Finite Group Databasebeta

← C2×#19536 #6354622

Subgroups — C2×#19536 #6354622

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C211yesmaximal, characteristic, complemented
2C2×#3212C211yesmaximal, characteristic
3C2×#3206C211yesmaximal, characteristic, complemented
4C2×#2993C211yesmaximal, characteristic, complemented
5#19536C2116yesmaximal, direct product factor
6C2×#1700C211yesmaximal, characteristic, complemented
7C2×#1686C211yesmaximal, characteristic, complemented
8C2×#1699C211yesmaximal, characteristic, complemented
9C2×#1685C211yesmaximal, characteristic, complemented
10C2×#1680C211yesmaximal, characteristic, complemented
11C2×#1666C211yesmaximal, characteristic, complemented
12C2×#1679C211yesmaximal, characteristic, complemented
13C2×#1665C211yesmaximal, characteristic, complemented
14C2×#1572C211yesmaximal, characteristic, complemented
15C2×#1557C211yesmaximal, characteristic, complemented
16C2×#1546C211yesmaximal, characteristic, complemented
17C2×#532C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20#3239—24no—
21#3239C2218yescomplemented
22#3212—24no—
23#3212C2218yes—
24#3206—24no—
25#3206C2218yescomplemented
26C2×#501—21no—
27C2×#495—21no—
28C2×#492C2211yescharacteristic
29C2×#492C2211yescharacteristic
30C2×#489—21no—
31C2×#488—21no—
32C2×#487C2211yescharacteristic
33C2×#487C2211yescharacteristic
34C2×#484—21no—
35C2×#481—21no—
36C2×#475C2211yescharacteristic
37C2×#475C2211yescharacteristic
38C2×#474C2211yescharacteristic
39C2×#474C2211yescharacteristic
40C2×#462C2211yescharacteristic
41C2×#460C2211yescharacteristic
42C2×#461—21no—
43C2×#461—21no—
44C2×#461C2211yescharacteristic
45C2×#459C2211yescharacteristic
46C2×#452—21no—
47C2×#449—21no—
48C2×#448—21no—
49C2×#448—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×#432C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#431C2211yescharacteristic
62C2×#424—21no—
63C2×#423C2211yescharacteristic
64#1700C2218yescomplemented
65#1699C2218yescomplemented
66#1686C2218yescomplemented
67#1685C2218yescomplemented
68#1680C2218yescomplemented
69#1679C2218yescomplemented
70#1666C2218yescomplemented
71#1665C2218yescomplemented
72#1572C2218yescomplemented
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1557C2218yescomplemented
78#1546C2218yescomplemented
79C2×#332C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×#332C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#331C2211yescharacteristic
86C2×#331C2211yescharacteristic
87C22×#41—21no—
88C23×Q8—21no—
89C22×Q16—21no—
90C22×SD16—21no—
91C22×SD16—21no—
92C22×SD16—21no—
93C22×SD16—21no—
94C22×SD16—21no—
95C22×SD16—21no—
96C22×D8—21no—
97C22×M16—21no—
98C22×M16—21no—
99C23×C8—21no—
100#532—22no—
101#532—22no—
102#532—22no—
103#532C2314yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2314yes—
114C2×#127C2311yescharacteristic
115#529—22no—
116#529—22no—
117#529D414yes—
118#529C2314yes—
119C2×#126—21no—
120C2×#125D411yescharacteristic
121C2×#124—21no—
122C2×#124D411yescharacteristic
123C2×#123—21no—
124C2×C22⋊Q8D411yescharacteristic
125C2×#121—21no—
126C2×#121—21no—
127C2×#121D411yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C42C2311yescharacteristic
137#501—28no—
138#495—28no—
139#492—22no—
140#492—22no—
141#492C2314yes—
142#492C2314yes—
143#489—28no—
144#488—28no—
145#487—22no—
146#487—22no—
147#487C2314yes—
148#487C2314yes—
149#484—28no—
150#481—28no—
151#475—22no—
152#475—22no—
153#475C2314yes—
154#475C2314yes—
155#474—22no—
156#474—22no—
157#474C2314yes—
158#474C2314yes—
159#462—22no—
160#462C2314yes—
161#461—22no—
162#461—28no—
163#461—28no—
164#461C2314yes—
165#460—22no—
166#460C2314yes—
167#459—22no—
168#459C2314yes—
169#452—28no—
170#449—28no—
171#448—28no—
172#448—28no—
173#446—22no—
174#446C2314yes—
175C8×Q8—22no—
176C8×Q8C2314yes—
177#436—28no—
178#436—22no—
179#436C2314yes—
180#435—28no—
181C8×D4—22no—
182C8×D4C2314yes—
183#433—28no—
184#432—22no—
185#432C2314yes—
186#431—22no—
187#431D414yes—
188#431C2314yes—
189#431C2314yes—
190C2×#107—21no—
191C2×#107—21no—
192C2×#106—21no—
193C2×#106—21no—
194#424—28no—
195#423—22no—
196#423C2314yes—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—21no—
199C2×C4⋊C8—21no—
200C2×C4⋊C8D411yescharacteristic
201C2×C4⋊C8D411yescharacteristic
202C2×C4⋊C8C2311yescharacteristic
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2311yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×#103—21no—
209C2×#103—21no—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#102—21no—
217C2×#102—21no—
218C2×#102—21no—
219C2×#102—21no—
220C2×C22⋊C8—21no—
221C2×C22⋊C8—21no—
222C2×C22⋊C8—21no—
223C2×C22⋊C8—21no—
224C2×C22⋊C8D411yescharacteristic
225C2×C22⋊C8D411yescharacteristic
226C2×#97—21no—
227C2×#97—21no—
228C2×C4×C8—21no—
229C2×C4×C8—21no—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232#335C2314yes—
233#335C2314yes—
234#334C2314yes—
235#334C2314yes—
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×Q8—24no—
252C22×Q8—42no—
253C22×Q8—41no—
254C22×Q8—82no—
255C22×Q8—82no—
256C22×Q8C2×D411yescharacteristic
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—41no—
262C22×D4C2×D411yescharacteristic
263C23×C4—21no—
264C23×C4—41no—
265C23×C4—41no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4#4111yescharacteristic
269C2×Q16—24no—
270C2×Q16—82no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—82no—
278C2×SD16—82no—
279C2×SD16—82no—
280C2×SD16—82no—
281C2×SD16—82no—
282C2×SD16—82no—
283C2×D8—24no—
284C2×D8—82no—
285C2×M16—24no—
286C2×M16—24no—
287C2×M16—82no—
288C2×M16—82no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—21no—
293C22×C8—21no—
294C22×C8—21no—
295C22×C8—24no—
296C22×C8—82no—
297C22×C8C2×D411yescharacteristic
298C22×C8C2×D411yescharacteristic
299C22×C8C2×D411yescharacteristic
300C22×C8C2×D411yescharacteristic
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—24no—
304C4⋊Q8—21no—
305C4⋊Q8—24no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—24no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—42no—
314C4⋊Q8—42no—
315C4⋊Q8—42no—
316C4⋊Q8C2412yes—
317C4⋊Q8C2412yes—
318C4⋊Q8C2412yes—
319#127—21no—
320#127—21no—
321#127—21no—
322#127C2412yes—
323#126—24no—
324#126—24no—
325#125—22no—
326#125C2×D414yes—
327#124—22no—
328#124—24no—
329#124—42no—
330#124C2×D414yes—
331#123—24no—
332#123—24no—
333C22⋊Q8—22no—
334C22⋊Q8C2×D414yes—
335#121—22no—
336#121—24no—
337#121—24no—
338#121—42no—
339#121—42no—
340#121C2×D414yes—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8C2×D412yes—
344C4×Q8C2412yes—
345C4×D4—24no—
346C4×D4—24no—
347C4×D4—21no—
348C4×D4—24no—
349C4×D4—21no—
350C4×D4—24no—
351C4×D4—42no—
352C4×D4—42no—
353C4×D4C2×D412yes—
354C4×D4C2412yes—
355#117—24no—
356#117—24no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
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×C42—21no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42#4111yescharacteristic
388C2×C42#4112yes—
389C2×C42C2411yescharacteristic
390C2×C42C2412yes—
391#107—24no—
392#107—24no—
393#106—24no—
394#106—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8C2×D414yes—
405C4⋊C8C2×D414yes—
406C4⋊C8C2412yes—
407C4⋊C8C2412yes—
408C4⋊C8C2412yes—
409C4⋊C8C2412yes—
410C4⋊C8C2412yes—
411C4⋊C8C2412yes—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#102—24no—
421#102—24no—
422#102—24no—
423#102—24no—
424C22⋊C8—24no—
425C22⋊C8—24no—
426C22⋊C8—24no—
427C22⋊C8—24no—
428C22⋊C8C2×D414yes—
429C22⋊C8C2×D414yes—
430#97—24no—
431#97—24no—
432C4×C8—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—44no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—84no—
461C2×Q8—84no—
462C2×Q8—84no—
463C2×Q8—88no—
464C2×Q8C22×D412yes—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—44no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4—84no—
483C2×D4C22×D412yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—44no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4#12112yes—
504C22×C4#12112yes—
505C22×C4#12111yescharacteristic
506C22×C4#12111yescharacteristic
507C22×C4#12111yescharacteristic
508C22×C4#12112yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C2×#4111yescharacteristic
514C22×C4C2×#4112yes—
515Q16—84no—
516SD16—84no—
517SD16—84no—
518SD16—84no—
519SD16—84no—
520SD16—84no—
521SD16—84no—
522D8—84no—
523M16—84no—
524M16—84no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—42no—
536C2×C8—42no—
537C2×C8—42no—
538C2×C8—42no—
539C2×C8—42no—
540C2×C8—42no—
541C2×C8—84no—
542C2×C8C22×D412yes—
543C2×C8C22×D412yes—
544C2×C8C22×D412yes—
545C2×C8C22×D412yes—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—44no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4—44no—
563C4⋊C4—41no—
564C4⋊C4—41no—
565C4⋊C4—41no—
566C4⋊C4—41no—
567C4⋊C4C22×D412yes—
568C4⋊C4C22×D412yes—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C42—22no—
578C42—22no—
579C42#12112yes—
580C42#12112yes—
581C42#12112yes—
582C42C22×D412yes—
583C42C22×D411yescharacteristic
584C42C22×D411yescharacteristic
585C42C22×D412yes—
586C42C2×#4112yes—
587C42C2×#4111yescharacteristic
588C42C2511yescharacteristic
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—82no—
598C23—82no—
599C23—82no—
600C23C2×#12112yes—
601C23C2×#12111yescharacteristic
602Q8—22no—
603Q8—22no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—88no—
612Q8—82no—
613Q8—82no—
614Q8—82no—
615D4—22no—
616D4—22no—
617D4—42no—
618D4—42no—
619D4—42no—
620D4—42no—
621D4—42no—
622D4—82no—
623D4—82no—
624D4—84no—
625D4—88no—
626D4—82no—
627D4—82no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—88no—
653C2×C4—88no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4C2×#12112yes—
658C2×C4C2×#12111yescharacteristic
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12111yescharacteristic
662C2×C4C2×#12111yescharacteristic
663C2×C4#57612yes—
664C2×C4#57712yes—
665C2×C4#57712yes—
666C2×C4#57712yes—
667C2×C4C23×D411yescharacteristic
668C2×C4C23×D411yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C8—22no—
671C8—22no—
672C8—22no—
673C8—22no—
674C8—42no—
675C8—42no—
676C8—42no—
677C8—42no—
678C8—42no—
679C8—42no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—84no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—164no—
692C22#304512yes—
693C22#305712yes—
694C22#306912yes—
695C22C22×#12111yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—21no—
700C4—21no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#304512yes—
708C4#305712yes—
709C4C2×#57611yescharacteristic
710C4C2×#57711yescharacteristic
711C4C2×#57711yescharacteristic
712C4C2×#57711yescharacteristic
713C2—42no—
714C2—82no—
715C2—162no—
716C2#1953614yesdirect product factor
717C2C2×#304511yescharacteristic
718C2C2×#305711yescharacteristic
719C2C2×#306911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3212C2
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×#3239C2semidirect
3. C2×#3206C2semidirect
4. C2×#2993C2semidirect
5. #19536C2direct (N × H)
6. C2×#1700C2semidirect
7. C2×#1686C2semidirect
8. C2×#1699C2semidirect
9. C2×#1685C2semidirect
10. C2×#1680C2semidirect
11. C2×#1666C2semidirect
12. C2×#1679C2semidirect
13. C2×#1665C2semidirect
14. C2×#1572C2semidirect
15. C2×#1557C2semidirect
16. C2×#1546C2semidirect
21. #3239C22semidirect
25. #3206C22semidirect
51. #2993C22semidirect
64. #1700C22semidirect
65. #1699C22semidirect
66. #1686C22semidirect
67. #1685C22semidirect
68. #1680C22semidirect
69. #1679C22semidirect
70. #1666C22semidirect
71. #1665C22semidirect
72. #1572C22semidirect
77. #1557C22semidirect
78. #1546C22semidirect
716. C2#19536direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32392
C2×#32122
C2×#32062
C2×#29932
#195362
C2×#17002
C2×#16862
C2×#16992
C2×#16852
C2×#16802
C2×#16662
C2×#16792
C2×#16652
C2×#15722
C2×#15572
C2×#15462