Finite Group Databasebeta

← #26221165

Subgroups — #26221165

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689178C211yesmaximal, characteristic
2#20245792C211yesmaximal, characteristic
3C4×C4⋊C4×D47C211yesmaximal, characteristic
4#20244150C211yesmaximal, characteristic, complemented
5#20244148C211yesmaximal, characteristic, complemented
6#20244145C211yesmaximal, characteristic, complemented
7#20244143C211yesmaximal, characteristic, complemented
8#20243582C211yesmaximal, characteristic
9#20243579C211yesmaximal, characteristic
10#20243577C211yesmaximal, characteristic
11#20243574C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13#20243286C211yesmaximal, characteristic, complemented
14#20243285C211yesmaximal, characteristic, complemented
15C47×#1776C211yesmaximal, characteristic, complemented
16C2×#13093981—21no—
17C2×C4⋊C4×D47C2211yescharacteristic
18C2×C4⋊C4×D47C2211yescharacteristic
19C2×C42×D47—21no—
20C2×C42×D47C2211yescharacteristic
21#15689178C418yescomplemented
22C2×#13093958C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093956C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28C2×Dic47⋊C8C2211yescharacteristic
29C2×Dic47⋊C8C2211yescharacteristic
30C2×(C2×D47)⋊C8C2211yescharacteristic
31C2×(C2×D47)⋊C8C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33#15688992C2211yescharacteristic
34#15688992C2211yescharacteristic
35#15688987—21no—
36#15688982—21no—
37#15688980C2211yescharacteristic
38#15688980C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4×Dic47⋊C4—21no—
42C4×Dic47⋊C4—21no—
43C4×C4⋊Dic47C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45#15688974—21no—
46#95×D47—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D411yescharacteristic
64C22×C4×D47C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13093981—44no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C4⋊C4×D47—22no—
79C4⋊C4×D47—22no—
80C4⋊C4×D47—22no—
81C4⋊C4×D47—22no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—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
91C42×D47—44no—
92C42×D47C2×C414yes—
93C42×D47Q814yes—
94#13093958—22no—
95#13093957—22no—
96#13093957—22no—
97#13093956—22no—
98#13093955—22no—
99#13093955—22no—
100Dic47⋊C8C2×C414yescomplemented
101Dic47⋊C8C2×C414yescomplemented
102(C2×D47)⋊C8C2×C414yes—
103(C2×D47)⋊C8C2×C414yes—
104#13093935C2×C414yescomplemented
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C42×C47—21no—
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120#107×C47—22no—
121#106×C47—22no—
122#97×C47C2×C414yescomplemented
123#95×C47—21no—
124#95×C47—21no—
125C23×D47C2×D411yescharacteristic
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C2×C4×D47—22no—
141C2×C4×D47—22no—
142C2×C4×D47—21no—
143C2×C4×D47—21no—
144C2×C4×D47—21no—
145C2×C4×D47—21no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—24no—
149C2×C4×D47—24no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47—42no—
153C2×C4×D47—42no—
154C2×C4×D47C4⋊C414yes—
155C2×C4×D47C4⋊C414yes—
156C2×C4×D47C22×C412yes—
157C2×C4×D47C2×Q812yes—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47C22×C412yes—
163C2×C8×C47C22×C412yes—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47#4112yes—
171C4⋊C4×C47#4112yes—
172C42×C47—42no—
173C42×C47—42no—
174C42×C47C22×C412yes—
175C42×C47C2×Q812yes—
176(C2×D47)⋊C4—42no—
177(C2×D47)⋊C4—42no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47#4112yes—
185C4⋊Dic47#4112yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—24no—
188Dic47⋊C4—42no—
189Dic47⋊C4—42no—
190Dic47⋊C4—42no—
191Dic47⋊C4—42no—
192C4×Dic47—42no—
193C4×Dic47—42no—
194C4×Dic47—42no—
195C4×Dic47—42no—
196C4×Dic47C4⋊C414yes—
197C4×Dic47C4⋊C414yes—
198C4×Dic47C22×C412yes—
199C4×Dic47C2×Q812yes—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8C22×C412yes—
205C2×C47⋊C8C22×C412yes—
206C23×C47C22×D411yescharacteristic
207C22×D47—21no—
208C22×D47—21no—
209C22×D47—21no—
210C22×D47—21no—
211C22×D47C2×C4⋊C412yes—
212C22×D47AGL(1,ℤ/8)12yes—
213C22×D47#13712yes—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
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×C47C2×C4⋊C412yes—
231C2×C4×C47C2×C4⋊C412yes—
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C22×Q811yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47C2×C4⋊C412yes—
255C2×Dic47C2×C4⋊C412yes—
256C2×Dic47C2×C4⋊C412yes—
257C2×Dic47AGL(1,ℤ/8)12yes—
258C2×Dic47#13712yes—
259C4×D47—28no—
260C4×D47—22no—
261C4×D47—22no—
262C4×D47—22no—
263C4×D47—22no—
264C4×D47—28no—
265C4×D47—48no—
266C4×D47—48no—
267C8×C47—22no—
268C8×C47—22no—
269C47⋊C8—22no—
270C47⋊C8—22no—
271#21365—471nomaximal, Sylow
272C22×C47C22×C4⋊C411yescharacteristic
273C22×C47#52511yescharacteristic
274C22×C47#52511yescharacteristic
275C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C47C2×#13711yescharacteristic
278C22×C47C2×#13711yescharacteristic
279C2×D47—22no—
280C2×D47—22no—
281C2×D47—22no—
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—22no—
285C2×D47#42814yes—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—42no—
289C4×C47—42no—
290C4×C47—81no—
291C4×C47—81no—
292C4×C47C4×#4111yescharacteristic
293C4×C47C4×#4111yescharacteristic
294C4×C47#53311yescharacteristic
295C4×C47#53311yescharacteristic
296Dic47—22no—
297Dic47—22no—
298Dic47—42no—
299Dic47—42no—
300Dic47—81no—
301Dic47—81no—
302Dic47#42814yes—
303C2×#432—471no—
304#2285—471no—
305C2×C4×C4⋊C4—471no—
306#1948—471no—
307#1948—471no—
308#1947—471no—
309#1947—471no—
310#1853—471no—
311#1853—471no—
312#1852—471no—
313#1852—471no—
314#1776—471no—
315#1776—471no—
316#1776—471no—
317#1776D4711yescharacteristic, complemented
318C2×C47#229611yescharacteristic
319C2×C47C2×#42811yescharacteristic
320C2×C47C2×#42811yescharacteristic
321C2×C47#310911yescharacteristic
322C2×C47#311011yescharacteristic
323C2×C47#311211yescharacteristic
324C2×C47#311311yescharacteristic
325D47—24no—
326C2×#117—941no—
327C22×C4⋊C4—471no—
328C22×C4⋊C4—471no—
329C22×C42—471no—
330C22×C42—941no—
331#432—478no—
332C2×#107—471no—
333C2×#107—471no—
334C2×#107—471no—
335C2×#107C2×D4711yescharacteristic
336C2×#106—471no—
337C2×#106—471no—
338C2×#106—471no—
339C2×#106C2×D4711yescharacteristic
340C2×C4⋊C8—471no—
341C2×C4⋊C8—471no—
342C2×C22⋊C8—471no—
343C2×C22⋊C8—471no—
344C2×#97—471no—
345C2×#97C2×D4711yescharacteristic
346#384—471no—
347#384—471no—
348#382—941no—
349#380—471no—
350#380—471no—
351#380—941no—
352C4×C4⋊C4—471no—
353C4×C4⋊C4—471no—
354C4×C4⋊C4—471no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—941no—
357C4×C4⋊C4—941no—
358C4×C4⋊C4—941no—
359C4×C4⋊C4C2×D4711yescharacteristic
360C4×C4⋊C4C2×D4711yescharacteristic
361C2×#95—941no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340C2×D4711yescharacteristic
369#340C2×D4711yescharacteristic
370C47#2136511yescharacteristic, complemented, Sylow
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—941no—
375C23×C4—941no—
376C22×C8—471no—
377C22×C8—471no—
378C22×C8C22×D4711yescharacteristic
379C22×C8C22×D4711yescharacteristic
380#117—1884no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42—471no—
406C2×C42—474no—
407C2×C42—474no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42—941no—
411C2×C42—941no—
412C2×C42—1884no—
413C2×C42C22×D4711yescharacteristic
414#107—22no—
415#107—942no—
416#107—942no—
417#107—942no—
418#106—22no—
419#106—942no—
420#106—942no—
421#106—942no—
422C4⋊C8—474no—
423C4⋊C8—474no—
424C22⋊C8—474no—
425C22⋊C8—474no—
426#97—474no—
427#97C4×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—471no—
443C22×C4—471no—
444C22×C4—474no—
445C22×C4—474no—
446C22×C4—472no—
447C22×C4—472no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—1882no—
463C22×C4—1882no—
464C22×C4—1882no—
465C22×C4—1882no—
466C22×C4D4×D4711yescharacteristic
467C22×C4D4×D4711yescharacteristic
468C22×C4C23×D4711yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—472no—
474C2×C8—472no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8C2×C4×D4712yes—
480C2×C8C2×C4×D4712yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—472no—
488C4⋊C4—472no—
489C4⋊C4—944no—
490C4⋊C4—944no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—941no—
497C4⋊C4—1882no—
498C4⋊C4—1882no—
499C4⋊C4—1882no—
500C4⋊C4—1882no—
501C4⋊C4#1062535012yes—
502C4⋊C4#1062535012yes—
503C22⋊C4—1882no—
504C22⋊C4—1882no—
505C42—42no—
506C42—42no—
507C42—474no—
508C42—474no—
509C42—472no—
510C42—472no—
511C42—1882no—
512C42—1882no—
513C42—1882no—
514C42—1882no—
515C42C2×C4×D4712yes—
516C42Q8×D4712yes—
517C23—472no—
518C23—472no—
519C23—472no—
520C23—941no—
521C23—941no—
522C23—941no—
523C23—941no—
524C23C2×D4×D4711yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—472no—
546C2×C4—948no—
547C2×C4—948no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—942no—
553C2×C4—942no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—1888no—
559C2×C4—1888no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4C4⋊C4×D4712yes—
571C2×C4C4⋊C4×D4712yes—
572C2×C4C22×C4×D4711yescharacteristic
573C2×C4C2×Q8×D4711yescharacteristic
574C2×C4#41×D4711yescharacteristic
575C2×C4#41×D4711yescharacteristic
576C2×C4C2×#1062535011yescharacteristic
577C2×C4C2×#1062535011yescharacteristic
578C8—22no—
579C8—22no—
580C8—942no—
581C8—942no—
582C22—474no—
583C22—942no—
584C22—942no—
585C22—942no—
586C22—942no—
587C22—942no—
588C22—942no—
589C22C2×C4⋊C4×D4711yescharacteristic
590C22#1568963211yescharacteristic
591C22#1568963211yescharacteristic
592C22AGL(1,ℤ/8)×D4711yescharacteristic
593C22AGL(1,ℤ/8)×D4711yescharacteristic
594C22#137×D4711yescharacteristic
595C22#137×D4711yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—474no—
603C4—942no—
604C4—942no—
605C4—3761no—
606C4—3761no—
607C4—1882no—
608C4—1882no—
609C4#1568959111yescharacteristic
610C4C4×#1062535011yescharacteristic
611C4#1568967711yescharacteristic
612C4#1568967711yescharacteristic
613C2—944no—
614C2#2024586811yescharacteristic
615C2#428×D4711yescharacteristic
616C2#428×D4711yescharacteristic
617C2#2025347711yescharacteristic
618C2#2025348511yescharacteristic
619C2#2025350811yescharacteristic
620C2#2025351611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#21365
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #20244150C2semidirect
5. #20244148C2semidirect
6. #20244145C2semidirect
7. #20244143C2semidirect
12. #20243286C2semidirect
13. #20243286C2semidirect
14. #20243285C2semidirect
15. C47×#1776C2semidirect
21. #15689178C4semidirect
100. Dic47⋊C8C2×C4semidirect
101. Dic47⋊C8C2×C4semidirect
104. #13093935C2×C4semidirect
122. #97×C47C2×C4semidirect
317. #1776D47semidirect
370. C47#21365semidirect
427. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891782
#202457922
C4×C4⋊C4×D472
#202441502
#202441482
#202441452
#202441432
#202435822
#202435792
#202435772
#202435742
#202432862
#202432852
C47×#17762
#2136547