Finite Group Databasebeta

← #26138172

Subgroups — #26138172

Order 12032.

2590 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
1C22×C4×C22⋊D47C211yesmaximal, characteristic
2C2×#15688983C211yesmaximal, characteristic, complemented
3C2×#15688980C212yesmaximal, complemented
4#20241635C212yesmaximal, complemented
5#20241627C212yesmaximal
6#20241626C212yesmaximal, complemented
7#20241621C212yesmaximal, complemented
8#20241617C212yesmaximal, complemented
9C22×#117×C47C211yesmaximal, characteristic, complemented
10C23×C22⋊D47C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C2×C4×C22⋊D47—24no—
13C2×C4×C22⋊D47—24no—
14C2×C4×C22⋊D47C418yescomplemented
15C22×C4⋊Dic47C2212yes—
16C22×Dic47⋊C4C2211yescharacteristic
17C22×C22⋊Dic47C2211yescharacteristic
18C22×C22⋊Dic47C2212yes—
19C22×(C2×D47)⋊C4C2214yes—
20C22×(C2×D47)⋊C4C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22#15688983—24no—
23#15688980—28no—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2214yes—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2214yes—
28C24×C4×C47C2211yescharacteristic
29C2×#117×C47—24no—
30C2×#117×C47—28no—
31C22×C4⋊C4×C47C2212yes—
32C22×C22⋊C4×C47C2212yes—
33C22×C42×C47C2212yes—
34C24×D47C2311yescharacteristic
35C22×C22⋊D47—22no—
36C22×C22⋊D47—22no—
37C22×C22⋊D47—24no—
38C22×C22⋊D47—24no—
39C22×C22⋊D47C2×C414yes—
40C23×Dic47C2314yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C22×C4×D47—24no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—24no—
48C22×C4×D47C2×C414yes—
49C4×C22⋊D47—232no—
50C2×C4⋊Dic47—24no—
51C2×C4⋊Dic47—24no—
52C2×C4⋊Dic47—24no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4C2×C414yes—
56C2×C22⋊Dic47—22no—
57C2×C22⋊Dic47—22no—
58C2×C22⋊Dic47—24no—
59C2×C22⋊Dic47—44no—
60C2×C22⋊Dic47C2×C414yes—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—22no—
63C2×(C2×D47)⋊C4—28no—
64C2×(C2×D47)⋊C4—28no—
65C2×(C2×D47)⋊C4—44no—
66C2×(C2×D47)⋊C4C2×C414yes—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47C2×C414yes—
70C47⋊#95—28no—
71C47⋊#95—24no—
72C47⋊#95—28no—
73C47⋊#95—44no—
74C25×C47C2311yescharacteristic
75C23×C4×C47—24no—
76C23×C4×C47—24no—
77C23×C4×C47C2×C414yes—
78C23×C4×C47D414yes—
79C23×C4×C47D414yes—
80C23×C4×C47C2314yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83#117×C47—232no—
84C2×C4⋊C4×C47—24no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47D418yescomplemented
87C2×C22⋊C4×C47—24no—
88C2×C22⋊C4×C47—28no—
89C2×C42×C47—28no—
90C2×C42×C47—24no—
91C24×C47—22no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47C22×C412yes—
98C24×C47C2×D412yes—
99C24×C47C2×D412yes—
100C24×C47C2411yescharacteristic
101C23×D47—22no—
102C23×D47—21no—
103C23×D47—21no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—22no—
107C23×D47—22no—
108C23×D47C22×C412yes—
109C23×D47#4114yes—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—216no—
113C22×C4×C47—28no—
114C22×C4×C47—216no—
115C22×C4×C47—28no—
116C22×C4×C47—24no—
117C22×C4×C47—24no—
118C22×C4×C47—24no—
119C22×C4×C47—24no—
120C22×C4×C47—24no—
121C22×C4×C47—24no—
122C22×C4×C47C22⋊C4116yescomplemented
123C22×C4×C47C22×C412yes—
124C22×C4×C47C22×C412yes—
125C22×C4×C47C2×D418yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D414yes—
131C22×C4×C47#4114yes—
132C2×C22⋊D47—216no—
133C2×C22⋊D47—216no—
134C2×C22⋊D47—28no—
135C2×C22⋊D47—28no—
136C2×C22⋊D47—28no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—24no—
150C22×Dic47—24no—
151C22×Dic47—24no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47C22×C412yes—
156C22×Dic47C22×C412yes—
157C22×Dic47C22×C412yes—
158C22×Dic47#4114yes—
159C2×C4×D47—28no—
160C2×C4×D47—216no—
161C2×C4×D47—44no—
162C2×C4×D47—48no—
163C2×C4×D47—44no—
164C4⋊C4×C47—216no—
165C22⋊C4×C47—216no—
166C42×C47—216no—
167C22⋊Dic47—28no—
168C22⋊Dic47—48no—
169(C2×D47)⋊C4—28no—
170(C2×D47)⋊C4—416no—
171C4⋊Dic47—48no—
172Dic47⋊C4—28no—
173C4×Dic47—28no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—24no—
177C23×C47—24no—
178C23×C47—22no—
179C23×C47—24no—
180C23×C47—22no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—24no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47C2×C22⋊C414yes—
195C23×C47C23×C411yescharacteristic
196C23×C47C22×D411yescharacteristic
197C23×C47C22×D412yes—
198C23×C47C22×D411yescharacteristic
199C23×C47C2×#4112yes—
200C23×C47C2×#4112yes—
201C23×C47C2×#4112yes—
202C23×C47H(5,𝔽2)12yes—
203C23×C472−1+412yes—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—41no—
215C22×D47—42no—
216C22×D47—41no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—42no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—42no—
223C22×D47—42no—
224C22×D47#11714yescomplemented
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—232no—
230C2×C4×C47—28no—
231C2×C4×C47—28no—
232C2×C4×C47—28no—
233C2×C4×C47—44no—
234C2×C4×C47—44no—
235C2×C4×C47—44no—
236C2×C4×C47—44no—
237C2×C4×C47C2×C22⋊C414yes—
238C2×C4×C47C2×C22⋊C414yes—
239C2×C4×C47C4×D418yes—
240C2×C4×C47#12014yes—
241C2×C4×C47#12118yes—
242C2×C4×C47#12314yes—
243C22⋊D47—232no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—41no—
253C2×Dic47—42no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—42no—
258C2×Dic47—44no—
259C2×Dic47—44no—
260C2×Dic47—44no—
261C2×Dic47—44no—
262C2×Dic47—44no—
263C2×Dic47—44no—
264C2×Dic47—44no—
265C2×Dic47#11714yescomplemented
266C4×D47—416no—
267#13871—471nomaximal, Sylow
268C22×C47—24no—
269C22×C47—24no—
270C22×C47—24no—
271C22×C47—44no—
272C22×C47—44no—
273C22×C47—44no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47C22×C22⋊C411yescharacteristic
278C22×C47C2×#11711yescharacteristic
279C22×C47C2×#11711yescharacteristic
280C22×C47C2×C4×D412yes—
281C22×C47#52012yes—
282C22×C47C2×#12011yescharacteristic
283C22×C47C2×#12112yes—
284C22×C47C2×#12311yescharacteristic
285C22×C47#52512yes—
286C22×C47#52512yes—
287C22×C47#53511yescharacteristic
288C22×C47#53611yescharacteristic
289C22×C47#53712yes—
290C22×C47#54612yes—
291C22×C47#54712yes—
292C22×C47#54812yes—
293C22×C47#55012yes—
294C22×C47#55212yes—
295C22×C47#55312yes—
296C22×C47#55512yes—
297C22×C47#55612yes—
298C2×D47—28no—
299C2×D47—24no—
300C2×D47—42no—
301C2×D47—44no—
302C2×D47—44no—
303C2×D47—44no—
304C2×D47—44no—
305C2×D47—44no—
306C2×D47—42no—
307C4×C47—24no—
308C4×C47—48no—
309C4×C47#38618yes—
310Dic47—24no—
311Dic47—42no—
312Dic47—42no—
313Dic47—84no—
314C22×C4×D4—471no—
315C22×#117D4711yescharacteristic, complemented
316C2×#380—471no—
317C2×#380—472no—
318#1440—472no—
319#1438—472no—
320#1437—472no—
321#1436—472no—
322C4⋊#95—472no—
323C2×C47—44no—
324C2×C47#228011yescharacteristic
325C2×C47C2×#38611yescharacteristic
326C2×C47#231111yescharacteristic
327C2×C47#235212yes—
328C2×C47#235412yes—
329C2×C47#240712yes—
330C2×C47#241412yes—
331C2×C47#241512yes—
332C2×C47#248512yes—
333D47—44no—
334C23×D4—471no—
335C24×C4—471no—
336C24×C4C2×D4711yescharacteristic
337C2×C4×D4—478no—
338C2×C4×D4—944no—
339C2×C4×D4—944no—
340C2×#117—24no—
341C2×#117—28no—
342C22×C4⋊C4—471no—
343C22×C4⋊C4—472no—
344C22×C4⋊C4C2×D4712yes—
345C22×C22⋊C4—471no—
346C22×C22⋊C4—471no—
347C22×C22⋊C4—474no—
348C22×C22⋊C4—472no—
349C22×C22⋊C4C2×D4712yes—
350C22×C42—471no—
351C22×C42C2×D4712yes—
352#380—944no—
353#380—948no—
354C2×#95—474no—
355C2×#95—472no—
356C2×#95—474no—
357C2×#95—474no—
358C47#1387111yescharacteristic, complemented, Sylow
359C25—471no—
360C25C22×D4711yescharacteristic
361C22×D4—474no—
362C22×D4—944no—
363C22×D4—944no—
364C22×D4—942no—
365C22×D4—942no—
366C23×C4—24no—
367C23×C4—24no—
368C23×C4—471no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—474no—
372C23×C4—474no—
373C23×C4—942no—
374C23×C4—944no—
375C23×C4—942no—
376C23×C4—944no—
377C23×C4C4×D4714yes—
378C23×C4D18814yes—
379C23×C4C22⋊D4714yes—
380C23×C4C22×D4714yes—
381C23×C4C22×D4711yescharacteristic
382C23×C4C22×D4711yescharacteristic
383C4×D4—9432no—
384#117—232no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—474no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4C22⋊D4718yescomplemented
394C2×C22⋊C4—24no—
395C2×C22⋊C4—28no—
396C2×C22⋊C4—474no—
397C2×C22⋊C4—474no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C22⋊C4—942no—
401C2×C22⋊C4—942no—
402C2×C22⋊C4—948no—
403C2×C22⋊C4—944no—
404C2×C22⋊C4—948no—
405C2×C22⋊C4—1884no—
406C2×C22⋊C4—1884no—
407C2×C42—28no—
408C2×C42—24no—
409C2×C42—474no—
410C2×C42—942no—
411C2×C42—942no—
412#95—948no—
413#95—944no—
414#95—948no—
415#95—1884no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—472no—
423C24—474no—
424C24—942no—
425C24—941no—
426C24—941no—
427C24—942no—
428C24—942no—
429C24—942no—
430C24—942no—
431C24C2×C4×D4712yes—
432C24C2×D18812yes—
433C24C2×C22⋊D4712yes—
434C24C23×D4711yescharacteristic
435C2×D4—948no—
436C2×D4—948no—
437C2×D4—9416no—
438C2×D4—9416no—
439C2×D4—948no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—216no—
443C22×C4—28no—
444C22×C4—216no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—472no—
453C22×C4—474no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—948no—
468C22×C4—9416no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—944no—
473C22×C4—944no—
474C22×C4—944no—
475C22×C4—944no—
476C22×C4—1884no—
477C22×C4—1888no—
478C22×C4—1884no—
479C22×C4(C2×D47)⋊C4116yescomplemented
480C22×C4C2×C4×D4712yes—
481C22×C4C2×C4×D4712yes—
482C22×C4C2×D18812yes—
483C22×C4C2×D18812yes—
484C22×C4#1062535014yes—
485C22×C4C2×C22⋊D4718yes—
486C22×C4C2×C22⋊D4712yes—
487C22×C4C2×C22⋊D4712yes—
488C22×C4C2×C22⋊D4714yes—
489C4⋊C4—216no—
490C4⋊C4—948no—
491C4⋊C4—1888no—
492C22⋊C4—216no—
493C22⋊C4—948no—
494C22⋊C4—948no—
495C22⋊C4—18816no—
496C22⋊C4—1888no—
497C42—216no—
498C42—948no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—474no—
520C23—944no—
521C23—944no—
522C23—944no—
523C23—944no—
524C23—942no—
525C23—942no—
526C23—944no—
527C23—944no—
528C23—944no—
529C23—944no—
530C23—1882no—
531C23—1882no—
532C23—1882no—
533C23—1882no—
534C23—1881no—
535C23—1882no—
536C23—1881no—
537C23—1881no—
538C23—1881no—
539C23—1882no—
540C23C2×(C2×D47)⋊C414yes—
541C23C22×C4×D4711yescharacteristic
542C23C22×C22⋊D4712yes—
543C23C22×C22⋊D4711yescharacteristic
544C23C22×D18811yescharacteristic
545C23#41×D4712yes—
546C23#41×D4712yes—
547C23C2×#1062535012yes—
548C23#1309407012yes—
549C23#1309407112yes—
550D4—9432no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—232no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—474no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—944no—
571C2×C4—944no—
572C2×C4—1884no—
573C2×C4—1884no—
574C2×C4—1884no—
575C2×C4—1884no—
576C2×C4—1884no—
577C2×C4—1884no—
578C2×C4—1884no—
579C2×C4—1881no—
580C2×C4—1882no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1882no—
585C2×C4—18816no—
586C2×C4C2×(C2×D47)⋊C414yes—
587C2×C4C2×(C2×D47)⋊C414yes—
588C2×C4C4×C22⋊D4718yes—
589C2×C4#1309399314yes—
590C2×C4#1309399818yes—
591C2×C4#1309400714yes—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—944no—
602C22—948no—
603C22—1884no—
604C22—1884no—
605C22—1882no—
606C22—1884no—
607C22—1884no—
608C22—1884no—
609C22—1882no—
610C22C22×(C2×D47)⋊C411yescharacteristic
611C22#117×D4711yescharacteristic
612C22#117×D4711yescharacteristic
613C22C2×C4×C22⋊D4712yes—
614C22#1568960312yes—
615C22C2×#1309399311yescharacteristic
616C22C2×#1309399812yes—
617C22C2×#1309400711yescharacteristic
618C22#1568963012yes—
619C22#1568963512yes—
620C22#1568969211yescharacteristic
621C22#1568969911yescharacteristic
622C22#1568970112yes—
623C22#1568976112yes—
624C22#1568977212yes—
625C22#1568978012yes—
626C22#1568979212yes—
627C22#1568981212yes—
628C22#1568981712yes—
629C22#1568983612yes—
630C22#1568984112yes—
631C4—24no—
632C4—48no—
633C4—944no—
634C4—3764no—
635C4—1882no—
636C4—1882no—
637C4#1568900218yes—
638C2—44no—
639C2—1884no—
640C2#2024576711yescharacteristic
641C2C2×#1568900211yescharacteristic
642C2#2024597611yescharacteristic
643C2#2024631812yes—
644C2#2024633812yes—
645C2#2024672712yes—
646C2#2024680012yes—
647C2#2024681112yes—
648C2#2024767912yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×#117×C47C2
socle, cosocleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC22×#117D47
hypocenterC47#13871
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#15688983C2semidirect
3. C2×#15688980C2semidirect
4. #20241635C2semidirect
6. #20241626C2semidirect
7. #20241621C2semidirect
8. #20241617C2semidirect
9. C22×#117×C47C2semidirect
14. C2×C4×C22⋊D47C4semidirect
86. C2×C4⋊C4×C47D4semidirect
122. C22×C4×C47C22⋊C4semidirect
224. C22×D47#117semidirect
265. C2×Dic47#117semidirect
315. C22×#117D47semidirect
358. C47#13871semidirect
393. C2×C4⋊C4C22⋊D47semidirect
479. C22×C4(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D472
C2×#156889832
C2×#156889802
#202416352
#202416272
#202416262
#202416212
#202416172
C22×#117×C472
#1387147