Finite Group Databasebeta

← #26223508

Subgroups — #26223508

Order 12032.

960 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689444C211yesmaximal, characteristic
2#121×Dic47C211yesmaximal, characteristic
3#117×Dic47C211yesmaximal, characteristic, complemented
4#20244018C211yesmaximal, characteristic, complemented
5#20244011C211yesmaximal, characteristic, complemented
6#20243969C211yesmaximal, characteristic, complemented
7#20243962C211yesmaximal, characteristic, complemented
8#20243587C211yesmaximal, characteristic, complemented
9#20243484C211yesmaximal, characteristic, complemented
10#20243284C211yesmaximal, characteristic, complemented
11#20243284C211yesmaximal, characteristic, complemented
12#20243215C211yesmaximal, characteristic, complemented
13#20243215C211yesmaximal, characteristic, complemented
14#20243152C211yesmaximal, characteristic, complemented
15C47×#1875C211yesmaximal, characteristic
16C2×D4×Dic47—21no—
17C2×D4×Dic47C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C22×C4×Dic47C2211yescharacteristic
20#15689444C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×#13093934C2211yescharacteristic
30#15689026C2211yescharacteristic
31#15689022C2211yescharacteristic
32#15689016—21no—
33#15689003—21no—
34#15688983C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C22⋊C4×Dic47—21no—
41C22⋊C4×Dic47—21no—
42C22⋊C4×Dic47—21no—
43#15688969—21no—
44C42×Dic47—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#121×C47C2211yescharacteristic
54C2×#117×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C2×C22⋊C8×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47—21no—
61C23×Dic47D411yescharacteristic
62C23×Dic47D411yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65D4×Dic47—22no—
66D4×Dic47—22no—
67D4×Dic47—44no—
68#13093983—22no—
69#13093983C2×C414yescomplemented
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92#13093957—22no—
93#13093957C2×C414yescomplemented
94#13093955—22no—
95#13093955C2×C414yescomplemented
96#13093945C2×C414yescomplemented
97#13093945C2×C414yescomplemented
98C22⋊(C47⋊C8)C2×C414yescomplemented
99#13093934—22no—
100#13093934—22no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×D4×C47—21no—
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113#121×C47—22no—
114#121×C47C2×C414yes—
115#117×C47—22no—
116#117×C47—22no—
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122C2×C42×C47—21no—
123#102×C47—22no—
124#102×C47—22no—
125C22⋊C8×C47—22no—
126C24×C47—21no—
127C24×C47C2×D411yescharacteristic
128C2×D4×C47—24no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—42no—
135C2×D4×C47—42no—
136C2×D4×C47C22×C412yes—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47#4112yes—
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—41no—
156C22×Dic47—41no—
157C22×Dic47—41no—
158C22×Dic47—42no—
159C22×Dic47—42no—
160C22×Dic47—42no—
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D411yescharacteristic
164C22×Dic47C2×D411yescharacteristic
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47C22×C412yes—
175C4⋊C4×C47#4112yes—
176C4⋊C4×C47#4112yes—
177C22⋊C4×C47—24no—
178C22⋊C4×C47—24no—
179C42×C47—24no—
180C22⋊Dic47—24no—
181C22⋊Dic47—42no—
182C22⋊Dic47—42no—
183C22⋊Dic47—42no—
184C22⋊Dic47—42no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—42no—
192C4⋊Dic47C22×C412yes—
193C4⋊Dic47C22×C412yes—
194C4×Dic47—24no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C4×Dic47—42no—
200C4×Dic47—42no—
201C4×Dic47—42no—
202C4×Dic47—42no—
203C4×Dic47—42no—
204C4×Dic47#4112yes—
205C4×Dic47#4112yes—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8C22×C412yes—
211C2×C47⋊C8C22×C412yes—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47—21no—
218C23×C47—22no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47C4×D412yes—
226C23×C47C22×D411yescharacteristic
227D4×C47—22no—
228D4×C47—22no—
229D4×C47—48no—
230D4×C47—41no—
231D4×C47—41no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
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—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47C4×D412yes—
248C2×C4×C47C23×C411yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—44no—
266C2×Dic47—44no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—84no—
278C2×Dic47#12312yes—
279C2×Dic47#12312yes—
280C2×Dic47AGL(1,ℤ/8)12yes—
281C8×C47—41no—
282C8×C47—41no—
283C47⋊C8—22no—
284C47⋊C8—22no—
285#21582—471nomaximal, Sylow
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47—81no—
299C22×C47—81no—
300C22×C47C2×C4×D411yescharacteristic
301C22×C47C2×#12311yescharacteristic
302C22×C47#52511yescharacteristic
303C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C47#57511yescharacteristic
305C22×C47#57511yescharacteristic
306C22×C47#57511yescharacteristic
307C4×C47—22no—
308C4×C47—42no—
309C4×C47—42no—
310C4×C47—42no—
311C4×C47C4×#4111yescharacteristic
312C4×C47C4×#4111yescharacteristic
313C4×C47#52911yescharacteristic
314C4×C47#52911yescharacteristic
315Dic47—22no—
316Dic47—22no—
317Dic47—22no—
318Dic47—41no—
319Dic47—42no—
320Dic47—42no—
321Dic47—81no—
322Dic47—81no—
323C2×#482—471no—
324C4×#121—471no—
325C4×#117—471no—
326#1926—471no—
327#1925—471no—
328#1919—471no—
329#1918—471no—
330#1875D4711yescharacteristic
331#1854—471no—
332#1825—471no—
333#1776—471no—
334#1776—471no—
335#1762—471no—
336#1762—471no—
337#1749—471no—
338C2×C47—42no—
339C2×C47—82no—
340C2×C47C4×#12311yescharacteristic
341C2×C47#294311yescharacteristic
342C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C47#309311yescharacteristic
344C2×C47#309411yescharacteristic
345C2×C47#311111yescharacteristic
346C2×C47#311511yescharacteristic
347C2×#121C2×D4711yescharacteristic
348C2×C4×D4—471no—
349C2×C4×D4—941no—
350C2×#117—471no—
351C2×#117C2×D4711yescharacteristic
352C22×C42—471no—
353#482—478no—
354C2×#107—471no—
355C2×#107—471no—
356C2×#106—471no—
357C2×#106—471no—
358C2×#102—471no—
359C2×#102—471no—
360C2×#102C2×D4711yescharacteristic
361C2×#102C2×D4711yescharacteristic
362C2×C22⋊C8—471no—
363C2×C22⋊C8C2×D4711yescharacteristic
364C2×#97—471no—
365C2×#97—471no—
366#390—471no—
367#389—471no—
368#388—941no—
369#386—941no—
370#380—471no—
371C4×C4⋊C4—471no—
372C4×C4⋊C4—471no—
373C4×C4⋊C4—471no—
374C4×C4⋊C4—471no—
375C4×C4⋊C4—471no—
376C4×C22⋊C4—941no—
377C4×C22⋊C4—941no—
378C4×C22⋊C4—941no—
379#376—941no—
380C43—941no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340C2×D4711yescharacteristic
388#340C2×D4711yescharacteristic
389#336—471no—
390#336—471no—
391C47#2158211yescharacteristic, complemented, Sylow
392C22×D4—21no—
393C22×D4C22×D4711yescharacteristic
394C23×C4—471no—
395C23×C4—471no—
396C23×C4—941no—
397C23×C4C22×D4711yescharacteristic
398C22×C8—471no—
399C22×C8—471no—
400C22×C8C22×D4711yescharacteristic
401C22×C8C22×D4711yescharacteristic
402#121—22no—
403#121C4×D4714yes—
404C4×D4—942no—
405C4×D4—942no—
406C4×D4—1884no—
407#117—22no—
408#117—22no—
409#117—474no—
410#117—942no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C42—21no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—942no—
432C2×C42—942no—
433C2×C42—941no—
434C2×C42—941no—
435C2×C42—941no—
436C2×C42—941no—
437C2×C42—941no—
438C2×C42—941no—
439#107—474no—
440#107—942no—
441#106—474no—
442#106—942no—
443#102—22no—
444#102—22no—
445#102—474no—
446#102—474no—
447C22⋊C8—22no—
448C22⋊C8—474no—
449#97—942no—
450#97—942no—
451#95—941no—
452#95—941no—
453#95—941no—
454#95—941no—
455#95—941no—
456#95—941no—
457#95—941no—
458C24—21no—
459C24D4×D4711yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×C4×D4712yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—1882no—
488C22×C4—1881no—
489C22×C4—1881no—
490C22×C4—1881no—
491C22×C4—1882no—
492C22×C4—1882no—
493C22×C4C2×C4×D4712yes—
494C22×C4D4×D4711yescharacteristic
495C22×C4#1062535212yes—
496C22×C4C23×D4711yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—472no—
504C2×C8—472no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—941no—
508C2×C8—941no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—472no—
513C4⋊C4—472no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—1882no—
521C4⋊C4C2×C4×D4712yes—
522C4⋊C4#1062535212yes—
523C4⋊C4#1062535212yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—944no—
527C22⋊C4—1882no—
528C22⋊C4—1882no—
529C22⋊C4—1882no—
530C22⋊C4—1882no—
531C42—24no—
532C42—472no—
533C42—472no—
534C42—942no—
535C42—942no—
536C42—944no—
537C42—941no—
538C42—941no—
539C42—1882no—
540C42—1882no—
541C42—1882no—
542C42—1882no—
543C42—1882no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23(C22⋊D47)⋊C412yes—
558C23C2×D4×D4711yescharacteristic
559D4—22no—
560D4—22no—
561D4—48no—
562D4—41no—
563D4—41no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—472no—
580C2×C4—472no—
581C2×C4—472no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—3764no—
594C2×C4—1884no—
595C2×C4—1884no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4(C22⋊D47)⋊C412yes—
607C2×C4C22×C4×D4711yescharacteristic
608C2×C4#41×D4711yescharacteristic
609C2×C4#41×D4711yescharacteristic
610C2×C4C2×#1062535211yescharacteristic
611C2×C4C2×#1062535211yescharacteristic
612C2×C4C2×#1062535211yescharacteristic
613C8—41no—
614C8—41no—
615C8—942no—
616C8—942no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×(C22⋊D47)⋊C411yescharacteristic
632C22#123×D4711yescharacteristic
633C22#1568962911yescharacteristic
634C22AGL(1,ℤ/8)×D4711yescharacteristic
635C22#1568991911yescharacteristic
636C22#1568992011yescharacteristic
637C22#1568992011yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—942no—
643C4—942no—
644C4—942no—
645C4—3761no—
646C4—3761no—
647C4—1882no—
648C4—1882no—
649C4—1881no—
650C4C4×#1062535211yescharacteristic
651C4C4×#1062535211yescharacteristic
652C4#1568965411yescharacteristic
653C4#1568965411yescharacteristic
654C2—42no—
655C2—82no—
656C2#2024591311yescharacteristic
657C2#2025218411yescharacteristic
658C2#2025219211yescharacteristic
659C2#2025331611yescharacteristic
660C2#2025332711yescharacteristic
661C2#2025349011yescharacteristic
662C2#2025353211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1875C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
cosocleC22×C4×C47C24
hypercenter#1875D47
hypocenterC47#21582
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
3. #117×Dic47C2semidirect
4. #20244018C2semidirect
5. #20244011C2semidirect
6. #20243969C2semidirect
7. #20243962C2semidirect
8. #20243587C2semidirect
9. #20243484C2semidirect
10. #20243284C2semidirect
11. #20243284C2semidirect
12. #20243215C2semidirect
13. #20243215C2semidirect
14. #20243152C2semidirect
20. #15689444C4semidirect
69. #13093983C2×C4semidirect
93. #13093957C2×C4semidirect
95. #13093955C2×C4semidirect
96. #13093945C2×C4semidirect
97. #13093945C2×C4semidirect
98. C22⋊(C47⋊C8)C2×C4semidirect
391. C47#21582semidirect

Maximal subgroups

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

subgroupindex
C2×#156894442
#121×Dic472
#117×Dic472
#202440182
#202440112
#202439692
#202439622
#202435872
#202434842
#202432842
#202432152
#202431522
C47×#18752
#2158247