Finite Group Databasebeta

← C2×#9053 #195470

Subgroups — C2×#9053 #195470

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic
2C2×#2935C211yesmaximal, characteristic, complemented
3C2×#1980C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic, complemented
6#9053C218yesmaximal, direct product factor
7C2×#1279C211yesmaximal, characteristic, complemented
8C2×#1278C211yesmaximal, characteristic, complemented
9C22×#131—21no—
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3169—22no—
13#3169—22no—
14#3169C414yescomplemented
15#3169C2214yes—
16C2×#500C411yescharacteristic, complemented
17C2×#493C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#486C411yescharacteristic, complemented
20C2×#466—22no—
21C2×C4×D8—22no—
22#2935—22no—
23#2935—22no—
24#2935—22no—
25#2935C2214yescomplemented
26C2×#436—21no—
27C2×#435—22no—
28C2×C8×D4—21no—
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#104—21no—
32C22×#104—21no—
33C2×#408—21no—
34C2×C4×M16C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1980—22no—
38#1980C2214yescomplemented
39#1502—22no—
40#1502C2214yescomplemented
41C2×#327—22no—
42#1480—22no—
43#1480C2214yescomplemented
44C2×D4⋊C8—21no—
45C2×D4⋊C8—21no—
46#1279C2214yescomplemented
47#1278C2214yescomplemented
48C22×#41—21no—
49C22×#41—21no—
50C22×D8—22no—
51C2×#131—24no—
52C2×#131—24no—
53C2×#131—44no—
54C2×#131—44no—
55C22×M16—22no—
56C22×M16D411yescharacteristic
57C22×M16D411yescharacteristic
58C23×C8—22no—
59C23×C8—21no—
60C23×C8—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532C2×C412yes—
72#532C2×C412yes—
73#532D412yes—
74#532D412yes—
75#532C2312yes—
76#532C2312yes—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#127C2×C411yescharacteristic
80C2×#127C2×C411yescharacteristic
81C2×#124—21no—
82C2×#124—21no—
83C2×#121—21no—
84C2×#121—21no—
85C2×#121—21no—
86C2×#121D412yes—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4D412yes—
91C22×C4⋊C4—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#500—22no—
95#500C2×C414yescomplemented
96#493—22no—
97#493C2×C414yescomplemented
98#486—22no—
99#486—22no—
100#486C2×C414yescomplemented
101#486C2×C414yescomplemented
102#466—28no—
103#466—28no—
104C4×D8—28no—
105C4×D8—28no—
106#436—24no—
107#436—24no—
108#435—28no—
109#435—28no—
110C8×D4—24no—
111C8×D4—24no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2312yes—
116C2×#107—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8D412yes—
124C2×C4⋊C8C2312yes—
125C2×#104—24no—
126C2×#104—24no—
127C2×#104—44no—
128C2×#104—44no—
129C2×#102—22no—
130C2×#102—22no—
131#408—24no—
132#408—24no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—22no—
135C2×C22⋊C8—21no—
136C4×M16—21no—
137C4×M16—21no—
138C4×M16D412yes—
139C4×M16C2312yes—
140C2×#97—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8C2×C411yescharacteristic
147C2×C4×C8C2×C411yescharacteristic
148C2×C4×C8C2×C412yes—
149C2×C4×C8C2312yes—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2312yes—
154C2×#95—21no—
155#327—28no—
156D4⋊C8—24no—
157D4⋊C8—24no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—41no—
183C22×D4C22⋊C412yes—
184C23×C4—22no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C2×D8—28no—
191C2×D8—48no—
192#131—48no—
193#131—416no—
194#131—48no—
195C2×M16—21no—
196C2×M16—22no—
197C2×M16—28no—
198C2×M16—21no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—48no—
202C2×M16D814yescomplemented
203C2×M16C2×D412yes—
204C2×M16C2×D412yes—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—28no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—44no—
217C22×C8—48no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—41no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8C22⋊C412yes—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229#127—21no—
230#127—21no—
231#127—21no—
232#127—21no—
233#127—21no—
234#127C22⋊C412yes—
235#127C22×C412yes—
236#127C22×C412yes—
237#124—22no—
238#124—22no—
239#124—22no—
240#124—22no—
241#124—22no—
242#124—22no—
243#124—41no—
244#124—41no—
245#121—22no—
246#121—22no—
247#121—22no—
248#121—22no—
249#121—22no—
250#121—22no—
251#121—22no—
252#121—22no—
253#121—22no—
254#121—22no—
255#121—41no—
256#121—41no—
257#121—41no—
258#121—41no—
259#121C22⋊C414yes—
260#121C2×D414yes—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4C22⋊C414yes—
276C4×D4C2×D414yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4C22⋊C412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C42—22no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—41no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305#107—28no—
306C4⋊C8—21no—
307C4⋊C8—22no—
308C4⋊C8—21no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—41no—
312C4⋊C8C2×D412yes—
313C4⋊C8C2×D412yes—
314#104—48no—
315#104—48no—
316#102—28no—
317#102—28no—
318C22⋊C8—28no—
319C22⋊C8—28no—
320C22⋊C8—22no—
321C22⋊C8—41no—
322#97—22no—
323#97—22no—
324C4×C8—22no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—41no—
332C4×C8C22×C412yes—
333C4×C8C22×C412yes—
334#95—22no—
335#95—41no—
336C24—22no—
337C24—41no—
338C24#12111yescharacteristic
339#41—42no—
340#41—42no—
341#41—42no—
342#41—42no—
343#41—44no—
344#41—44no—
345#41—44no—
346#41—44no—
347#41—44no—
348#41—44no—
349#41—42no—
350#41—42no—
351#41—42no—
352#41—42no—
353#41—44no—
354#41—44no—
355#41—82no—
356#41—82no—
357#41—82no—
358#41—82no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4C2×C22⋊C414yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4#12011yescharacteristic
421C22×C4#12011yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12311yescharacteristic
428C22×C4#12311yescharacteristic
429C22×C4C2×D812yes—
430C22×C4#13512yes—
431C22×C4AGL(1,ℤ/8)12yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4111yescharacteristic
436D8—416no—
437M16—22no—
438M16—24no—
439M16—22no—
440M16—22no—
441M16—416no—
442M16—42no—
443M16C2×D814yescomplemented
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—416no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—44no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—82no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4C2×C22⋊C414yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C42—42no—
484C42—42no—
485C42—82no—
486C42C2×C22⋊C411yescharacteristic
487C42C2×C22⋊C411yescharacteristic
488C42C4×D411yescharacteristic
489C42C4×D411yescharacteristic
490C42C4×D411yescharacteristic
491C42C4×D411yescharacteristic
492C42#12011yescharacteristic
493C42#12011yescharacteristic
494C42#12111yescharacteristic
495C42#12111yescharacteristic
496C42#12311yescharacteristic
497C42#12311yescharacteristic
498C42C23×C411yescharacteristic
499C42C22×D411yescharacteristic
500C42C22×D411yescharacteristic
501C42C2×#4111yescharacteristic
502C23—22no—
503C23—22no—
504C23—24no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23#38611yescharacteristic
513C23#38611yescharacteristic
514C23#45912yes—
515C23#46312yes—
516C23C2×#12111yescharacteristic
517Q8—44no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—82no—
523D4—44no—
524D4—44no—
525D4—48no—
526D4—44no—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—82no—
539D4—82no—
540D4—84no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—88no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—84no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4#38611yescharacteristic
573C2×C4#38611yescharacteristic
574C2×C4#38611yescharacteristic
575C2×C4#38611yescharacteristic
576C2×C4#38611yescharacteristic
577C2×C4#38611yescharacteristic
578C2×C4C4×D812yes—
579C2×C4C4×D812yes—
580C2×C4#44212yes—
581C2×C4#44212yes—
582C2×C4#44312yes—
583C2×C4#44412yes—
584C2×C4#44712yes—
585C2×C4#44712yes—
586C2×C4#44912yes—
587C2×C4#44912yes—
588C2×C4#45912yes—
589C2×C4#46312yes—
590C2×C4C22×C22⋊C411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×#12011yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4C22×D811yescharacteristic
597C2×C4C2×#13511yescharacteristic
598C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
599C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
600C8—24no—
601C8—24no—
602C8—42no—
603C8—42no—
604C8—42no—
605C8—42no—
606C8—82no—
607C22—24no—
608C22—24no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—88no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—164no—
621C22#187612yes—
622C22#187912yes—
623C22#187912yes—
624C22#188412yes—
625C22C2×#38611yescharacteristic
626C22C2×#45911yescharacteristic
627C22C2×#46311yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—44no—
635C4—84no—
636C4—82no—
637C4#187612yes—
638C4#187912yes—
639C4#187912yes—
640C4C2×C4×D811yescharacteristic
641C4C2×#44211yescharacteristic
642C4C2×#44311yescharacteristic
643C4C2×#44411yescharacteristic
644C4C2×#44711yescharacteristic
645C4C2×#44911yescharacteristic
646C2—21no—
647C2—22no—
648C2—21no—
649C2—84no—
650C2—162no—
651C2#905314yesdirect product factor
652C2C2×#187611yescharacteristic
653C2C2×#187911yescharacteristic
654C2C2×#188411yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC2×#3169C2
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×#2935C2semidirect
3. C2×#1980C2semidirect
4. C2×#1502C2semidirect
5. C2×#1480C2semidirect
6. #9053C2direct (N × H)
7. C2×#1279C2semidirect
8. C2×#1278C2semidirect
14. #3169C4semidirect
16. C2×#500C4semidirect
17. C2×#493C4semidirect
18. C2×#486C4semidirect
19. C2×#486C4semidirect
25. #2935C22semidirect
38. #1980C22semidirect
40. #1502C22semidirect
43. #1480C22semidirect
46. #1279C22semidirect
47. #1278C22semidirect
95. #500C2×C4semidirect
97. #493C2×C4semidirect
100. #486C2×C4semidirect
101. #486C2×C4semidirect
202. C2×M16D8semidirect
443. M16C2×D8semidirect
651. C2#9053direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#29352
C2×#19802
C2×#15022
C2×#14802
#90532
C2×#12792
C2×#12782