Finite Group Databasebeta

← #26139994

Subgroups — #26139994

Order 12032.

2006 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688995C211yesmaximal, characteristic
2C2×#15688995C211yesmaximal, characteristic
3C2×C4×C22⋊Dic47C211yesmaximal, characteristic
4C22×C47⋊#95C211yesmaximal, characteristic
5#20241643C212yesmaximal, complemented
6#20241638C212yesmaximal, complemented
7#20241620C212yesmaximal, complemented
8Dic47⋊#95C212yesmaximal, complemented
9#20241610C211yesmaximal, characteristic
10#20241610C211yesmaximal, characteristic
11C2×C47×#385C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C24×Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16C22×C22⋊Dic47C2211yescharacteristic
17C22×C22⋊Dic47C2211yescharacteristic
18C22×C4×Dic47C2212yes—
19#15688995—24no—
20#15688995C418yescomplemented
21C4×C22⋊Dic47—24no—
22C2×C47⋊#95—24no—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C24×C4×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C47×#385—24no—
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2211yescharacteristic
40C23×Dic47—22no—
41C23×Dic47—22no—
42C23×Dic47—22no—
43C23×Dic47—24no—
44C23×Dic47—22no—
45C23×Dic47—22no—
46C23×Dic47—42no—
47C23×Dic47—42no—
48C23×Dic47—42no—
49C23×Dic47C2×C414yes—
50C23×Dic47C2312yes—
51C23×Dic47C2312yes—
52C23×Dic47C2312yes—
53C23×Dic47C2312yes—
54C2×Dic47⋊C4—24no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—42no—
57C2×Dic47⋊C4—42no—
58C2×Dic47⋊C4—42no—
59C2×Dic47⋊C4C2×C418yescomplemented
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47C2×C414yes—
63C2×C22⋊Dic47D414yes—
64C2×C22⋊Dic47Q814yes—
65C2×C22⋊Dic47Q814yes—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—42no—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—42no—
75C47⋊#95—42no—
76C47⋊#95—42no—
77C47⋊#95C2×C418yescomplemented
78C25×C47C2311yescharacteristic
79C23×C4×C47—22no—
80C23×C4×C47—22no—
81C23×C4×C47—24no—
82C23×C4×C47—42no—
83C23×C4×C47D414yes—
84C23×C4×C47C2311yescharacteristic
85C23×C4×C47C2311yescharacteristic
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2311yescharacteristic
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—24no—
93C2×C4⋊C4×C47—42no—
94C2×C22⋊C4×C47—22no—
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47—22no—
97C2×C22⋊C4×C47—22no—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47C2×C414yes—
100#95×C47—22no—
101#95×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47—22no—
106C24×C47C22×C412yes—
107C24×C47C2×D412yes—
108C24×C47C2×D412yes—
109C24×C47C2×Q812yes—
110C24×C47C2×Q812yes—
111C24×C47#4112yes—
112C24×C47#4112yes—
113C24×C47C2411yescharacteristic
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—28no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—28no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—44no—
142C22×C4×C47—48no—
143C22×C4×C47—48no—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47#4114yes—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—28no—
154C22×Dic47—28no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—22no—
172C22×Dic47—22no—
173C22×Dic47—48no—
174C22×Dic47—48no—
175C22×Dic47—48no—
176C22×Dic47—44no—
177C22×Dic47—44no—
178C22×Dic47—44no—
179C22×Dic47—48no—
180C22×Dic47—44no—
181C22×Dic47C22×C414yes—
182C22×Dic47C22×C414yes—
183C22×Dic47C2×D414yes—
184C22×Dic47C2×Q814yes—
185C22×Dic47C2×Q814yes—
186C22×Dic47#4114yes—
187C4⋊C4×C47—44no—
188C4⋊C4×C47—44no—
189C22⋊C4×C47—28no—
190C22⋊C4×C47—44no—
191C22⋊Dic47—28no—
192C22⋊Dic47C4⋊C4116yescomplemented
193Dic47⋊C4—48no—
194Dic47⋊C4—48no—
195C4×Dic47—48no—
196C23×C47—24no—
197C23×C47—24no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—24no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—24no—
210C23×C47—24no—
211C23×C47—24no—
212C23×C47C2×C4⋊C414yes—
213C23×C47C4×D414yes—
214C23×C47C4×Q814yes—
215C23×C47#12114yes—
216C23×C47C22⋊Q814yes—
217C23×C47#12514yes—
218C23×C47C4⋊Q814yes—
219C23×C47C23×C411yescharacteristic
220C23×C47C22×D411yescharacteristic
221C23×C47C22×D411yescharacteristic
222C23×C47C22×Q811yescharacteristic
223C23×C47C22×Q811yescharacteristic
224C23×C47C2×#4112yes—
225C23×C47C2×#4111yescharacteristic
226C23×C47C2×#4112yes—
227C23×C47C2×#4111yescharacteristic
228C23×C47H(5,𝔽2)12yes—
229C23×C47H(5,𝔽2)12yes—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—416no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—42no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—42no—
257C2×C4×C47—42no—
258C2×C4×C47—42no—
259C2×C4×C47—42no—
260C2×C4×C47#12314yes—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—24no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47—24no—
275C2×Dic47—24no—
276C2×Dic47—416no—
277C2×Dic47—416no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47—42no—
285C2×Dic47—42no—
286C2×Dic47—42no—
287C2×Dic47—42no—
288C2×Dic47—42no—
289C2×Dic47—42no—
290C2×Dic47C2×C4⋊C418yes—
291#14044—471nomaximal, Sylow
292C22×C47—24no—
293C22×C47—24no—
294C22×C47—28no—
295C22×C47—28no—
296C22×C47—24no—
297C22×C47—24no—
298C22×C47—28no—
299C22×C47—24no—
300C22×C47—28no—
301C22×C47—24no—
302C22×C47#38918yes—
303C22×C47C22×C4⋊C411yescharacteristic
304C22×C47C2×C4×D411yescharacteristic
305C22×C47C2×C4×Q811yescharacteristic
306C22×C47C4×#4112yes—
307C22×C47#51812yes—
308C22×C47C2×#12111yescharacteristic
309C22×C47C2×C22⋊Q811yescharacteristic
310C22×C47C2×#12311yescharacteristic
311C22×C47#52511yescharacteristic
312C22×C47C2×#12511yescharacteristic
313C22×C47#52912yes—
314C22×C47C2×C4⋊Q811yescharacteristic
315C22×C47#53411yescharacteristic
316C22×C47#53511yescharacteristic
317C22×C47#53812yes—
318C22×C47#54612yes—
319C22×C47#54612yes—
320C22×C47#55112yes—
321C22×C47#55112yes—
322C22×C47#55412yes—
323C22×C47#55412yes—
324C22×C47#55412yes—
325C22×C47#55412yes—
326C4×C47—24no—
327C4×C47—42no—
328C4×C47—42no—
329C4×C47—42no—
330C4×C47—81no—
331C4×C47—81no—
332Dic47—28no—
333Dic47—44no—
334Dic47—44no—
335Dic47—44no—
336C2×#385—471no—
337C2×#385—471no—
338C2×#385D4711yescharacteristic
339C2×C4×C22⋊C4—471no—
340C22×#95—471no—
341#1442—472no—
342#1441—472no—
343#1436—472no—
344C4⋊#95—472no—
345C22⋊#95—471no—
346C22⋊#95—471no—
347C2×C47—28no—
348C2×C47C2×#38911yescharacteristic
349C2×C47C4×#12311yescharacteristic
350C2×C47#232111yescharacteristic
351C2×C47#234912yes—
352C2×C47#236812yes—
353C2×C47#241111yescharacteristic
354C2×C47#241211yescharacteristic
355C2×C47#245012yes—
356C2×C47#250211yescharacteristic
357C2×C47#250512yes—
358C2×C47#258311yescharacteristic
359C24×C4—471no—
360C24×C4—471no—
361C24×C4C2×D4711yescharacteristic
362C22×C4⋊C4—472no—
363C22×C4⋊C4—472no—
364C22×C4⋊C4C2×D4711yescharacteristic
365C22×C4⋊C4C2×D4711yescharacteristic
366C22×C22⋊C4—471no—
367C22×C22⋊C4—471no—
368C22×C22⋊C4C2×D4711yescharacteristic
369C22×C22⋊C4C2×D4711yescharacteristic
370C22×C42—472no—
371#385—24no—
372#385—478no—
373#385—944no—
374C4×C22⋊C4—944no—
375C2×#95—472no—
376C2×#95—472no—
377C2×#95—472no—
378C2×#95—472no—
379C2×#95—472no—
380C2×#95—472no—
381C2×#95—472no—
382C2×#95—472no—
383C2×#95—472no—
384C2×#95—944no—
385C2×#95C2×D4711yescharacteristic
386C2×#95C2×D4711yescharacteristic
387C47#1404411yescharacteristic, complemented, Sylow
388C25C22×D4711yescharacteristic
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—24no—
392C23×C4—42no—
393C23×C4—472no—
394C23×C4—472no—
395C23×C4—472no—
396C23×C4—472no—
397C23×C4—474no—
398C23×C4—944no—
399C23×C4—942no—
400C23×C4—942no—
401C23×C4—942no—
402C23×C4—942no—
403C23×C4—942no—
404C23×C4—1882no—
405C23×C4—1882no—
406C23×C4—1882no—
407C23×C4C22⋊D4714yes—
408C23×C4C22×D4711yescharacteristic
409C23×C4C22×D4711yescharacteristic
410C23×C4C22×D4711yescharacteristic
411C23×C4C22×D4711yescharacteristic
412C23×C4C22×D4711yescharacteristic
413C23×C4C22×D4711yescharacteristic
414C2×C4⋊C4—24no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—42no—
418C2×C4⋊C4—478no—
419C2×C4⋊C4—944no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—1882no—
422C2×C4⋊C4—1882no—
423C2×C4⋊C4—1882no—
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—474no—
431C2×C22⋊C4—474no—
432C2×C22⋊C4—474no—
433C2×C22⋊C4—942no—
434C2×C22⋊C4—942no—
435C2×C22⋊C4C4×D4714yes—
436C2×C42—944no—
437C2×C42—944no—
438C2×C42—1882no—
439#95—22no—
440#95—22no—
441#95—478no—
442#95—944no—
443#95—944no—
444#95—944no—
445#95—944no—
446#95—944no—
447#95—1882no—
448#95—1882no—
449#95—1882no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24C2×C4×D4712yes—
455C24#1062535012yes—
456C24D4×D4712yes—
457C24#1062535212yes—
458C24Q8×D4712yes—
459C24Q8×D4712yes—
460C24C2×C22⋊D4712yes—
461C24C23×D4711yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—474no—
493C22×C4—474no—
494C22×C4—474no—
495C22×C4—474no—
496C22×C4—474no—
497C22×C4—474no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—948no—
503C22×C4—948no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—942no—
521C22×C4—942no—
522C22×C4—1888no—
523C22×C4—1888no—
524C22×C4—1888no—
525C22×C4—1884no—
526C22×C4—1884no—
527C22×C4—1884no—
528C22×C4—1888no—
529C22×C4—1884no—
530C22×C4C2×C4×D4712yes—
531C22×C4C2×C4×D4712yes—
532C22×C4#1062535214yes—
533C22×C4C2×C22⋊D4712yes—
534C22×C4C2×C22⋊D4712yes—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—1888no—
538C4⋊C4—1888no—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C22⋊C4—4716no—
542C22⋊C4—948no—
543C42—1888no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23C4⋊C4×D4714yes—
561C23C4×C22⋊D4714yes—
562C23#1309398914yes—
563C23#1309399614yes—
564C23#1309400214yes—
565C23#1309401414yes—
566C23#1309402214yes—
567C23C22×C4×D4711yescharacteristic
568C23C2×D4×D4711yescharacteristic
569C23C22×C22⋊D4711yescharacteristic
570C23C2×Q8×D4711yescharacteristic
571C23C2×Q8×D4711yescharacteristic
572C23#41×D4712yes—
573C23C2×#1062535212yes—
574C23C2×#1062535211yescharacteristic
575C23C2×#1062535011yescharacteristic
576C23#1309406912yes—
577C23#1309407012yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—478no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—944no—
612C2×C4—944no—
613C2×C4—944no—
614C2×C4—944no—
615C2×C4—944no—
616C2×C4—944no—
617C2×C4—944no—
618C2×C4—944no—
619C2×C4—944no—
620C2×C4—944no—
621C2×C4—944no—
622C2×C4—944no—
623C2×C4—944no—
624C2×C4—18816no—
625C2×C4—18816no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4—1882no—
629C2×C4—1882no—
630C2×C4—1882no—
631C2×C4—1882no—
632C2×C4—1882no—
633C2×C4—1882no—
634C2×C4—1882no—
635C2×C4—1882no—
636C2×C4—1882no—
637C2×C4—1882no—
638C2×C4#1309400514yes—
639C22—24no—
640C22—28no—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—24no—
647C22—28no—
648C22—24no—
649C22#1568901818yes—
650C22C2×C4⋊C4×D4711yescharacteristic
651C22C2×C4×C22⋊D4711yescharacteristic
652C22C2×#1309398911yescharacteristic
653C22C4×#1062535212yes—
654C22#1568959512yes—
655C22C2×#1309399611yescharacteristic
656C22C2×#1309400211yescharacteristic
657C22C2×#1309400511yescharacteristic
658C22#1568963011yescharacteristic
659C22C2×#1309401411yescharacteristic
660C22#1568965312yes—
661C22C2×#1309402211yescharacteristic
662C22#1568968211yescharacteristic
663C22#1568968711yescharacteristic
664C22#1568970612yes—
665C22#1568976012yes—
666C22#1568976512yes—
667C22#1568979612yes—
668C22#1568980412yes—
669C22#1568982712yes—
670C22#1568982712yes—
671C22#1568983112yes—
672C22#1568983112yes—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—81no—
678C4—81no—
679C4—948no—
680C4—1884no—
681C4—1884no—
682C4—1884no—
683C2—28no—
684C2C2×#1568901811yescharacteristic
685C2C4×#1309400511yescharacteristic
686C2#2024604211yescharacteristic
687C2#2024629412yes—
688C2#2024646912yes—
689C2#2024677411yescharacteristic
690C2#2024678511yescharacteristic
691C2#2024721412yes—
692C2#2024786711yescharacteristic
693C2#2024789112yes—
694C2#2024876911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#385C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#385D47
hypocenterC47#14044
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #20241643C2semidirect
6. #20241638C2semidirect
7. #20241620C2semidirect
8. Dic47⋊#95C2semidirect
20. #15688995C4semidirect
59. C2×Dic47⋊C4C2×C4semidirect
77. C47⋊#95C2×C4semidirect
192. C22⋊Dic47C4⋊C4semidirect
387. C47#14044semidirect

Maximal subgroups

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

subgroupindex
C2×#156889952
C2×C4×C22⋊Dic472
C22×C47⋊#952
#202416432
#202416382
#202416202
Dic47⋊#952
#202416102
C2×C47×#3852
#1404447