Finite Group Databasebeta

← #26147488

Subgroups — #26147488

Order 12032.

6494 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4×D47C211yesmaximal, characteristic
2C2×#15688998C211yesmaximal, characteristic, complemented
3C2×#15688997C214yesmaximal, complemented
4C2×#15688996C211yesmaximal, characteristic, complemented
5C2×#15688978C211yesmaximal, characteristic
6C2×#15688978C211yesmaximal, characteristic
7#20241610C211yesmaximal, characteristic, complemented
8(C2×D47)⋊#95C214yesmaximal, complemented
9C47×C22⋊#95C211yesmaximal, characteristic, complemented
10C25×D47C2211yescharacteristic
11C24×Dic47C2211yescharacteristic
12C23×C4×D47—24no—
13C23×C4×D47—24no—
14C23×C4×D47—216no—
15C23×C4×D47—24no—
16C23×C4×D47C2214yes—
17C22×C4⋊Dic47C2212yescomplemented
18C22×Dic47⋊C4C2212yescomplemented
19C22×C22⋊Dic47C2211yescharacteristic
20C22×C22⋊Dic47C2211yescharacteristic
21C22×(C2×D47)⋊C4C2212yes—
22C22×(C2×D47)⋊C4C2212yes—
23C22×(C2×D47)⋊C4C2214yes—
24#15688998—24no—
25#15688997—216no—
26#15688996—24no—
27#15688978—24no—
28#15688978C418yescomplemented
29C2×C47⋊#95C2212yescomplemented
30C2×C47⋊#95C2214yescomplemented
31C2×C47⋊#95C2212yescomplemented
32C24×C4×C47C2211yescharacteristic
33C22×C22⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47C2211yescharacteristic
35C2×#95×C47C2214yescomplemented
36C24×D47—22no—
37C24×D47—22no—
38C24×D47—22no—
39C24×D47—22no—
40C24×D47—28no—
41C24×D47—28no—
42C24×D47C2×C414yes—
43C24×D47D414yes—
44C24×D47Q814yes—
45C24×D47C2312yes—
46C23×Dic47—22no—
47C23×Dic47—22no—
48C23×Dic47—28no—
49C23×Dic47D414yes—
50C23×Dic47C2311yescharacteristic
51C23×Dic47C2311yescharacteristic
52C23×Dic47C2312yes—
53C23×Dic47C2312yes—
54C22×C4×D47—232no—
55C22×C4×D47—264no—
56C22×C4×D47—264no—
57C22×C4×D47—264no—
58C22×C4×D47—28no—
59C22×C4×D47—28no—
60C22×C4×D47—28no—
61C22×C4×D47—28no—
62C22×C4×D47—28no—
63C22×C4×D47—28no—
64C22×C4×D47D4116yes—
65C2×C4⋊Dic47—28no—
66C2×C4⋊Dic47—24no—
67C2×Dic47⋊C4—24no—
68C2×Dic47⋊C4—28no—
69C2×C22⋊Dic47—22no—
70C2×C22⋊Dic47—22no—
71C2×C22⋊Dic47—22no—
72C2×C22⋊Dic47—22no—
73C2×C22⋊Dic47—22no—
74C2×C22⋊Dic47C2×C414yescomplemented
75C2×(C2×D47)⋊C4—28no—
76C2×(C2×D47)⋊C4—28no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—24no—
79C2×(C2×D47)⋊C4—28no—
80C2×(C2×D47)⋊C4—28no—
81C2×(C2×D47)⋊C4C2×C4116yescomplemented
82C47⋊#95—24no—
83C47⋊#95—28no—
84C47⋊#95—24no—
85C25×C47C2311yescharacteristic
86C23×C4×C47—22no—
87C23×C4×C47—22no—
88C23×C4×C47—28no—
89C23×C4×C47D414yes—
90C23×C4×C47C2312yes—
91C23×C4×C47C2312yes—
92C23×C4×C47C2312yes—
93C2×C22⋊C4×C47—22no—
94C2×C22⋊C4×C47—22no—
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47—22no—
97C2×C22⋊C4×C47—22no—
98C2×C22⋊C4×C47C2×C414yescomplemented
99#95×C47—28no—
100C24×C47—24no—
101C24×C47—24no—
102C24×C47C22×C412yes—
103C24×C47C2×D412yes—
104C24×C47C2×D412yes—
105C24×C47C2×D412yes—
106C24×C47C2×Q812yes—
107C24×C47#4112yes—
108C24×C47#4112yes—
109C24×C47C2411yescharacteristic
110C23×D47—216no—
111C23×D47—216no—
112C23×D47—232no—
113C23×D47—232no—
114C23×D47—216no—
115C23×D47—216no—
116C23×D47—28no—
117C23×D47—28no—
118C23×D47—28no—
119C23×D47—28no—
120C23×D47—28no—
121C23×D47—28no—
122C23×D47—24no—
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—24no—
126C23×D47—22no—
127C23×D47—22no—
128C23×D47—232no—
129C23×D47—216no—
130C23×D47—232no—
131C23×D47—216no—
132C23×D47C4⋊C4116yescomplemented
133C23×D47C22×C414yes—
134C23×D47C2×D414yes—
135C23×D47C2×D418yes—
136C23×D47C2×Q814yes—
137C23×D47#4118yes—
138C22×C4×C47—216no—
139C22×C4×C47—216no—
140C22×C4×C47—28no—
141C22×C4×C47—22no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—216no—
145C22×C4×C47—22no—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—24no—
149C22×C4×C47—22no—
150C22×C4×C47—24no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—24no—
154C22×C4×C47C22×C414yescomplemented
155C22×C4×C47C2×D414yes—
156C22×C4×C47C2×D418yes—
157C22×Dic47—216no—
158C22×Dic47—216no—
159C22×Dic47—28no—
160C22×Dic47—22no—
161C22×Dic47—24no—
162C22×Dic47—22no—
163C22×Dic47—216no—
164C22×Dic47—24no—
165C22×Dic47—22no—
166C22×Dic47—24no—
167C22×Dic47—22no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C22×Dic47—21no—
171C22×Dic47—21no—
172C22×Dic47—24no—
173C22×Dic47—21no—
174C22×Dic47—21no—
175C22×Dic47—24no—
176C22×Dic47C22×C414yescomplemented
177C22×Dic47C2×D414yes—
178C22×Dic47C2×D418yes—
179C2×C4×D47—2256no—
180C2×C4×D47—232no—
181C2×C4×D47—232no—
182C2×C4×D47—232no—
183C2×C4×D47—232no—
184C2×C4×D47—232no—
185C2×C4×D47—232no—
186C2×C4×D47—232no—
187C22⋊C4×C47—28no—
188C22⋊C4×C47—44no—
189C22⋊Dic47—28no—
190C22⋊Dic47—44no—
191(C2×D47)⋊C4—232no—
192(C2×D47)⋊C4—48no—
193(C2×D47)⋊C4—48no—
194C4⋊Dic47—48no—
195Dic47⋊C4—48no—
196C23×C47—28no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—28no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—28no—
205C23×C47—28no—
206C23×C47C2×C4⋊C414yes—
207C23×C47C4×D414yes—
208C23×C47C4×D414yes—
209C23×C47#12014yes—
210C23×C47C22⋊Q814yes—
211C23×C47C22⋊Q814yes—
212C23×C47#12314yes—
213C23×C47C23×C411yescharacteristic
214C23×C47C22×D411yescharacteristic
215C23×C47C22×D414yes—
216C23×C47C22×D411yescharacteristic
217C23×C47C22×D411yescharacteristic
218C23×C47C22×Q811yescharacteristic
219C23×C47C2×#4114yes—
220C23×C47C2×#4111yescharacteristic
221C23×C47C2×#4111yescharacteristic
222C22×D47—232no—
223C22×D47—264no—
224C22×D47—264no—
225C22×D47—264no—
226C22×D47—232no—
227C22×D47—264no—
228C22×D47—264no—
229C22×D47—28no—
230C22×D47—24no—
231C22×D47—24no—
232C22×D47—28no—
233C22×D47—24no—
234C22×D47—28no—
235C22×D47—24no—
236C22×D47—28no—
237C22×D47—24no—
238C22×D47—24no—
239C22×D47—24no—
240C22×D47—24no—
241C22×D47—28no—
242C22×D47—28no—
243C22×D47—28no—
244C22×D47—28no—
245C22×D47—28no—
246C22×D47—28no—
247C22×D47—232no—
248C22×D47—264no—
249C22×D47C2×C4⋊C418yes—
250C22×D47C4×D4116yes—
251C22×D47#12018yes—
252C22×D47C22⋊Q8116yes—
253C22×D47#12318yes—
254C2×C4×C47—232no—
255C2×C4×C47—28no—
256C2×C4×C47—28no—
257C2×C4×C47—24no—
258C2×C4×C47—24no—
259C2×C4×C47—28no—
260C2×C4×C47—24no—
261C2×C4×C47—42no—
262C2×C4×C47—42no—
263C2×C4×C47—44no—
264C2×C4×C47—42no—
265C2×C4×C47—42no—
266C2×C4×C47—42no—
267C2×C4×C47—44no—
268C2×C4×C47—44no—
269C2×C4×C47#12018yes—
270C2×Dic47—232no—
271C2×Dic47—24no—
272C2×Dic47—28no—
273C2×Dic47—28no—
274C2×Dic47—24no—
275C2×Dic47—28no—
276C2×Dic47—24no—
277C2×Dic47—42no—
278C2×Dic47—44no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47—42no—
286C2×Dic47—42no—
287C2×Dic47—42no—
288C2×Dic47#12018yes—
289C4×D47—2128no—
290#14693—471nomaximal, Sylow
291C22×C47—24no—
292C22×C47—28no—
293C22×C47—28no—
294C22×C47—24no—
295C22×C47—28no—
296C22×C47—28no—
297C22×C47—28no—
298C22×C47—28no—
299C22×C47#38518yescomplemented
300C22×C47C22×C4⋊C411yescharacteristic
301C22×C47C2×C4×D414yes—
302C22×C47C2×C4×D411yescharacteristic
303C22×C47C2×C4×D411yescharacteristic
304C22×C47C2×#12012yes—
305C22×C47C2×#12012yes—
306C22×C47C2×#12012yes—
307C22×C47C2×#12011yescharacteristic
308C22×C47C2×C22⋊Q814yes—
309C22×C47C2×C22⋊Q811yescharacteristic
310C22×C47C2×C22⋊Q811yescharacteristic
311C22×C47C2×#12312yes—
312C22×C47C2×#12311yescharacteristic
313C22×C47#52512yes—
314C22×C47#52514yes—
315C22×C47#52514yes—
316C22×C47#52512yes—
317C2×D47—2128no—
318C2×D47—28no—
319C2×D47—216no—
320C2×D47—216no—
321C2×D47—28no—
322C2×D47—28no—
323C2×D47—28no—
324C2×D47—28no—
325C2×D47—28no—
326C2×D47—216no—
327C2×D47—216no—
328C2×D47#385116yescomplemented
329C4×C47—28no—
330C4×C47—44no—
331C4×C47—82no—
332Dic47—28no—
333Dic47—44no—
334Dic47—81no—
335Dic47—81no—
336C25×C4—471no—
337C2×#385—474no—
338C2×#385—471no—
339C2×#385—471no—
340C2×#379—471no—
341C2×#379—471no—
342C22⋊#95—474no—
343C22⋊#95—471no—
344C22⋊#95D4711yescharacteristic, complemented
345C2×C47—28no—
346C2×C47C2×#38512yes—
347C2×C47C2×#38511yescharacteristic
348C2×C47C4×#12012yes—
349C2×C47C4×#12012yes—
350C2×C47#284611yescharacteristic
351C2×C47#284812yes—
352C2×C47#284812yes—
353C2×C47#284912yes—
354C2×C47#284911yescharacteristic
355D47—216no—
356C26—471no—
357C24×C4—471no—
358C24×C4—474no—
359C24×C4—944no—
360C24×C4—9416no—
361C24×C4—944no—
362C24×C4—944no—
363C24×C4C2×D4711yescharacteristic
364C22×C4⋊C4—472no—
365C22×C4⋊C4—472no—
366C22×C22⋊C4—471no—
367C22×C22⋊C4—474no—
368C22×C22⋊C4—471no—
369C22×C22⋊C4—472no—
370C22×C22⋊C4—472no—
371C22×C22⋊C4C2×D4711yescharacteristic
372C22×C22⋊C4C2×D4711yescharacteristic
373#385—9416no—
374#385—944no—
375#385—944no—
376#379—478no—
377#379—944no—
378C2×#95—474no—
379C2×#95—472no—
380C2×#95—472no—
381C2×#95C2×D4714yescomplemented
382C47#1469311yescharacteristic, complemented, Sylow
383C25—472no—
384C25—474no—
385C25—474no—
386C25—474no—
387C25—948no—
388C25—948no—
389C25—942no—
390C25—942no—
391C25—942no—
392C25—942no—
393C25C22×D4711yescharacteristic
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—28no—
397C23×C4—472no—
398C23×C4—474no—
399C23×C4—472no—
400C23×C4—4716no—
401C23×C4—471no—
402C23×C4—471no—
403C23×C4—948no—
404C23×C4—948no—
405C23×C4—948no—
406C23×C4—948no—
407C23×C4—9464no—
408C23×C4—9464no—
409C23×C4—9432no—
410C23×C4—9464no—
411C23×C4—948no—
412C23×C4—942no—
413C23×C4—942no—
414C23×C4—948no—
415C23×C4—948no—
416C23×C4D18814yes—
417C23×C4C22×D4712yes—
418C23×C4C22×D4712yes—
419C23×C4C22×D4712yes—
420C2×C4⋊C4—948no—
421C2×C4⋊C4—944no—
422C2×C4⋊C4—944no—
423C2×C4⋊C4—948no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—474no—
430C2×C22⋊C4—4716no—
431C2×C22⋊C4—948no—
432C2×C22⋊C4—948no—
433C2×C22⋊C4—942no—
434C2×C22⋊C4—942no—
435C2×C22⋊C4—948no—
436C2×C22⋊C4—944no—
437C2×C22⋊C4—944no—
438C2×C22⋊C4—948no—
439C2×C22⋊C4—942no—
440C2×C22⋊C4—942no—
441C2×C22⋊C4—942no—
442C2×C22⋊C4C4×D4714yescomplemented
443#95—28no—
444#95—944no—
445#95—948no—
446#95—944no—
447C24—24no—
448C24—24no—
449C24—474no—
450C24—478no—
451C24—474no—
452C24—474no—
453C24—478no—
454C24—4716no—
455C24—9416no—
456C24—9432no—
457C24—9432no—
458C24—9416no—
459C24—9416no—
460C24—9416no—
461C24—9416no—
462C24—9432no—
463C24—9432no—
464C24—9416no—
465C24—948no—
466C24—948no—
467C24—948no—
468C24—948no—
469C24—948no—
470C24—948no—
471C24—944no—
472C24—942no—
473C24—942no—
474C24—944no—
475C24—942no—
476C24—942no—
477C24C2×C4×D4712yes—
478C24C2×D18812yes—
479C24#1062535012yes—
480C24D4×D4712yes—
481C24D4×D4712yes—
482C24#1062535212yes—
483C24Q8×D4712yes—
484C24C23×D4711yescharacteristic
485C22×C4—216no—
486C22×C4—216no—
487C22×C4—28no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—216no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—478no—
502C22×C4—474no—
503C22×C4—474no—
504C22×C4—9432no—
505C22×C4—9432no—
506C22×C4—94256no—
507C22×C4—9432no—
508C22×C4—9432no—
509C22×C4—9432no—
510C22×C4—9432no—
511C22×C4—9416no—
512C22×C4—9416no—
513C22×C4—9416no—
514C22×C4—948no—
515C22×C4—942no—
516C22×C4—944no—
517C22×C4—942no—
518C22×C4—9432no—
519C22×C4—944no—
520C22×C4—942no—
521C22×C4—944no—
522C22×C4—942no—
523C22×C4—941no—
524C22×C4—941no—
525C22×C4—941no—
526C22×C4—944no—
527C22×C4—941no—
528C22×C4—941no—
529C22×C4—941no—
530C22×C4—944no—
531C22×C4C2×C4×D4714yescomplemented
532C22×C4C2×D18814yes—
533C22×C4D4×D4718yes—
534C4⋊C4—1888no—
535C4⋊C4—1888no—
536C22⋊C4—28no—
537C22⋊C4—44no—
538C22⋊C4—9432no—
539C22⋊C4—948no—
540C22⋊C4—1888no—
541C22⋊C4—1888no—
542C22⋊C4—1884no—
543C23—28no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—24no—
553C23—478no—
554C23—4716no—
555C23—478no—
556C23—4716no—
557C23—478no—
558C23—9464no—
559C23—9432no—
560C23—9464no—
561C23—9464no—
562C23—9464no—
563C23—9432no—
564C23—9464no—
565C23—9464no—
566C23—9432no—
567C23—948no—
568C23—944no—
569C23—944no—
570C23—948no—
571C23—944no—
572C23—948no—
573C23—944no—
574C23—948no—
575C23—944no—
576C23—944no—
577C23—944no—
578C23—944no—
579C23—948no—
580C23—948no—
581C23—948no—
582C23—948no—
583C23—948no—
584C23—948no—
585C23C4⋊C4×D4714yes—
586C23(C22⋊D47)⋊C414yes—
587C23C4×D18814yes—
588C23#1309399414yes—
589C23#1309400214yes—
590C23#1309400414yes—
591C23#1309400614yes—
592C23C22×C4×D4711yescharacteristic
593C23C2×D4×D4711yescharacteristic
594C23C2×D4×D4714yes—
595C23C2×D4×D4711yescharacteristic
596C23C22×D18811yescharacteristic
597C23C2×Q8×D4711yescharacteristic
598C23#41×D4714yes—
599C23C2×#1062535211yescharacteristic
600C23C2×#1062535011yescharacteristic
601C2×C4—232no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—478no—
617C2×C4—94128no—
618C2×C4—9432no—
619C2×C4—944no—
620C2×C4—948no—
621C2×C4—948no—
622C2×C4—944no—
623C2×C4—948no—
624C2×C4—944no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4—1881no—
628C2×C4—1881no—
629C2×C4—1882no—
630C2×C4—1882no—
631C2×C4—1882no—
632C2×C4—1882no—
633C2×C4—1882no—
634C2×C4—1884no—
635C2×C4—1882no—
636C2×C4#1309399418yes—
637C22—24no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—4716no—
646C22—94128no—
647C22—948no—
648C22—9416no—
649C22—9416no—
650C22—948no—
651C22—948no—
652C22—948no—
653C22—948no—
654C22—948no—
655C22—9416no—
656C22—9416no—
657C22#1568899718yescomplemented
658C22C2×C4⋊C4×D4711yescharacteristic
659C22C4×D4×D4714yes—
660C22C2×(C22⋊D47)⋊C411yescharacteristic
661C22C2×C4×D18811yescharacteristic
662C22#120×D4712yes—
663C22#120×D4712yes—
664C22C2×#1309399412yes—
665C22C2×#1309399411yescharacteristic
666C22C22⋊Q8×D4714yes—
667C22C2×#1309400211yescharacteristic
668C22C2×#1309400411yescharacteristic
669C22#123×D4712yes—
670C22C2×#1309400611yescharacteristic
671C22#1568963112yes—
672C22#1568963214yes—
673C22#1568963314yes—
674C22#1568963512yes—
675C4—28no—
676C4—44no—
677C4—82no—
678C4—948no—
679C4—3761no—
680C4—3761no—
681C4—1884no—
682C2—28no—
683C2—9416no—
684C2#385×D4712yes—
685C2C2×#1568899711yescharacteristic
686C2C4×#1309399412yes—
687C2#2024589312yes—
688C2#2025166611yescharacteristic
689C2#2025167412yes—
690C2#2025167612yes—
691C2#2025167912yes—
692C2#2025168111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C22⋊#95C2
socle, cosocleC24×C47C24
prime radicalC25×D47C22
hypercenterC22⋊#95D47
hypocenterC47#14693
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#15688998C2semidirect
3. C2×#15688997C2semidirect
4. C2×#15688996C2semidirect
7. #20241610C2semidirect
8. (C2×D47)⋊#95C2semidirect
9. C47×C22⋊#95C2semidirect
17. C22×C4⋊Dic47C22semidirect
18. C22×Dic47⋊C4C22semidirect
28. #15688978C4semidirect
29. C2×C47⋊#95C22semidirect
30. C2×C47⋊#95C22semidirect
31. C2×C47⋊#95C22semidirect
35. C2×#95×C47C22semidirect
74. C2×C22⋊Dic47C2×C4semidirect
81. C2×(C2×D47)⋊C4C2×C4semidirect
98. C2×C22⋊C4×C47C2×C4semidirect
132. C23×D47C4⋊C4semidirect
154. C22×C4×C47C22×C4semidirect
176. C22×Dic47C22×C4semidirect
299. C22×C47#385semidirect
328. C2×D47#385semidirect
344. C22⋊#95D47semidirect
381. C2×#95C2×D47semidirect
382. C47#14693semidirect
442. C2×C22⋊C4C4×D47semidirect
531. C22×C4C2×C4×D47semidirect
657. C22#15688997semidirect

Maximal subgroups

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

subgroupindex
C24×C4×D472
C2×#156889982
C2×#156889972
C2×#156889962
C2×#156889782
#202416102
(C2×D47)⋊#952
C47×C22⋊#952
#1469347