Finite Group Databasebeta

← C2×#19543 #6354629

Subgroups — C2×#19543 #6354629

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3251C211yesmaximal, characteristic
2C2×#3248C211yesmaximal, characteristic, complemented
3C2×#3210C211yesmaximal, characteristic, complemented
4C2×#2993C211yesmaximal, characteristic, complemented
5#19543C2116yesmaximal, direct product factor
6C2×#1700C211yesmaximal, characteristic, complemented
7C2×#1686C211yesmaximal, characteristic, complemented
8C2×#1699C211yesmaximal, characteristic, complemented
9C2×#1685C211yesmaximal, characteristic, complemented
10C2×#1682C211yesmaximal, characteristic, complemented
11C2×#1670C211yesmaximal, characteristic, complemented
12C2×#1681C211yesmaximal, characteristic, complemented
13C2×#1669C211yesmaximal, characteristic, complemented
14C2×#1571C211yesmaximal, characteristic, complemented
15C2×#1561C211yesmaximal, characteristic, complemented
16C2×#1547C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3251—24no—
21#3251C2218yes—
22#3248—24no—
23#3248C2218yescomplemented
24#3210—24no—
25#3210C2218yescomplemented
26C2×#494C2211yescharacteristic
27C2×#492C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#486C2211yescharacteristic
30C2×#479—21no—
31C2×#475C2211yescharacteristic
32C2×#475C2211yescharacteristic
33C2×#478—21no—
34C2×#474C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#469—21no—
37C2×#468—21no—
38C2×#467—21no—
39C2×#465—21no—
40C2×#464—21no—
41C2×#464—21no—
42C2×#464—21no—
43C2×#462C2211yescharacteristic
44C2×#460C2211yescharacteristic
45C2×#463—21no—
46C2×#461C2211yescharacteristic
47C2×#459C2211yescharacteristic
48C2×#452—21no—
49C2×#452—21no—
50#2993—24no—
51#2993C2218yescomplemented
52C2×#446C2211yescharacteristic
53C2×C8×Q8C2211yescharacteristic
54C2×#436—21no—
55C2×#436C2211yescharacteristic
56C2×#435—21no—
57C2×C8×D4C2211yescharacteristic
58C2×#433—21no—
59C2×#431C2211yescharacteristic
60C2×#424—21no—
61C2×#423C2211yescharacteristic
62C2×#423C2211yescharacteristic
63C22×C4⋊C8C2211yescharacteristic
64#1700C2218yescomplemented
65#1699C2218yescomplemented
66#1686C2218yescomplemented
67#1685C2218yescomplemented
68#1682C2218yescomplemented
69#1681C2218yescomplemented
70#1670C2218yescomplemented
71#1669C2218yescomplemented
72#1571C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#335C2211yescharacteristic
76C2×#335C2211yescharacteristic
77#1561C2218yescomplemented
78#1547C2218yescomplemented
79C2×#332C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#331C2211yescharacteristic
86C2×#331C2211yescharacteristic
87C22×#41—21no—
88C22×Q16—21no—
89C22×Q16—21no—
90C22×Q16—21no—
91C22×SD16—21no—
92C22×SD16—21no—
93C22×SD16—21no—
94C22×SD16—21no—
95C22×D8—21no—
96C22×M16—21no—
97C22×M16—21no—
98C22×M16—21no—
99C23×C8—21no—
100C23×C8—21no—
101#533—22no—
102#533—22no—
103#533—22no—
104#533C2314yes—
105#532—22no—
106#532—22no—
107#532—22no—
108#532C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2311yescharacteristic
113#529—22no—
114#529—22no—
115#529D414yes—
116#529C2314yes—
117C2×#126—21no—
118C2×#125—21no—
119C2×#125D411yescharacteristic
120C2×#124—21no—
121C2×#124D411yescharacteristic
122C2×#123—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8D411yescharacteristic
126C2×#121—21no—
127C2×#121—21no—
128C2×#121D411yescharacteristic
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2311yescharacteristic
136C2×#117—21no—
137C2×#117—21no—
138C22×C42C2311yescharacteristic
139#494—22no—
140#494C2314yes—
141#492—22no—
142#492C2314yes—
143#487—22no—
144#487C2314yes—
145#486—22no—
146#486C2314yes—
147#479—28no—
148#478—28no—
149#475—22no—
150#475—22no—
151#475C2314yes—
152#475C2314yes—
153#474—22no—
154#474—22no—
155#474C2314yes—
156#474C2314yes—
157#469—28no—
158#468—28no—
159#467—28no—
160#465—28no—
161#464—28no—
162#464—28no—
163#464—28no—
164#463—28no—
165#462—22no—
166#462C2314yes—
167#461—22no—
168#461C2314yes—
169#460—22no—
170#460C2314yes—
171#459—22no—
172#459C2314yes—
173#452—28no—
174#452—28no—
175#446—22no—
176#446C2314yes—
177C8×Q8—22no—
178C8×Q8C2314yes—
179#436—28no—
180#436—22no—
181#436C2314yes—
182#435—28no—
183C8×D4—22no—
184C8×D4C2314yes—
185#433—28no—
186#431D414yes—
187#431C2314yes—
188C2×#107—21no—
189C2×#107—21no—
190C2×#106—21no—
191C2×#106—21no—
192#424—28no—
193#423—22no—
194#423—22no—
195#423C2314yes—
196#423C2314yes—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—21no—
199C2×C4⋊C8—21no—
200C2×C4⋊C8—21no—
201C2×C4⋊C8—22no—
202C2×C4⋊C8—21no—
203C2×C4⋊C8D411yescharacteristic
204C2×C4⋊C8D411yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2311yescharacteristic
209C2×C4⋊C8C2311yescharacteristic
210C2×C4⋊C8C2311yescharacteristic
211C2×C4⋊C8C2314yes—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#102—21no—
219C2×#102—21no—
220C2×#102—21no—
221C2×#102—21no—
222C2×#102—21no—
223C2×#102—21no—
224C2×C22⋊C8—21no—
225C2×C22⋊C8—21no—
226C2×C22⋊C8D411yescharacteristic
227C2×C22⋊C8D411yescharacteristic
228C2×#97—21no—
229C2×C4×C8—21no—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232#335C2314yes—
233#335C2314yes—
234#335C2314yes—
235#335C2314yes—
236#332C2314yes—
237#332C2314yes—
238#332C2314yes—
239#332C2314yes—
240#332C2314yes—
241#332C2314yes—
242#331C2314yes—
243#331C2314yes—
244C2×#41—24no—
245C2×#41—42no—
246C2×#41—82no—
247C2×#41—82no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8C2×D411yescharacteristic
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—41no—
257C22×D4C2×D411yescharacteristic
258C23×C4—21no—
259C23×C4—41no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4#4111yescharacteristic
263C2×Q16—24no—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—82no—
267C2×Q16—82no—
268C2×Q16—82no—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—82no—
274C2×SD16—82no—
275C2×SD16—82no—
276C2×SD16—82no—
277C2×D8—24no—
278C2×D8—82no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16—24no—
282C2×M16—82no—
283C2×M16—82no—
284C2×M16—82no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—24no—
292C22×C8—24no—
293C22×C8—82no—
294C22×C8—82no—
295C22×C8C2×D411yescharacteristic
296C22×C8C2×D411yescharacteristic
297C22×C8C2×D411yescharacteristic
298C22×C8C2×D411yescharacteristic
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8C2412yes—
309C4⋊Q8C2412yes—
310C4⋊Q8C2412yes—
311#127—21no—
312#127—21no—
313#127—21no—
314#127C2412yes—
315#126—24no—
316#126—24no—
317#125—22no—
318#125—24no—
319#125—42no—
320#125C2×D414yes—
321#124—22no—
322#124—24no—
323#124—42no—
324#124C2×D414yes—
325#123—24no—
326#123—24no—
327C22⋊Q8—22no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330C22⋊Q8—42no—
331C22⋊Q8—42no—
332C22⋊Q8C2×D414yes—
333#121—22no—
334#121—24no—
335#121—24no—
336#121—42no—
337#121—42no—
338#121C2×D414yes—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—24no—
342C4×Q8—24no—
343C4×Q8—42no—
344C4×Q8—42no—
345C4×Q8C2×D412yes—
346C4×Q8C2412yes—
347C4×D4—24no—
348C4×D4—24no—
349C4×D4—21no—
350C4×D4—24no—
351C4×D4—21no—
352C4×D4—24no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4C2×D412yes—
356C4×D4C2412yes—
357#117—24no—
358#117—24no—
359#117—24no—
360#117—42no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4C2×D411yescharacteristic
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—41no—
381C2×C22⋊C4—41no—
382C2×C22⋊C4—41no—
383C2×C42—21no—
384C2×C42—41no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D412yes—
388C2×C42C2×D412yes—
389C2×C42#4111yescharacteristic
390C2×C42#4112yes—
391C2×C42C2411yescharacteristic
392C2×C42C2412yes—
393#107—24no—
394#107—24no—
395#106—24no—
396#106—24no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8C2×D414yes—
409C4⋊C8C2×D414yes—
410C4⋊C8C2412yes—
411C4⋊C8C2412yes—
412C4⋊C8C2412yes—
413C4⋊C8C2412yes—
414C4⋊C8C2412yes—
415C4⋊C8C2412yes—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—24no—
422#102—24no—
423#102—24no—
424#102—24no—
425#102—24no—
426#102—24no—
427#102—24no—
428C22⋊C8—24no—
429C22⋊C8—24no—
430C22⋊C8C2×D414yes—
431C22⋊C8C2×D414yes—
432#97—24no—
433C4×C8—24no—
434C4×C8C2×D412yes—
435C4×C8C2×D412yes—
436C4×C8C2412yes—
437C4×C8C2412yes—
438C24—21no—
439C24—41no—
440C24#12111yescharacteristic
441#41—84no—
442#41—84no—
443#41—88no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8—41no—
452C2×Q8—41no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8C22×D412yes—
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—44no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—82no—
476C2×D4—82no—
477C2×D4—84no—
478C2×D4C22×D412yes—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—44no—
495C22×C4—84no—
496C22×C4#12112yes—
497C22×C4#12112yes—
498C22×C4#12111yescharacteristic
499C22×C4#12111yescharacteristic
500C22×C4#12111yescharacteristic
501C22×C4#12112yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4111yescharacteristic
507C22×C4C2×#4112yes—
508Q16—84no—
509Q16—84no—
510Q16—84no—
511SD16—84no—
512SD16—84no—
513SD16—84no—
514SD16—84no—
515D8—84no—
516M16—84no—
517M16—84no—
518M16—84no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—42no—
530C2×C8—42no—
531C2×C8—42no—
532C2×C8—42no—
533C2×C8—42no—
534C2×C8—42no—
535C2×C8—84no—
536C2×C8—84no—
537C2×C8C22×D412yes—
538C2×C8C22×D412yes—
539C2×C8C22×D412yes—
540C2×C8C22×D412yes—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4C22×D412yes—
563C4⋊C4C22×D412yes—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C42—22no—
574C42—22no—
575C42—44no—
576C42#12112yes—
577C42#12112yes—
578C42#12112yes—
579C42C22×D411yescharacteristic
580C42C22×D412yes—
581C42C22×D411yescharacteristic
582C42C22×D412yes—
583C42C2×#4112yes—
584C42C2×#4111yescharacteristic
585C42C2511yescharacteristic
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—82no—
595C23—82no—
596C23—82no—
597C23C2×#12112yes—
598C23C2×#12111yescharacteristic
599Q8—22no—
600Q8—22no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—82no—
609Q8—82no—
610Q8—82no—
611D4—22no—
612D4—22no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—82no—
619D4—82no—
620D4—84no—
621D4—88no—
622D4—82no—
623D4—82no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—88no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4C2×#12112yes—
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×#12112yes—
655C2×C4C2×#12112yes—
656C2×C4C2×#12111yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4#57612yes—
659C2×C4#57612yes—
660C2×C4#57712yes—
661C2×C4#57812yes—
662C2×C4C23×D411yescharacteristic
663C2×C4C23×D411yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C8—22no—
666C8—22no—
667C8—22no—
668C8—22no—
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C8—42no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—84no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—164no—
687C22#304512yes—
688C22#306112yes—
689C22#306912yes—
690C22C22×#12111yescharacteristic
691C4—22no—
692C4—22no—
693C4—22no—
694C4—21no—
695C4—21no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#304512yes—
703C4#306112yes—
704C4C2×#57611yescharacteristic
705C4C2×#57611yescharacteristic
706C4C2×#57711yescharacteristic
707C4C2×#57811yescharacteristic
708C2—42no—
709C2—82no—
710C2—162no—
711C2#1954314yesdirect product factor
712C2C2×#304511yescharacteristic
713C2C2×#306111yescharacteristic
714C2C2×#306911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3251C2
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
2. C2×#3248C2semidirect
3. C2×#3210C2semidirect
4. C2×#2993C2semidirect
5. #19543C2direct (N × H)
6. C2×#1700C2semidirect
7. C2×#1686C2semidirect
8. C2×#1699C2semidirect
9. C2×#1685C2semidirect
10. C2×#1682C2semidirect
11. C2×#1670C2semidirect
12. C2×#1681C2semidirect
13. C2×#1669C2semidirect
14. C2×#1571C2semidirect
15. C2×#1561C2semidirect
16. C2×#1547C2semidirect
23. #3248C22semidirect
25. #3210C22semidirect
51. #2993C22semidirect
64. #1700C22semidirect
65. #1699C22semidirect
66. #1686C22semidirect
67. #1685C22semidirect
68. #1682C22semidirect
69. #1681C22semidirect
70. #1670C22semidirect
71. #1669C22semidirect
72. #1571C22semidirect
77. #1561C22semidirect
78. #1547C22semidirect
711. C2#19543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32512
C2×#32482
C2×#32102
C2×#29932
#195432
C2×#17002
C2×#16862
C2×#16992
C2×#16852
C2×#16822
C2×#16702
C2×#16812
C2×#16692
C2×#15712
C2×#15612
C2×#15472