13093982—24no—
82#13093981—28no—
83#13093980—28no—
84C2×C4⋊Dic47C2312yes—
85C2×Dic47⋊C4—22no—
86C4⋊C4×D47—24no—
87C4⋊C4×D47—24no—
88C2×C22⋊Dic47—21no—
89C2×(C2×D47)⋊C4—22no—
90C2×(C2×D47)⋊C4—28no—
91C2×(C2×D47)⋊C4—22no—
92C2×(C2×D47)⋊C4—22no—
93C22⋊C4×D47—28no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—22no—
96C42×D47—28no—
97#13093946—28no—
98#13093944—28no—
99(C2×D47)⋊C8—24no—
100(C2×D47)⋊C8—24no—
101C23×C4×C47—21no—
102C23×C4×C47C2311yescharacteristic
103C2×M16×C47—22no—
104C2×M16×C47—22no—
105C2×M16×C47—22no—
106C22×C8×C47C2312yes—
107#117×C47—22no—
108#117×C47D414yescomplemented
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—22no—
111C2×C4⋊C4×C47C2×C414yescomplemented
112C2×C4⋊C4×C47C2312yes—
113C2×C4⋊C4×C47C2312yes—
114C2×C22⋊C4×C47—21no—
115C2×C42×C47—21no—
116C24×C47C2×D411yescharacteristic
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—24no—
120C23×D47—24no—
121C23×D47—22no—
122C23×D47—42no—
123C23×D47—44no—
124C23×D47—44no—
125C23×D47—42no—
126C23×D47—42no—
127C23×D47C22⋊C414yes—
128C23×D47C2×D411yescharacteristic
129C23×D47C2×D411yescharacteristic
130C22×C4×C47—24no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—44no—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47#4112yes—
140C22×C4×C47#4112yes—
141C22×C4×C47C2411yescharacteristic
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47C22⋊C414yes—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C2×D188—28no—
149C2×D188—28no—
150C2×D188—22no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—28no—
155C2×D188—28no—
156C2×D188—24no—
157C2×D188—22no—
158C2×D188—48no—
159C2×D188—432no—
160C2×D188—48no—
161C2×D188—42no—
162C2×D188—44no—
163C2×D188—44no—
164C2×D188—42no—
165C2×D188C22×C414yes—
166C2×C4×D47—28no—
167C2×C4×D47—28no—
168C2×C4×D47—22no—
169C2×C4×D47—22no—
170C2×C4×D47—28no—
171C2×C4×D47—28no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—21no—
175C2×C4×D47—21no—
176C2×C4×D47—21no—
177C2×C4×D47—21no—
178C2×C4×D47—21no—
179C2×C4×D47—21no—
180C2×C4×D47—21no—
181C2×C4×D47—21no—
182C2×C4×D47—28no—
183C2×C4×D47—44no—
184C2×C4×D47—44no—
185C2×C4×D47C22⋊C414yes—
186C2×C4×D47C22⋊C414yes—
187C2×C4×D47C22×C412yes—
188C2×C4×D47C22×C412yes—
189C2×C4×D47C2×D412yes—
190C2×C4×D47C2×D412yes—
191M16×C47—44no—
192C2×C8×C47—22no—
193C2×C8×C47—22no—
194C2×C8×C47—22no—
195C4⋊C4×C47—24no—
196C4⋊C4×C47—22no—
197C4⋊C4×C47—22no—
198C4⋊C4×C47C22×C414yes—
199C4⋊C4×C47C2×D414yescomplemented
200C22⋊C4×C47—24no—
201C42×C47—24no—
202C22⋊Dic47—24no—
203(C2×D47)⋊C4—28no—
204(C2×D47)⋊C4—28no—
205(C2×D47)⋊C4—28no—
206C4⋊Dic47—22no—
207C4⋊Dic47C2×D414yescomplemented
208Dic47⋊C4—28no—
209C4×Dic47—28no—
210C4×Dic47—24no—
211#10625329—44no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—22no—
214C2×C47⋊C8—22no—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47C2×C22⋊C412yes—
218C23×C47#12012yes—
219C23×C47#12312yes—
220C23×C47C22×D411yescharacteristic
221C22×D47—21no—
222C22×D47—21no—
223C22×D47—22no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—28no—
229C22×D47—28no—
230C22×D47—24no—
231C22×D47—44no—
232C22×D47—48no—
233C22×D47—48no—
234C22×D47—48no—
235C22×D47—48no—
236C22×D47—42no—
237C22×D47—42no—
238C22×D47—44no—
239C22×D47—44no—
240C22×D47—48no—
241C22×D47—88no—
242C22×D47—88no—
243C22×D47C2×C22⋊C412yes—
244C22×D47C2×C22⋊C412yes—
245C22×D47AGL(1,ℤ/8)14yes—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—48no—
249C2×C4×C47—42no—
250C2×C4×C47—44no—
251C2×C4×C47—42no—
252C2×C4×C47C2×C22⋊C412yes—
253C2×C4×C47C4×D414yes—
254C2×C4×C47C4×D414yes—
255C2×C4×C47#12012yes—
256C2×C4×C47#12114yes—
257C2×C4×C47#12114yes—
258C2×C4×C47#12312yes—
259C2×C4×C47C23×C411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—28no—
270C2×Dic47—24no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47C2×C22⋊C412yes—
274C2×Dic47C2×C22⋊C412yes—
275D188—24no—
276D188—24no—
277D188—416no—
278D188—416no—
279D188—42no—
280D188—42no—
281D188—48no—
282C4×D47—22no—
283C4×D47—22no—
284C4×D47—22no—
285C4×D47—22no—
286C4×D47—22no—
287C4×D47—22no—
288C4×D47—22no—
289C4×D47—22no—
290C4×D47—28no—
291C4×D47—28no—
292C4×D47—416no—
293C8×C47—42no—
294C47⋊C8—42no—
295#20912—471nomaximal, Sylow
296C22×C47—24no—
297C22×C47—22no—
298C22×C47—22no—
299C22×C47—24no—
300C22×C47#38614yes—
301C22×C47C22×C22⋊C411yescharacteristic
302C22×C47C2×#12011yescharacteristic
303C22×C47C2×#12311yescharacteristic
304C22×C47C2×AGL(1,ℤ/8)12yes—
305C22×C47#57512yes—
306C2×D47—24no—
307C2×D47—22no—
308C2×D47—24no—
309C2×D47—22no—
310C2×D47—22no—
311C2×D47—22no—
312C2×D47—22no—
313C2×D47—416no—
314C2×D47—42no—
315C2×D47—42no—
316C2×D47—44no—
317C2×D47—44no—
318C2×D47—44no—
319C2×D47—816no—
320C2×D47—84no—
321C2×D47—84no—
322C2×D47#41814yescomplemented
323C4×C47—44no—
324C4×C47—44no—
325C4×C47#38614yes—
326C4×C47C2×C4×D411yescharacteristic
327C4×C47C2×C4×D411yescharacteristic
328C4×C47C2×#12111yescharacteristic
329C4×C47C2×#12111yescharacteristic
330Dic47—22no—
331Dic47—22no—
332Dic47—44no—
333C22×#121—471no—
334C22×#117—471no—
335C2×#418—471no—
336C2×#418—471no—
337C2×#407—471no—
338#1875—474no—
339#1874—474no—
340#1736—471no—
341#1736D4711yescharacteristic, complemented
342C2×C47—24no—
343C2×C47C2×#38611yescharacteristic
344C2×C47C2×#41811yescharacteristic
345C2×C47#289611yescharacteristic
346C2×C47#300612yes—
347C2×C47#309412yes—
348D47—24no—
349D47—42no—
350D47—84no—
351C23×D4—471no—
352C23×D4—941no—
353C24×C4—471no—
354C22×M16—471no—
355C22×M16C2×D4711yescharacteristic
356C2×#121—478no—
357C2×#121—472no—
358C2×#121—472no—
359C2×#121—941no—
360C2×#121—944no—
361C2×#121—941no—
362C2×#117—471no—
363C2×#117—472no—
364C2×#117—941no—
365C2×#117—941no—
366C2×#117—944no—
367C2×#117—944no—
368C2×#117C2×D4711yescharacteristic
369C22×C4⋊C4—472no—
370C22×C4⋊C4C2×D4711yescharacteristic
371C22×C22⋊C4—941no—
372C22×C22⋊C4—941no—
373C22×C42—941no—
374#418—944no—
375#418—944no—
376C2×#102—474no—
377C2×#102—474no—
378#407—944no—
379C2×C22⋊C8—472no—
380C2×C22⋊C8—472no—
381#340—474no—
382#340C2×D4714yescomplemented
383C47#2091211yescharacteristic, complemented, Sylow
384C25—471no—
385C25—941no—
386C22×D4—474no—
387C22×D4—472no—
388C22×D4—948no—
389C22×D4—942no—
390C22×D4—942no—
391C22×D4—942no—
392C22×D4—942no—
393C22×D4—942no—
394C22×D4—941no—
395C22×D4—942no—
396C22×D4—942no—
397C22×D4—1888no—
398C23×C4—21no—
399C23×C4—471no—
400C23×C4—474no—
401C23×C4—471no—
402C23×C4—474no—
403C23×C4—471no—
404C23×C4—471no—
405C23×C4—471no—
406C23×C4—942no—
407C23×C4—942no—
408C23×C4—942no—
409C23×C4—942no—
410C23×C4—942no—
411C23×C4C22×D4711yescharacteristic
412C2×M16—22no—
413C2×M16—22no—
414C2×M16—22no—
415C2×M16—942no—
416C2×M16—942no—
417C2×M16—942no—
418C22×C8—472no—
419C22×C8C22×D4712yes—
420#121—478no—
421#121—478no—
422#121—948no—
423#121—948no—
424#121—944no—
425#121—944no—
426#117—22no—
427#117—474no—
428#117—948no—
429#117—948no—
430#117—944no—
431#117—942no—
432#117—944no—
433#117D18814yescomplemented
434C2×C4⋊C4—21no—
435C2×C4⋊C4—22no—
436C2×C4⋊C4—472no—
437C2×C4⋊C4—944no—
438C2×C4⋊C4—942no—
439C2×C4⋊C4—944no—
440C2×C4⋊C4C4×D4714yescomplemented
441C2×C4⋊C4C22×D4712yes—
442C2×C4⋊C4C22×D4712yes—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—948no—
445C2×C22⋊C4—948no—
446C2×C22⋊C4—942no—
447C2×C22⋊C4—942no—
448C2×C22⋊C4—942no—
449C2×C22⋊C4—941no—
450C2×C42—21no—
451C2×C42—948no—
452C2×C42—942no—
453C2×C42—941no—
454#102—948no—
455#102—948no—
456C22⋊C8—944no—
457C22⋊C8—944no—
458C24—471no—
459C24—471no—
460C24—474no—
461C24—944no—
462C24—942no—
463C24—942no—
464C24—944no—
465C24—942no—
466C24—1882no—
467C24—1884no—
468C24—1884no—
469C24—1882no—
470C24—1882no—
471C24D4×D4711yescharacteristic
472C2×D4—474no—
473C2×D4—948no—
474C2×D4—948no—
475C2×D4—944no—
476C2×D4—948no—
477C2×D4—948no—
478C2×D4—942no—
479C2×D4—942no—
480C2×D4—942no—
481C2×D4—942no—
482C2×D4—942no—
483C2×D4—1888no—
484C2×D4—18832no—
485C2×D4—1888no—
486C2×D4—1882no—
487C2×D4—1884no—
488C2×D4—1884no—
489C2×D4—1882no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—44no—
494C22×C4—474no—
495C22×C4—471no—
496C22×C4—471no—
497C22×C4—474no—
498C22×C4—474no—
499C22×C4—472no—
500C22×C4—472no—
501C22×C4—472no—
502C22×C4—472no—
503C22×C4—948no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—948no—
507C22×C4—948no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—948no—
511C22×C4—948no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—941no—
515C22×C4—941no—
516C22×C4—941no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4—941no—
520C22×C4—941no—
521C22×C4—941no—
522C22×C4—942no—
523C22×C4—1884no—
524C22×C4—1884no—
525C22×C4C2×C4×D4712yes—
526C22×C4C2×D18812yes—
527C22×C4#1062535012yes—
528C22×C4D4×D4711yescharacteristic
529C22×C4D4×D4712yes—
530C22×C4D4×D4712yes—
531C22×C4#1062535412yes—
532C22×C4C23×D4711yescharacteristic
533M16—44no—
534M16—1884no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—942no—
539C2×C8—942no—
540C2×C8—942no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—474no—
545C4⋊C4—948no—
546C4⋊C4—942no—
547C4⋊C4C2×C4×D4714yes—
548C4⋊C4C2×D18814yescomplemented
549C22⋊C4—24no—
550C22⋊C4—948no—
551C22⋊C4—948no—
552C22⋊C4—948no—
553C22⋊C4—944no—
554C42—24no—
555C42—948no—
556C42—944no—
557C23—22no—
558C23—22no—
559C23—472no—
560C23—474no—
561C23—472no—
562C23—948no—
563C23—941no—
564C23—941no—
565C23—942no—
566C23—948no—
567C23—941no—
568C23—941no—
569C23—942no—
570C23—942no—
571C23—944no—
572C23—3768no—
573C23—3768no—
574C23—1884no—
575C23—1888no—
576C23—1888no—
577C23—1888no—
578C23—1888no—
579C23—1888no—
580C23—1882no—
581C23—1882no—
582C23—1884no—
583C23—1884no—
584C23C22⋊C4×D4712yes—
585C23#1309399412yes—
586C23#1309400912yes—
587C23C2×D4×D4711yescharacteristic
588D4—944no—
589D4—944no—
590D4—18816no—
591D4—18816no—
592D4—1888no—
593D4—1882no—
594D4—1882no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—472no—
602C2×C4—472no—
603C2×C4—948no—
604C2×C4—941no—
605C2×C4—941no—
606C2×C4—941no—
607C2×C4—941no—
608C2×C4—948no—
609C2×C4—948no—
610C2×C4—942no—
611C2×C4—942no—
612C2×C4—942no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—942no—
616C2×C4—942no—
617C2×C4—942no—
618C2×C4—944no—
619C2×C4—18816no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4C22⋊C4×D4712yes—
623C2×C4#1309398614yes—
624C2×C4C4×D18814yes—
625C2×C4#1309399412yes—
626C2×C4#1309399714yes—
627C2×C4#1309399914yes—
628C2×C4#1309400912yes—
629C2×C4C22×C4×D4711yescharacteristic
630C2×C4C2×D4×D4711yescharacteristic
631C2×C4C2×D4×D4711yescharacteristic
632C2×C4C22×D18811yescharacteristic
633C2×C4C2×#1062535011yescharacteristic
634C2×C4C2×#1062535411yescharacteristic
635C8—42no—
636C8—1882no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—22no—
641C22—474no—
642C22—942no—
643C22—942no—
644C22—944no—
645C22—944no—
646C22—942no—
647C22—942no—
648C22—942no—
649C22—37616no—
650C22—3764no—
651C22—3764no—
652C22—18816no—
653C22—1884no—
654C22—1884no—
655C22—1884no—
656C22—1882no—
657C22—1882no—
658C22#1568900114yes—
659C22C2×C22⋊C4×D4711yescharacteristic
660C22C2×#1309399411yescharacteristic
661C22C2×#1309400911yescharacteristic
662C22AGL(1,ℤ/8)×D4712yes—
663C22#1568992212yes—
664C4—44no—
665C4—44no—
666C4—942no—
667C4—942no—
668C4—1884no—
669C4#1568900114yes—
670C4C2×#1309398611yescharacteristic
671C4C2×C4×D18811yescharacteristic
672C4C2×#1309399711yescharacteristic
673C4C2×#1309399911yescharacteristic
674C2—24no—
675C2—944no—
676C2—3764no—
677C2—1882no—
678C2C2×#1568900111yescharacteristic
679C2#418×D4711yescharacteristic
680C2#2025190611yescharacteristic
681C2#2025264112yes—
682C2#2025333012yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC22×#13093999C2
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20912
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#117×D47C2semidirect
3. C2×#15689129C2semidirect
4. C2×#15689127C2semidirect
5. C2×#15689086C2semidirect
6. #20243709C2semidirect
7. #20243704C2semidirect
8. #20243107C2semidirect
9. C47×#1736C2semidirect
17. C2×#13093999C4semidirect
21. C2×#13093982C22semidirect
26. C2×C4⋊C4×D47C22semidirect
32. C2×#13093946C22semidirect
33. C2×#13093944C22semidirect
35. C2×(C2×D47)⋊C8C22semidirect
36. C2×(C2×D47)⋊C8C22semidirect
37. #15688885C22semidirect
41. C47×#340C22semidirect
54. C22×D188C2×C4semidirect
77. #13093999C2×C4semidirect
79. #13093983D4semidirect
108. #117×C47D4semidirect
111. C2×C4⋊C4×C47C2×C4semidirect
199. C4⋊C4×C47C2×D4semidirect
207. C4⋊Dic47C2×D4semidirect
322. C2×D47#418semidirect
341. #1736D47semidirect
382. #340C2×D47semidirect
383. C47#20912semidirect
433. #117D188semidirect
440. C2×C4⋊C4C4×D47semidirect
548. C4⋊C4C2×D188semidirect

Maximal subgroups

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

subgroupindex
C22×#130939992
C2×#117×D472
C2×#156891292
C2×#156891272
C2×#156890862
#202437092
#202437042
#202431072
C47×#17362
#2091247