Finite Group Databasebeta

← #26218451

Subgroups — #26218451

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689162C211yesmaximal, characteristic
2#20245869C211yesmaximal, characteristic
3C4×C4⋊C4×D47C211yesmaximal, characteristic
4#20243979C211yesmaximal, characteristic, complemented
5#20243978C211yesmaximal, characteristic, complemented
6#20243970C211yesmaximal, characteristic, complemented
7#20243969C211yesmaximal, characteristic, complemented
8#20243968C211yesmaximal, characteristic, complemented
9#20243960C211yesmaximal, characteristic, complemented
10#20243959C211yesmaximal, characteristic, complemented
11#20243106C211yesmaximal, characteristic
12#20243106C211yesmaximal, characteristic
13#20243106C211yesmaximal, characteristic
14#20243080C211yesmaximal, characteristic
15C47×#1920C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×C4⋊C4×D47C2211yescharacteristic
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C42×D47—21no—
23C2×C42×D47C2211yescharacteristic
24#15689162C418yescomplemented
25C2×#13093958C2211yescharacteristic
26C2×#13093957C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28#15689022C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689019C2211yescharacteristic
31#15688997—21no—
32#15688994—21no—
33C4⋊C4×Dic47—21no—
34C4⋊C4×Dic47C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic
36C4×Dic47⋊C4—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38#15688974—21no—
39#15688974—21no—
40#95×D47—21no—
41#15688886C2211yescharacteristic
42#15688886C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688878C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#106×C47C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C47×#340C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D411yescharacteristic
64C22×C4×D47C2311yescharacteristic
65C2×#10625324—22no—
66C2×#10625324—22no—
67C2×#10625324—22no—
68C2×#10625324—22no—
69C2×#10625324—22no—
70C2×#10625324C2×C414yes—
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73#13093982—22no—
74#13093982—22no—
75#13093982—22no—
76#13093982C2×C414yes—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C4⋊C4×D47—22no—
87C4⋊C4×D47—22no—
88C4⋊C4×D47C2×C414yes—
89C4⋊C4×D47Q814yes—
90C2×(C2×D47)⋊C4—21no—
91C2×(C2×D47)⋊C4—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47—21no—
97C2×C4×Dic47D411yescharacteristic
98C2×C4×Dic47D411yescharacteristic
99C2×C4×Dic47C2311yescharacteristic
100C42×D47—22no—
101C42×D47—22no—
102C42×D47—44no—
103#13093958C2×C414yescomplemented
104#13093957C2×C414yescomplemented
105#13093955C2×C414yescomplemented
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#106×C47C2×C414yescomplemented
122#95×C47—21no—
123#95×C47—21no—
124C23×D47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C2×C4×D47—22no—
140C2×C4×D47—22no—
141C2×C4×D47—22no—
142C2×C4×D47—24no—
143C2×C4×D47—24no—
144C2×C4×D47—22no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47C22×C412yes—
152C2×C4×D47C2×Q812yes—
153C2×C4×D47#4112yes—
154C2×C4×D47#4112yes—
155C2×C4×D47#4112yes—
156C2×C4×D47#4112yes—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47C22×C412yes—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—42no—
169C4⋊C4×C47C22×C412yes—
170C4⋊C4×C47C22×C412yes—
171C4⋊C4×C47C2×Q812yes—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—42no—
175(C2×D47)⋊C4—24no—
176(C2×D47)⋊C4—42no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—42no—
183C4⋊Dic47C22×C412yes—
184C4⋊Dic47C22×C412yes—
185C4⋊Dic47C2×Q812yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—24no—
188Dic47⋊C4—42no—
189Dic47⋊C4—42no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—24no—
193C4×Dic47—22no—
194C4×Dic47—22no—
195C4×Dic47—42no—
196C4×Dic47—42no—
197C4×Dic47—42no—
198C4×Dic47—42no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8C22×C412yes—
205#10625324—28no—
206#10625324—44no—
207C23×C47C22×D411yescharacteristic
208C22×D47—21no—
209C22×D47—21no—
210C22×D47—21no—
211C22×D47—21no—
212C22×D47C4×D412yes—
213C22×D47C22⋊Q812yes—
214C22×D47#12312yes—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47AGL(1,ℤ/8)12yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C22×Q811yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47C4×D412yes—
257C2×Dic47C22⋊Q812yes—
258C2×Dic47#12312yes—
259C2×Dic47#13712yes—
260C4×D47—28no—
261C4×D47—48no—
262C4×D47—44no—
263C4×D47—48no—
264C4×D47#11714yes—
265C4×D47C4×Q814yes—
266C4×D47#12514yes—
267C4×D47#12614yes—
268C8×C47—22no—
269C8×C47—41no—
270C47⋊C8—22no—
271C47⋊C8—41no—
272#21033—471nomaximal, Sylow
273C22×C47C2×C4×D411yescharacteristic
274C22×C47C2×C22⋊Q811yescharacteristic
275C22×C47C2×#12311yescharacteristic
276C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C47C2×#13711yescharacteristic
278C22×C47#57511yescharacteristic
279C22×C47#57511yescharacteristic
280C2×D47—22no—
281C2×D47—22no—
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—22no—
285C2×D47—22no—
286C2×D47#38714yes—
287C4×C47—22no—
288C4×C47—42no—
289C4×C47—41no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47—81no—
293C4×C47C2×#11711yescharacteristic
294C4×C47C2×C4×Q811yescharacteristic
295C4×C47C2×#12511yescharacteristic
296C4×C47C2×#12611yescharacteristic
297Dic47—22no—
298Dic47—42no—
299Dic47—41no—
300Dic47—42no—
301Dic47—42no—
302Dic47—81no—
303Dic47#38714yes—
304C2×#428—471no—
305#2296—471no—
306C2×C4×C4⋊C4—471no—
307#1921—471no—
308#1921—471no—
309#1920—471no—
310#1920D4711yescharacteristic, complemented
311#1919—471no—
312#1919—471no—
313#1918—471no—
314#1918—471no—
315#1736—471no—
316#1736—471no—
317#1736—471no—
318#1728—471no—
319C2×C47C2×#38711yescharacteristic
320C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
321C2×C47C4×#13711yescharacteristic
322C2×C47#307711yescharacteristic
323C2×C47#307811yescharacteristic
324C2×C47#309211yescharacteristic
325C2×C47#309311yescharacteristic
326D47—24no—
327C22×M16—471no—
328C22×M16—471no—
329C2×#117—471no—
330C2×#117—471no—
331C22×C4⋊C4—471no—
332C22×C4⋊C4—471no—
333C22×C42—471no—
334C22×C42—941no—
335#428—478no—
336C2×#107—471no—
337C2×#107—471no—
338C2×#106—471no—
339C2×#106C2×D4711yescharacteristic
340#389—471no—
341#389—471no—
342#389—471no—
343#389C2×D4711yescharacteristic
344#385—941no—
345#385—941no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—941no—
350C4×C4⋊C4—941no—
351C4×C4⋊C4—941no—
352C4×C4⋊C4—941no—
353C4×C4⋊C4C2×D4711yescharacteristic
354C2×#95—941no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340C2×D4711yescharacteristic
365#340C2×D4711yescharacteristic
366#340C2×D4711yescharacteristic
367#336—471no—
368#336—471no—
369#336—471no—
370#336C2×D4711yescharacteristic
371C47#2103311yescharacteristic, complemented, Sylow
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C23×C4—941no—
376C23×C4—941no—
377C2×M16—474no—
378C2×M16—942no—
379C2×M16—942no—
380C2×M16—942no—
381C2×M16—942no—
382C2×M16—942no—
383C22×C8—471no—
384C22×C8—471no—
385C22×C8C22×D4711yescharacteristic
386C22×C8C22×D4711yescharacteristic
387#117—474no—
388#117—942no—
389#117—942no—
390#117—942no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—474no—
396C2×C4⋊C4—474no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—942no—
399C2×C4⋊C4—942no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4C22×D4711yescharacteristic
406C2×C4⋊C4C22×D4711yescharacteristic
407C2×C4⋊C4C22×D4711yescharacteristic
408C2×C4⋊C4C22×D4711yescharacteristic
409C2×C22⋊C4—941no—
410C2×C22⋊C4—941no—
411C2×C42—21no—
412C2×C42—471no—
413C2×C42—471no—
414C2×C42—471no—
415C2×C42—942no—
416C2×C42—942no—
417C2×C42—941no—
418C2×C42—941no—
419C2×C42—941no—
420C2×C42—941no—
421C2×C42—941no—
422C2×C42—1884no—
423C2×C42C22×D4711yescharacteristic
424#107—474no—
425#107—474no—
426#106—474no—
427#106C4×D4714yescomplemented
428#95—21no—
429#95—21no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436C24—471no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—471no—
442C22×C4—472no—
443C22×C4—472no—
444C22×C4—472no—
445C22×C4—472no—
446C22×C4—472no—
447C22×C4—472no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—944no—
454C22×C4—944no—
455C22×C4—942no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—1881no—
461C22×C4—1881no—
462C22×C4—1882no—
463C22×C4—1882no—
464C22×C4—1882no—
465C22×C4—1882no—
466C22×C4D4×D4711yescharacteristic
467C22×C4D4×D4711yescharacteristic
468C22×C4C23×D4711yescharacteristic
469M16—948no—
470M16—1884no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—472no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8C2×C4×D4712yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—472no—
490C4⋊C4—472no—
491C4⋊C4—472no—
492C4⋊C4—944no—
493C4⋊C4—944no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—1882no—
500C4⋊C4—1882no—
501C4⋊C4—1882no—
502C4⋊C4C2×C4×D4712yes—
503C4⋊C4C2×C4×D4712yes—
504C4⋊C4Q8×D4712yes—
505C22⋊C4—944no—
506C22⋊C4—1882no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—942no—
511C42—942no—
512C42—944no—
513C42—941no—
514C42—941no—
515C42—1882no—
516C42—1882no—
517C42—1882no—
518C42—1882no—
519C23—472no—
520C23—472no—
521C23—472no—
522C23—941no—
523C23—941no—
524C23—941no—
525C23—941no—
526C23C2×D4×D4711yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—472no—
545C2×C4—472no—
546C2×C4—472no—
547C2×C4—474no—
548C2×C4—474no—
549C2×C4—474no—
550C2×C4—474no—
551C2×C4—472no—
552C2×C4—948no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—1888no—
562C2×C4—1888no—
563C2×C4—1884no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4#1309405112yes—
574C2×C4C22×C4×D4711yescharacteristic
575C2×C4C2×Q8×D4711yescharacteristic
576C2×C4#41×D4711yescharacteristic
577C2×C4#41×D4711yescharacteristic
578C2×C4#41×D4711yescharacteristic
579C2×C4#41×D4711yescharacteristic
580C8—22no—
581C8—41no—
582C8—942no—
583C8—1881no—
584C22—474no—
585C22—942no—
586C22—942no—
587C22—942no—
588C22—942no—
589C22—942no—
590C22—942no—
591C22C4×D4×D4711yescharacteristic
592C22C22⋊Q8×D4711yescharacteristic
593C22#123×D4711yescharacteristic
594C22C2×#1309405111yescharacteristic
595C22#137×D4711yescharacteristic
596C22#1568991911yescharacteristic
597C22#1568992011yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—474no—
605C4—942no—
606C4—3761no—
607C4—1882no—
608C4—1882no—
609C4—1882no—
610C4—1881no—
611C4#117×D4711yescharacteristic
612C4C4×Q8×D4711yescharacteristic
613C4#125×D4711yescharacteristic
614C4#126×D4711yescharacteristic
615C2—944no—
616C2#387×D4711yescharacteristic
617C2C4×#1309405111yescharacteristic
618C2#2025221011yescharacteristic
619C2#2025318611yescharacteristic
620C2#2025319111yescharacteristic
621C2#2025331211yescharacteristic
622C2#2025331611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1920C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1920D47
hypocenterC47#21033
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. #20243979C2semidirect
5. #20243978C2semidirect
6. #20243970C2semidirect
7. #20243969C2semidirect
8. #20243968C2semidirect
9. #20243960C2semidirect
10. #20243959C2semidirect
15. C47×#1920C2semidirect
24. #15689162C4semidirect
103. #13093958C2×C4semidirect
104. #13093957C2×C4semidirect
105. #13093955C2×C4semidirect
121. #106×C47C2×C4semidirect
310. #1920D47semidirect
371. C47#21033semidirect
427. #106C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891622
#202458692
C4×C4⋊C4×D472
#202439792
#202439782
#202439702
#202439692
#202439682
#202439602
#202439592
#202431062
#202430802
C47×#19202
#2103347