Finite Group Databasebeta

← C2×C4×#485 #6373030

Subgroups — C2×C4×#485 #6373030

Order 512.

2586 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×#117C211yesmaximal, characteristic
4C4×#485C2116yesmaximal, direct product factor
5C2×#1928C211yesmaximal, characteristic, complemented
6C2×#1927C211yesmaximal, characteristic, complemented
7C2×#1919C211yesmaximal, characteristic, complemented
8C2×#1918C211yesmaximal, characteristic, complemented
9C2×#1873C211yesmaximal, characteristic
10C2×#1854C211yesmaximal, characteristic
11C2×#1825C211yesmaximal, characteristic
12C2×C4×#107C211yesmaximal, characteristic, complemented
13C2×C4×#106C211yesmaximal, characteristic, complemented
14C2×C4×#103C211yesmaximal, characteristic, complemented
15C2×C4×#103C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊C8C211yesmaximal, characteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#485C4116yesdirect product factor
23C2×#485C2218yes—
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36C4×C22⋊Q8—24no—
37C4×C22⋊Q8C2218yes—
38C2×#391C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#385—21no—
42C2×#380C2211yescharacteristic
43C4×#117—24no—
44C4×#117C2218yes—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4—21no—
54C2×#376—21no—
55C2×C43—21no—
56#1928C2218yescomplemented
57#1927C2218yescomplemented
58#1919C2218yescomplemented
59#1918C2218yescomplemented
60#1873C2218yes—
61#1854C2218yes—
62#1825C2218yes—
63C4×#107C2218yescomplemented
64C4×#106C2218yescomplemented
65C4×#103C2218yescomplemented
66C4×#103C2218yescomplemented
67C4×C22⋊C8C2218yes—
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2×C418yes—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8C2314yes—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117C2×C418yes—
100C2×#117C2314yes—
101C2×#117C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42D411yescharacteristic
127C22×C42D414yes—
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132#485C2×C4164yesdirect product factor
133C2×#107—24no—
134C2×#107C2×C418yescomplemented
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106C2×C418yescomplemented
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#103—24no—
142C2×#103—24no—
143C2×#103C2×C418yescomplemented
144C2×#103C2×C418yescomplemented
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8—24no—
150C2×C22⋊C8C2×C418yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8—24no—
154C2×C4×C8—24no—
155C2×C4×C8C2314yes—
156C2×C4×C8C2314yes—
157#391—22no—
158#391C2314yes—
159#389—22no—
160#389C2314yes—
161#387—28no—
162#385—28no—
163#380—22no—
164#380C2314yes—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4—28no—
177C4×C22⋊C4—28no—
178C4×C22⋊C4—28no—
179#376—28no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C43—28no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C22×C414yes—
208C22×Q8C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—42no—
227C23×C4C22×C414yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4#4114yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8C22×C414yes—
250C22×C8C22×C414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22⋊Q8—216no—
256C22⋊Q8—28no—
257C22⋊Q8C22×C4116yes—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×Q8—48no—
261#117—216no—
262#117—216no—
263#117—28no—
264#117—28no—
265#117—28no—
266#117C22×C4116yes—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C22×C414yes—
290C2×C4⋊C4C22×C414yes—
291C2×C4⋊C4C22×C414yes—
292C2×C4⋊C4#4114yes—
293C2×C4⋊C4#4114yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—216no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—28no—
341#107C22×C4116yescomplemented
342#106—28no—
343#106C22×C4116yescomplemented
344#103—28no—
345#103—28no—
346#103C22×C4116yescomplemented
347#103C22×C4116yescomplemented
348C22⋊C8—28no—
349C22⋊C8C22×C4116yes—
350C4×C8—28no—
351C4×C8—28no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C4×D414yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8C23×C414yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—416no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—416no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4C4×D414yes—
443C22×C4C4×D418yes—
444C22×C4#12314yes—
445C22×C4#12314yes—
446C22×C4#13514yes—
447C22×C4#13514yes—
448C22×C4#13714yes—
449C22×C4C23×C414yes—
450C22×C4C23×C412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2511yescharacteristic
464C2×C8—28no—
465C2×C8—28no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8C23×C414yes—
479C2×C8C23×C414yes—
480C4⋊C4—216no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—48no—
499C4⋊C4C23×C414yes—
500C4⋊C4C23×C414yes—
501C4⋊C4C23×C414yes—
502C4⋊C4C2×#4114yes—
503C4⋊C4C2×#4114yes—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—48no—
508C42—216no—
509C42—216no—
510C42—28no—
511C42—28no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—48no—
516C42—48no—
517C42—48no—
518C42—48no—
519C42—48no—
520C42—48no—
521C42#12318yes—
522C42C2×#4114yes—
523C42C2×#4114yes—
524C23—22no—
525C23—22no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C2×C4×D414yes—
538C23C2×C4×D412yes—
539C23C2×#12312yes—
540C23#52512yes—
541C23C2×#13512yes—
542C23C2×#13512yes—
543C23C2×#13712yes—
544C23#57512yes—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—42no—
549Q8—42no—
550Q8—48no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—216no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4#485116yesdirect product factor
604C2×C4C2×C4×D414yes—
605C2×C4C2×C4×D412yes—
606C2×C4C4×#4114yes—
607C2×C4C4×#4114yes—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12312yes—
611C2×C4#52512yes—
612C2×C4#52914yes—
613C2×C4#52914yes—
614C2×C4C2×#13514yes—
615C2×C4C2×#13514yes—
616C2×C4C2×#13714yes—
617C2×C4C24×C411yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C8—24no—
624C8—24no—
625C8—42no—
626C8—42no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—416no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22C4×#12314yes—
636C22C4×#13514yes—
637C22C4×#13714yes—
638C22C2×#48514yes—
639C22#309114yes—
640C22#310214yes—
641C22#310714yes—
642C22C22×C4×D411yescharacteristic
643C22C22×#12311yescharacteristic
644C22C2×#52511yescharacteristic
645C22C22×#13511yescharacteristic
646C22C22×#13511yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57511yescharacteristic
649C4—24no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4C4×#12314yes—
662C4C2×#48518yesdirect product factor
663C4C2×C4×#4111yescharacteristic
664C4C2×C4×#4111yescharacteristic
665C4C2×#52911yescharacteristic
666C4C2×#52911yescharacteristic
667C2—44no—
668C2C4×#48518yesdirect product factor
669C2C2×C4×#12311yescharacteristic
670C2C2×C4×#13511yescharacteristic
671C2C2×C4×#13711yescharacteristic
672C2C22×#48511yescharacteristic
673C2C2×#309111yescharacteristic
674C2C2×#310211yescharacteristic
675C2C2×#310711yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C4×#485C2direct (N × H)
5. C2×#1928C2semidirect
6. C2×#1927C2semidirect
7. C2×#1919C2semidirect
8. C2×#1918C2semidirect
12. C2×C4×#107C2semidirect
13. C2×C4×#106C2semidirect
14. C2×C4×#103C2semidirect
15. C2×C4×#103C2semidirect
22. C2×#485C4direct (N × H)
56. #1928C22semidirect
57. #1927C22semidirect
58. #1919C22semidirect
59. #1918C22semidirect
63. C4×#107C22semidirect
64. C4×#106C22semidirect
65. C4×#103C22semidirect
66. C4×#103C22semidirect
132. #485C2×C4direct (N × H)
134. C2×#107C2×C4semidirect
138. C2×#106C2×C4semidirect
143. C2×#103C2×C4semidirect
144. C2×#103C2×C4semidirect
341. #107C22×C4semidirect
343. #106C22×C4semidirect
346. #103C22×C4semidirect
347. #103C22×C4semidirect
603. C2×C4#485direct (N × H)
662. C4C2×#485direct (N × H)
668. C2C4×#485direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C2×C4×C22⋊Q82
C2×C4×#1172
C4×#4852
C2×#19282
C2×#19272
C2×#19192
C2×#19182
C2×#18732
C2×#18542
C2×#18252
C2×C4×#1072
C2×C4×#1062
C2×C4×#1032
C2×C4×C22⋊C82