Finite Group Databasebeta

← #26138671

Subgroups — #26138671

Order 12032.

2166 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D47C211yesmaximal, characteristic
2C2×#15688982C211yesmaximal, characteristic
3C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
4#20241640C212yesmaximal, complemented
5#20241639C212yesmaximal, complemented
6#20241623C211yesmaximal, characteristic, complemented
7#20241621C211yesmaximal, characteristic, complemented
8#20241620C211yesmaximal, characteristic, complemented
9(C2×D47)⋊#95C212yesmaximal
10(C2×D47)⋊#95C212yesmaximal
11C47×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C23×C4×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2211yescharacteristic
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2212yes—
19C2×C4⋊C4×D47—24no—
20C2×C4⋊C4×D47—24no—
21C2×C4⋊C4×D47C418yescomplemented
22C22×(C2×D47)⋊C4C2212yes—
23C22×(C2×D47)⋊C4C2212yes—
24C22×C4×Dic47C2211yescharacteristic
25#15688982—24no—
26C4×(C2×D47)⋊C4—24no—
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C42×C47C2211yescharacteristic
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C24×D47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—24no—
49C22×C4×D47—24no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—42no—
53C22×C4×D47—42no—
54C22×C4×D47—42no—
55C22×C4×D47—42no—
56C22×C4×D47C2×C414yes—
57C22×C4×D47C2×C414yes—
58C22×C4×D47C2×C414yes—
59C22×C4×D47D414yes—
60C22×C4×D47D414yes—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47C2×C414yes—
64C2×Dic47⋊C4—22no—
65C2×Dic47⋊C4—22no—
66C2×Dic47⋊C4—22no—
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4—24no—
69C2×Dic47⋊C4—42no—
70C2×Dic47⋊C4—42no—
71C2×Dic47⋊C4C2×C414yes—
72C2×Dic47⋊C4C2×C414yes—
73C4⋊C4×D47—232no—
74C2×(C2×D47)⋊C4—24no—
75C2×(C2×D47)⋊C4—24no—
76C2×(C2×D47)⋊C4—24no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—24no—
79C2×(C2×D47)⋊C4Q818yes—
80C2×C4×Dic47—22no—
81C2×C4×Dic47—42no—
82C47⋊#95—22no—
83C47⋊#95—24no—
84C47⋊#95—24no—
85C47⋊#95—24no—
86C47⋊#95—24no—
87C47⋊#95—42no—
88C47⋊#95—42no—
89C47⋊#95—42no—
90C23×C4×C47C2312yes—
91C23×C4×C47C2311yescharacteristic
92C23×C4×C47C2312yes—
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47C2×C414yes—
98C2×C42×C47—22no—
99C2×C42×C47—42no—
100#95×C47—22no—
101#95×C47—24no—
102#95×C47—42no—
103C24×C47C2411yescharacteristic
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—22no—
107C23×D47—22no—
108C23×D47C22×C412yes—
109C23×D47C2×D412yes—
110C23×D47C2×D412yes—
111C23×D47C2×Q814yes—
112C23×D47#4114yes—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
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×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C22×C412yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×Q814yes—
141C22×C4×C47#4114yes—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
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×Dic47C22×C412yes—
165C22×Dic47C22×C412yes—
166C22×Dic47C22×C412yes—
167C22×Dic47C2×D412yes—
168C22×Dic47C2×D412yes—
169C22×Dic47C2×Q814yes—
170C22×Dic47#4114yes—
171C2×C4×D47—216no—
172C2×C4×D47—216no—
173C2×C4×D47—28no—
174C2×C4×D47—28no—
175C2×C4×D47—28no—
176C2×C4×D47—28no—
177C2×C4×D47—48no—
178C2×C4×D47—48no—
179C2×C4×D47—48no—
180C2×C4×D47—44no—
181C2×C4×D47—44no—
182C2×C4×D47—48no—
183C2×C4×D47—44no—
184C2×C4×D47—44no—
185C2×C4×D47C22⋊C4116yes—
186C4⋊C4×C47—28no—
187C42×C47—44no—
188(C2×D47)⋊C4—216no—
189(C2×D47)⋊C4—48no—
190C4⋊Dic47—28no—
191Dic47⋊C4—28no—
192Dic47⋊C4—28no—
193Dic47⋊C4—48no—
194C4×Dic47—44no—
195C23×C47C23×C411yescharacteristic
196C23×C47C22×D411yescharacteristic
197C23×C47C22×D411yescharacteristic
198C23×C47C22×Q812yes—
199C23×C47C2×#4112yes—
200C23×C47C2×#4112yes—
201C23×C47C2×#4112yes—
202C23×C47H(5,𝔽2)12yes—
203C23×C47H(5,𝔽2)12yes—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47C2×C22⋊C414yes—
221C22×D47C4×Q818yes—
222C22×D47C22⋊Q818yes—
223C22×D47#12414yes—
224C22×D47C4⋊Q814yes—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47C2×C22⋊C414yes—
253C2×C4×C47#11714yes—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47C2×C22⋊C414yes—
282C2×Dic47#11714yes—
283C4×D47—232no—
284C4×D47—416no—
285C4×D47—416no—
286#13923—471nomaximal, Sylow
287C22×C47C22×C22⋊C411yescharacteristic
288C22×C47C2×#11711yescharacteristic
289C22×C47C2×#11711yescharacteristic
290C22×C47C2×C4×Q812yes—
291C22×C47#51812yes—
292C22×C47C2×C22⋊Q812yes—
293C22×C47#52512yes—
294C22×C47#52512yes—
295C22×C47C2×#12411yescharacteristic
296C22×C47C2×C4⋊Q811yescharacteristic
297C22×C47#53412yes—
298C22×C47#53411yescharacteristic
299C22×C47#53411yescharacteristic
300C22×C47#55312yes—
301C22×C47#55312yes—
302C22×C47#55312yes—
303C22×C47#55312yes—
304C22×C47#55412yes—
305C22×C47#55412yes—
306C22×C47#55412yes—
307C22×C47#55412yes—
308C2×D47—24no—
309C2×D47—28no—
310C2×D47—28no—
311C2×D47—28no—
312C2×D47—28no—
313C2×D47—28no—
314C2×D47—28no—
315C2×D47—24no—
316C2×D47#39118yes—
317C4×C47—24no—
318C4×C47—24no—
319C4×C47—42no—
320C4×C47—44no—
321C4×C47—82no—
322Dic47—24no—
323Dic47—24no—
324Dic47—42no—
325Dic47—44no—
326Dic47—82no—
327C23×C4⋊C4—471no—
328C2×#380—471no—
329C2×C4×C22⋊C4—471no—
330#1441—472no—
331#1441—472no—
332#1436—471no—
333#1436—471no—
334#1436—471no—
335#1436D4711yescharacteristic, complemented
336C22⋊#95—472no—
337C22⋊#95—472no—
338C2×C47#228011yescharacteristic
339C2×C47C2×#39111yescharacteristic
340C2×C47#230811yescharacteristic
341C2×C47#234512yes—
342C2×C47#234512yes—
343C2×C47#257912yes—
344C2×C47#257912yes—
345C2×C47#258312yes—
346C2×C47#259112yes—
347D47—28no—
348C24×C4—471no—
349C24×C4—471no—
350C24×C4—471no—
351C22×C4⋊C4—471no—
352C22×C4⋊C4—471no—
353C22×C4⋊C4—471no—
354C22×C4⋊C4—478no—
355C22×C4⋊C4—472no—
356C22×C4⋊C4—944no—
357C22×C4⋊C4—944no—
358C22×C4⋊C4C2×D4711yescharacteristic
359C22×C22⋊C4—472no—
360C22×C22⋊C4—472no—
361C22×C42—471no—
362C22×C42C2×D4711yescharacteristic
363#380—944no—
364C4×C22⋊C4—944no—
365C2×#95—472no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—471no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95C2×D4712yes—
374C2×#95C2×D4711yescharacteristic
375C2×#95C2×D4712yes—
376C47#1392311yescharacteristic, complemented, Sylow
377C25—471no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—474no—
381C23×C4—474no—
382C23×C4—474no—
383C23×C4—472no—
384C23×C4—471no—
385C23×C4—474no—
386C23×C4—474no—
387C23×C4—472no—
388C23×C4—944no—
389C23×C4—942no—
390C23×C4—942no—
391C23×C4—942no—
392C23×C4—942no—
393C23×C4—944no—
394C23×C4—1882no—
395C23×C4—1882no—
396C23×C4—1882no—
397C23×C4—1882no—
398C23×C4C22×D4712yes—
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4712yes—
401C23×C4C22×D4711yescharacteristic
402C23×C4C22×D4711yescharacteristic
403C2×C4⋊C4—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—474no—
407C2×C4⋊C4—474no—
408C2×C4⋊C4—9432no—
409C2×C4⋊C4—942no—
410C2×C4⋊C4—942no—
411C2×C4⋊C4—942no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—944no—
416C2×C4⋊C4—1882no—
417C2×C4⋊C4—1882no—
418C2×C4⋊C4C4×D4714yes—
419C2×C22⋊C4—478no—
420C2×C22⋊C4—944no—
421C2×C22⋊C4—944no—
422C2×C22⋊C4—944no—
423C2×C22⋊C4—944no—
424C2×C22⋊C4—944no—
425C2×C42—22no—
426C2×C42—42no—
427C2×C42—942no—
428C2×C42—1882no—
429#95—24no—
430#95—22no—
431#95—42no—
432#95—944no—
433#95—944no—
434#95—944no—
435#95—944no—
436#95—942no—
437#95—1882no—
438#95—1882no—
439#95—1882no—
440C24—472no—
441C24—472no—
442C24—472no—
443C24—474no—
444C24—474no—
445C24—942no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24C23×D4711yescharacteristic
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—472no—
473C22×C4—472no—
474C22×C4—472no—
475C22×C4—472no—
476C22×C4—474no—
477C22×C4—4716no—
478C22×C4—472no—
479C22×C4—474no—
480C22×C4—948no—
481C22×C4—948no—
482C22×C4—9416no—
483C22×C4—9416no—
484C22×C4—948no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—948no—
491C22×C4—942no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—1888no—
509C22×C4—1888no—
510C22×C4—1884no—
511C22×C4—1884no—
512C22×C4—1888no—
513C22×C4—1888no—
514C22×C4—1884no—
515C22×C4—1884no—
516C22×C4C2×C4×D4712yes—
517C22×C4C2×C4×D4712yes—
518C22×C4C2×C4×D4712yes—
519C22×C4#1062535014yes—
520C22×C4D4×D4712yes—
521C22×C4D4×D4712yes—
522C22×C4Q8×D4714yes—
523C4⋊C4—28no—
524C4⋊C4—948no—
525C4⋊C4—948no—
526C4⋊C4—948no—
527C4⋊C4—1888no—
528C22⋊C4—9416no—
529C22⋊C4—1888no—
530C42—44no—
531C42—1884no—
532C23—474no—
533C23—478no—
534C23—478no—
535C23—474no—
536C23—474no—
537C23—944no—
538C23—944no—
539C23—944no—
540C23—944no—
541C23—944no—
542C23—944no—
543C23—944no—
544C23—942no—
545C23—942no—
546C23—944no—
547C23—944no—
548C23—944no—
549C23—944no—
550C23—944no—
551C23—942no—
552C23—942no—
553C23C22×C4×D4711yescharacteristic
554C23C2×D4×D4711yescharacteristic
555C23C2×D4×D4711yescharacteristic
556C23C2×Q8×D4712yes—
557C23#41×D4712yes—
558C23#41×D4712yes—
559C23C2×#1062535012yes—
560C23#1309406912yes—
561C23#1309407012yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—474no—
590C2×C4—474no—
591C2×C4—9432no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—18816no—
606C2×C4—18816no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4—1881no—
619C2×C4—1882no—
620C2×C4—1881no—
621C2×C4C22⋊C4×D4714yes—
622C2×C4#1309398114yes—
623C22—478no—
624C22—944no—
625C22—948no—
626C22—948no—
627C22—948no—
628C22—948no—
629C22—944no—
630C22—948no—
631C22—948no—
632C22C2×C22⋊C4×D4711yescharacteristic
633C22#117×D4711yescharacteristic
634C22C2×#1309398111yescharacteristic
635C22C4×Q8×D4712yes—
636C22#1568959512yes—
637C22C22⋊Q8×D4712yes—
638C22#1568963212yes—
639C22#1568963212yes—
640C22#124×D4711yescharacteristic
641C22C4⋊Q8×D4711yescharacteristic
642C22#1568968112yes—
643C22#1568968311yescharacteristic
644C22#1568968411yescharacteristic
645C22#1568981712yes—
646C22#1568981812yes—
647C22#1568982312yes—
648C22#1568982412yes—
649C22#1568982712yes—
650C22#1568982812yes—
651C22#1568982912yes—
652C22#1568983312yes—
653C4—24no—
654C4—24no—
655C4—44no—
656C4—42no—
657C4—82no—
658C4—944no—
659C4—944no—
660C4—3762no—
661C4—1884no—
662C4—1882no—
663C2—948no—
664C2#2024576911yescharacteristic
665C2#391×D4711yescharacteristic
666C2#2024596211yescharacteristic
667C2#2024626112yes—
668C2#2024626212yes—
669C2#2024873312yes—
670C2#2024873912yes—
671C2#2024876612yes—
672C2#2024886612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1436C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1436D47
hypocenterC47#13923
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
4. #20241640C2semidirect
5. #20241639C2semidirect
6. #20241623C2semidirect
7. #20241621C2semidirect
8. #20241620C2semidirect
11. C47×#1436C2semidirect
21. C2×C4⋊C4×D47C4semidirect
335. #1436D47semidirect
376. C47#13923semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D472
C2×#156889822
C2×C4×(C2×D47)⋊C42
#202416402
#202416392
#202416232
#202416212
#202416202
(C2×D47)⋊#952
C47×#14362
#1392347