Finite Group Databasebeta

← C127×#8331 #44572004

Subgroups — C127×#8331 #44572004

Order 32512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525×C127C211yesmaximal, characteristic
2C127×#1779C211yesmaximal, characteristic, complemented
3C2×#328×C127C212yesmaximal, complemented
4C2×#327×C127C212yesmaximal, complemented
5C127×C22⋊#95C211yesmaximal, characteristic, complemented
6C22×#41×C127C411yescharacteristic, complemented
7C24×C4×C127C2211yescharacteristic
8#525×C127—24no—
9#525×C127—24no—
10C2×#123×C127—21no—
11C2×C22⋊Q8×C127C411yescharacteristic, complemented
12C2×C22⋊Q8×C127C2211yescharacteristic
13C2×#121×C127C411yescharacteristic, complemented
14C2×#121×C127C2211yescharacteristic
15C2×#120×C127—21no—
16C2×C4×D4×C127—21no—
17C2×C4×D4×C127—21no—
18C2×#117×C127C411yescharacteristic, complemented
19C22×C22⋊C4×C127—21no—
20C2×C22⋊C8×C127—21no—
21C2×C22⋊C8×C127—21no—
22C2×C22⋊C8×C127C2212yes—
23C2×#95×C127—21no—
24C2×#95×C127C2212yes—
25#328×C127—24no—
26#327×C127—24no—
27C25×C127D411yescharacteristic
28C2×#41×C127—22no—
29C2×#41×C127—22no—
30C2×#41×C127—42no—
31C2×#41×C127—42no—
32C22×Q8×C127C2×C411yescharacteristic
33C22×D4×C127—21no—
34C22×D4×C127—21no—
35C22×D4×C127—21no—
36C22×D4×C127C2×C411yescharacteristic
37C23×C4×C127—21no—
38C23×C4×C127—22no—
39C23×C4×C127—22no—
40C23×C4×C127—22no—
41C23×C4×C127—42no—
42C23×C4×C127—42no—
43C23×C4×C127C2×C411yescharacteristic
44C23×C4×C127C2×C411yescharacteristic
45C23×C4×C127D411yescharacteristic
46C23×C4×C127D411yescharacteristic
47C23×C4×C127D411yescharacteristic
48C23×C4×C127C2311yescharacteristic
49C22×C8×C127—22no—
50#123×C127—28no—
51C22⋊Q8×C127—22no—
52C22⋊Q8×C127—22no—
53C22⋊Q8×C127—22no—
54C22⋊Q8×C127D414yes—
55#121×C127—22no—
56#121×C127—22no—
57#121×C127—22no—
58#121×C127D414yes—
59#120×C127—28no—
60C4×D4×C127—28no—
61C4×D4×C127—28no—
62#117×C127—22no—
63#117×C127—22no—
64C2×C4⋊C4×C127—21no—
65C2×C4⋊C4×C127—21no—
66C2×C4⋊C4×C127C2×C411yescharacteristic
67C2×C4⋊C4×C127C2×C411yescharacteristic
68C2×C22⋊C4×C127—21no—
69C2×C22⋊C4×C127—21no—
70C2×C22⋊C4×C127—21no—
71C2×C22⋊C4×C127—21no—
72C2×C22⋊C4×C127—21no—
73C2×C22⋊C4×C127—24no—
74C2×C22⋊C4×C127—42no—
75C2×C22⋊C4×C127—42no—
76C2×C42×C127—21no—
77C22⋊C8×C127—22no—
78C22⋊C8×C127—22no—
79C22⋊C8×C127—42no—
80C22⋊C8×C127—42no—
81#95×C127—22no—
82#95×C127—22no—
83#95×C127—42no—
84C24×C127—22no—
85C24×C127—22no—
86C24×C127—21no—
87C24×C127—21no—
88C24×C127—21no—
89C24×C127—22no—
90C24×C127—22no—
91C24×C127—41no—
92C24×C127—41no—
93C24×C127C22⋊C411yescharacteristic
94C24×C127C22⋊C411yescharacteristic
95C24×C127C2×D411yescharacteristic
96#41×C127—48no—
97#41×C127—48no—
98C2×Q8×C127—21no—
99C2×Q8×C127—41no—
100C2×Q8×C127—41no—
101C2×Q8×C127C22⋊C412yescomplemented
102C2×D4×C127—24no—
103C2×D4×C127—24no—
104C2×D4×C127—24no—
105C2×D4×C127—21no—
106C2×D4×C127—41no—
107C2×D4×C127—41no—
108C2×D4×C127—42no—
109C2×D4×C127—42no—
110C2×D4×C127—42no—
111C2×D4×C127—42no—
112C2×D4×C127—42no—
113C2×D4×C127—41no—
114C2×D4×C127—41no—
115C2×D4×C127C22⋊C412yescomplemented
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—21no—
119C22×C4×C127—21no—
120C22×C4×C127—24no—
121C22×C4×C127—22no—
122C22×C4×C127—21no—
123C22×C4×C127—21no—
124C22×C4×C127—21no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
127C22×C4×C127—22no—
128C22×C4×C127—21no—
129C22×C4×C127—21no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—24no—
133C22×C4×C127—41no—
134C22×C4×C127—41no—
135C22×C4×C127—41no—
136C22×C4×C127—41no—
137C22×C4×C127—41no—
138C22×C4×C127—41no—
139C22×C4×C127—41no—
140C22×C4×C127—48no—
141C22×C4×C127—48no—
142C22×C4×C127—42no—
143C22×C4×C127—42no—
144C22×C4×C127—42no—
145C22×C4×C127—42no—
146C22×C4×C127—42no—
147C22×C4×C127—42no—
148C22×C4×C127—42no—
149C22×C4×C127—42no—
150C22×C4×C127C22⋊C411yescharacteristic
151C22×C4×C127C22⋊C411yescharacteristic
152C22×C4×C127C22×C411yescharacteristic
153C22×C4×C127C2×D411yescharacteristic
154C22×C4×C127C2×D412yes—
155C22×C4×C127C2×D412yes—
156C2×C8×C127—42no—
157C2×C8×C127—42no—
158C2×C8×C127—42no—
159C4⋊C4×C127—24no—
160C4⋊C4×C127—21no—
161C4⋊C4×C127—21no—
162C4⋊C4×C127—24no—
163C4⋊C4×C127C22⋊C412yescomplemented
164C4⋊C4×C127C22⋊C412yescomplemented
165C22⋊C4×C127—24no—
166C22⋊C4×C127—24no—
167C22⋊C4×C127—24no—
168C22⋊C4×C127—24no—
169C22⋊C4×C127—24no—
170C22⋊C4×C127—48no—
171C42×C127—24no—
172C23×C127—21no—
173C23×C127—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127—24no—
177C23×C127—24no—
178C23×C127—21no—
179C23×C127—21no—
180C23×C127—21no—
181C23×C127—21no—
182C23×C127—21no—
183C23×C127—21no—
184C23×C127—21no—
185C23×C127—21no—
186C23×C127—21no—
187C23×C127—21no—
188C23×C127—21no—
189C23×C127—21no—
190C23×C127—22no—
191C23×C127—44no—
192C23×C127—44no—
193C23×C127—44no—
194C23×C127—44no—
195C23×C127—42no—
196C23×C127—42no—
197C23×C127—42no—
198C23×C127—42no—
199C23×C127—41no—
200C23×C127—41no—
201C23×C127—41no—
202C23×C127—41no—
203C23×C127—41no—
204C23×C127—41no—
205C23×C127#9912yes—
206C23×C127#10412yes—
207C23×C127C2×C22⋊C411yescharacteristic
208C23×C127#12012yes—
209C23×C127#13512yes—
210C23×C127AGL(1,ℤ/8)11yescharacteristic
211C23×C127#13711yescharacteristic
212Q8×C127—42no—
213Q8×C127—42no—
214D4×C127—42no—
215D4×C127—42no—
216D4×C127—48no—
217D4×C127—48no—
218D4×C127—44no—
219D4×C127—44no—
220C2×C4×C127—24no—
221C2×C4×C127—24no—
222C2×C4×C127—21no—
223C2×C4×C127—21no—
224C2×C4×C127—21no—
225C2×C4×C127—21no—
226C2×C4×C127—21no—
227C2×C4×C127—21no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—21no—
231C2×C4×C127—22no—
232C2×C4×C127—21no—
233C2×C4×C127—21no—
234C2×C4×C127—42no—
235C2×C4×C127—42no—
236C2×C4×C127—44no—
237C2×C4×C127—44no—
238C2×C4×C127—44no—
239C2×C4×C127—42no—
240C2×C4×C127—42no—
241C2×C4×C127—48no—
242C2×C4×C127—41no—
243C2×C4×C127—41no—
244C2×C4×C127—41no—
245C2×C4×C127—41no—
246C2×C4×C127—41no—
247C2×C4×C127—41no—
248C2×C4×C127—44no—
249C2×C4×C127—44no—
250C2×C4×C127—44no—
251C2×C4×C127—82no—
252C2×C4×C127—82no—
253C2×C4×C127—82no—
254C2×C4×C127—82no—
255C2×C4×C127#9912yes—
256C2×C4×C127C2×C22⋊C411yescharacteristic
257C2×C4×C127C2×C22⋊C411yescharacteristic
258C2×C4×C127#12012yes—
259C8×C127—82no—
260C22×C127—22no—
261C22×C127—21no—
262C22×C127—21no—
263C22×C127—21no—
264C22×C127—21no—
265C22×C127—22no—
266C22×C127—22no—
267C22×C127—22no—
268C22×C127—22no—
269C22×C127—22no—
270C22×C127—22no—
271C22×C127—22no—
272C22×C127—22no—
273C22×C127—22no—
274C22×C127—48no—
275C22×C127—48no—
276C22×C127—41no—
277C22×C127—41no—
278C22×C127—41no—
279C22×C127—41no—
280C22×C127—41no—
281C22×C127—41no—
282C22×C127—41no—
283C22×C127—41no—
284C22×C127—42no—
285C22×C127—42no—
286C22×C127—42no—
287C22×C127—81no—
288C22×C127—81no—
289C22×C127—81no—
290C22×C127—81no—
291C22×C127#37911yescharacteristic
292C22×C127C2×#9911yescharacteristic
293C22×C127C2×#9911yescharacteristic
294C22×C127#41611yescharacteristic
295C22×C127#41711yescharacteristic
296C22×C127C2×#10411yescharacteristic
297C22×C127#42111yescharacteristic
298C22×C127#44912yes—
299C22×C127#45212yes—
300C4×C127—22no—
301C4×C127—22no—
302C4×C127—21no—
303C4×C127—21no—
304C4×C127—41no—
305C4×C127—42no—
306C4×C127—42no—
307C4×C127—42no—
308C4×C127—84no—
309#8331C12711yesmaximal, characteristic, direct product factor, Sylow
310C2×C127—22no—
311C2×C127—22no—
312C2×C127—42no—
313C2×C127—42no—
314C2×C127—82no—
315C2×C127#150111yescharacteristic
316C2×C127#150411yescharacteristic
317C2×C127#151311yescharacteristic
318C2×C127#151611yescharacteristic
319C2×C127#178111yescharacteristic
320C2×C127#178811yescharacteristic
321C2×C127#178911yescharacteristic
322C2×#525C2×C12711yescharacteristic
323#1779C2×C12711yescharacteristic, complemented
324C2×#328C2×C12712yescomplemented
325C2×#327C2×C12712yescomplemented
326C22⋊#95C2×C12711yescharacteristic, complemented
327C127#833111yescharacteristic, direct product factor, Sylow
328C22×#41C4×C12711yescharacteristic, complemented
329C24×C4C22×C12711yescharacteristic
330#525—24no—
331#525—24no—
332C2×#123—21no—
333C2×C22⋊Q8C4×C12711yescharacteristic, complemented
334C2×C22⋊Q8C22×C12711yescharacteristic
335C2×#121C4×C12711yescharacteristic, complemented
336C2×#121C22×C12711yescharacteristic
337C2×#120—21no—
338C2×C4×D4—21no—
339C2×C4×D4—21no—
340C2×#117C4×C12711yescharacteristic, complemented
341C22×C22⋊C4—21no—
342C2×C22⋊C8—21no—
343C2×C22⋊C8—21no—
344C2×C22⋊C8C22×C12712yes—
345C2×#95—21no—
346C2×#95C22×C12712yes—
347#328—24no—
348#327—24no—
349C25D4×C12711yescharacteristic
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—42no—
353C2×#41—42no—
354C22×Q8C2×C4×C12711yescharacteristic
355C22×D4—21no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4C2×C4×C12711yescharacteristic
359C23×C4—21no—
360C23×C4—22no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—42no—
364C23×C4—42no—
365C23×C4C2×C4×C12711yescharacteristic
366C23×C4C2×C4×C12711yescharacteristic
367C23×C4D4×C12711yescharacteristic
368C23×C4D4×C12711yescharacteristic
369C23×C4D4×C12711yescharacteristic
370C23×C4C23×C12711yescharacteristic
371C22×C8—22no—
372#123—28no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8D4×C12714yes—
377#121—22no—
378#121—22no—
379#121—22no—
380#121D4×C12714yes—
381#120—28no—
382C4×D4—28no—
383C4×D4—28no—
384#117—22no—
385#117—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4C2×C4×C12711yescharacteristic
389C2×C4⋊C4C2×C4×C12711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C42—21no—
399C22⋊C8—22no—
400C22⋊C8—22no—
401C22⋊C8—42no—
402C22⋊C8—42no—
403#95—22no—
404#95—22no—
405#95—42no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—22no—
413C24—41no—
414C24—41no—
415C24C22⋊C4×C12711yescharacteristic
416C24C22⋊C4×C12711yescharacteristic
417C24C2×D4×C12711yescharacteristic
418#41—48no—
419#41—48no—
420C2×Q8—21no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8C22⋊C4×C12712yescomplemented
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4C22⋊C4×C12712yescomplemented
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4C22⋊C4×C12711yescharacteristic
473C22×C4C22⋊C4×C12711yescharacteristic
474C22×C4C22×C4×C12711yescharacteristic
475C22×C4C2×D4×C12711yescharacteristic
476C22×C4C2×D4×C12712yes—
477C22×C4C2×D4×C12712yes—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4C22⋊C4×C12712yescomplemented
486C4⋊C4C22⋊C4×C12712yescomplemented
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—48no—
493C42—24no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—24no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23#99×C12712yes—
528C23#104×C12712yes—
529C23C2×C22⋊C4×C12711yescharacteristic
530C23#120×C12712yes—
531C23#135×C12712yes—
532C23AGL(1,ℤ/8)×C12711yescharacteristic
533C23#137×C12711yescharacteristic
534Q8—42no—
535Q8—42no—
536D4—42no—
537D4—42no—
538D4—48no—
539D4—48no—
540D4—44no—
541D4—44no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4#99×C12712yes—
578C2×C4C2×C22⋊C4×C12711yescharacteristic
579C2×C4C2×C22⋊C4×C12711yescharacteristic
580C2×C4#120×C12712yes—
581C8—82no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—48no—
597C22—48no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#379×C12711yescharacteristic
614C22C2×#99×C12711yescharacteristic
615C22C2×#99×C12711yescharacteristic
616C22#416×C12711yescharacteristic
617C22#417×C12711yescharacteristic
618C22C2×#104×C12711yescharacteristic
619C22#421×C12711yescharacteristic
620C22#449×C12712yes—
621C22#452×C12712yes—
622C4—22no—
623C4—22no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—84no—
631C2—22no—
632C2—22no—
633C2—42no—
634C2—42no—
635C2—82no—
636C2C127×#150111yescharacteristic
637C2C127×#150411yescharacteristic
638C2C127×#151311yescharacteristic
639C2C127×#151611yescharacteristic
640C2C127×#178111yescharacteristic
641C2C127×#178811yescharacteristic
642C2C127×#178911yescharacteristic
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC2×#525×C127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1779C2semidirect
3. C2×#328×C127C2semidirect
4. C2×#327×C127C2semidirect
5. C127×C22⋊#95C2semidirect
6. C22×#41×C127C4semidirect
11. C2×C22⋊Q8×C127C4semidirect
13. C2×#121×C127C4semidirect
18. C2×#117×C127C4semidirect
101. C2×Q8×C127C22⋊C4semidirect
115. C2×D4×C127C22⋊C4semidirect
163. C4⋊C4×C127C22⋊C4semidirect
164. C4⋊C4×C127C22⋊C4semidirect
309. #8331C127direct (N × H)
323. #1779C2×C127semidirect
324. C2×#328C2×C127semidirect
325. C2×#327C2×C127semidirect
326. C22⋊#95C2×C127semidirect
327. C127#8331direct (N × H)
328. C22×#41C4×C127semidirect
333. C2×C22⋊Q8C4×C127semidirect
335. C2×#121C4×C127semidirect
340. C2×#117C4×C127semidirect
423. C2×Q8C22⋊C4×C127semidirect
437. C2×D4C22⋊C4×C127semidirect
485. C4⋊C4C22⋊C4×C127semidirect
486. C4⋊C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#525×C1272
C127×#17792
C2×#328×C1272
C2×#327×C1272
C127×C22⋊#952
#8331127