Finite Group Databasebeta

← #26137415

Subgroups — #26137415

Order 12032.

2294 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093980C211yesmaximal, characteristic
2C2×C22⋊C4×Dic47C212yesmaximal
3C2×#15688968C211yesmaximal, characteristic, complemented
4C2×C42×Dic47C211yesmaximal, characteristic, complemented
5#20241625C211yesmaximal, characteristic
6#20241625C212yesmaximal
7#20241623C211yesmaximal, characteristic, complemented
8#20241621C211yesmaximal, characteristic, complemented
9#20241619C212yesmaximal, complemented
10#95×Dic47C212yesmaximal, complemented
11C47×#1437C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C2×#13093980—24no—
14C2×#13093980—24no—
15C2×#13093980C418yescomplemented
16C22×Dic47⋊C4C2211yescharacteristic
17C22×Dic47⋊C4C2211yescharacteristic
18C22×C22⋊Dic47C2211yescharacteristic
19C22×C22⋊Dic47C2212yes—
20C22×C4×Dic47C2212yes—
21C22×C4×Dic47C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23C22×C4×Dic47C2212yes—
24C22×C4×Dic47C2212yes—
25C22×C4×Dic47C2211yescharacteristic
26C22⋊C4×Dic47C4116yescomplemented
27#15688968—24no—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2211yescharacteristic
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C42×Dic47—24no—
36C22×C22⋊C4×C47C2211yescharacteristic
37C22×C22⋊C4×C47C2212yes—
38C22×C42×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C2×#95×C47C2211yescharacteristic
41C23×Dic47—22no—
42C23×Dic47—22no—
43C23×Dic47—24no—
44C23×Dic47C2×C418yes—
45C23×Dic47C2×C414yes—
46C23×Dic47C2312yes—
47C23×Dic47C2311yescharacteristic
48C23×Dic47C2311yescharacteristic
49C23×Dic47C2311yescharacteristic
50C23×Dic47C2312yes—
51C23×Dic47C2311yescharacteristic
52#13093980—232no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4C2×C414yes—
58C2×Dic47⋊C4C2×C414yes—
59C2×C22⋊Dic47—22no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—24no—
62C2×C22⋊Dic47—24no—
63C2×C22⋊Dic47C2×C418yes—
64C2×C22⋊Dic47C2×C414yes—
65C2×C4×Dic47—22no—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—24no—
69C2×C4×Dic47—24no—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—24no—
73C2×C4×Dic47—24no—
74C2×C4×Dic47—24no—
75C2×C4×Dic47—24no—
76C2×C4×Dic47C2×C418yes—
77C2×C4×Dic47C2×C418yes—
78C2×C4×Dic47C2×C414yes—
79C2×C4×Dic47C2×C414yes—
80C2×C4×Dic47D414yes—
81C2×C4×Dic47D414yes—
82C47⋊#95—22no—
83C47⋊#95—24no—
84C47⋊#95—24no—
85C47⋊#95—24no—
86C47⋊#95—24no—
87C47⋊#95C2×C418yes—
88C47⋊#95C2×C418yes—
89C25×C47C2311yescharacteristic
90C23×C4×C47C2312yes—
91C23×C4×C47C2311yescharacteristic
92C23×C4×C47C2311yescharacteristic
93C23×C4×C47C2312yes—
94C2×C22⋊C4×C47—22no—
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47—24no—
97C2×C22⋊C4×C47—24no—
98C2×C22⋊C4×C47C2×C418yes—
99C2×C22⋊C4×C47C2×C414yes—
100C2×C42×C47—22no—
101C2×C42×C47—24no—
102#95×C47—22no—
103#95×C47—24no—
104C24×C47—21no—
105C24×C47—21no—
106C24×C47—22no—
107C24×C47—22no—
108C24×C47—22no—
109C24×C47—22no—
110C24×C47—22no—
111C24×C47C22×C412yes—
112C24×C47C22×C414yes—
113C24×C47C2411yescharacteristic
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47C22×C414yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47#4114yes—
137C22×C4×C47#4114yes—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—28no—
148C22×Dic47—28no—
149C22×Dic47—216no—
150C22×Dic47—216no—
151C22×Dic47—22no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47C42116yescomplemented
162C22×Dic47C22×C414yes—
163C22×Dic47C22×C412yes—
164C22×Dic47C22×C414yes—
165C22×Dic47C22×C412yes—
166C22×Dic47C22×C412yes—
167C22×Dic47C22×C414yes—
168C22×Dic47C22×C412yes—
169C22×Dic47C22×C414yes—
170C22×Dic47C2×D412yes—
171C22×Dic47C2×D412yes—
172C22×Dic47C2×D412yes—
173C22×Dic47C2×D412yes—
174C22×Dic47#4114yes—
175C22×Dic47#4114yes—
176C22×Dic47#4114yes—
177C22⋊C4×C47—28no—
178C22⋊C4×C47—216no—
179C42×C47—28no—
180C22⋊Dic47—28no—
181C22⋊Dic47—216no—
182Dic47⋊C4—28no—
183Dic47⋊C4—28no—
184C4×Dic47—216no—
185C4×Dic47—28no—
186C4×Dic47—28no—
187C4×Dic47—216no—
188C4×Dic47—216no—
189C4×Dic47C22⋊C4116yescomplemented
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47—22no—
194C23×C47—24no—
195C23×C47—24no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47C2×C4214yes—
211C23×C47C23×C411yescharacteristic
212C23×C47C23×C412yes—
213C23×C47C22×D411yescharacteristic
214C23×C47C22×D411yescharacteristic
215C23×C47C2×#4112yes—
216C23×C47C2×#4112yes—
217C23×C47C2×#4112yes—
218C23×C47C2×#4112yes—
219C23×C47C2×#4112yes—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47C2×C22⋊C414yes—
243C2×C4×C47#11718yescomplemented
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—232no—
259C2×Dic47—22no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47C2×C4214yes—
272C2×Dic47C2×C4214yes—
273C2×Dic47C2×C22⋊C414yes—
274C2×Dic47C2×C22⋊C414yes—
275C2×Dic47#11714yes—
276C2×Dic47#11718yescomplemented
277C2×Dic47C4×D418yes—
278C2×Dic47C4×D418yes—
279#13761—471nomaximal, Sylow
280C22×C47—24no—
281C22×C47—28no—
282C22×C47—44no—
283C22×C47—44no—
284C22×C47—44no—
285C22×C47—44no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—44no—
289C22×C47C22×C4211yescharacteristic
290C22×C47C22×C22⋊C411yescharacteristic
291C22×C47C2×#11711yescharacteristic
292C22×C47C2×#11712yes—
293C22×C47C2×#11712yes—
294C22×C47C2×C4×D412yes—
295C22×C47C2×C4×D412yes—
296C22×C47C4×#4112yes—
297C22×C47C4×#4112yes—
298C22×C47C4×#4112yes—
299C22×C47C4×#4112yes—
300C22×C47#52511yescharacteristic
301C22×C47#52511yescharacteristic
302C22×C47#52512yes—
303C22×C47#52912yes—
304C22×C47#52912yes—
305C22×C47#52912yes—
306C22×C47#52912yes—
307C22×C47#53212yes—
308C22×C47#53211yescharacteristic
309C22×C47#53211yescharacteristic
310C4×C47—28no—
311C4×C47—24no—
312C4×C47—44no—
313C4×C47—42no—
314Dic47—24no—
315Dic47—28no—
316Dic47—24no—
317Dic47—44no—
318Dic47—42no—
319Dic47C4×C22⋊C418yes—
320C22×#117—471no—
321C2×C4×C22⋊C4—472no—
322C2×#376—471no—
323C2×C43—471no—
324#1437—472no—
325#1437—471no—
326#1437D4711yescharacteristic
327#1436—471no—
328#1436—472no—
329#1436—471no—
330C4×#95—472no—
331C2×C47—44no—
332C2×C47C2×C4×C22⋊C411yescharacteristic
333C2×C47C4×#11712yes—
334C2×C47#229012yes—
335C2×C47#229012yes—
336C2×C47#229312yes—
337C2×C47#229312yes—
338C2×C47#229711yescharacteristic
339C2×C47#229711yescharacteristic
340C2×C47#229812yes—
341C24×C4—471no—
342C2×#117—478no—
343C2×#117—944no—
344C2×#117—944no—
345C22×C4⋊C4—471no—
346C22×C4⋊C4—471no—
347C22×C22⋊C4—471no—
348C22×C22⋊C4—472no—
349C22×C22⋊C4C2×D4711yescharacteristic
350C22×C22⋊C4C2×D4712yes—
351C22×C42—471no—
352C22×C42—471no—
353C22×C42—472no—
354C22×C42—472no—
355C22×C42—472no—
356C22×C42—471no—
357C22×C42C2×D4711yescharacteristic
358C4×C22⋊C4—4716no—
359#376—944no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—472no—
363C2×#95—472no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95—471no—
367C2×#95C2×D4712yes—
368C2×#95C2×D4711yescharacteristic
369C43—944no—
370C47#1376111yescharacteristic, complemented, Sylow
371C25C22×D4711yescharacteristic
372C23×C4—471no—
373C23×C4—474no—
374C23×C4—478no—
375C23×C4—471no—
376C23×C4—472no—
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—472no—
380C23×C4—942no—
381C23×C4—942no—
382C23×C4—944no—
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4712yes—
387#117—9432no—
388C2×C4⋊C4—474no—
389C2×C4⋊C4—474no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—942no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—474no—
399C2×C22⋊C4—478no—
400C2×C22⋊C4—942no—
401C2×C22⋊C4—942no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—944no—
404C2×C22⋊C4C4×D4714yes—
405C2×C22⋊C4C4×D4718yes—
406C2×C42—22no—
407C2×C42—24no—
408C2×C42—474no—
409C2×C42—474no—
410C2×C42—474no—
411C2×C42—474no—
412C2×C42—478no—
413C2×C42—478no—
414C2×C42—942no—
415C2×C42—942no—
416C2×C42—944no—
417C2×C42—944no—
418C2×C42—944no—
419C2×C42—944no—
420C2×C42—944no—
421C2×C42—944no—
422C2×C42—944no—
423C2×C42—942no—
424C2×C42—944no—
425#95—22no—
426#95—24no—
427#95—478no—
428#95—478no—
429#95—944no—
430#95—944no—
431#95—944no—
432#95—942no—
433#95—944no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24C2×C4×D4712yes—
442C24C2×C4×D4714yes—
443C24C23×D4711yescharacteristic
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—472no—
461C22×C4—474no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—474no—
465C22×C4—4716no—
466C22×C4—474no—
467C22×C4—474no—
468C22×C4—472no—
469C22×C4—472no—
470C22×C4—472no—
471C22×C4—472no—
472C22×C4—474no—
473C22×C4—474no—
474C22×C4—474no—
475C22×C4—472no—
476C22×C4—948no—
477C22×C4—948no—
478C22×C4—9416no—
479C22×C4—9416no—
480C22×C4—942no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4C2×C4×D4714yes—
500C22×C4C2×C4×D4712yes—
501C22×C4C2×C4×D4712yes—
502C22×C4C2×C4×D4714yes—
503C22×C4D4×D4712yes—
504C22×C4D4×D4712yes—
505C22×C4#1062535214yes—
506C22×C4#1062535214yes—
507C4⋊C4—948no—
508C4⋊C4—948no—
509C22⋊C4—28no—
510C22⋊C4—216no—
511C22⋊C4—9416no—
512C22⋊C4—948no—
513C42—28no—
514C42—4716no—
515C42—9416no—
516C42—9416no—
517C42—948no—
518C42—948no—
519C42—9416no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23C42×D4714yes—
541C23C22×C4×D4711yescharacteristic
542C23C22×C4×D4712yes—
543C23C2×D4×D4711yescharacteristic
544C23C2×D4×D4711yescharacteristic
545C23#41×D4712yes—
546C23#41×D4712yes—
547C23#41×D4712yes—
548C23C2×#1062535212yes—
549C23C2×#1062535212yes—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—474no—
573C2×C4—474no—
574C2×C4—474no—
575C2×C4—478no—
576C2×C4—478no—
577C2×C4—474no—
578C2×C4—474no—
579C2×C4—478no—
580C2×C4—9432no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—944no—
589C2×C4—944no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—944no—
593C2×C4—942no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—944no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4C22⋊C4×D4714yes—
608C2×C4#1309398018yescomplemented
609C22—24no—
610C22—28no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22C2×C42×D4711yescharacteristic
619C22C2×C22⋊C4×D4711yescharacteristic
620C22#117×D4712yes—
621C22#117×D4711yescharacteristic
622C22C2×#1309398012yes—
623C22C4×D4×D4712yes—
624C22C4×D4×D4712yes—
625C22C4×#1062535212yes—
626C22C4×#1062535212yes—
627C22C4×#1062535212yes—
628C22C4×#1062535212yes—
629C22#1568962911yescharacteristic
630C22#1568963111yescharacteristic
631C22#1568963312yes—
632C22#1568965012yes—
633C22#1568965212yes—
634C22#1568965412yes—
635C22#1568965612yes—
636C22#1568966812yes—
637C22#1568967111yescharacteristic
638C22#1568967211yescharacteristic
639C4—28no—
640C4—24no—
641C4—44no—
642C4—42no—
643C4—478no—
644C4—948no—
645C4—944no—
646C4—944no—
647C4—1884no—
648C4—1882no—
649C2—44no—
650C2C4×C22⋊C4×D4711yescharacteristic
651C2C4×#1309398012yes—
652C2#2024582812yes—
653C2#2024583012yes—
654C2#2024584812yes—
655C2#2024585012yes—
656C2#2024587311yescharacteristic
657C2#2024587511yescharacteristic
658C2#2024588212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC47×#1437C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1437D47
hypocenterC47#13761
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×#15688968C2semidirect
4. C2×C42×Dic47C2semidirect
7. #20241623C2semidirect
8. #20241621C2semidirect
9. #20241619C2semidirect
10. #95×Dic47C2semidirect
15. C2×#13093980C4semidirect
26. C22⋊C4×Dic47C4semidirect
161. C22×Dic47C42semidirect
189. C4×Dic47C22⋊C4semidirect
243. C2×C4×C47#117semidirect
276. C2×Dic47#117semidirect
370. C47#13761semidirect
608. C2×C4#13093980semidirect

Maximal subgroups

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

subgroupindex
C22×#130939802
C2×C22⋊C4×Dic472
C2×#156889682
C2×C42×Dic472
#202416252
#202416232
#202416212
#202416192
#95×Dic472
C47×#14372
#1376147