Finite Group Databasebeta

← C4×#1996 #354722

Subgroups — C4×#1996 #354722

Order 512.

1958 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3C4×#380C211yesmaximal, characteristic
4C42×C22⋊C4C211yesmaximal, characteristic
5#10912C214yesmaximal
6#10854C212yesmaximal
7C4×#342C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C4×C22⋊Q8—21no—
12C4×C22⋊Q8—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C2×#391C411yescharacteristic, complemented
20C2×#389C411yescharacteristic, complemented
21C2×#387C412yescomplemented
22C2×#385C2212yes—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C412yescomplemented
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C43—21no—
34#1996C4116yesdirect product factor
35C2×#342C2218yes—
36C2×#342C2218yes—
37C4⋊#95—21no—
38#1434—21no—
39C22⋊#95C2212yes—
40C4×#95—21no—
41C4×#95—21no—
42C4×#95—21no—
43C4×#95—21no—
44C23×Q8C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8C2×C418yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—44no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2312yes—
69C22×C22⋊C4C2314yes—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C412yes—
77C22×C42C2×C412yes—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89#391—21no—
90#391—21no—
91#391—21no—
92#391—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—24no—
110#380C2×C418yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—44no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4C2×C418yes—
129C4×C22⋊C4Q818yescomplemented
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95C2×C412yes—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C43—44no—
143#342—216no—
144#342C2×C4132yes—
145C25C2411yescharacteristic
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8C4214yescomplemented
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C414yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×Q814yes—
185C23×C4#4114yes—
186C23×C4#4114yes—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8—44no—
194C4×Q8—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C4214yescomplemented
217C2×C4⋊C4C4218yescomplemented
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C22×C418yes—
231C2×C22⋊C4C22×C418yes—
232C2×C22⋊C4C22×C418yes—
233C2×C22⋊C4C2×Q818yes—
234C2×C22⋊C4#4118yes—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—28no—
244C2×C42—28no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281#95—28no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—44no—
296#95—44no—
297#95—44no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24C23×C411yescharacteristic
303C24C23×C412yes—
304C24C22×D411yescharacteristic
305C24C22×D411yescharacteristic
306C24C22×Q812yes—
307C24C2×#4112yes—
308C24C2×#4112yes—
309C24C2×#4114yes—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—42no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—416no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C2×C4212yes—
396C22×C4C2×C4214yes—
397C22×C4C2×C4214yes—
398C22×C4C2×C4212yes—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4#11712yes—
412C22×C4#11712yes—
413C22×C4C4×D414yes—
414C22×C4C4×D414yes—
415C22×C4C4×D414yes—
416C22×C4C4×D414yes—
417C22×C4C4×Q814yes—
418C22×C4C4×Q814yes—
419C22×C4C22⋊Q814yes—
420C22×C4C22⋊Q814yes—
421C22×C4#12412yes—
422C22×C4#12412yes—
423C22×C4C4⋊Q812yes—
424C22×C4C4⋊Q812yes—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—42no—
430C4⋊C4—416no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4C4×Q8116yescomplemented
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—416no—
464C42—44no—
465C42—416no—
466C42—416no—
467C42—44no—
468C42—44no—
469C42—44no—
470C42—88no—
471C42#9914yes—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23C22×C4211yescharacteristic
495C23C22×C22⋊C411yescharacteristic
496C23C2×#11711yescharacteristic
497C23C2×C4×D412yes—
498C23C2×C4×D412yes—
499C23C2×C4×Q812yes—
500C23C2×C4×Q814yes—
501C23C4×#4112yes—
502C23C4×#4114yes—
503C23C2×C22⋊Q812yes—
504C23#52512yes—
505C23C2×#12411yescharacteristic
506C23#52914yes—
507C23C2×C4⋊Q811yescharacteristic
508C23#53212yes—
509C23#53314yes—
510Q8—48no—
511Q8—48no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—216no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C4×C22⋊C412yes—
589C2×C4C4×C22⋊C412yes—
590C2×C4C4×C22⋊C412yes—
591C2×C4C4×C22⋊C412yes—
592C2×C4#39112yes—
593C2×C4#39112yes—
594C2×C4#39112yes—
595C2×C4#39112yes—
596C2×C4C2×#9912yes—
597C2×C4C2×#9912yes—
598C2×C4C2×#9912yes—
599C2×C4#41012yes—
600C2×C4#41012yes—
601C2×C4#41012yes—
602C22—22no—
603C22—21no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—22no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22C2×C4×C22⋊C411yescharacteristic
629C22C42×Q812yes—
630C22C2×#39111yescharacteristic
631C22#229512yes—
632C22#229711yescharacteristic
633C22C4×C22⋊Q814yes—
634C22C4×#12412yes—
635C22C4×C4⋊Q812yes—
636C22C22×#9911yescharacteristic
637C22C2×#41011yescharacteristic
638C22C2×#41011yescharacteristic
639C22C2×#41011yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—416no—
649C4—44no—
650C4—44no—
651C4—41no—
652C4—41no—
653C4—88no—
654C4—82no—
655C4—82no—
656C4C4×#9914yes—
657C4#199614yesdirect product factor
658C2—24no—
659C2—22no—
660C2—22no—
661C2—22no—
662C2—22no—
663C2C4×#39111yescharacteristic
664C2C2×C4×#9911yescharacteristic
665C2C4×#41011yescharacteristic
666C2C2×#199611yescharacteristic
667C2#2133211yescharacteristic
668C2#2142011yescharacteristic
669C2#2142111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C42
Frattini, prime radical, cosocleC25C24
socleC23C22×C22⋊C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×C4×Q8C4semidirect
19. C2×#391C4semidirect
20. C2×#389C4semidirect
21. C2×#387C4semidirect
26. C2×C4×C4⋊C4C4semidirect
27. C2×C4×C4⋊C4C4semidirect
34. #1996C4direct (N × H)
129. C4×C22⋊C4Q8semidirect
151. C22×Q8C42semidirect
216. C2×C4⋊C4C42semidirect
217. C2×C4⋊C4C42semidirect
447. C22⋊C4C4×Q8semidirect
657. C4#1996direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#19962
C4×#3802
C42×C22⋊C42
#109122
#108542
C4×#3422