Finite Group Databasebeta

← C127×#10970 #44574643

Subgroups — C127×#10970 #44574643

Order 32512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C127C211yesmaximal, characteristic
2C22×#103×C127C212yesmaximal, complemented
3C22×C22⋊C8×C127C211yesmaximal, characteristic
4C2×#380×C127C211yesmaximal, characteristic
5C2×#340×C127C212yesmaximal, complemented
6C23×Q8×C127C2211yescharacteristic
7C24×C4×C127C2211yescharacteristic
8C23×C8×C127C2212yes—
9C2×C22⋊Q8×C127—24no—
10C2×C22⋊Q8×C127—22no—
11C2×C22⋊Q8×C127—24no—
12C2×C22⋊Q8×C127C414yescomplemented
13C22×C4⋊C4×C127—21no—
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127—21no—
17C22×C22⋊C4×C127—21no—
18C2×#103×C127—24no—
19C2×#103×C127—24no—
20C2×#103×C127—24no—
21C2×C22⋊C8×C127—22no—
22C2×C22⋊C8×C127—24no—
23#380×C127—24no—
24C2×#95×C127—21no—
25#340×C127—24no—
26C25×C127D411yescharacteristic
27C22×Q8×C127—21no—
28C22×Q8×C127—22no—
29C22×Q8×C127—22no—
30C22×Q8×C127—22no—
31C22×Q8×C127—22no—
32C22×Q8×C127—22no—
33C22×Q8×C127C2×C412yes—
34C22×Q8×C127D414yes—
35C23×C4×C127—21no—
36C23×C4×C127—21no—
37C23×C4×C127—22no—
38C23×C4×C127—22no—
39C23×C4×C127—22no—
40C23×C4×C127—22no—
41C23×C4×C127—22no—
42C23×C4×C127—22no—
43C23×C4×C127—22no—
44C23×C4×C127C2×C412yes—
45C23×C4×C127D411yescharacteristic
46C23×C4×C127D412yes—
47C23×C4×C127C2311yescharacteristic
48C22×C8×C127—22no—
49C22×C8×C127—22no—
50C22×C8×C127—22no—
51C22×C8×C127—22no—
52C22×C8×C127—22no—
53C22×C8×C127—22no—
54C22×C8×C127D414yes—
55C22⋊Q8×C127—216no—
56C22⋊Q8×C127—28no—
57C22⋊Q8×C127—28no—
58C2×C4⋊C4×C127—28no—
59C2×C4⋊C4×C127—24no—
60C2×C4⋊C4×C127—22no—
61C2×C4⋊C4×C127—21no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127—22no—
65C2×C4⋊C4×C127—22no—
66C2×C4⋊C4×C127—22no—
67C2×C4⋊C4×C127C2×C412yes—
68C2×C4⋊C4×C127D414yes—
69C2×C4⋊C4×C127D414yes—
70C2×C22⋊C4×C127—28no—
71C2×C22⋊C4×C127—24no—
72C2×C22⋊C4×C127—24no—
73C2×C22⋊C4×C127—42no—
74C2×C22⋊C4×C127—42no—
75#103×C127—216no—
76C22⋊C8×C127—28no—
77#95×C127—24no—
78C24×C127—22no—
79C24×C127—21no—
80C24×C127—22no—
81C24×C127—21no—
82C24×C127—22no—
83C24×C127—21no—
84C24×C127—21no—
85C24×C127—22no—
86C24×C127C22⋊C412yes—
87C24×C127SD1612yes—
88C24×C127Q1612yes—
89C24×C127C2×D411yescharacteristic
90C2×Q8×C127—22no—
91C2×Q8×C127—24no—
92C2×Q8×C127—24no—
93C2×Q8×C127—24no—
94C2×Q8×C127—24no—
95C2×Q8×C127—24no—
96C2×Q8×C127—24no—
97C2×Q8×C127—24no—
98C2×Q8×C127—41no—
99C2×Q8×C127—42no—
100C2×Q8×C127—42no—
101C2×Q8×C127—42no—
102C2×Q8×C127—41no—
103C2×Q8×C127—42no—
104C2×Q8×C127—42no—
105C2×Q8×C127C22⋊C414yescomplemented
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—21no—
109C22×C4×C127—24no—
110C22×C4×C127—24no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—21no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—24no—
117C22×C4×C127—24no—
118C22×C4×C127—22no—
119C22×C4×C127—24no—
120C22×C4×C127—24no—
121C22×C4×C127—22no—
122C22×C4×C127—24no—
123C22×C4×C127—41no—
124C22×C4×C127—42no—
125C22×C4×C127—41no—
126C22×C4×C127—42no—
127C22×C4×C127—41no—
128C22×C4×C127—41no—
129C22×C4×C127—42no—
130C22×C4×C127—42no—
131C22×C4×C127—42no—
132C22×C4×C127—42no—
133C22×C4×C127—42no—
134C22×C4×C127—42no—
135C22×C4×C127—42no—
136C22×C4×C127—42no—
137C22×C4×C127—42no—
138C22×C4×C127—42no—
139C22×C4×C127—42no—
140C22×C4×C127—42no—
141C22×C4×C127—42no—
142C22×C4×C127—42no—
143C22×C4×C127C22⋊C412yes—
144C22×C4×C127C22×C411yescharacteristic
145C22×C4×C127C2×D412yes—
146C22×C4×C127C2×D412yes—
147C22×C4×C127C2×D412yes—
148C22×C4×C127C2×D412yes—
149C22×C4×C127C2×D411yescharacteristic
150C22×C4×C127#4112yes—
151C22×C4×C127#4112yes—
152C2×C8×C127—24no—
153C2×C8×C127—24no—
154C2×C8×C127—24no—
155C2×C8×C127—42no—
156C2×C8×C127—42no—
157C2×C8×C127—42no—
158C2×C8×C127—42no—
159C4⋊C4×C127—216no—
160C4⋊C4×C127—22no—
161C4⋊C4×C127—24no—
162C4⋊C4×C127—24no—
163C4⋊C4×C127—24no—
164C4⋊C4×C127—42no—
165C4⋊C4×C127—42no—
166C4⋊C4×C127C22⋊C414yescomplemented
167C22⋊C4×C127—216no—
168C22⋊C4×C127—48no—
169C23×C127—22no—
170C23×C127—22no—
171C23×C127—22no—
172C23×C127—22no—
173C23×C127—22no—
174C23×C127—22no—
175C23×C127—24no—
176C23×C127—22no—
177C23×C127—22no—
178C23×C127—24no—
179C23×C127—24no—
180C23×C127—24no—
181C23×C127—22no—
182C23×C127—22no—
183C23×C127—42no—
184C23×C127—42no—
185C23×C127—42no—
186C23×C127—42no—
187C23×C127—42no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127#10314yescomplemented
192C23×C127C2×C22⋊C411yescharacteristic
193C23×C127#12012yes—
194C23×C127#12011yescharacteristic
195C23×C127#12112yes—
196C23×C127#12311yescharacteristic
197C23×C127C2×SD1611yescharacteristic
198C23×C127C2×Q1611yescharacteristic
199C23×C127#13512yes—
200C23×C127AGL(1,ℤ/8)11yescharacteristic
201C23×C127#13711yescharacteristic
202C23×C127#13712yes—
203Q8×C127—28no—
204Q8×C127—44no—
205Q8×C127—44no—
206Q8×C127—44no—
207C2×C4×C127—24no—
208C2×C4×C127—24no—
209C2×C4×C127—22no—
210C2×C4×C127—22no—
211C2×C4×C127—22no—
212C2×C4×C127—42no—
213C2×C4×C127—42no—
214C2×C4×C127—42no—
215C2×C4×C127—44no—
216C2×C4×C127—42no—
217C2×C4×C127—44no—
218C2×C4×C127—42no—
219C2×C4×C127—42no—
220C2×C4×C127—44no—
221C2×C4×C127—44no—
222C2×C4×C127—44no—
223C2×C4×C127—44no—
224C2×C4×C127—44no—
225C2×C4×C127—44no—
226C2×C4×C127—44no—
227C2×C4×C127—44no—
228C2×C4×C127—44no—
229C2×C4×C127—82no—
230C2×C4×C127—82no—
231C2×C4×C127—82no—
232C2×C4×C127—82no—
233C2×C4×C127C2×C22⋊C411yescharacteristic
234C2×C4×C127C2×C22⋊C411yescharacteristic
235C2×C4×C127C4×D412yes—
236C2×C4×C127C4×D412yes—
237C2×C4×C127#12012yes—
238C2×C4×C127#12011yescharacteristic
239C2×C4×C127#12012yes—
240C2×C4×C127#12112yes—
241C2×C4×C127#12112yes—
242C2×C4×C127#12112yes—
243C2×C4×C127#12112yes—
244C2×C4×C127#12112yes—
245C2×C4×C127#12312yes—
246C2×C4×C127#12311yescharacteristic
247C2×C4×C127#12412yes—
248C2×C4×C127#12712yes—
249C8×C127—44no—
250C22×C127—24no—
251C22×C127—24no—
252C22×C127—24no—
253C22×C127—42no—
254C22×C127—42no—
255C22×C127—42no—
256C22×C127—44no—
257C22×C127—44no—
258C22×C127—44no—
259C22×C127—44no—
260C22×C127—42no—
261C22×C127#37911yescharacteristic
262C22×C127#38611yescharacteristic
263C22×C127#38611yescharacteristic
264C22×C127#39212yes—
265C22×C127#39412yes—
266C22×C127C2×#10311yescharacteristic
267C22×C127#41611yescharacteristic
268C22×C127#41711yescharacteristic
269C22×C127#41911yescharacteristic
270C22×C127#44812yes—
271C22×C127#44912yes—
272C22×C127#45011yescharacteristic
273C22×C127#45111yescharacteristic
274C22×C127#45112yes—
275C22×C127#45212yes—
276C22×C127#45212yes—
277C22×C127#46512yes—
278C22×C127#46712yes—
279C22×C127#46812yes—
280C22×C127#47012yes—
281C22×C127#48311yescharacteristic
282C22×C127#48411yescharacteristic
283C22×C127#48512yes—
284C4×C127—42no—
285C4×C127—44no—
286C4×C127—44no—
287C4×C127—84no—
288C4×C127#38612yes—
289C4×C127#39012yes—
290C4×C127#39212yes—
291C4×C127#39412yes—
292#10970C12711yesmaximal, characteristic, direct product factor, Sylow
293C2×C127—44no—
294C2×C127#143811yescharacteristic
295C2×C127#178711yescharacteristic
296C2×C127#178811yescharacteristic
297C2×C127#187211yescharacteristic
298C2×C127#187311yescharacteristic
299C2×C127#189211yescharacteristic
300C2×C127#189411yescharacteristic
301C2×C127#200012yes—
302C2×C127#200112yes—
303C2×C127#203512yes—
304C2×C127#203612yes—
305C22×C22⋊Q8C2×C12711yescharacteristic
306C22×#103C2×C12712yescomplemented
307C22×C22⋊C8C2×C12711yescharacteristic
308C2×#380C2×C12711yescharacteristic
309C2×#340C2×C12712yescomplemented
310C127#1097011yescharacteristic, direct product factor, Sylow
311C23×Q8C22×C12711yescharacteristic
312C24×C4C22×C12711yescharacteristic
313C23×C8C22×C12712yes—
314C2×C22⋊Q8—24no—
315C2×C22⋊Q8—22no—
316C2×C22⋊Q8—24no—
317C2×C22⋊Q8C4×C12714yescomplemented
318C22×C4⋊C4—21no—
319C22×C4⋊C4C22×C12712yes—
320C22×C4⋊C4C22×C12711yescharacteristic
321C22×C22⋊C4—21no—
322C22×C22⋊C4—21no—
323C2×#103—24no—
324C2×#103—24no—
325C2×#103—24no—
326C2×C22⋊C8—22no—
327C2×C22⋊C8—24no—
328#380—24no—
329C2×#95—21no—
330#340—24no—
331C25D4×C12711yescharacteristic
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C2×C4×C12712yes—
339C22×Q8D4×C12714yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—22no—
348C23×C4—22no—
349C23×C4C2×C4×C12712yes—
350C23×C4D4×C12711yescharacteristic
351C23×C4D4×C12712yes—
352C23×C4C23×C12711yescharacteristic
353C22×C8—22no—
354C22×C8—22no—
355C22×C8—22no—
356C22×C8—22no—
357C22×C8—22no—
358C22×C8—22no—
359C22×C8D4×C12714yes—
360C22⋊Q8—216no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4C2×C4×C12712yes—
373C2×C4⋊C4D4×C12714yes—
374C2×C4⋊C4D4×C12714yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380#103—216no—
381C22⋊C8—28no—
382#95—24no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24C22⋊C4×C12712yes—
392C24SD16×C12712yes—
393C24Q16×C12712yes—
394C24C2×D4×C12711yescharacteristic
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—41no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C22⋊C4×C12714yescomplemented
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4C22⋊C4×C12712yes—
449C22×C4C22×C4×C12711yescharacteristic
450C22×C4C2×D4×C12712yes—
451C22×C4C2×D4×C12712yes—
452C22×C4C2×D4×C12712yes—
453C22×C4C2×D4×C12712yes—
454C22×C4C2×D4×C12711yescharacteristic
455C22×C4#41×C12712yes—
456C22×C4#41×C12712yes—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C4⋊C4—216no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4C22⋊C4×C12714yescomplemented
472C22⋊C4—216no—
473C22⋊C4—48no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#103×C12714yescomplemented
497C23C2×C22⋊C4×C12711yescharacteristic
498C23#120×C12712yes—
499C23#120×C12711yescharacteristic
500C23#121×C12712yes—
501C23#123×C12711yescharacteristic
502C23C2×SD16×C12711yescharacteristic
503C23C2×Q16×C12711yescharacteristic
504C23#135×C12712yes—
505C23AGL(1,ℤ/8)×C12711yescharacteristic
506C23#137×C12711yescharacteristic
507C23#137×C12712yes—
508Q8—28no—
509Q8—44no—
510Q8—44no—
511Q8—44no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4C2×C22⋊C4×C12711yescharacteristic
539C2×C4C2×C22⋊C4×C12711yescharacteristic
540C2×C4C4×D4×C12712yes—
541C2×C4C4×D4×C12712yes—
542C2×C4#120×C12712yes—
543C2×C4#120×C12711yescharacteristic
544C2×C4#120×C12712yes—
545C2×C4#121×C12712yes—
546C2×C4#121×C12712yes—
547C2×C4#121×C12712yes—
548C2×C4#121×C12712yes—
549C2×C4#121×C12712yes—
550C2×C4#123×C12712yes—
551C2×C4#123×C12711yescharacteristic
552C2×C4#124×C12712yes—
553C2×C4#127×C12712yes—
554C8—44no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22#379×C12711yescharacteristic
567C22#386×C12711yescharacteristic
568C22#386×C12711yescharacteristic
569C22#392×C12712yes—
570C22#394×C12712yes—
571C22C2×#103×C12711yescharacteristic
572C22#416×C12711yescharacteristic
573C22#417×C12711yescharacteristic
574C22#419×C12711yescharacteristic
575C22#448×C12712yes—
576C22#449×C12712yes—
577C22#450×C12711yescharacteristic
578C22#451×C12711yescharacteristic
579C22#451×C12712yes—
580C22#452×C12712yes—
581C22#452×C12712yes—
582C22#465×C12712yes—
583C22#467×C12712yes—
584C22#468×C12712yes—
585C22#470×C12712yes—
586C22#483×C12711yescharacteristic
587C22#484×C12711yescharacteristic
588C22#485×C12712yes—
589C4—42no—
590C4—44no—
591C4—44no—
592C4—84no—
593C4#386×C12712yes—
594C4#390×C12712yes—
595C4#392×C12712yes—
596C4#394×C12712yes—
597C2—44no—
598C2C127×#143811yescharacteristic
599C2C127×#178711yescharacteristic
600C2C127×#178811yescharacteristic
601C2C127×#187211yescharacteristic
602C2C127×#187311yescharacteristic
603C2C127×#189211yescharacteristic
604C2C127×#189411yescharacteristic
605C2C127×#200012yes—
606C2C127×#200112yes—
607C2C127×#203512yes—
608C2C127×#203612yes—
subgroupG/H
center, socleC24×C127C2×D4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC25×C127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#103×C127C2semidirect
5. C2×#340×C127C2semidirect
12. C2×C22⋊Q8×C127C4semidirect
105. C2×Q8×C127C22⋊C4semidirect
166. C4⋊C4×C127C22⋊C4semidirect
191. C23×C127#103semidirect
292. #10970C127direct (N × H)
306. C22×#103C2×C127semidirect
309. C2×#340C2×C127semidirect
310. C127#10970direct (N × H)
317. C2×C22⋊Q8C4×C127semidirect
410. C2×Q8C22⋊C4×C127semidirect
471. C4⋊C4C22⋊C4×C127semidirect
496. C23#103×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C1272
C22×#103×C1272
C22×C22⋊C8×C1272
C2×#380×C1272
C2×#340×C1272
#10970127