Finite Group Databasebeta

← #26139156

Subgroups — #26139156

Order 12032.

2558 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689026C211yesmaximal, characteristic
2C2×#15689026C211yesmaximal, characteristic
3C2×#15689021C211yesmaximal, characteristic, complemented
4C2×#15688996C214yesmaximal, complemented
5C2×#15688992C211yesmaximal, characteristic, complemented
6#20241631C214yesmaximal, complemented
7#20241617C211yesmaximal, characteristic, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C22×C4×D4×C47C211yesmaximal, characteristic
10C22×C4⋊Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4C2212yes—
13C22×Dic47⋊C4C2214yes—
14C22×C22⋊Dic47C2214yes—
15C22×C22⋊Dic47C2214yes—
16C22×C4×Dic47C2211yescharacteristic
17C22×C4×Dic47C2211yescharacteristic
18#15689026—24no—
19#15689026C418yescomplemented
20#15689021—24no—
21#15688996—216no—
22#15688992—24no—
23C2×C47⋊#95C2214yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C23×D4×C47C2211yescharacteristic
28C24×C4×C47C2212yes—
29C2×C4×D4×C47—24no—
30C2×C4×D4×C47—24no—
31C2×C4×D4×C47—24no—
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C22⋊C4×C47C2212yes—
34C22×C42×C47C2211yescharacteristic
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C23×Dic47C2312yes—
39C2×C4⋊Dic47—22no—
40C2×C4⋊Dic47—22no—
41C2×C4⋊Dic47—22no—
42C2×C4⋊Dic47—22no—
43C2×C4⋊Dic47—22no—
44C2×C4⋊Dic47C2×C414yescomplemented
45C2×Dic47⋊C4—28no—
46C2×Dic47⋊C4—28no—
47C2×Dic47⋊C4—24no—
48C2×Dic47⋊C4—28no—
49C2×Dic47⋊C4—44no—
50C2×C22⋊Dic47—28no—
51C2×C22⋊Dic47—28no—
52C2×C22⋊Dic47—28no—
53C2×C22⋊Dic47—28no—
54C2×C22⋊Dic47—44no—
55C2×C22⋊Dic47C2×C4116yescomplemented
56C2×C4×Dic47—22no—
57C2×C4×Dic47—22no—
58C2×C4×Dic47—22no—
59C2×C4×Dic47—22no—
60C2×C4×Dic47—22no—
61C2×C4×Dic47C2×C414yescomplemented
62C47⋊#95—24no—
63C47⋊#95—28no—
64C47⋊#95—24no—
65C47⋊#95D418yescomplemented
66C25×C47C2312yes—
67C22×D4×C47—22no—
68C22×D4×C47—22no—
69C22×D4×C47—22no—
70C22×D4×C47—22no—
71C22×D4×C47C2×C414yes—
72C22×D4×C47D414yes—
73C22×D4×C47Q814yes—
74C23×C4×C47—24no—
75C23×C4×C47—24no—
76C23×C4×C47—24no—
77C23×C4×C47—28no—
78C23×C4×C47—44no—
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C4×D4×C47—232no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47D414yes—
87C2×C22⋊C4×C47—24no—
88C2×C22⋊C4×C47—24no—
89C2×C22⋊C4×C47—24no—
90C2×C42×C47—22no—
91C2×C42×C47—22no—
92C2×C42×C47D414yes—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—24no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—24no—
99C24×C47—24no—
100C24×C47—24no—
101C24×C47C22×C414yes—
102C24×C47C2×D414yes—
103C24×C47C2×Q814yes—
104C24×C47C2411yescharacteristic
105C2×D4×C47—28no—
106C2×D4×C47—28no—
107C2×D4×C47—28no—
108C2×D4×C47—28no—
109C2×D4×C47—28no—
110C2×D4×C47—28no—
111C2×D4×C47C4⋊C4116yescomplemented
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—216no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—24no—
123C22×C4×C47—22no—
124C22×C4×C47—416no—
125C22×C4×C47—416no—
126C22×C4×C47—48no—
127C22×C4×C47—48no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D414yes—
131C22×C4×C47C2×D414yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×Q812yes—
137C22×C4×C47#4112yes—
138C22×C4×C47#4112yes—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—24no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47C22×C414yes—
159C22×Dic47C22×C414yes—
160C22×Dic47C2×D414yes—
161C22×Dic47C2×D414yes—
162C22×Dic47#4114yes—
163C4⋊C4×C47—28no—
164C22⋊C4×C47—216no—
165C42×C47—28no—
166C22⋊Dic47—232no—
167C22⋊Dic47—416no—
168C4⋊Dic47—28no—
169C4⋊Dic47—44no—
170Dic47⋊C4—416no—
171Dic47⋊C4—48no—
172C4×Dic47—28no—
173C4×Dic47—44no—
174C23×C47—28no—
175C23×C47—24no—
176C23×C47—24no—
177C23×C47—28no—
178C23×C47—24no—
179C23×C47—24no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—28no—
183C23×C47—28no—
184C23×C47—28no—
185C23×C47—28no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—44no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—44no—
192C23×C47—44no—
193C23×C47—44no—
194C23×C47—44no—
195C23×C47—44no—
196C23×C47C2×C4⋊C418yes—
197C23×C47C23×C411yescharacteristic
198C23×C47C22×D412yes—
199C23×C47C22×D411yescharacteristic
200C23×C47C22×D411yescharacteristic
201C23×C47C22×D411yescharacteristic
202C23×C47C22×Q811yescharacteristic
203C23×C47C2×#4112yes—
204C23×C47C2×#4111yescharacteristic
205C23×C47C2×#4111yescharacteristic
206C23×C47H(5,𝔽2)12yes—
207C23×C472−1+412yes—
208D4×C47—232no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—22no—
214C2×C4×C47—24no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—28no—
218C2×C4×C47—432no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—44no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47C2×C4⋊C414yes—
225C2×C4×C47C4×D414yes—
226C2×C4×C47C4×D414yes—
227C2×C4×C47#12014yes—
228C2×C4×C47C22⋊Q814yes—
229C2×C4×C47C22⋊Q814yes—
230C2×C4×C47#12314yes—
231C2×C4×C47#12714yes—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—28no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—44no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—44no—
254C2×Dic47—44no—
255C2×Dic47C4×D418yescomplemented
256#13977—471nomaximal, Sylow
257C22×C47—216no—
258C22×C47—28no—
259C22×C47—44no—
260C22×C47—44no—
261C22×C47—44no—
262C22×C47—48no—
263C22×C47—48no—
264C22×C47—44no—
265C22×C47—44no—
266C22×C47—48no—
267C22×C47—44no—
268C22×C47C22×C4⋊C411yescharacteristic
269C22×C47C2×C4×D412yes—
270C22×C47C2×C4×D411yescharacteristic
271C22×C47C2×C4×D411yescharacteristic
272C22×C47#52012yes—
273C22×C47C2×#12011yescharacteristic
274C22×C47C2×C22⋊Q811yescharacteristic
275C22×C47C2×C22⋊Q811yescharacteristic
276C22×C47C2×#12311yescharacteristic
277C22×C47C2×#12711yescharacteristic
278C22×C47#53211yescharacteristic
279C22×C47#53212yes—
280C22×C47#53511yescharacteristic
281C22×C47#53611yescharacteristic
282C22×C47#54212yes—
283C22×C47D4212yes—
284C22×C47D4212yes—
285C22×C47#54712yes—
286C22×C47#54712yes—
287C22×C47#54812yes—
288C22×C47D4×Q812yes—
289C22×C47#55312yes—
290C22×C47#55412yes—
291C4×C47—24no—
292C4×C47—42no—
293C4×C47—44no—
294C4×C47#38518yescomplemented
295Dic47—28no—
296Dic47—44no—
297Dic47—44no—
298Dic47—82no—
299C22×C4×D4D4711yescharacteristic
300C2×#390—471no—
301C2×#390—471no—
302C2×#389—471no—
303C2×#385—474no—
304C2×#384—471no—
305#1439—474no—
306C4⋊#95—471no—
307C4×#95—471no—
308C2×C47—48no—
309C2×C47C2×#38511yescharacteristic
310C2×C47C4×#12711yescharacteristic
311C2×C47#232611yescharacteristic
312C2×C47D4×C4⋊C412yes—
313C2×C47#236112yes—
314C2×C47#242212yes—
315C2×C47#243211yescharacteristic
316C2×C47#243311yescharacteristic
317C2×C47#245212yes—
318C2×C47#254211yescharacteristic
319C2×C47#256611yescharacteristic
320C23×D4C2×D4711yescharacteristic
321C24×C4C2×D4712yes—
322C2×C4×D4—24no—
323C2×C4×D4—24no—
324C2×C4×D4—24no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4—472no—
327C22×C4⋊C4—471no—
328C22×C4⋊C4—474no—
329C22×C4⋊C4C2×D4711yescharacteristic
330C22×C22⋊C4—474no—
331C22×C22⋊C4—474no—
332C22×C22⋊C4C2×D4712yes—
333C22×C42—471no—
334C22×C42—471no—
335C22×C42C2×D4711yescharacteristic
336#390—478no—
337#390—944no—
338#389—944no—
339#385—9416no—
340#384—944no—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—474no—
344C2×#95—472no—
345C47#1397711yescharacteristic, complemented, Sylow
346C25C22×D4712yes—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4Dic9414yes—
352C22×D4C4×D4714yes—
353C22×D4C22⋊D4714yes—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—24no—
357C23×C4—28no—
358C23×C4—44no—
359C23×C4—472no—
360C23×C4—472no—
361C23×C4—472no—
362C23×C4—472no—
363C23×C4C22×D4711yescharacteristic
364C23×C4C22×D4712yes—
365C23×C4C22×D4711yescharacteristic
366C23×C4C22×D4711yescharacteristic
367C4×D4—232no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4—948no—
374C2×C4⋊C4—948no—
375C2×C4⋊C4—948no—
376C2×C4⋊C4—944no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—942no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—1884no—
381C2×C4⋊C4C22⋊D4714yes—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—4716no—
386C2×C22⋊C4—948no—
387C2×C22⋊C4—948no—
388C2×C22⋊C4—948no—
389C2×C22⋊C4—948no—
390C2×C22⋊C4—1884no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—474no—
394C2×C42—942no—
395C2×C42—942no—
396C2×C42—942no—
397C2×C42—942no—
398C2×C42—942no—
399C2×C42C22⋊D4714yes—
400#95—478no—
401#95—944no—
402#95—948no—
403#95—944no—
404C24—24no—
405C24—22no—
406C24—22no—
407C24—24no—
408C24—22no—
409C24—22no—
410C24—24no—
411C24—24no—
412C24C2×Dic9414yes—
413C24C2×C4×D4714yes—
414C24C2×C22⋊D4714yes—
415C24C23×D4711yescharacteristic
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4Dic47⋊C4116yescomplemented
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—474no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—944no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—944no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—944no—
463C22×C4C2×Dic9412yes—
464C22×C4C2×C4×D4712yes—
465C22×C4#1062535012yes—
466C22×C4#1062535012yes—
467C22×C4D4×D4714yes—
468C22×C4C2×C22⋊D4712yes—
469C22×C4C2×C22⋊D4714yes—
470C22×C4C2×C22⋊D4712yes—
471C22×C4C2×C22⋊D4712yes—
472C22×C4C2×C22⋊D4712yes—
473C22×C4C2×C22⋊D4712yes—
474C4⋊C4—28no—
475C4⋊C4—948no—
476C4⋊C4—18816no—
477C4⋊C4—1884no—
478C4⋊C4—1888no—
479C22⋊C4—216no—
480C22⋊C4—9432no—
481C22⋊C4—18816no—
482C42—28no—
483C42—948no—
484C42—1884no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23C2×Dic47⋊C418yes—
508C23C22×C4×D4711yescharacteristic
509C23C2×D4×D4712yes—
510C23C22×C22⋊D4711yescharacteristic
511C23C22×C22⋊D4711yescharacteristic
512C23C22×C22⋊D4711yescharacteristic
513C23C22×Dic9411yescharacteristic
514C23#41×D4712yes—
515C23C2×#1062535011yescharacteristic
516C23C2×#1062535011yescharacteristic
517C23#1309406912yes—
518C23#1309407112yes—
519D4—232no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—432no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—478no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—944no—
543C2×C4—944no—
544C2×C4—948no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—1884no—
548C2×C4—1884no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4—1882no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4—1884no—
559C2×C4C2×Dic47⋊C414yes—
560C2×C4C4×C22⋊D4714yes—
561C2×C4C4×C22⋊D4714yes—
562C2×C4#1309399314yes—
563C2×C4#1309400314yes—
564C2×C4#1309400314yes—
565C2×C4#1309400714yes—
566C2×C4#1309402014yes—
567C22—216no—
568C22—28no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—48no—
574C22—44no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22C22×Dic47⋊C411yescharacteristic
579C22C4×D4×D4712yes—
580C22C2×C4×C22⋊D4711yescharacteristic
581C22C2×C4×C22⋊D4711yescharacteristic
582C22#1568960212yes—
583C22C2×#1309399311yescharacteristic
584C22C2×#1309400311yescharacteristic
585C22C2×#1309400311yescharacteristic
586C22C2×#1309400711yescharacteristic
587C22C2×#1309402011yescharacteristic
588C22#1568966911yescharacteristic
589C22#1568967012yes—
590C22#1568969011yescharacteristic
591C22#1568969411yescharacteristic
592C22#1568973812yes—
593C22D4×C22⋊D4712yes—
594C22D4×C22⋊D4712yes—
595C22#1568977212yes—
596C22#1568977812yes—
597C22#1568978012yes—
598C22D4×Dic9412yes—
599C22#1568982312yes—
600C22#1568983212yes—
601C4—24no—
602C4—42no—
603C4—44no—
604C4—948no—
605C4—3762no—
606C4—1884no—
607C4—1884no—
608C4#1568899618yescomplemented
609C2—48no—
610C2C2×#1568899611yescharacteristic
611C2C4×#1309402011yescharacteristic
612C2#2024610011yescharacteristic
613C2D4×Dic47⋊C412yes—
614C2#2024640012yes—
615C2#2024689612yes—
616C2#2024701211yescharacteristic
617C2#2024702111yescharacteristic
618C2#2024723212yes—
619C2#2024834111yescharacteristic
620C2#2024858411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×C4×D4×C47C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC22×C4×D4D47
hypocenterC47#13977
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
3. C2×#15689021C2semidirect
4. C2×#15688996C2semidirect
5. C2×#15688992C2semidirect
6. #20241631C2semidirect
7. #20241617C2semidirect
8. C4×C47⋊#95C2semidirect
19. #15689026C4semidirect
44. C2×C4⋊Dic47C2×C4semidirect
55. C2×C22⋊Dic47C2×C4semidirect
61. C2×C4×Dic47C2×C4semidirect
65. C47⋊#95D4semidirect
111. C2×D4×C47C4⋊C4semidirect
255. C2×Dic47C4×D4semidirect
294. C4×C47#385semidirect
345. C47#13977semidirect
422. C2×D4Dic47⋊C4semidirect
608. C4#15688996semidirect

Maximal subgroups

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

subgroupindex
C2×#156890262
C2×#156890212
C2×#156889962
C2×#156889922
#202416312
#202416172
C4×C47⋊#952
C22×C4×D4×C472
#1397747