Finite Group Databasebeta

← C2×#9181 #197225

Subgroups — C2×#9181 #197225

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2C2×#2357C211yesmaximal, characteristic, complemented
3C2×#1517C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic, complemented
6#9181C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)—21no—
10C2×#532C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3036—22no—
13#3036—22no—
14#3036C414yescomplemented
15#3036C2214yes—
16C2×#463—22no—
17C2×#461C412yescomplemented
18C2×#459C412yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417—21no—
23#2357—22no—
24#2357—22no—
25#2357—22no—
26#2357C2214yescomplemented
27C2×#390—21no—
28C2×#388—22no—
29C2×#382C2211yescharacteristic
30C2×#381—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34#1517—22no—
35#1517C2214yescomplemented
36#1502—22no—
37#1502C2214yescomplemented
38C2×#327—21no—
39C2×#327—21no—
40C2×#327—22no—
41#1478—22no—
42#1478C2214yescomplemented
43C2×D4⋊C8—22no—
44#1303C2214yescomplemented
45#1286C2214yescomplemented
46C22×#41—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C2×AGL(1,ℤ/8)—24no—
50C2×AGL(1,ℤ/8)—24no—
51C2×AGL(1,ℤ/8)—44no—
52C2×AGL(1,ℤ/8)—44no—
53C22×SD16—22no—
54C22×D8—22no—
55C22×M16—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532C2×C412yes—
63#532C2312yes—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#124—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4C2312yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C42—21no—
90C22×C42—22no—
91C22×C42C2311yescharacteristic
92#463—28no—
93#463—28no—
94#461—24no—
95#461C2×C418yescomplemented
96#459—24no—
97#459C2×C418yescomplemented
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431C2312yes—
105#431C2312yes—
106#423—21no—
107#423—21no—
108#423C2×C412yes—
109#423C2312yes—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2×C412yes—
114#417—24no—
115#417—24no—
116C2×#103—22no—
117C2×#102—22no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#390—24no—
123#390—24no—
124#388—28no—
125#388—28no—
126#382—21no—
127#382—21no—
128#382D412yes—
129#382C2312yes—
130#381—24no—
131#381—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4D412yes—
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—24no—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144C2×#95D412yes—
145#327—24no—
146#327—24no—
147#327—28no—
148D4⋊C8—28no—
149C25—21no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41—42no—
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—41no—
170C22×D4C22⋊C412yes—
171C22×D4C22⋊C412yes—
172C22×D4C2×D412yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—41no—
184C23×C4—84no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188AGL(1,ℤ/8)—48no—
189AGL(1,ℤ/8)—48no—
190AGL(1,ℤ/8)—416no—
191C2×SD16—28no—
192C2×SD16—48no—
193C2×D8—28no—
194C2×D8—48no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—42no—
198C2×M16—42no—
199C22×C8—22no—
200C22×C8—41no—
201C22×C8—41no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8C22×C412yes—
206#127—21no—
207#127—21no—
208#127—21no—
209#127C22×C412yes—
210#124—22no—
211#124—22no—
212#124—41no—
213#124—41no—
214#121—22no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—41no—
219#121—41no—
220#121—41no—
221#121—41no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4C22⋊C414yes—
243C4×D4C22⋊C414yes—
244C4×D4C22×C414yes—
245C4×D4C2×D414yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—84no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—28no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—84no—
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42#4111yescharacteristic
289C2×C42#4111yescharacteristic
290C2×C42C2411yescharacteristic
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8C22×C414yes—
299#103—28no—
300#102—28no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—41no—
308C4×C8—41no—
309#95—28no—
310#95—22no—
311#95—21no—
312#95—21no—
313#95—22no—
314#95—22no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D414yes—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—44no—
322C24—44no—
323C24—42no—
324C24—42no—
325C24—41no—
326C24#12011yescharacteristic
327#41—42no—
328#41—42no—
329#41—42no—
330#41—42no—
331#41—44no—
332#41—44no—
333#41—82no—
334#41—82no—
335#41—82no—
336#41—82no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4C2×C22⋊C414yes—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4#12011yescharacteristic
413C22×C4#12011yescharacteristic
414C22×C4#12011yescharacteristic
415C22×C4#12111yescharacteristic
416C22×C4#12111yescharacteristic
417C22×C4#12111yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12311yescharacteristic
420C22×C4#12311yescharacteristic
421C22×C4#13512yes—
422C22×C4#13512yes—
423C22×C4AGL(1,ℤ/8)12yes—
424C22×C4AGL(1,ℤ/8)12yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428SD16—416no—
429D8—416no—
430M16—42no—
431M16—42no—
432M16—44no—
433C2×C8—24no—
434C2×C8—44no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—82no—
438C2×C8—82no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—44no—
451C4⋊C4C2×C22⋊C414yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—416no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—88no—
462C42—42no—
463C42—42no—
464C42—44no—
465C42—88no—
466C42C2×C22⋊C411yescharacteristic
467C42C2×C22⋊C411yescharacteristic
468C42C4×D411yescharacteristic
469C42C4×D411yescharacteristic
470C42C4×D411yescharacteristic
471C42C4×D411yescharacteristic
472C42#12111yescharacteristic
473C42#12111yescharacteristic
474C42#12111yescharacteristic
475C42#12111yescharacteristic
476C42#12311yescharacteristic
477C42#12311yescharacteristic
478C42C23×C411yescharacteristic
479C42C22×D411yescharacteristic
480C42C2×#4111yescharacteristic
481C42C2×#4111yescharacteristic
482C23—22no—
483C23—22no—
484C23—24no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—88no—
497C23—88no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23#38611yescharacteristic
502C23#38611yescharacteristic
503C23#44912yes—
504C23#44912yes—
505C23C2×#12011yescharacteristic
506Q8—42no—
507Q8—42no—
508Q8—82no—
509D4—24no—
510D4—24no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—82no—
518D4—82no—
519D4—82no—
520D4—82no—
521D4—84no—
522D4—82no—
523D4—82no—
524D4—82no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—88no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4#38611yescharacteristic
563C2×C4#38611yescharacteristic
564C2×C4#38611yescharacteristic
565C2×C4#38611yescharacteristic
566C2×C4#38611yescharacteristic
567C2×C4#38611yescharacteristic
568C2×C4#41612yes—
569C2×C4#41612yes—
570C2×C4#41812yes—
571C2×C4#41812yes—
572C2×C4#44912yes—
573C2×C4#44912yes—
574C2×C4#48212yes—
575C2×C4#48212yes—
576C2×C4#48212yes—
577C2×C4#48212yes—
578C2×C4C22×C22⋊C411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×#12311yescharacteristic
584C2×C4C2×#13511yescharacteristic
585C2×C4C2×#13511yescharacteristic
586C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
587C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
588C8—44no—
589C8—82no—
590C8—82no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—24no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—84no—
601C22—84no—
602C22—88no—
603C22—88no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—164no—
608C22#187512yes—
609C22#187512yes—
610C22#190412yes—
611C22#190612yes—
612C22C2×#38611yescharacteristic
613C22C2×#44911yescharacteristic
614C22C2×#44911yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4#187512yes—
629C4#187512yes—
630C4C2×#41611yescharacteristic
631C4C2×#41811yescharacteristic
632C4C2×#48211yescharacteristic
633C4C2×#48211yescharacteristic
634C2—21no—
635C2—21no—
636C2—22no—
637C2—84no—
638C2—162no—
639C2#918114yesdirect product factor
640C2C2×#187511yescharacteristic
641C2C2×#190411yescharacteristic
642C2C2×#190611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2357C2semidirect
3. C2×#1517C2semidirect
4. C2×#1502C2semidirect
5. C2×#1478C2semidirect
6. #9181C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1286C2semidirect
14. #3036C4semidirect
17. C2×#461C4semidirect
18. C2×#459C4semidirect
26. #2357C22semidirect
35. #1517C22semidirect
37. #1502C22semidirect
42. #1478C22semidirect
44. #1303C22semidirect
45. #1286C22semidirect
95. #461C2×C4semidirect
97. #459C2×C4semidirect
639. C2#9181direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30362
C2×#23572
C2×#15172
C2×#15022
C2×#14782
#91812
C2×#13032
C2×#12862