Finite Group Databasebeta

← C2×#9320 #199331

Subgroups — C2×#9320 #199331

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2C2×#2989C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6#9320C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1292C211yesmaximal, characteristic, complemented
9C22×#137—21no—
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3039—22no—
13#3039—22no—
14#3039C414yescomplemented
15#3039C2214yes—
16C2×#462—21no—
17C2×#462C412yescomplemented
18C2×#461—21no—
19C2×#461C412yescomplemented
20#2989—22no—
21#2989—22no—
22#2989—22no—
23#2989C2214yes—
24C2×#446—21no—
25C2×#436—21no—
26C2×#435—22no—
27C2×#432—21no—
28C2×#431C2211yescharacteristic
29C2×#423—21no—
30C2×#423C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#104—21no—
34C2×#419—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1988—22no—
37#1988C2214yescomplemented
38#1806—22no—
39#1806C2214yes—
40C2×#340—22no—
41C2×#339—21no—
42#1546—22no—
43#1546C2214yescomplemented
44C2×#332—21no—
45C2×#332—21no—
46#1303C2214yescomplemented
47#1292C2214yescomplemented
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C2×#137—24no—
51C2×#137—24no—
52C2×#137—44no—
53C2×#137—44no—
54C22×Q16—22no—
55C22×SD16—22no—
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—21no—
59C23×C8D412yes—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8C2×C412yes—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8C2312yes—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41C2×C412yes—
77C4×#41D412yes—
78C4×#41D412yes—
79C4×#41C2312yes—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8D411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#462—24no—
95#462—24no—
96#462—24no—
97#462C2×C418yescomplemented
98#461—24no—
99#461—24no—
100#461—24no—
101#461C2×C418yescomplemented
102#446—24no—
103#446—24no—
104#436—24no—
105#436—24no—
106#435—28no—
107#435—28no—
108#432—24no—
109#432—24no—
110#431—21no—
111#431—21no—
112#431D412yes—
113#431C2312yes—
114#423—24no—
115#423—21no—
116#423—21no—
117#423—24no—
118#423—21no—
119#423—21no—
120#423—21no—
121#423C2×C412yes—
122#423C2312yes—
123#423C2312yes—
124C2×C4⋊C8—22no—
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⋊C8—21no—
132C2×C4⋊C8C2×C412yes—
133C2×C4⋊C8D412yes—
134C2×C4⋊C8D411yescharacteristic
135C2×C4⋊C8D411yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×#104—24no—
138C2×#104—44no—
139#419—24no—
140#419—24no—
141C2×#103—22no—
142C2×#103—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—21no—
145C2×#97—22no—
146C2×C4×C8—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2312yes—
151C2×#95—21no—
152#340—28no—
153#339—24no—
154#332—24no—
155#332—24no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41C22⋊C412yes—
163C2×#41C2×D412yes—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C22⋊C411yescharacteristic
175C22×D4—21no—
176C22×D4C22⋊C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185#137—48no—
186#137—48no—
187#137—416no—
188C2×Q16—28no—
189C2×Q16—48no—
190C2×SD16—28no—
191C2×SD16—48no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—28no—
195C2×M16—22no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16—41no—
199C2×M16—42no—
200C2×M16—42no—
201C2×M16—84no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C22×C8—44no—
208C22×C8—41no—
209C22×C8C2×D414yes—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—41no—
220C4⋊Q8—41no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223C4⋊Q8—41no—
224C4⋊Q8—41no—
225C4⋊Q8C22×C414yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C22⋊C412yes—
231C4×Q8C22⋊C412yes—
232C4×Q8C22×C412yes—
233C4×Q8C2×D412yes—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C22⋊C412yes—
247C4×D4C22⋊C412yes—
248C4×D4C22×C412yes—
249C4×D4C2×D412yes—
250#117—21no—
251#117—21no—
252#117—22no—
253#117—22no—
254#117—22no—
255#117—22no—
256#117C22⋊C412yes—
257#117C2×D412yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—41no—
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42C2411yescharacteristic
304C4⋊C8—28no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318C4⋊C8C22×C414yes—
319C4⋊C8C2×D412yes—
320C4⋊C8C2×D412yes—
321#104—48no—
322#103—28no—
323#103—28no—
324C22⋊C8—28no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327#97—28no—
328C4×C8—22no—
329C4×C8—22no—
330#95—22no—
331#95—41no—
332C24—21no—
333C24#12111yescharacteristic
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—24no—
339#41—24no—
340#41—42no—
341#41—42no—
342#41—42no—
343#41—42no—
344C2×Q8—21no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—82no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8C2×C22⋊C412yes—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4C2×C22⋊C412yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4#12011yescharacteristic
407C22×C4#12011yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12311yescharacteristic
414C22×C4#12311yescharacteristic
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4AGL(1,ℤ/8)12yes—
418C22×C4#13712yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422Q16—416no—
423SD16—416no—
424M16—42no—
425M16—42no—
426M16—44no—
427M16—42no—
428M16—42no—
429M16—88no—
430M16—82no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—44no—
440C2×C8—42no—
441C2×C8—82no—
442C2×C8—82no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4C2×C22⋊C412yes—
461C4⋊C4C2×C22⋊C412yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—42no—
475C42—42no—
476C42—82no—
477C42C2×C22⋊C411yescharacteristic
478C42C2×C22⋊C411yescharacteristic
479C42C4×D411yescharacteristic
480C42C4×D411yescharacteristic
481C42C4×D411yescharacteristic
482C42C4×D411yescharacteristic
483C42#12011yescharacteristic
484C42#12011yescharacteristic
485C42#12111yescharacteristic
486C42#12111yescharacteristic
487C42#12311yescharacteristic
488C42#12311yescharacteristic
489C42C23×C411yescharacteristic
490C42C22×D411yescharacteristic
491C42C22×D411yescharacteristic
492C42C2×#4111yescharacteristic
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#38611yescharacteristic
500C23#38611yescharacteristic
501C23#46812yes—
502C23#46912yes—
503C23C2×#12111yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—42no—
507Q8—44no—
508Q8—44no—
509Q8—84no—
510Q8—84no—
511Q8—82no—
512Q8—82no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—42no—
517D4—42no—
518D4—42no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—88no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#38611yescharacteristic
556C2×C4#38611yescharacteristic
557C2×C4#38611yescharacteristic
558C2×C4#38611yescharacteristic
559C2×C4#38611yescharacteristic
560C2×C4#38611yescharacteristic
561C2×C4#41712yes—
562C2×C4#41712yes—
563C2×C4#41812yes—
564C2×C4#41812yes—
565C2×C4#45512yes—
566C2×C4#45612yes—
567C2×C4#46812yes—
568C2×C4#46812yes—
569C2×C4#46812yes—
570C2×C4#46912yes—
571C2×C4#46912yes—
572C2×C4#46912yes—
573C2×C4C22×C22⋊C411yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×#12011yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×#12311yescharacteristic
579C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
580C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
581C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
582C2×C4C2×#13711yescharacteristic
583C8—44no—
584C8—44no—
585C8—42no—
586C8—42no—
587C8—82no—
588C22—21no—
589C22—21no—
590C22—24no—
591C22—24no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—84no—
598C22#188212yes—
599C22#189512yes—
600C22#189512yes—
601C22#190212yes—
602C22C2×#38611yescharacteristic
603C22C2×#46811yescharacteristic
604C22C2×#46911yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—84no—
615C4—84no—
616C4#188212yes—
617C4#189512yes—
618C4#189512yes—
619C4C2×#41811yescharacteristic
620C4C2×#41711yescharacteristic
621C4C2×#45511yescharacteristic
622C4C2×#45611yescharacteristic
623C4C2×#46811yescharacteristic
624C4C2×#46911yescharacteristic
625C2—21no—
626C2—22no—
627C2—21no—
628C2—82no—
629C2#932014yesdirect product factor
630C2C2×#188211yescharacteristic
631C2C2×#189511yescharacteristic
632C2C2×#190211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
3. C2×#1988C2semidirect
5. C2×#1546C2semidirect
6. #9320C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1292C2semidirect
14. #3039C4semidirect
17. C2×#462C4semidirect
19. C2×#461C4semidirect
37. #1988C22semidirect
43. #1546C22semidirect
46. #1303C22semidirect
47. #1292C22semidirect
97. #462C2×C4semidirect
101. #461C2×C4semidirect
629. C2#9320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30392
C2×#29892
C2×#19882
C2×#18062
C2×#15462
#93202
C2×#13032
C2×#12922