Finite Group Databasebeta

← C4×#1637 #209689

Subgroups — C4×#1637 #209689

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Q8C211yesmaximal, characteristic
2C4×#487C211yesmaximal, characteristic, complemented
3C4×#477C211yesmaximal, characteristic, complemented
4C4×#476C211yesmaximal, characteristic
5C2×#1637C211yesmaximal, characteristic
6#9065C211yesmaximal, characteristic, complemented
7#8975C211yesmaximal, characteristic, complemented
8#8973C211yesmaximal, characteristic
9#8926C211yesmaximal, characteristic, complemented
10#8896C211yesmaximal, characteristic, complemented
11#8890C211yesmaximal, characteristic
12#8447C211yesmaximal, characteristic
13C4×#332C211yesmaximal, characteristic, complemented
14C4×Q8⋊C8C211yesmaximal, characteristic, complemented
15C4×D4⋊C8C211yesmaximal, characteristic
16C2×D4×Q8C2211yescharacteristic
17C22×C4×Q8—21no—
18C2×#487C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#476C2211yescharacteristic
21#2365—21no—
22#2365C2211yescharacteristic
23#2360—22no—
24#2360—21no—
25Q8×C4⋊C4C2211yescharacteristic
26D4×C4⋊C4—21no—
27D4×C4⋊C4C2211yescharacteristic
28Q8×C22⋊C4—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×C22⋊Q8—22no—
33C4×C22⋊Q8—21no—
34C42×Q8C2211yescharacteristic
35C42×D4—21no—
36C42×D4C2211yescharacteristic
37#1990C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1956C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1933—22no—
44#1921—21no—
45#1920—21no—
46#1864—21no—
47#1863—21no—
48#1834C2211yescharacteristic
49#1803C2211yescharacteristic
50#1802C2211yescharacteristic
51C4×#107—21no—
52C4×#106—21no—
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×#103—22no—
56C4×#103—21no—
57C4×#102—21no—
58C42×C8C2211yescharacteristic
59#1637C418yesdirect product factor
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C23×Q8—21no—
73D4×Q8—22no—
74D4×Q8—22no—
75D4×Q8—22no—
76D4×Q8C2×C414yes—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—44no—
89C2×C4×Q8—44no—
90C2×C4×Q8D411yescharacteristic
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4D411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C42—22no—
101C22×C42—21no—
102#487—22no—
103#487C2×C414yescomplemented
104#477—22no—
105#477C2×C414yescomplemented
106#476—22no—
107#476C2×C414yes—
108C2×#107—21no—
109C2×#107—21no—
110C2×#106—21no—
111C2×#106—21no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#103—21no—
117C2×#103—22no—
118C2×#103—21no—
119C2×#103—22no—
120C2×#102—21no—
121C2×#102—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391—22no—
129#391—22no—
130#391—21no—
131#391—21no—
132#391—21no—
133#389—22no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—22no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389D412yes—
142#387—22no—
143#387—22no—
144#387—22no—
145#385—22no—
146#385—22no—
147#385—22no—
148#384—21no—
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#380—22no—
152#380—21no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4D411yescharacteristic
163C4×C4⋊C4D412yes—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4C2311yescharacteristic
168C4×C4⋊C4C2311yescharacteristic
169C4×C22⋊C4—22no—
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C43—21no—
176C43C2311yescharacteristic
177#340—21no—
178#340—21no—
179#340—22no—
180#340—21no—
181#340—21no—
182#340—21no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—22no—
187#336—21no—
188#336—21no—
189#332—22no—
190#332C2×C414yescomplemented
191Q8⋊C8—22no—
192Q8⋊C8C2×C414yescomplemented
193D4⋊C8—22no—
194D4⋊C8C2×C414yes—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×D411yescharacteristic
205C22×D4C2×D411yescharacteristic
206C23×C4—22no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C22×C412yes—
226C4⋊Q8#4112yes—
227C22⋊Q8—24no—
228C22⋊Q8—28no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C4×Q8—22no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—22no—
235C4×Q8—41no—
236C4×Q8—48no—
237C4×Q8—44no—
238C4×Q8—42no—
239C4×Q8—48no—
240C4×Q8—42no—
241C4×Q8—42no—
242C4×Q8—42no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×Q8SD1614yes—
246C4×Q8C22×C412yes—
247C4×Q8#4112yes—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—41no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4Q1614yes—
258C4×D4C22×C412yes—
259C4×D4#4112yes—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—48no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—41no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—24no—
318#107—42no—
319#106—24no—
320#106—42no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8C22×C412yes—
328C4⋊C8C22×C412yes—
329#103—24no—
330#103—28no—
331#103—44no—
332#103—42no—
333#102—24no—
334#102—42no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—42no—
339C4×C8—42no—
340C4×C8—42no—
341C4×C8—42no—
342C4×C8C22×C412yes—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—21no—
355#95—41no—
356#95—41no—
357C24—21no—
358C2×Q8—28no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8C4×D412yes—
379C2×Q8C2×SD1612yes—
380C2×Q8#13512yes—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4C4×D412yes—
385C2×D4C2×Q1612yes—
386C2×D4#13512yes—
387C22×C4—21no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4#12011yescharacteristic
417C22×C4#12011yescharacteristic
418C22×C4#12011yescharacteristic
419C22×C4#12011yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—44no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C4×D414yes—
478C4⋊C4C4×D412yes—
479C4⋊C4C4×D412yes—
480C4⋊C4C2×SD1612yes—
481C4⋊C4C2×Q1612yes—
482C22⋊C4—28no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—24no—
495C42—22no—
496C42—41no—
497C42—42no—
498C42—44no—
499C42—44no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—42no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—81no—
509C42—82no—
510C42C2×SD1612yes—
511C42C2×Q1612yes—
512C42AGL(1,ℤ/8)12yes—
513C42#13712yes—
514C42C23×C411yescharacteristic
515C42C2×#4111yescharacteristic
516C42C2×#4111yescharacteristic
517C42C2×#4111yescharacteristic
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×#12011yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8—416no—
530Q8—44no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—82no—
536Q8C4×SD1614yes—
537D4—22no—
538D4—22no—
539D4—22no—
540D4C4×Q1614yes—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—416no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—88no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4#45012yes—
592C2×C4#45012yes—
593C2×C4#45112yes—
594C2×C4#45112yes—
595C2×C4#45412yes—
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4C22×SD1611yescharacteristic
603C2×C4C22×Q1611yescharacteristic
604C2×C4C2×#13511yescharacteristic
605C2×C4C2×#13511yescharacteristic
606C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
607C2×C4C2×#13711yescharacteristic
608C2×C4#57511yescharacteristic
609C2×C4#57511yescharacteristic
610C8—42no—
611C8—42no—
612C8—42no—
613C8—42no—
614C8—81no—
615C8—81no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C4×#12011yescharacteristic
624C22C2×#45011yescharacteristic
625C22C2×#45111yescharacteristic
626C22#300211yescharacteristic
627C22#300211yescharacteristic
628C22C2×#45411yescharacteristic
629C22#301811yescharacteristic
630C4—22no—
631C4—22no—
632C4—24no—
633C4—21no—
634C4—21no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4—161no—
646C4#163714yesdirect product factor
647C4C2×C4×SD1611yescharacteristic
648C4C2×C4×Q1611yescharacteristic
649C4C4×AGL(1,ℤ/8)11yescharacteristic
650C4C4×#13711yescharacteristic
651C4#310111yescharacteristic
652C4#310411yescharacteristic
653C4#310511yescharacteristic
654C4#310811yescharacteristic
655C2—44no—
656C2C2×#163711yescharacteristic
657C2#1849411yescharacteristic
658C2#1851711yescharacteristic
659C2#1851911yescharacteristic
660C2C4×#45011yescharacteristic
661C2C4×#45111yescharacteristic
662C2C4×#45411yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×#120
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#487C2semidirect
3. C4×#477C2semidirect
6. #9065C2semidirect
7. #8975C2semidirect
9. #8926C2semidirect
10. #8896C2semidirect
13. C4×#332C2semidirect
14. C4×Q8⋊C8C2semidirect
59. #1637C4direct (N × H)
103. #487C2×C4semidirect
105. #477C2×C4semidirect
190. #332C2×C4semidirect
192. Q8⋊C8C2×C4semidirect
646. C4#1637direct (N × H)

Maximal subgroups

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

subgroupindex
C4×D4×Q82
C4×#4872
C4×#4772
C4×#4762
C2×#16372
#90652
#89752
#89732
#89262
#88962
#88902
#84472
C4×#3322
C4×Q8⋊C82
C4×D4⋊C82