Finite Group Databasebeta

← C2×#9208 #197573

Subgroups — C2×#9208 #197573

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3035C211yesmaximal, characteristic
2C2×C8×#41C211yesmaximal, characteristic, complemented
3C2×#1806C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic, complemented
6#9208C218yesmaximal, direct product factor
7C2×#1301C211yesmaximal, characteristic, complemented
8C2×#1279C211yesmaximal, characteristic, complemented
9C22×#135—21no—
10C2×#532C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3035—22no—
13#3035—22no—
14#3035C414yescomplemented
15#3035C2214yes—
16C2×#464—21no—
17C2×#462C411yescharacteristic, complemented
18C2×#460C411yescharacteristic, complemented
19C2×#463—21no—
20C2×#461C411yescharacteristic, complemented
21C2×#459C411yescharacteristic, complemented
22C8×#41—22no—
23C8×#41—22no—
24C8×#41—22no—
25C8×#41C2214yescomplemented
26C2×C8×Q8—21no—
27C2×C8×D4—22no—
28C2×C8×D4—21no—
29C2×#431—21no—
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#104—21no—
34C2×#416—21no—
35C22×C4×C8—21no—
36C22×C4×C8C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1806—22no—
39#1806C2214yescomplemented
40C2×#339—21no—
41#1502—22no—
42#1502C2214yescomplemented
43C2×#327—22no—
44#1492—22no—
45#1492C2214yescomplemented
46C2×D4⋊C8—21no—
47C2×D4⋊C8—21no—
48#1301C2214yescomplemented
49#1279C2214yescomplemented
50C22×#41—21no—
51C22×#41D411yescharacteristic
52C2×#135—24no—
53C2×#135—24no—
54C2×#135—44no—
55C2×#135—44no—
56C22×Q16—21no—
57C22×SD16—21no—
58C22×SD16—21no—
59C22×D8—21no—
60C22×M16—21no—
61C23×C8—22no—
62C23×C8—21no—
63C23×C8—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532C2×C412yes—
71#532C2312yes—
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×#127C2×C411yescharacteristic
74C2×#124—21no—
75C2×#121—21no—
76C2×#121—21no—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41—21no—
80C4×#41—21no—
81C4×#41C2×C412yes—
82C4×#41D412yes—
83C4×#41D412yes—
84C4×#41C2312yes—
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8D411yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×#117—21no—
94C2×#117D411yescharacteristic
95C22×C4⋊C4—21no—
96C22×C42—21no—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#464—24no—
100#464—24no—
101#463—24no—
102#463—24no—
103#462—22no—
104#462C2×C414yescomplemented
105#461—22no—
106#461C2×C414yescomplemented
107#460—22no—
108#460C2×C414yescomplemented
109#459—22no—
110#459C2×C414yescomplemented
111C8×Q8—24no—
112C8×Q8—24no—
113C8×D4—28no—
114C8×D4—24no—
115C8×D4—24no—
116C8×D4—28no—
117#431—24no—
118#431—21no—
119#431—21no—
120#431—24no—
121#431D412yes—
122#431C2312yes—
123#423—21no—
124#423—21no—
125#423—21no—
126#423C2312yes—
127C2×C4⋊C8—22no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8C2×C411yescharacteristic
135C2×C4⋊C8C2×C411yescharacteristic
136C2×C4⋊C8C2×C412yes—
137C2×C4⋊C8C2312yes—
138C2×#104—24no—
139C2×#104—44no—
140#416—24no—
141#416—24no—
142C2×#103—21no—
143C2×#103—21no—
144C2×#102—21no—
145C2×#102—21no—
146C2×C22⋊C8—22no—
147C2×C22⋊C8D412yes—
148C2×C4×C8—22no—
149C2×C4×C8—24no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154C2×C4×C8—24no—
155C2×C4×C8D411yescharacteristic
156C2×C4×C8D411yescharacteristic
157C2×C4×C8D412yes—
158C2×C4×C8C2312yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4C2312yes—
163C2×#95—21no—
164#339—24no—
165#327—28no—
166D4⋊C8—24no—
167D4⋊C8—24no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—41no—
175C2×#41—41no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—42no—
180C2×#41C22⋊C412yes—
181C2×#41D814yes—
182C2×#41SD1614yes—
183C2×#41C2×D412yes—
184C22×Q8—21no—
185C22×Q8C22⋊C411yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—41no—
191C22×D4C22⋊C411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4#4111yescharacteristic
200#135—416no—
201#135—48no—
202#135—48no—
203C2×Q16—24no—
204C2×Q16—44no—
205C2×SD16—24no—
206C2×SD16—24no—
207C2×SD16—44no—
208C2×SD16—44no—
209C2×D8—24no—
210C2×D8—44no—
211C2×M16—22no—
212C2×M16—41no—
213C2×M16—42no—
214C2×M16—42no—
215C22×C8—21no—
216C22×C8—28no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—44no—
226C22×C8—48no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8—41no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×C412yes—
234#127—21no—
235#127—21no—
236#127—21no—
237#127C22×C412yes—
238#124—22no—
239#124—22no—
240#124—41no—
241#124—41no—
242#121—22no—
243#121—22no—
244#121—22no—
245#121—22no—
246#121—41no—
247#121—41no—
248#121—41no—
249#121—41no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C22⋊C412yes—
255C4×Q8C22⋊C412yes—
256C4×Q8C22×C412yes—
257C4×Q8C2×D412yes—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4C22⋊C412yes—
279C4×D4C22⋊C412yes—
280C4×D4C22×C412yes—
281C4×D4C2×D412yes—
282#117—21no—
283#117—21no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117C22⋊C412yes—
289#117C2×D412yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—41no—
298C2×C4⋊C4C22⋊C411yescharacteristic
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—41no—
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42#4111yescharacteristic
326C2×C42#4111yescharacteristic
327C2×C42C2411yescharacteristic
328C4⋊C8—28no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—41no—
338C4⋊C8—41no—
339C4⋊C8C22×C412yes—
340C4⋊C8C22×C412yes—
341#104—48no—
342#103—24no—
343#103—24no—
344#102—24no—
345#102—24no—
346C22⋊C8—28no—
347C22⋊C8—22no—
348C22⋊C8C2×D414yes—
349C4×C8—28no—
350C4×C8—21no—
351C4×C8—22no—
352C4×C8—21no—
353C4×C8—22no—
354C4×C8—22no—
355C4×C8—22no—
356C4×C8C2×D412yes—
357C4×C8C2×D412yes—
358#95—22no—
359#95—41no—
360C24—21no—
361C24—41no—
362C24#12111yescharacteristic
363#41—22no—
364#41—22no—
365#41—24no—
366#41—24no—
367#41—24no—
368#41—24no—
369#41—24no—
370#41—42no—
371#41—42no—
372#41—42no—
373#41—42no—
374#41—44no—
375#41—44no—
376#41—82no—
377#41—82no—
378#41—82no—
379#41—82no—
380#41#10218yes—
381#41C2×D814yes—
382#41C2×SD1614yes—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8#10214yes—
390C2×Q8C2×C22⋊C412yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4#10214yes—
413C2×D4C2×C22⋊C412yes—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4#12011yescharacteristic
447C22×C4#12011yescharacteristic
448C22×C4#12111yescharacteristic
449C22×C4#12111yescharacteristic
450C22×C4#12111yescharacteristic
451C22×C4#12111yescharacteristic
452C22×C4#12111yescharacteristic
453C22×C4#12311yescharacteristic
454C22×C4#12311yescharacteristic
455C22×C4C2×D812yes—
456C22×C4C2×SD1612yes—
457C22×C4#13512yes—
458C22×C4#13512yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4111yescharacteristic
462Q16—48no—
463SD16—48no—
464SD16—48no—
465D8—48no—
466M16—42no—
467M16—42no—
468M16—82no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—416no—
478C2×C8—44no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—44no—
485C2×C8—42no—
486C2×C8—82no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4C2×C22⋊C412yes—
501C4⋊C4C2×C22⋊C412yes—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—42no—
519C42—42no—
520C42—82no—
521C42C2×C22⋊C411yescharacteristic
522C42C2×C22⋊C411yescharacteristic
523C42C4×D411yescharacteristic
524C42C4×D411yescharacteristic
525C42C4×D411yescharacteristic
526C42C4×D411yescharacteristic
527C42#12011yescharacteristic
528C42#12011yescharacteristic
529C42#12111yescharacteristic
530C42#12111yescharacteristic
531C42#12311yescharacteristic
532C42#12311yescharacteristic
533C42C23×C411yescharacteristic
534C42C22×D411yescharacteristic
535C42C22×D411yescharacteristic
536C42C2×#4111yescharacteristic
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23#38611yescharacteristic
548C23#38611yescharacteristic
549C23#46512yes—
550C23#46612yes—
551C23C2×#12111yescharacteristic
552Q8—22no—
553Q8—24no—
554Q8—42no—
555Q8—42no—
556Q8—82no—
557Q8C2×#10214yes—
558D4—28no—
559D4—24no—
560D4—24no—
561D4—22no—
562D4—24no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—82no—
568D4—82no—
569D4—84no—
570D4—82no—
571D4—82no—
572D4—82no—
573D4—82no—
574D4C2×#10214yes—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—82no—
605C2×C4—88no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—84no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4#38611yescharacteristic
612C2×C4#38611yescharacteristic
613C2×C4#38611yescharacteristic
614C2×C4#38611yescharacteristic
615C2×C4#38611yescharacteristic
616C2×C4#38611yescharacteristic
617C2×C4C2×#10212yes—
618C2×C4C2×#10212yes—
619C2×C4#41612yes—
620C2×C4#41612yes—
621C2×C4#44312yes—
622C2×C4#44412yes—
623C2×C4#46512yes—
624C2×C4#46512yes—
625C2×C4#46512yes—
626C2×C4#46612yes—
627C2×C4#46612yes—
628C2×C4#46612yes—
629C2×C4C22×C22⋊C411yescharacteristic
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C2×#12011yescharacteristic
633C2×C4C2×#12111yescharacteristic
634C2×C4C2×#12311yescharacteristic
635C2×C4C22×D811yescharacteristic
636C2×C4C22×SD1611yescharacteristic
637C2×C4C2×#13511yescharacteristic
638C2×C4C2×#13511yescharacteristic
639C8—24no—
640C8—44no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—82no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—48no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—164no—
659C22#187612yes—
660C22#189312yes—
661C22#189312yes—
662C22#190612yes—
663C22C2×#38611yescharacteristic
664C22C2×#46511yescharacteristic
665C22C2×#46611yescharacteristic
666C4—21no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—84no—
676C4—82no—
677C4#187612yes—
678C4#189312yes—
679C4#189312yes—
680C4C22×#10211yescharacteristic
681C4C2×#41611yescharacteristic
682C4C2×#44311yescharacteristic
683C4C2×#44411yescharacteristic
684C4C2×#46511yescharacteristic
685C4C2×#46611yescharacteristic
686C2—21no—
687C2—22no—
688C2—21no—
689C2—44no—
690C2—162no—
691C2#920814yesdirect product factor
692C2C2×#187611yescharacteristic
693C2C2×#189311yescharacteristic
694C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC2×#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C8×#41C2semidirect
3. C2×#1806C2semidirect
4. C2×#1502C2semidirect
5. C2×#1492C2semidirect
6. #9208C2direct (N × H)
7. C2×#1301C2semidirect
8. C2×#1279C2semidirect
14. #3035C4semidirect
17. C2×#462C4semidirect
18. C2×#460C4semidirect
20. C2×#461C4semidirect
21. C2×#459C4semidirect
25. C8×#41C22semidirect
39. #1806C22semidirect
42. #1502C22semidirect
45. #1492C22semidirect
48. #1301C22semidirect
49. #1279C22semidirect
104. #462C2×C4semidirect
106. #461C2×C4semidirect
108. #460C2×C4semidirect
110. #459C2×C4semidirect
691. C2#9208direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30352
C2×C8×#412
C2×#18062
C2×#15022
C2×#14922
#92082
C2×#13012
C2×#12792