Finite Group Databasebeta

← C4×#3206 #6377712

Subgroups — C4×#3206 #6377712

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2C4×#532C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#22268C212yesmaximal, complemented
5#22218C212yesmaximal, complemented
6#22146C214yesmaximal, complemented
7#22041C214yesmaximal, complemented
8#21981C211yesmaximal, characteristic, complemented
9#21859C211yesmaximal, characteristic
10#21804C211yesmaximal, characteristic, complemented
11C4×#492C212yesmaximal, complemented
12C4×#487C212yesmaximal, complemented
13C4×#461C214yesmaximal, complemented
14C4×#448C214yesmaximal, complemented
15C4×#431C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41—21no—
19C22×C4×Q8C2212yes—
20#3206C4116yesdirect product factor
21C2×#492C2212yes—
22C2×#487C2212yes—
23C2×#461C2214yes—
24C2×#448C2214yes—
25C2×C4×SD16C2218yes—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×#127C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#124—21no—
41C4×#124—21no—
42C4×#121—21no—
43C4×#121—21no—
44C4×#121C2212yes—
45C4×#121C2212yes—
46#2297—21no—
47C2×#391C2212yes—
48#2293—21no—
49#2293C2212yes—
50C2×#389—21no—
51C2×#389—21no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C42×D4—21no—
55C42×D4C2212yes—
56C2×C4×C4⋊C4—21no—
57C2×C4×C4⋊C4—21no—
58C2×C43C2211yescharacteristic
59#1990C2212yes—
60#1981C2214yes—
61#1968C2218yes—
62#1957C2212yes—
63#1956C2212yes—
64#1956C2212yes—
65#1948C2214yes—
66#1933C2218yes—
67#1927C2218yes—
68#1880C2218yes—
69#1877C2218yes—
70#1874C2214yes—
71#1865C2218yes—
72#1853C2214yes—
73#1844C2212yes—
74#1834C2212yes—
75#1802C2214yes—
76#1787C2214yes—
77C4×C4⋊C8C2212yes—
78C4×C4⋊C8C2212yes—
79C4×#103C2218yes—
80C4×C22⋊C8C2212yes—
81C4×C22⋊C8C2212yes—
82#1744C2212yes—
83C42×C8C2212yes—
84C22×#41—21no—
85C23×Q8C2312yes—
86C22×SD16C2318yes—
87#532—24no—
88#532C2×C418yes—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8C2×C418yescomplemented
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#124—21no—
98C2×#121—21no—
99C2×#121C2312yes—
100C4×#41—48no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—48no—
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42—22no—
127C22×C42—22no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#492C2×C4116yescomplemented
136#487C2×C4116yescomplemented
137#461C2×C4132yescomplemented
138#448C2×C4132yescomplemented
139C4×SD16—416no—
140#431—24no—
141#431C2×C418yescomplemented
142C2×#106C2318yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2318yes—
146C2×#103C2318yes—
147C2×#103C2318yes—
148C2×#102C2318yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153C2×C4×C8C2314yes—
154C2×C4×C8C2312yes—
155#391—22no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391C2314yes—
160#391C2314yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—22no—
184#386—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#380—21no—
193#380C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C2×#95—22no—
215C2×#95—22no—
216C43D411yescharacteristic
217C43D411yescharacteristic
218C43D411yescharacteristic
219C43D411yescharacteristic
220C43C2311yescharacteristic
221C43C2311yescharacteristic
222C43C2311yescharacteristic
223C43C2311yescharacteristic
224#340C2318yes—
225#340C2318yes—
226#336C2314yes—
227#336C2314yes—
228C2×#41—28no—
229C2×#41—44no—
230C2×#41—44no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—24no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8C22×C418yes—
237C22×Q8C2414yes—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—48no—
255C2×SD16—48no—
256C2×SD16C22×C4132yes—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8C22×C418yes—
265C4⋊Q8C22×C418yes—
266C4⋊Q8C22×C414yes—
267#127—22no—
268#127C22×C414yes—
269#124—28no—
270#121—28no—
271#121—24no—
272#121C22×C418yes—
273C4×Q8—216no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×Q8—48no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4C22×C418yes—
290#117—44no—
291#117—48no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—48no—
340C2×C42—48no—
341C2×C42SD1618yescomplemented
342C2×C42C22×C414yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42#4114yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#106—48no—
373C4⋊C8—24no—
374C4⋊C8C22×C418yes—
375#103—48no—
376#103—48no—
377#103C22×C4132yes—
378#102—48no—
379C22⋊C8—24no—
380C22⋊C8C22×C418yes—
381C4×C8—24no—
382C4×C8—28no—
383C4×C8—44no—
384C4×C8C22×C418yes—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24C22×D411yescharacteristic
397#41—432no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—416no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—416no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8C23×C418yes—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C23×C414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4C4×D414yes—
458C22×C4C4×D414yes—
459C22×C4C2×SD1614yes—
460C22×C4C2×SD1614yes—
461C22×C4C2×SD1614yes—
462C22×C4#13514yes—
463C22×C4C23×C412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479SD16—432no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8C23×C418yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4C23×C414yes—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—48no—
537C42—48no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42C4×D414yes—
542C42C4×D414yes—
543C42C2×SD1614yes—
544C42C2×SD1614yes—
545C42C2×SD1614yes—
546C42C2×SD1614yes—
547C42#13714yes—
548C42C23×C412yes—
549C42C23×C412yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C42H(5,𝔽2)14yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—24no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C2×C4×D412yes—
565C23C22×SD1612yes—
566C23C2×#13512yes—
567C23C23×D411yescharacteristic
568Q8—216no—
569Q8—416no—
570Q8—48no—
571Q8—48no—
572D4—28no—
573D4—416no—
574D4—48no—
575D4—48no—
576D4—44no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—416no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—88no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4C4×SD16116yescomplemented
625C2×C4C2×C4×D412yes—
626C2×C4C2×C4×D412yes—
627C2×C4C2×C4×D412yes—
628C2×C4C2×C4×D412yes—
629C2×C4C2×C4×D412yes—
630C2×C4#53512yes—
631C2×C4#53512yes—
632C2×C4C22×SD1612yes—
633C2×C4C22×SD1612yes—
634C2×C4C22×SD1612yes—
635C2×C4C22×SD1612yes—
636C2×C4C22×SD1612yes—
637C2×C4C22×SD1612yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13512yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4#57512yes—
644C2×C4C24×C411yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4#58512yes—
648C8—28no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22C2×C4×SD1614yes—
664C22C22×C4×D411yescharacteristic
665C22C2×#53511yescharacteristic
666C22#344911yescharacteristic
667C22C23×SD1611yescharacteristic
668C22C22×#13511yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57511yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—48no—
680C4—41no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4C2×C4×SD1614yes—
686C4C2×C4×SD1614yes—
687C4C4×#13714yes—
688C4#320614yesdirect product factor
689C4C4×H(5,𝔽2)12yes—
690C4#346012yes—
691C2—24no—
692C2—84no—
693C2C4×#53511yescharacteristic
694C2C22×C4×SD1611yescharacteristic
695C2C2×C4×#13711yescharacteristic
696C2C2×#320611yescharacteristic
697C2#6078211yescharacteristic
698C2#6094111yescharacteristic
699C2#6094711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×C4⋊Q8C2semidirect
4. #22268C2semidirect
5. #22218C2semidirect
6. #22146C2semidirect
7. #22041C2semidirect
8. #21981C2semidirect
10. #21804C2semidirect
11. C4×#492C2semidirect
12. C4×#487C2semidirect
13. C4×#461C2semidirect
14. C4×#448C2semidirect
15. C4×#431C2semidirect
20. #3206C4direct (N × H)
92. C2×C4⋊Q8C2×C4semidirect
135. #492C2×C4semidirect
136. #487C2×C4semidirect
137. #461C2×C4semidirect
138. #448C2×C4semidirect
141. #431C2×C4semidirect
341. C2×C42SD16semidirect
624. C2×C4C4×SD16semidirect
688. C4#3206direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32062
C4×#5322
C2×C4×C4⋊Q82
#222682
#222182
#221462
#220412
#219812
#218592
#218042
C4×#4922
C4×#4872
C4×#4612
C4×#4482
C4×#4312