Finite Group Databasebeta

← C2×#8996 #194365

Subgroups — C2×#8996 #194365

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3035C211yesmaximal, characteristic
2C2×#2925C211yesmaximal, characteristic, complemented
3C2×#1492C211yesmaximal, characteristic, complemented
4C2×#1480C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic, complemented
6#8996C218yesmaximal, direct product factor
7C2×#1301C211yesmaximal, characteristic, complemented
8C2×#1277C211yesmaximal, characteristic, complemented
9C22×#135—21no—
10C2×#532C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3035—22no—
13#3035—22no—
14#3035C414yes—
15#3035C2214yes—
16C2×#464—21no—
17C2×#462C411yescharacteristic
18C2×#460C411yescharacteristic
19C2×#463—21no—
20C2×#461C411yescharacteristic
21C2×#459C411yescharacteristic
22#2925—22no—
23#2925—22no—
24#2925—22no—
25#2925C2214yescomplemented
26C2×#433—22no—
27C2×#432—21no—
28C2×#431—21no—
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C2×#416—21no—
34C2×#408—21no—
35C2×#405—21no—
36C22×C4×C8C2211yescharacteristic
37#1492—22no—
38#1492C2214yescomplemented
39#1480—22no—
40#1480C2214yescomplemented
41#1476—22no—
42#1476C2214yescomplemented
43C2×Q8⋊C8—21no—
44C2×D4⋊C8—21no—
45C2×D4⋊C8—21no—
46C2×D4⋊C8—22no—
47C2×D4⋊C8—21no—
48#1301C2214yescomplemented
49#1277C2214yescomplemented
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—21no—
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×C42—21no—
96C22×C42C2311yescharacteristic
97#464—24no—
98#464—24no—
99#463—24no—
100#463—24no—
101#462—22no—
102#462C2×C414yes—
103#461—22no—
104#461C2×C414yes—
105#460—22no—
106#460C2×C414yes—
107#459—22no—
108#459C2×C414yes—
109#433—28no—
110#433—28no—
111#432—24no—
112#432—24no—
113#431—24no—
114#431—24no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431C2312yes—
119#423—21no—
120#423—21no—
121#423D412yes—
122#423C2312yes—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8C2×C411yescharacteristic
132C2×C4⋊C8C2×C411yescharacteristic
133C2×C4⋊C8C2×C412yes—
134C2×C4⋊C8D411yescharacteristic
135C2×C4⋊C8D411yescharacteristic
136C2×C4⋊C8D412yes—
137C2×C4⋊C8D412yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140#416—24no—
141#416—24no—
142C2×#103—21no—
143C2×#103—21no—
144C2×#102—21no—
145C2×#102—21no—
146#408—24no—
147#408—24no—
148C2×C22⋊C8—22no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8—21no—
151#405—24no—
152#405—24no—
153C2×#97—22no—
154C2×C4×C8—22no—
155C2×C4×C8—21no—
156C2×C4×C8—21no—
157C2×C4×C8—21no—
158C2×C4×C8—21no—
159C2×C4×C8—21no—
160C2×C4×C8—21no—
161C2×C4×C8C2312yes—
162Q8⋊C8—24no—
163D4⋊C8—24no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166D4⋊C8—28no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—41no—
176C2×#41—41no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—42no—
180C2×#41—42no—
181C2×#41C22⋊C412yes—
182C2×#41C2×D412yes—
183C22×Q8—21no—
184C22×Q8C22⋊C411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—41no—
190C22×D4C22⋊C411yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—41no—
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4#4111yescharacteristic
198#135—48no—
199#135—48no—
200#135—416no—
201C2×Q16—24no—
202C2×Q16—44no—
203C2×SD16—24no—
204C2×SD16—24no—
205C2×SD16—44no—
206C2×SD16—44no—
207C2×D8—24no—
208C2×D8—44no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—42no—
212C2×M16—42no—
213C22×C8—24no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—42no—
225C22×C8—41no—
226C22×C8—41no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8—41no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C22×C412yes—
236#127—21no—
237#127—21no—
238#127—21no—
239#127C22×C412yes—
240#124—22no—
241#124—22no—
242#124—41no—
243#124—41no—
244#121—22no—
245#121—22no—
246#121—22no—
247#121—22no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—41no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8C22⋊C412yes—
257C4×Q8C22⋊C412yes—
258C4×Q8C22×C412yes—
259C4×Q8C2×D412yes—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—41no—
280C4×D4C22⋊C412yes—
281C4×D4C22⋊C412yes—
282C4×D4C22×C412yes—
283C4×D4C2×D412yes—
284#117—21no—
285#117—21no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117C22⋊C412yes—
291#117C2×D412yes—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42#4111yescharacteristic
320C2×C42#4111yescharacteristic
321C2×C42C2411yescharacteristic
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—22no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—41no—
335C4⋊C8C22×C412yes—
336C4⋊C8C22×C412yes—
337C4⋊C8C2×D412yes—
338C4⋊C8C2×D412yes—
339C4⋊C8C2×D414yes—
340#103—24no—
341#103—24no—
342#102—24no—
343#102—24no—
344C22⋊C8—28no—
345C22⋊C8—28no—
346C22⋊C8—22no—
347C22⋊C8—41no—
348#97—28no—
349C4×C8—28no—
350C4×C8—22no—
351C4×C8—22no—
352C4×C8—22no—
353C4×C8—41no—
354C4×C8—41no—
355C4×C8—41no—
356C24—21no—
357C24—41no—
358C24#12311yescharacteristic
359#41—22no—
360#41—22no—
361#41—22no—
362#41—22no—
363#41—24no—
364#41—24no—
365#41—42no—
366#41—42no—
367#41—42no—
368#41—42no—
369#41—42no—
370#41—42no—
371#41—42no—
372#41—42no—
373#41—44no—
374#41—44no—
375#41—82no—
376#41—82no—
377#41—82no—
378#41—82no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8C2×C22⋊C412yes—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4C2×C22⋊C412yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4#12011yescharacteristic
439C22×C4#12011yescharacteristic
440C22×C4#12111yescharacteristic
441C22×C4#12111yescharacteristic
442C22×C4#12111yescharacteristic
443C22×C4#12111yescharacteristic
444C22×C4#12311yescharacteristic
445C22×C4#12311yescharacteristic
446C22×C4#12311yescharacteristic
447C22×C4#13512yes—
448C22×C4#13512yes—
449C22×C4AGL(1,ℤ/8)12yes—
450C22×C4AGL(1,ℤ/8)12yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454Q16—48no—
455SD16—48no—
456SD16—48no—
457D8—48no—
458M16—42no—
459M16—42no—
460M16—44no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—44no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—42no—
483C2×C8—82no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4C2×C22⋊C412yes—
495C4⋊C4C2×C22⋊C412yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42C2×C22⋊C411yescharacteristic
513C42C2×C22⋊C411yescharacteristic
514C42C4×D411yescharacteristic
515C42C4×D411yescharacteristic
516C42C4×D411yescharacteristic
517C42C4×D411yescharacteristic
518C42#12011yescharacteristic
519C42#12011yescharacteristic
520C42#12111yescharacteristic
521C42#12111yescharacteristic
522C42#12111yescharacteristic
523C42#12111yescharacteristic
524C42C23×C411yescharacteristic
525C42C22×D411yescharacteristic
526C42C22×D411yescharacteristic
527C42C22×D411yescharacteristic
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23#38611yescharacteristic
539C23#38611yescharacteristic
540C23#48212yes—
541C23#48212yes—
542C23C2×#12311yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—82no—
549D4—22no—
550D4—22no—
551D4—44no—
552D4—42no—
553D4—42no—
554D4—44no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—84no—
564D4—82no—
565D4—82no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4#38611yescharacteristic
599C2×C4#38611yescharacteristic
600C2×C4#38611yescharacteristic
601C2×C4#38611yescharacteristic
602C2×C4#38611yescharacteristic
603C2×C4#38611yescharacteristic
604C2×C4#41612yes—
605C2×C4#41612yes—
606C2×C4#41812yes—
607C2×C4#41812yes—
608C2×C4#44312yes—
609C2×C4#44412yes—
610C2×C4#44912yes—
611C2×C4#44912yes—
612C2×C4#44912yes—
613C2×C4#44912yes—
614C2×C4#48212yes—
615C2×C4#48212yes—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×#12011yescharacteristic
620C2×C4C2×#12111yescharacteristic
621C2×C4C2×#12111yescharacteristic
622C2×C4C2×#13511yescharacteristic
623C2×C4C2×#13511yescharacteristic
624C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
625C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
626C8—44no—
627C8—42no—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—44no—
632C8—82no—
633C22—24no—
634C22—24no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—164no—
647C22#187512yes—
648C22#187512yes—
649C22#187612yes—
650C22#188112yes—
651C22C2×#38611yescharacteristic
652C22C2×#48211yescharacteristic
653C22C2×#48211yescharacteristic
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4#187512yes—
665C4#187512yes—
666C4#187612yes—
667C4C2×#41611yescharacteristic
668C4C2×#41811yescharacteristic
669C4C2×#44311yescharacteristic
670C4C2×#44411yescharacteristic
671C4C2×#44911yescharacteristic
672C4C2×#44911yescharacteristic
673C2—21no—
674C2—21no—
675C2—22no—
676C2—82no—
677C2—162no—
678C2#899614yesdirect product factor
679C2C2×#187511yescharacteristic
680C2C2×#187611yescharacteristic
681C2C2×#188111yescharacteristic
subgroupG/H
centerC22×C4#120
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×#2925C2semidirect
3. C2×#1492C2semidirect
4. C2×#1480C2semidirect
5. C2×#1476C2semidirect
6. #8996C2direct (N × H)
7. C2×#1301C2semidirect
8. C2×#1277C2semidirect
25. #2925C22semidirect
38. #1492C22semidirect
40. #1480C22semidirect
42. #1476C22semidirect
48. #1301C22semidirect
49. #1277C22semidirect
678. C2#8996direct (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×#29252
C2×#14922
C2×#14802
C2×#14762
#89962
C2×#13012
C2×#12772