Finite Group Databasebeta

← C2×#21059 #6371635

Subgroups — C2×#21059 #6371635

Order 512.

3874 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C22×#417C211yesmaximal, characteristic
3C22×#417C211yesmaximal, characteristic
4C2×#2297C211yesmaximal, characteristic
5C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
6#21059C2116yesmaximal, direct product factor
7C2×#1928C212yesmaximal, complemented
8C2×#1926C212yesmaximal, complemented
9C2×#1927C212yesmaximal, complemented
10C2×#1925C212yesmaximal, complemented
11C2×#1736C211yesmaximal, characteristic, complemented
12C2×#1728C211yesmaximal, characteristic, complemented
13C23×#41C2211yescharacteristic
14C23×M16C2211yescharacteristic
15C23×M16C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42—21no—
20C23×C42C2211yescharacteristic
21C2×#428—28no—
22C2×#428C2218yescomplemented
23C22×#107C2212yes—
24C22×#106C2212yes—
25C2×#417—28no—
26C2×#417C4116yescomplemented
27C2×#417C2218yes—
28C2×#417C2218yes—
29C22×#103C2212yes—
30C22×#103C2212yes—
31C22×#102C2212yes—
32C22×#102C2212yes—
33#2297—24no—
34#2297C2218yes—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#386—21no—
40C2×#386—21no—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4—24no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2218yescomplemented
47C2×C4×C22⋊C4—21no—
48C2×C4×C22⋊C4—21no—
49C22×#95—21no—
50#1928C22116yescomplemented
51#1927C22116yescomplemented
52#1926C22116yescomplemented
53#1925C22116yescomplemented
54#1736C2218yescomplemented
55C2×#340C2212yes—
56C2×#340C2212yes—
57#1728C2218yescomplemented
58C2×#336C2212yes—
59C2×#336C2212yes—
60C22×#41—22no—
61C22×#41—24no—
62C22×#41—24no—
63C22×#41—24no—
64C22×#41—24no—
65C22×#41—24no—
66C22×#41C2×C418yes—
67C22×#41C2314yes—
68C23×Q8C2311yescharacteristic
69C23×D4—21no—
70C23×D4C2311yescharacteristic
71C24×C4—21no—
72C24×C4—21no—
73C24×C4—21no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4C2311yescharacteristic
77C22×M16—24no—
78C22×M16—24no—
79C22×M16—24no—
80C22×M16—24no—
81C22×M16—24no—
82C22×M16C2×C418yes—
83C22×M16C2314yes—
84C22×M16C2314yes—
85C23×C8C2312yes—
86C23×C8C2312yes—
87C2×#117—22no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117C2314yes—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4C2×C418yes—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C42—24no—
111C22×C42—24no—
112C22×C42—22no—
113C22×C42—22no—
114C22×C42—28no—
115C22×C42—21no—
116C22×C42—48no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D414yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2314yes—
123#428—232no—
124C2×#107—28no—
125C2×#107C2318yes—
126C2×#106—28no—
127C2×#106C2318yes—
128#417—232no—
129#417C2×C4164yescomplemented
130C2×#103—28no—
131C2×#103C2×C4116yescomplemented
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×#102—28no—
135C2×#102C2×C4116yescomplemented
136C2×#102C2318yes—
137C2×#102C2318yes—
138#391—22no—
139#391—22no—
140#391C2314yes—
141#391C2314yes—
142#390—22no—
143#390—22no—
144#390C2314yes—
145#390C2314yes—
146#386—28no—
147#386—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4C2318yes—
153C4×C4⋊C4C2318yes—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—28no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95—21no—
162C2×#95—21no—
163#340C2318yes—
164#340C2318yes—
165#336C2318yes—
166#336C2318yes—
167C25—21no—
168C25C2×D411yescharacteristic
169C2×#41—216no—
170C2×#41—28no—
171C2×#41—232no—
172C2×#41—232no—
173C2×#41—28no—
174C2×#41—28no—
175C2×#41—28no—
176C2×#41—28no—
177C2×#41—48no—
178C2×#41—48no—
179C2×#41—416no—
180C2×#41—416no—
181C2×#41C22×C4116yes—
182C22×Q8—21no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C22×C414yes—
189C22×Q8C2412yes—
190C22×D4—28no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4C22×C414yes—
201C22×D4C2412yes—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C22×C414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D414yes—
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D414yes—
235C23×C4#4114yes—
236C23×C4#4114yes—
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C2×M16—232no—
242C2×M16—28no—
243C2×M16—28no—
244C2×M16—28no—
245C2×M16—28no—
246C2×M16—28no—
247C2×M16—416no—
248C2×M16C22×C4116yes—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8C22×C418yes—
255C22×C8C2414yes—
256C22×C8C2414yes—
257#117—28no—
258#117—28no—
259#117—48no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C4116yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D4116yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2414yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—216no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—28no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—48no—
309C2×C42—432no—
310C2×C42—44no—
311C2×C42—48no—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318#107—216no—
319#106—216no—
320#103—216no—
321#103C22×C4132yescomplemented
322#102—216no—
323#102C22×C4132yescomplemented
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24C4×D414yes—
343C24#12114yes—
344C24#12314yes—
345C24C22×D412yes—
346C24C22×D411yescharacteristic
347#41—232no—
348#41—232no—
349#41—416no—
350#41—416no—
351#41—464no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—28no—
355C2×Q8—28no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8C23×C414yes—
365C2×D4—216no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—432no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4C23×C414yes—
382C22×C4—232no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—416no—
413C22×C4—432no—
414C22×C4—432no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—432no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4#117116yes—
433C22×C4C4×D4116yes—
434C22×C4C4×D414yes—
435C22×C4C4×D418yes—
436C22×C4#12114yes—
437C22×C4#12118yes—
438C22×C4#12314yes—
439C22×C4#12318yes—
440C22×C4#124116yes—
441C22×C4#126116yes—
442C22×C4AGL(1,ℤ/8)14yes—
443C22×C4#13714yes—
444C22×C4C23×C412yes—
445C22×C4C23×C414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4114yes—
462C22×C4C2511yescharacteristic
463M16—232no—
464M16—416no—
465C2×C8—216no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—48no—
472C2×C8C23×C418yes—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4C23×C418yes—
486C4⋊C4C22×D418yes—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C42—28no—
493C42—28no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—416no—
501C42—48no—
502C42—416no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23#388116yes—
528C23C2×C4×D414yes—
529C23C2×C4×D412yes—
530C23C2×#12114yes—
531C23C2×#12112yes—
532C23C2×#12314yes—
533C23C2×#12312yes—
534C23C2×AGL(1,ℤ/8)12yes—
535C23C2×#13712yes—
536C23#57512yes—
537C23#57512yes—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—24no—
541Q8—42no—
542Q8—42no—
543Q8—48no—
544D4—24no—
545D4—24no—
546D4—432no—
547D4—42no—
548D4—42no—
549D4—48no—
550C2×C4—232no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—416no—
567C2×C4—432no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—432no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#388116yes—
590C2×C4#388132yes—
591C2×C4C2×#11714yes—
592C2×C4C2×#11718yes—
593C2×C4C2×C4×D414yes—
594C2×C4C2×C4×D418yes—
595C2×C4C2×C4×D414yes—
596C2×C4C2×C4×D412yes—
597C2×C4C2×#12114yes—
598C2×C4C2×#12112yes—
599C2×C4C2×#12314yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12414yes—
602C2×C4C2×#12418yes—
603C2×C4C2×#12614yes—
604C2×C4C2×#12618yes—
605C2×C4C2×AGL(1,ℤ/8)14yes—
606C2×C4C2×#13714yes—
607C2×C4C24×C411yescharacteristic
608C2×C4C23×D411yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C8—28no—
614C8—44no—
615C22—232no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×#38818yes—
631C22C2×#38814yes—
632C22C4×AGL(1,ℤ/8)14yes—
633C22C4×#13714yes—
634C22#303614yes—
635C22#303714yes—
636C22#309214yes—
637C22#309314yes—
638C22C22×C4×D411yescharacteristic
639C22C22×#12111yescharacteristic
640C22C22×#12311yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C22×#13711yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57511yescharacteristic
645C4—24no—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—48no—
651C4—48no—
652C4—82no—
653C4—82no—
654C4C2×#38818yes—
655C4C2×#38814yes—
656C4C22×#11711yescharacteristic
657C4C22×C4×D411yescharacteristic
658C4C22×#12411yescharacteristic
659C4C22×#12611yescharacteristic
660C2—28no—
661C2—84no—
662C2#2105918yesdirect product factor
663C2C22×#38811yescharacteristic
664C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
665C2C2×C4×#13711yescharacteristic
666C2C2×#303611yescharacteristic
667C2C2×#303711yescharacteristic
668C2C2×#309211yescharacteristic
669C2C2×#309311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#428C2semidirect
5. C22×C4×C4⋊C4C2semidirect
6. #21059C2direct (N × H)
7. C2×#1928C2semidirect
8. C2×#1926C2semidirect
9. C2×#1927C2semidirect
10. C2×#1925C2semidirect
11. C2×#1736C2semidirect
12. C2×#1728C2semidirect
22. C2×#428C22semidirect
26. C2×#417C4semidirect
46. C2×C4×C4⋊C4C22semidirect
50. #1928C22semidirect
51. #1927C22semidirect
52. #1926C22semidirect
53. #1925C22semidirect
54. #1736C22semidirect
57. #1728C22semidirect
129. #417C2×C4semidirect
131. C2×#103C2×C4semidirect
135. C2×#102C2×C4semidirect
321. #103C22×C4semidirect
323. #102C22×C4semidirect
662. C2#21059direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4282
C22×#4172
C2×#22972
C22×C4×C4⋊C42
#210592
C2×#19282
C2×#19262
C2×#19272
C2×#19252
C2×#17362
C2×#17282