Finite Group Databasebeta

← C2×C4×#478 #6373147

Subgroups — C2×C4×#478 #6373147

Order 512.

2802 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic, complemented
3C2×C42×D4C211yesmaximal, characteristic
4C4×#478C2116yesmaximal, direct product factor
5C2×#1989C211yesmaximal, characteristic, complemented
6C2×#1949C211yesmaximal, characteristic, complemented
7C2×#1921C211yesmaximal, characteristic, complemented
8C2×#1919C211yesmaximal, characteristic, complemented
9C2×#1864C211yesmaximal, characteristic
10C2×#1862C211yesmaximal, characteristic
11C2×#1804C211yesmaximal, characteristic
12C2×C4×#107C211yesmaximal, characteristic, complemented
13C2×C4×#106C211yesmaximal, characteristic, complemented
14C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
15C2×C4×#102C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C42—21no—
22C2×#478C4116yesdirect product factor
23C2×#478C2218yes—
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36C4×#125—24no—
37C4×#125C2218yescomplemented
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#387—21no—
42C2×#382C2211yescharacteristic
43C42×D4—24no—
44C42×D4C2218yes—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4—21no—
54C2×C4×C22⋊C4—21no—
55C2×C43C2211yescharacteristic
56#1989C2218yescomplemented
57#1949C2218yescomplemented
58#1921C2218yescomplemented
59#1919C2218yescomplemented
60#1864C2218yes—
61#1862C2218yes—
62#1804C2218yes—
63C4×#107C2218yescomplemented
64C4×#106C2218yescomplemented
65C4×C4⋊C8C2218yescomplemented
66C4×#102C2218yes—
67C4×#102C2218yes—
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#125—22no—
87C2×#125—24no—
88C2×#125C2×C418yescomplemented
89C2×#125C2314yes—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4C2×C418yes—
98C2×C4×D4Q818yes—
99C2×C4×D4C2314yes—
100C2×C4×D4C2314yes—
101C2×C4×D4C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—28no—
122C22×C42—48no—
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#478C2×C4164yesdirect product factor
131C2×#107—24no—
132C2×#107C2×C418yescomplemented
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106—24no—
136C2×#106C2×C418yescomplemented
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8C2×C418yescomplemented
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#102—24no—
144C2×#102—24no—
145C2×#102C2×C418yes—
146C2×#102C2×C418yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C4×C8—24no—
152C2×C4×C8—24no—
153C2×C4×C8C2314yes—
154C2×C4×C8C2314yes—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#387—28no—
160#387—28no—
161#382—22no—
162#382C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—28no—
176C4×C22⋊C4—28no—
177C4×C22⋊C4—28no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C43D414yes—
188C43C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C22×D4—21no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C22×C414yes—
205C22×D4C2×Q814yes—
206C22×D4#4114yes—
207C22×D4C2412yes—
208C23×C4—28no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8C22×C414yes—
245C22×C8C22×C414yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250#125—216no—
251#125—28no—
252#125C22×C4116yescomplemented
253C4×D4—216no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—216no—
257C4×D4—216no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4C22×C4116yes—
261C4×D4C2×Q8116yes—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
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—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C22×C414yes—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C414yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—432no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C22×C414yes—
322C2×C42C2×D412yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×Q814yes—
330C2×C42#4114yes—
331C2×C42#4114yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—28no—
336#107C22×C4116yescomplemented
337#106—28no—
338#106C22×C4116yescomplemented
339C4⋊C8—28no—
340C4⋊C8C22×C4116yescomplemented
341#102—28no—
342#102—28no—
343#102C22×C4116yes—
344#102C22×C4116yes—
345C4×C8—28no—
346C4×C8—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24C22×D411yescharacteristic
365C2×D4—24no—
366C2×D4—28no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4C4×Q8116yes—
375C2×D4C23×C414yes—
376C2×D4C22×Q814yes—
377C2×D4C2×#4114yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—432no—
407C22×C4—432no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—432no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4C4×D414yes—
430C22×C4C4×D414yes—
431C22×C4C22⋊Q814yes—
432C22×C4C22⋊Q814yes—
433C22×C4#13514yes—
434C22×C4#13514yes—
435C22×C4AGL(1,ℤ/8)14yes—
436C22×C4C23×C412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×Q812yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—44no—
463C2×C8—44no—
464C2×C8C23×C414yes—
465C2×C8C23×C414yes—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—48no—
486C4⋊C4C23×C414yes—
487C4⋊C4C23×C414yes—
488C4⋊C4C23×C414yes—
489C4⋊C4C22×Q814yes—
490C4⋊C4C2×#4114yes—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C42—24no—
495C42—28no—
496C42—28no—
497C42—28no—
498C42—22no—
499C42—22no—
500C42—44no—
501C42—48no—
502C42—48no—
503C42—48no—
504C42—48no—
505C42—48no—
506C42C4×D418yes—
507C42C22⋊Q818yes—
508C42C23×C414yes—
509C42C22×Q814yes—
510C42C2×#4114yes—
511C42C2×#4114yes—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—216no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C2×C4×D412yes—
527C23C2×C22⋊Q812yes—
528C23#52512yes—
529C23C2×#13512yes—
530C23C2×#13512yes—
531C23C2×AGL(1,ℤ/8)12yes—
532C23#57512yes—
533C23C23×D411yescharacteristic
534D4—24no—
535D4—24no—
536D4—24no—
537D4—216no—
538D4C2×C4×Q818yes—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—432no—
560C2×C4—432no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#478116yesdirect product factor
587C2×C4C2×C4×D414yes—
588C2×C4C2×C4×D414yes—
589C2×C4C2×C4×D412yes—
590C2×C4C2×C4×Q814yes—
591C2×C4C4×#4114yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4C2×C22⋊Q814yes—
594C2×C4C2×C22⋊Q812yes—
595C2×C4#52512yes—
596C2×C4#52914yes—
597C2×C4#53314yes—
598C2×C4C2×#13514yes—
599C2×C4C2×#13514yes—
600C2×C4C2×AGL(1,ℤ/8)14yes—
601C2×C4C24×C411yescharacteristic
602C2×C4C23×Q811yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C8—24no—
608C8—24no—
609C8—42no—
610C8—42no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22C4×C22⋊Q814yes—
620C22C4×#13514yes—
621C22C4×AGL(1,ℤ/8)14yes—
622C22C2×#47814yes—
623C22#307614yes—
624C22#310214yes—
625C22#310514yes—
626C22C22×C4×D411yescharacteristic
627C22C22×C22⋊Q811yescharacteristic
628C22C2×#52511yescharacteristic
629C22C22×#13511yescharacteristic
630C22C22×#13511yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57511yescharacteristic
633C4—24no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—82no—
644C4—82no—
645C4C4×C22⋊Q814yes—
646C4C2×#47818yesdirect product factor
647C4C22×C4×Q811yescharacteristic
648C4C2×C4×#4111yescharacteristic
649C4C2×#52911yescharacteristic
650C4C2×#53311yescharacteristic
651C2—48no—
652C2C4×#47818yesdirect product factor
653C2C2×C4×C22⋊Q811yescharacteristic
654C2C2×C4×#13511yescharacteristic
655C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
656C2C22×#47811yescharacteristic
657C2C2×#307611yescharacteristic
658C2C2×#310211yescharacteristic
659C2C2×#310511yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#125C2semidirect
4. C4×#478C2direct (N × H)
5. C2×#1989C2semidirect
6. C2×#1949C2semidirect
7. C2×#1921C2semidirect
8. C2×#1919C2semidirect
12. C2×C4×#107C2semidirect
13. C2×C4×#106C2semidirect
14. C2×C4×C4⋊C8C2semidirect
22. C2×#478C4direct (N × H)
37. C4×#125C22semidirect
56. #1989C22semidirect
57. #1949C22semidirect
58. #1921C22semidirect
59. #1919C22semidirect
63. C4×#107C22semidirect
64. C4×#106C22semidirect
65. C4×C4⋊C8C22semidirect
88. C2×#125C2×C4semidirect
130. #478C2×C4direct (N × H)
132. C2×#107C2×C4semidirect
136. C2×#106C2×C4semidirect
140. C2×C4⋊C8C2×C4semidirect
252. #125C22×C4semidirect
336. #107C22×C4semidirect
338. #106C22×C4semidirect
340. C4⋊C8C22×C4semidirect
586. C2×C4#478direct (N × H)
646. C4C2×#478direct (N × H)
652. C2C4×#478direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C2×C4×#1252
C2×C42×D42
C4×#4782
C2×#19892
C2×#19492
C2×#19212
C2×#19192
C2×#18642
C2×#18622
C2×#18042
C2×C4×#1072
C2×C4×#1062
C2×C4×C4⋊C82
C2×C4×#1022