Finite Group Databasebeta

← C2×#19456 #6354521

Subgroups — C2×#19456 #6354521

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic, complemented
3C22×#459C211yesmaximal, characteristic
4C2×#2974C211yesmaximal, characteristic, complemented
5#19456C2116yesmaximal, direct product factor
6C2×#1701C212yesmaximal, complemented
7C2×#1685C212yesmaximal, complemented
8C2×#1681C212yesmaximal, complemented
9C2×#1665C212yesmaximal, complemented
10C2×#1573C211yesmaximal, characteristic, complemented
11C2×#1555C211yesmaximal, characteristic, complemented
12C2×#1552C211yesmaximal, characteristic, complemented
13C23×D8—21no—
14C2×#533C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3208—24no—
18#3208C2218yescomplemented
19C2×#492C2212yescomplemented
20C2×#486C2212yescomplemented
21#3110—24no—
22#3110C2218yescomplemented
23C2×#484—21no—
24C2×#474C2211yescharacteristic, complemented
25C2×#474C2211yescharacteristic, complemented
26C2×#460—22no—
27C2×#461C2211yescharacteristic, complemented
28C2×#461C2211yescharacteristic, complemented
29C2×#459—24no—
30C2×#459—21no—
31C2×#459—21no—
32C2×#459C2218yes—
33C2×#459C2212yes—
34C2×#459C2212yes—
35C2×#450—21no—
36C2×#450—21no—
37C2×#450—21no—
38#2974—24no—
39#2974C2218yescomplemented
40C2×#440—21no—
41C2×#440—21no—
42C2×#436C2212yescomplemented
43C2×#435—22no—
44C2×C8×D4C2212yescomplemented
45C2×#432C2211yescharacteristic
46C2×#432C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C22×C4⋊C8C2211yescharacteristic
49C22×#102—21no—
50C2×#407—21no—
51C22×C22⋊C8—21no—
52#1701C22116yescomplemented
53#1685C22116yescomplemented
54#1681C22116yescomplemented
55#1665C22116yescomplemented
56#1573C2218yescomplemented
57C2×#335C2212yescomplemented
58C2×#334C2212yescomplemented
59#1555C2218yescomplemented
60#1552C2218yescomplemented
61C2×#332C2212yescomplemented
62C2×#332C2212yescomplemented
63C2×#331C2212yescomplemented
64C2×#331C2212yescomplemented
65C23×D4—21no—
66C23×D4—21no—
67C23×D4—21no—
68C23×D4D411yescharacteristic
69C24×C4—21no—
70C22×SD16—21no—
71C22×SD16—21no—
72C22×SD16—22no—
73C22×SD16—22no—
74C22×D8—22no—
75C22×D8—28no—
76C22×D8—22no—
77C22×D8—48no—
78C22×M16—22no—
79C23×C8—21no—
80C23×C8—22no—
81#533—22no—
82#533—22no—
83#533—22no—
84#533C2314yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127—22no—
88C2×#127—21no—
89C2×#127—22no—
90C2×#127—21no—
91C2×#127—21no—
92C2×#127—22no—
93C2×#127C2312yes—
94C2×#127C2314yes—
95C2×#125—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×D4—21no—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—22no—
104C2×C4×D4D414yes—
105C2×C4×D4D411yescharacteristic
106C2×C4×D4D411yescharacteristic
107C2×C4×D4C2314yes—
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×#117—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C22⋊C4—21no—
113C22×C42C2311yescharacteristic
114#492—24no—
115#492C2318yescomplemented
116#486—24no—
117#486C2318yescomplemented
118#484—28no—
119#474—22no—
120#474—22no—
121#474C2314yescomplemented
122#474C2314yescomplemented
123#461—22no—
124#461—22no—
125#461C2314yescomplemented
126#461C2314yescomplemented
127#460—216no—
128#459—28no—
129#459—28no—
130#459—24no—
131#459—24no—
132#459C2318yes—
133#459C2318yes—
134#450—28no—
135#450—28no—
136#450—28no—
137#440—28no—
138#440—28no—
139#436—24no—
140#436C2318yescomplemented
141#435—216no—
142C8×D4—24no—
143C8×D4C2318yescomplemented
144#432—22no—
145#432—22no—
146#432C2314yes—
147#432C2314yes—
148#431D414yes—
149#431C2314yes—
150C2×#107—21no—
151C2×#107—21no—
152C2×C4⋊C8—21no—
153C2×C4⋊C8—22no—
154C2×C4⋊C8D412yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×#103—21no—
161C2×#103—21no—
162C2×#102—21no—
163C2×#102—21no—
164C2×#102—22no—
165C2×#102—22no—
166C2×#102—22no—
167C2×#102—22no—
168C2×#102—28no—
169#407—28no—
170C2×C22⋊C8—21no—
171C2×C22⋊C8—21no—
172C2×C22⋊C8—22no—
173C2×C22⋊C8—22no—
174C2×C22⋊C8—28no—
175C2×C22⋊C8D411yescharacteristic
176C2×C22⋊C8D411yescharacteristic
177C2×#97—21no—
178C2×#97—21no—
179C2×C4×C8C2312yes—
180#335C2318yescomplemented
181#334C2318yescomplemented
182#332C2318yescomplemented
183#332C2318yescomplemented
184#331C2318yescomplemented
185#331C2318yescomplemented
186C25—21no—
187C25—41no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—84no—
209C22×D4—84no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2×D414yes—
212C22×D4C2×D411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—84no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4#4111yescharacteristic
224C2×SD16—24no—
225C2×SD16—24no—
226C2×SD16—28no—
227C2×SD16—28no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×SD16—84no—
231C2×SD16—84no—
232C2×D8—28no—
233C2×D8—28no—
234C2×D8—432no—
235C2×D8—48no—
236C2×D8—48no—
237C2×M16—28no—
238C2×M16—84no—
239C22×C8—22no—
240C22×C8—24no—
241C22×C8—22no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—28no—
245C22×C8—44no—
246C22×C8—84no—
247C22×C8C2×D412yes—
248C22×C8C2×D412yes—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C2412yes—
256C4⋊Q8C2412yes—
257#127—24no—
258#127—22no—
259#127—24no—
260#127—22no—
261#127—24no—
262#127—22no—
263#127—42no—
264#127—42no—
265#127—42no—
266#127C2414yes—
267#125—24no—
268#125—42no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—42no—
272C22⋊Q8—42no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—21no—
282C4×D4—24no—
283C4×D4—21no—
284C4×D4—24no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4C2×D414yes—
288C4×D4C2×D414yes—
289C4×D4C2×D412yes—
290C4×D4C2×D412yes—
291C4×D4C2412yes—
292C4×D4C2412yes—
293#117—24no—
294#117—42no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—41no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D414yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—41no—
313C2×C42—41no—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42#4111yescharacteristic
319C2×C42#4112yes—
320C2×C42C2411yescharacteristic
321C2×C42C2412yes—
322#107—24no—
323#107—24no—
324C4⋊C8—24no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8C2×D418yes—
330C4⋊C8C2414yes—
331C4⋊C8C2412yes—
332C4⋊C8C2412yes—
333C4⋊C8C2414yes—
334#103—24no—
335#103—24no—
336#102—24no—
337#102—24no—
338#102—28no—
339#102—28no—
340#102—28no—
341#102—28no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C22⋊C8—28no—
345C22⋊C8—28no—
346C22⋊C8C2×D414yes—
347C22⋊C8C2×D414yes—
348#97—24no—
349#97—24no—
350C4×C8C2×D414yes—
351C4×C8C2414yes—
352C24—21no—
353C24—21no—
354C24—24no—
355C24—44no—
356C24—44no—
357C24—44no—
358C24—42no—
359C24—82no—
360C24—84no—
361C24—84no—
362C24—84no—
363C24#12111yescharacteristic
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—44no—
371C2×D4—28no—
372C2×D4—21no—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—84no—
402C2×D4—88no—
403C2×D4—84no—
404C2×D4—88no—
405C2×D4AGL(1,ℤ/8)14yes—
406C2×D4C22×D412yes—
407C2×D4C22×D412yes—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4#12112yes—
431C22×C4#12112yes—
432C22×C4#12112yes—
433C22×C4#12111yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4#12111yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4111yescharacteristic
442SD16—48no—
443SD16—48no—
444SD16—88no—
445SD16—88no—
446D8—416no—
447D8—416no—
448M16—88no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—48no—
456C2×C8—44no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—44no—
460C2×C8—88no—
461C2×C8C22×D414yes—
462C2×C8C22×D414yes—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4C22×D412yes—
477C4⋊C4C22×D412yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—44no—
485C42—44no—
486C42#12112yes—
487C42#12112yes—
488C42#12112yes—
489C42C22×D411yescharacteristic
490C42C22×D412yes—
491C42C22×D412yes—
492C42C22×D411yescharacteristic
493C42C2×#4112yes—
494C42C2×#4111yescharacteristic
495C42C2511yescharacteristic
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—48no—
507C23—42no—
508C23—44no—
509C23—84no—
510C23—816no—
511C23—88no—
512C23—88no—
513C23—84no—
514C23—84no—
515C23—88no—
516C23—84no—
517C23—1616no—
518C23#47114yes—
519C23C2×#12112yes—
520C23C2×#12111yescharacteristic
521Q8—42no—
522Q8—42no—
523Q8—44no—
524Q8—82no—
525Q8—82no—
526D4—24no—
527D4—22no—
528D4—22no—
529D4—22no—
530D4—48no—
531D4—42no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—44no—
536D4—84no—
537D4—88no—
538D4—88no—
539D4—88no—
540D4—84no—
541D4—84no—
542D4—84no—
543D4C2×AGL(1,ℤ/8)14yes—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4C2×#12112yes—
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12112yes—
573C2×C4C2×#12112yes—
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4#57512yes—
578C2×C4#57614yes—
579C2×C4C23×D411yescharacteristic
580C2×C4C23×D411yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C8—24no—
583C8—24no—
584C8—44no—
585C8—42no—
586C8—42no—
587C8—44no—
588C22—28no—
589C22—22no—
590C22—22no—
591C22—24no—
592C22—48no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—816no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—1616no—
604C22—168no—
605C22#304112yes—
606C22#305512yes—
607C22C2×#47114yes—
608C22C2×#47112yes—
609C22C22×#12111yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—82no—
618C4—82no—
619C4#304112yes—
620C4#305512yes—
621C4C22×AGL(1,ℤ/8)11yescharacteristic
622C4C2×#57511yescharacteristic
623C4C2×#57612yes—
624C2—24no—
625C2—44no—
626C2—82no—
627C2—164no—
628C2#1945614yesdirect product factor
629C2C2×#304111yescharacteristic
630C2C2×#305511yescharacteristic
631C2C22×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC22×#459C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3208C2semidirect
2. C2×#3110C2semidirect
4. C2×#2974C2semidirect
5. #19456C2direct (N × H)
6. C2×#1701C2semidirect
7. C2×#1685C2semidirect
8. C2×#1681C2semidirect
9. C2×#1665C2semidirect
10. C2×#1573C2semidirect
11. C2×#1555C2semidirect
12. C2×#1552C2semidirect
18. #3208C22semidirect
19. C2×#492C22semidirect
20. C2×#486C22semidirect
22. #3110C22semidirect
24. C2×#474C22semidirect
25. C2×#474C22semidirect
27. C2×#461C22semidirect
28. C2×#461C22semidirect
39. #2974C22semidirect
42. C2×#436C22semidirect
44. C2×C8×D4C22semidirect
52. #1701C22semidirect
53. #1685C22semidirect
54. #1681C22semidirect
55. #1665C22semidirect
56. #1573C22semidirect
57. C2×#335C22semidirect
58. C2×#334C22semidirect
59. #1555C22semidirect
60. #1552C22semidirect
61. C2×#332C22semidirect
62. C2×#332C22semidirect
63. C2×#331C22semidirect
64. C2×#331C22semidirect
115. #492C23semidirect
117. #486C23semidirect
121. #474C23semidirect
122. #474C23semidirect
125. #461C23semidirect
126. #461C23semidirect
140. #436C23semidirect
143. C8×D4C23semidirect
180. #335C23semidirect
181. #334C23semidirect
182. #332C23semidirect
183. #332C23semidirect
184. #331C23semidirect
185. #331C23semidirect
628. C2#19456direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32082
C2×#31102
C22×#4592
C2×#29742
#194562
C2×#17012
C2×#16852
C2×#16812
C2×#16652
C2×#15732
C2×#15552
C2×#15522