Finite Group Databasebeta

← #26137979

Subgroups — #26137979

Order 12032.

1814 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Dic47C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic47C211yesmaximal, characteristic
3C2×C4×C22⋊Dic47C211yesmaximal, characteristic
4C2×#15688969C211yesmaximal, characteristic, complemented
5#20241650C212yesmaximal, complemented
6#20241645C212yesmaximal, complemented
7#20241637C212yesmaximal
8#20241633C212yesmaximal
9C4×C47⋊#95C211yesmaximal, characteristic, complemented
10C4×C47⋊#95C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C47C211yesmaximal, characteristic
12C22×C4⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C4×Dic47C2211yescharacteristic
16C22×C4×Dic47C2211yescharacteristic
17C22×C4×Dic47C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19C4×C4⋊Dic47—24no—
20C4×C22⋊Dic47—24no—
21C4×C22⋊Dic47—24no—
22#15688969—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×C22⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47C2211yescharacteristic
35C22×C42×C47C2211yescharacteristic
36C22×C42×C47C2211yescharacteristic
37C4×C22⋊C4×C47C418yescomplemented
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2211yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C23×Dic47C2312yes—
44C2×C4⋊Dic47—24no—
45C2×C4⋊Dic47—24no—
46C2×C4⋊Dic47—24no—
47C2×C22⋊Dic47—24no—
48C2×C22⋊Dic47—24no—
49C2×C22⋊Dic47—24no—
50C2×C22⋊Dic47—24no—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47—24no—
53C2×C4×Dic47—22no—
54C2×C4×Dic47—22no—
55C2×C4×Dic47—22no—
56C2×C4×Dic47—22no—
57C2×C4×Dic47—22no—
58C2×C4×Dic47—22no—
59C2×C4×Dic47—22no—
60C2×C4×Dic47—22no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47D414yes—
64C2×C4×Dic47D414yes—
65C47⋊#95—24no—
66C47⋊#95—24no—
67C47⋊#95—24no—
68C47⋊#95—24no—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C25×C47C2311yescharacteristic
75C23×C4×C47—22no—
76C23×C4×C47—22no—
77C23×C4×C47—22no—
78C23×C4×C47—22no—
79C23×C4×C47—22no—
80C23×C4×C47—22no—
81C23×C4×C47C2×C414yes—
82C23×C4×C47C2311yescharacteristic
83C23×C4×C47C2311yescharacteristic
84C23×C4×C47C2311yescharacteristic
85C23×C4×C47C2311yescharacteristic
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C2×C22⋊C4×C47—24no—
89C2×C22⋊C4×C47—24no—
90C2×C22⋊C4×C47C2×C414yes—
91C2×C22⋊C4×C47C2×C414yes—
92C2×C42×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47—22no—
95C2×C42×C47—22no—
96C2×C42×C47C2×C414yes—
97C2×C42×C47C2×C414yes—
98#95×C47C2×C414yes—
99#95×C47C2×C414yes—
100C24×C47—21no—
101C24×C47—21no—
102C24×C47—22no—
103C24×C47—21no—
104C24×C47—21no—
105C24×C47—21no—
106C24×C47—21no—
107C24×C47—22no—
108C24×C47—22no—
109C24×C47C22×C412yes—
110C24×C47#4114yes—
111C24×C47C2411yescharacteristic
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
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—22no—
125C22×C4×C47—22no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—28no—
134C22×C4×C47—28no—
135C22×C4×C47—28no—
136C22×C4×C47—44no—
137C22×C4×C47—44no—
138C22×C4×C47—44no—
139C22×C4×C47—44no—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C22×C412yes—
142C22×C4×C47C22×C412yes—
143C22×C4×C47C22×C412yes—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47#4114yes—
149C22×C4×C47#4112yes—
150C22×C4×C47#4112yes—
151C22×C4×C47#4112yes—
152C22×C4×C47#4112yes—
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×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—22no—
174C22×Dic47—22no—
175C22×Dic47—22no—
176C22×Dic47—22no—
177C22×Dic47C2×D414yes—
178C22×Dic47C2×D414yes—
179C22×Dic47#4114yes—
180C22×Dic47#4114yes—
181C22⋊C4×C47—28no—
182C22⋊C4×C47—28no—
183C42×C47—28no—
184C42×C47—28no—
185C22⋊Dic47—216no—
186C22⋊Dic47—216no—
187C4⋊Dic47—216no—
188C4×Dic47—28no—
189C4×Dic47—28no—
190C4×Dic47—44no—
191C4×Dic47—44no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—24no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—42no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—42no—
212C23×C47—42no—
213C23×C47—42no—
214C23×C47#11714yes—
215C23×C47C23×C411yescharacteristic
216C23×C47C22×D411yescharacteristic
217C23×C47C22×D411yescharacteristic
218C23×C47C2×#4112yes—
219C23×C47C2×#4112yes—
220C23×C47C2×#4111yescharacteristic
221C23×C47C2×#4111yescharacteristic
222C23×C47C2×#4111yescharacteristic
223C23×C47C2×#4111yescharacteristic
224C23×C47C2×#4112yes—
225C23×C47C2×#4112yes—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—416no—
253C2×C4×C47#11714yes—
254C2×C4×C47#11714yes—
255C2×C4×C47C4×D414yes—
256C2×C4×C47C4×D414yes—
257C2×C4×C47#12114yes—
258C2×C4×C47#12314yes—
259C2×C4×C47#12414yes—
260C2×C4×C47#12614yes—
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—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—42no—
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—
289#13842—471nomaximal, Sylow
290C22×C47—24no—
291C22×C47—24no—
292C22×C47—24no—
293C22×C47—44no—
294C22×C47—44no—
295C22×C47—44no—
296C22×C47—44no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—42no—
300C22×C47—42no—
301C22×C47C2×#11711yescharacteristic
302C22×C47C2×#11711yescharacteristic
303C22×C47C2×#11711yescharacteristic
304C22×C47C2×C4×D411yescharacteristic
305C22×C47C2×C4×D411yescharacteristic
306C22×C47C4×#4111yescharacteristic
307C22×C47C4×#4111yescharacteristic
308C22×C47C2×#12111yescharacteristic
309C22×C47C2×#12311yescharacteristic
310C22×C47#52512yes—
311C22×C47#52512yes—
312C22×C47#52512yes—
313C22×C47C2×#12411yescharacteristic
314C22×C47C2×#12611yescharacteristic
315C22×C47#52912yes—
316C22×C47#52912yes—
317C22×C47#52912yes—
318C22×C47#52912yes—
319C22×C47#52912yes—
320C22×C47#52912yes—
321C22×C47#52912yes—
322C22×C47#52912yes—
323C22×C47#53212yes—
324C4×C47—24no—
325C4×C47—42no—
326C4×C47—42no—
327C4×C47—42no—
328C4×C47—42no—
329C4×C47#38818yes—
330Dic47—44no—
331Dic47—44no—
332Dic47—44no—
333Dic47—44no—
334C2×C4×C4⋊C4—471no—
335C2×C4×C22⋊C4—471no—
336C2×C4×C22⋊C4—471no—
337C2×C4×C22⋊C4D4711yescharacteristic
338C2×#376—471no—
339#1443—472no—
340#1442—472no—
341#1440—472no—
342#1439—472no—
343C4×#95—471no—
344C4×#95—471no—
345C2×C47—44no—
346C2×C47#228311yescharacteristic
347C2×C47C2×#38811yescharacteristic
348C2×C47#229011yescharacteristic
349C2×C47#229011yescharacteristic
350C2×C47#229011yescharacteristic
351C2×C47#229311yescharacteristic
352C2×C47#229811yescharacteristic
353C2×C47#239212yes—
354C2×C47#239512yes—
355C2×C47#239812yes—
356C2×C47#240112yes—
357C24×C4C2×D4711yescharacteristic
358C22×C4⋊C4—472no—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—472no—
361C22×C22⋊C4C2×D4711yescharacteristic
362C22×C22⋊C4C2×D4711yescharacteristic
363C22×C42—471no—
364C22×C42—471no—
365C22×C42—471no—
366C22×C42—471no—
367C22×C42C2×D4711yescharacteristic
368C22×C42C2×D4711yescharacteristic
369C4×C4⋊C4—944no—
370C4×C22⋊C4—944no—
371C4×C22⋊C4—944no—
372C4×C22⋊C4Dic4718yescomplemented
373#376—944no—
374C2×#95—472no—
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×#95C2×D4711yescharacteristic
384C2×#95C2×D4711yescharacteristic
385C47#1384211yescharacteristic, complemented, Sylow
386C25C22×D4711yescharacteristic
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—472no—
394C23×C4—472no—
395C23×C4—472no—
396C23×C4—472no—
397C23×C4C2×Dic4714yes—
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401C23×C4C22×D4711yescharacteristic
402C23×C4C22×D4711yescharacteristic
403C23×C4C22×D4711yescharacteristic
404C2×C4⋊C4—944no—
405C2×C4⋊C4—944no—
406C2×C4⋊C4—944no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—944no—
410C2×C22⋊C4—944no—
411C2×C22⋊C4—944no—
412C2×C22⋊C4—944no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—944no—
415C2×C22⋊C4C2×Dic4714yes—
416C2×C22⋊C4C2×Dic4714yes—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42—474no—
422C2×C42—474no—
423C2×C42—942no—
424C2×C42—942no—
425C2×C42—942no—
426C2×C42—942no—
427C2×C42—942no—
428C2×C42—942no—
429C2×C42—942no—
430C2×C42—942no—
431C2×C42—942no—
432C2×C42—942no—
433C2×C42C2×Dic4714yes—
434C2×C42C2×Dic4714yes—
435#95—944no—
436#95—944no—
437#95—944no—
438#95—944no—
439#95—944no—
440#95—944no—
441#95—944no—
442#95—944no—
443#95—944no—
444#95C2×Dic4714yes—
445#95C2×Dic4714yes—
446C24—21no—
447C24—21no—
448C24—22no—
449C24—22no—
450C24—21no—
451C24—21no—
452C24—22no—
453C24—21no—
454C24—21no—
455C24#1062535014yes—
456C24C22×Dic4712yes—
457C24C23×D4711yescharacteristic
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—474no—
487C22×C4—474no—
488C22×C4—474no—
489C22×C4—474no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
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—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4#1062535014yes—
515C22×C4#1062535012yes—
516C22×C4#1062535012yes—
517C22×C4D4×D4712yes—
518C22×C4D4×D4712yes—
519C22×C4#1062535212yes—
520C22×C4#1062535212yes—
521C22×C4C22×Dic4712yes—
522C22×C4C22×Dic4712yes—
523C22×C4C22×Dic4712yes—
524C22×C4C22×Dic4712yes—
525C22×C4C22×Dic4712yes—
526C22×C4C22×Dic4712yes—
527C4⋊C4—9416no—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—9416no—
531C22⋊C4—9416no—
532C42—28no—
533C42—28no—
534C42—948no—
535C42—948no—
536C42—1884no—
537C42—1884no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23#1309398314yes—
561C23C23×Dic4711yescharacteristic
562C23C2×D4×D4711yescharacteristic
563C23C2×D4×D4711yescharacteristic
564C23#41×D4712yes—
565C23#41×D4712yes—
566C23C2×#1062535211yescharacteristic
567C23C2×#1062535211yescharacteristic
568C23C2×#1062535011yescharacteristic
569C23C2×#1062535011yescharacteristic
570C23C2×#1062535012yes—
571C23C2×#1062535012yes—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—416no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—944no—
606C2×C4—944no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4—1882no—
619C2×C4—1882no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1882no—
623C2×C4—1882no—
624C2×C4—1882no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4#1309398314yes—
628C2×C4#1309398314yes—
629C2×C4D4×Dic4714yes—
630C2×C4D4×Dic4714yes—
631C2×C4#1309399714yes—
632C2×C4#1309400614yes—
633C2×C4#1309401014yes—
634C2×C4#1309401714yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22C2×#1309398311yescharacteristic
647C22C2×#1309398311yescharacteristic
648C22C2×#1309398311yescharacteristic
649C22C2×D4×Dic4711yescharacteristic
650C22C2×D4×Dic4711yescharacteristic
651C22#41×Dic4711yescharacteristic
652C22#41×Dic4711yescharacteristic
653C22C2×#1309399711yescharacteristic
654C22C2×#1309400611yescharacteristic
655C22#1568963212yes—
656C22#1568963212yes—
657C22#1568963212yes—
658C22C2×#1309401011yescharacteristic
659C22C2×#1309401711yescharacteristic
660C22#1568965112yes—
661C22#1568965112yes—
662C22#1568965312yes—
663C22#1568965312yes—
664C22#1568965312yes—
665C22#1568965512yes—
666C22#1568965712yes—
667C22#1568965712yes—
668C22#1568967012yes—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—1884no—
675C4—1884no—
676C4—1884no—
677C4—1884no—
678C4#1568901618yes—
679C2—44no—
680C2#2024578611yescharacteristic
681C2C2×#1568901611yescharacteristic
682C2#2024583311yescharacteristic
683C2#2024583311yescharacteristic
684C2#2024583311yescharacteristic
685C2#2024585311yescharacteristic
686C2#2024588611yescharacteristic
687C2#2024663212yes—
688C2#2024664912yes—
689C2#2024667512yes—
690C2#2024668812yes—
subgroupG/H
centerC23×C4C22×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C4×C22⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#13842
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊Dic47C2semidirect
4. C2×#15688969C2semidirect
5. #20241650C2semidirect
6. #20241645C2semidirect
9. C4×C47⋊#95C2semidirect
10. C4×C47⋊#95C2semidirect
37. C4×C22⋊C4×C47C4semidirect
372. C4×C22⋊C4Dic47semidirect
385. C47#13842semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Dic472
C2×C4×C22⋊Dic472
C2×#156889692
#202416502
#202416452
#202416372
#202416332
C4×C47⋊#952
C2×C4×C22⋊C4×C472
#1384247