Finite Group Databasebeta

← C4×#1520 #182638

Subgroups — C4×#1520 #182638

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4C4×#380C211yesmaximal, characteristic
5#8221C214yesmaximal, complemented
6C4×#328C214yesmaximal, complemented
7#8023C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12C22×C4×C8—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C412yescomplemented
19C4×C22⋊Q8C2212yes—
20C4×C22⋊Q8C2212yes—
21C2×#391C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#387—21no—
24C2×#385—21no—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C4×C22⋊C8—21no—
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yes—
35C4×C22⋊C8C2212yes—
36C2×#336—21no—
37#1520C4116yesdirect product factor
38C2×#328C2218yes—
39C4⋊#95—21no—
40#1434C2212yes—
41C22⋊#95—21no—
42C4×#95C2212yes—
43#1294C2218yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8C2×C418yes—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—44no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—22no—
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8C2×C418yes—
89C2×C22⋊C8C2314yes—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—44no—
93#391—21no—
94#391—21no—
95#391C2×C412yes—
96#391C2×C412yes—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#380—21no—
110#380—21no—
111#380—24no—
112#380C2×C412yes—
113#380C2×C418yes—
114#380C2312yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—44no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—44no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95C2314yes—
138#336—22no—
139#336—22no—
140#328C2×C4132yescomplemented
141C25C2×D411yescharacteristic
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8C4214yescomplemented
147C22×Q8C22×C412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4SD1614yes—
169C23×C4Q1614yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C414yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4114yes—
181C23×C4C2411yescharacteristic
182C22×C8—28no—
183C22×C8—24no—
184C22×C8—44no—
185C22×C8—44no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—24no—
190C22⋊Q8C4218yescomplemented
191C22⋊Q8C22×C418yes—
192C4×Q8—44no—
193C4×Q8—44no—
194C4×Q8—44no—
195C4×Q8—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C4214yescomplemented
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C22⋊C8—28no—
265C22⋊C8—24no—
266C22⋊C8C22×C418yes—
267C4×C8—44no—
268C4×C8—44no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—24no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95C22×C418yes—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C4×D414yes—
291C24C2×SD1612yes—
292C24C2×Q1612yes—
293C24#13514yes—
294C24C22×D411yescharacteristic
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8C2×C4214yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
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—28no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—416no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4#10314yes—
375C22×C4#10314yes—
376C22×C4#10314yes—
377C22×C4#10314yes—
378C22×C4C2×C4212yes—
379C22×C4C2×C4212yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4#11712yes—
391C22×C4#11712yes—
392C22×C4C4×D414yes—
393C22×C4C4×D414yes—
394C22×C4C4×D414yes—
395C22×C4C2×SD1612yes—
396C22×C4C2×SD1612yes—
397C22×C4C2×Q1612yes—
398C22×C4C2×Q1612yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C23×C412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C2×C8—28no—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—416no—
407C4⋊C4—22no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—416no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4C2×C4214yes—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—416no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—44no—
442C42—44no—
443C42—44no—
444C42—44no—
445C42—88no—
446C42#9914yes—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23C4×C22⋊C412yes—
468C23C4×C22⋊C412yes—
469C23C2×#10312yes—
470C23C2×#10312yes—
471C23#41612yes—
472C23#41612yes—
473C23C4×SD1618yes—
474C23C4×Q1618yes—
475C23C22×C22⋊C411yescharacteristic
476C23C2×C4×D412yes—
477C23C22×SD1611yescharacteristic
478C23C22×Q1611yescharacteristic
479C23C2×#13512yes—
480Q8—44no—
481Q8—44no—
482Q8—48no—
483C2×C4—28no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—48no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4C4×C22⋊C412yes—
552C2×C4C4×C22⋊C412yes—
553C2×C4C2×#9912yes—
554C2×C4C2×#9912yes—
555C2×C4C2×#9912yes—
556C2×C4#41012yes—
557C2×C4C2×#10312yes—
558C2×C4C2×#10312yes—
559C2×C4C2×#10312yes—
560C2×C4C2×#10312yes—
561C2×C4#42112yes—
562C2×C4#42112yes—
563C2×C4C22×C4211yescharacteristic
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×C4×D412yes—
566C8—48no—
567C22—21no—
568C22—21no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—48no—
588C22—48no—
589C22C4×#10314yes—
590C22C4×#10314yes—
591C22C2×C4×C22⋊C411yescharacteristic
592C22C22×#9911yescharacteristic
593C22C2×#41011yescharacteristic
594C22C22×#10311yescharacteristic
595C22C2×#41611yescharacteristic
596C22C2×#42111yescharacteristic
597C22C2×#42111yescharacteristic
598C22C2×C4×SD1612yes—
599C22C2×C4×Q1612yes—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—21no—
607C4—22no—
608C4—21no—
609C4—41no—
610C4—41no—
611C4—44no—
612C4—44no—
613C4—88no—
614C4—82no—
615C4—82no—
616C4#152014yesdirect product factor
617C4C4×#9914yes—
618C2—22no—
619C2—22no—
620C2—24no—
621C2—24no—
622C2C2×#152011yescharacteristic
623C2#1756711yescharacteristic
624C2#1757211yescharacteristic
625C2#1757511yescharacteristic
626C2C2×C4×#9911yescharacteristic
627C2C4×#42111yescharacteristic
628C2C2×C4×#10311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8221C2semidirect
6. C4×#328C2semidirect
9. C22×C4×Q8C4semidirect
17. C4×C22⋊Q8C4semidirect
18. C4×C22⋊Q8C4semidirect
21. C2×#391C4semidirect
22. C2×#389C4semidirect
29. C2×C4×C4⋊C4C4semidirect
37. #1520C4direct (N × H)
140. #328C2×C4semidirect
146. C22×Q8C42semidirect
190. C22⋊Q8C42semidirect
214. C2×C4⋊C4C42semidirect
616. C4#1520direct (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×C4×C22⋊C82
C2×#15202
C4×#3802
#82212
C4×#3282
#80232