Finite Group Databasebeta

← C2×#20188 #6359539

Subgroups — C2×#20188 #6359539

Order 512.

1671 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3392C211yesmaximal, characteristic
2C2×#3387C211yesmaximal, characteristic, complemented
3C2×#3385C211yesmaximal, characteristic, complemented
4C2×#3333C211yesmaximal, characteristic
5#20188C2116yesmaximal, direct product factor
6C2×#1714C211yesmaximal, characteristic, complemented
7C2×#1698C211yesmaximal, characteristic, complemented
8C2×#1686C211yesmaximal, characteristic, complemented
9C2×#1691C211yesmaximal, characteristic, complemented
10C2×#1676C211yesmaximal, characteristic, complemented
11C2×#1623C211yesmaximal, characteristic, complemented
12C2×#1631C211yesmaximal, characteristic
13C2×#1636C211yesmaximal, characteristic, complemented
14C2×#1643C211yesmaximal, characteristic, complemented
15C2×#1592C211yesmaximal, characteristic, complemented
16C2×#1584C211yesmaximal, characteristic, complemented
17C2×Q82C2211yescharacteristic
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×#550C2211yescharacteristic
21#3392—24no—
22#3392C2218yes—
23#3387—24no—
24#3387C2218yescomplemented
25#3385—24no—
26#3385C2218yescomplemented
27#3333—24no—
28#3333C2218yes—
29C2×#501—21no—
30C2×#501—21no—
31C2×#500C2211yescharacteristic
32C2×#498C2211yescharacteristic
33C2×#492—21no—
34C2×#492C2211yescharacteristic
35C2×#492C2211yescharacteristic
36C2×#487C2211yescharacteristic
37C2×#486C2211yescharacteristic
38C2×#480—21no—
39C2×#479—21no—
40C2×#479—21no—
41C2×#477C2211yescharacteristic
42C2×#477C2211yescharacteristic
43C2×#477C2211yescharacteristic
44C2×#475—21no—
45C2×#475C2211yescharacteristic
46C2×#476C2211yescharacteristic
47C2×#474—21no—
48C2×#468—21no—
49C2×#460—21no—
50C2×#460C2211yescharacteristic
51C2×#460C2211yescharacteristic
52C2×#461—21no—
53C2×#461C2211yescharacteristic
54C2×#452—21no—
55C2×#446C2211yescharacteristic
56C2×#446C2211yescharacteristic
57C2×C8×Q8C2211yescharacteristic
58C2×#441—21no—
59C2×#440—21no—
60C2×C4×Q16C2211yescharacteristic
61C2×C4×SD16—21no—
62C2×C4×SD16C2211yescharacteristic
63C2×#436C2211yescharacteristic
64C2×#417—21no—
65#1714C2218yescomplemented
66#1698C2218yescomplemented
67#1691C2218yescomplemented
68#1686C2218yescomplemented
69#1676C2218yescomplemented
70#1643C2218yescomplemented
71#1636C2218yescomplemented
72#1631C2218yes—
73#1623C2218yescomplemented
74#1592C2218yescomplemented
75#1584C2218yescomplemented
76C2×#335C2211yescharacteristic
77C2×#335C2211yescharacteristic
78C2×#333C2211yescharacteristic
79C2×#333C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#331C2211yescharacteristic
82C2×Q8⋊C8C2211yescharacteristic
83C2×Q8⋊C8C2211yescharacteristic
84C2×Q8⋊C8C2211yescharacteristic
85C2×D4⋊C8C2211yescharacteristic
86C2×#322C2211yescharacteristic
87C22×#41—21no—
88C22×Q16D411yescharacteristic
89C22×SD16—21no—
90C22×SD16—21no—
91C22×SD16—21no—
92C22×SD16—21no—
93C22×SD16—21no—
94C22×SD16—41no—
95C22×SD16D411yescharacteristic
96C22×M16—21no—
97Q82—22no—
98Q82—22no—
99Q82—41no—
100Q82C2314yes—
101#557—22no—
102#557—22no—
103#557—41no—
104#557C2314yes—
105#554—22no—
106#554—22no—
107#554—41no—
108#554C2314yes—
109#550—22no—
110#550—22no—
111#550—41no—
112#550C2314yes—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—41no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#127—21no—
121C2×#127C2311yescharacteristic
122C2×#125—21no—
123C2×#125—21no—
124C2×#125—41no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—41no—
127C2×#121—21no—
128C2×#121—41no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8C2311yescharacteristic
134C2×C4×Q8C2311yescharacteristic
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2311yescharacteristic
139C22×C4⋊C4—21no—
140#501—28no—
141#501—28no—
142#500—22no—
143#500C2314yes—
144#498—22no—
145#498C2314yes—
146#492—28no—
147#492—22no—
148#492—41no—
149#492C2314yes—
150#492C2314yes—
151#487—41no—
152#487C2314yes—
153#486—22no—
154#486C2314yes—
155#480—28no—
156#479—28no—
157#479—28no—
158#477—22no—
159#477—22no—
160#477—22no—
161#477C2314yes—
162#477C2314yes—
163#477C2314yes—
164#476—22no—
165#476C2314yes—
166#475—22no—
167#475—28no—
168#475C2314yes—
169#474—28no—
170#468—28no—
171#461—22no—
172#461—28no—
173#461C2314yes—
174#460—22no—
175#460—22no—
176#460—28no—
177#460C2314yes—
178#460C2314yes—
179#452—28no—
180#446—22no—
181#446—22no—
182#446C2314yes—
183#446C2314yes—
184C8×Q8—22no—
185C8×Q8C2314yes—
186#441—28no—
187#440—28no—
188C4×Q16D414yes—
189C4×Q16C2314yes—
190C4×SD16—28no—
191C4×SD16D414yes—
192C4×SD16C2314yes—
193#436—22no—
194#436C2314yes—
195C2×#107—21no—
196C2×#107—21no—
197C2×#107—21no—
198C2×#107—21no—
199C2×#107—21no—
200C2×#107D411yescharacteristic
201C2×#106—21no—
202C2×#106—21no—
203C2×#106—21no—
204C2×#106—21no—
205C2×#106D411yescharacteristic
206C2×C4⋊C8—21no—
207C2×C4⋊C8—21no—
208C2×C4⋊C8—21no—
209C2×C4⋊C8C2311yescharacteristic
210C2×C4⋊C8C2311yescharacteristic
211C2×C4⋊C8C2311yescharacteristic
212C2×C4⋊C8C2311yescharacteristic
213#417—28no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#103—41no—
220C2×#103D411yescharacteristic
221C2×#103D411yescharacteristic
222C2×#103D411yescharacteristic
223C2×#102—21no—
224C2×#102—21no—
225C2×#102—21no—
226C2×#102—21no—
227C2×#102—21no—
228C2×#102D411yescharacteristic
229C2×C22⋊C8—21no—
230C2×#97—21no—
231C2×#97—21no—
232C2×C4×C8—21no—
233C2×C4×C8C2311yescharacteristic
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236#335C2314yes—
237#335C2314yes—
238#333C2314yes—
239#333C2314yes—
240#332C2314yes—
241#331C2314yes—
242Q8⋊C8C2314yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2314yes—
245D4⋊C8C2314yes—
246#322C2314yes—
247C2×#41—24no—
248C2×#41—42no—
249C2×#41—44no—
250C2×#41—82no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—41no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2×D411yescharacteristic
257C22×Q8C2×D411yescharacteristic
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—41no—
262C22×D4—41no—
263C22×D4C2×D411yescharacteristic
264C23×C4—21no—
265C23×C4—41no—
266C23×C4—41no—
267C2×Q16—24no—
268C2×Q16C2×D414yes—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—44no—
276C2×SD16—44no—
277C2×SD16—82no—
278C2×SD16—82no—
279C2×SD16—82no—
280C2×SD16—82no—
281C2×SD16—84no—
282C2×SD16C2×D414yes—
283C2×M16—24no—
284C2×M16—82no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D411yescharacteristic
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—42no—
307C4⋊Q8—42no—
308C4⋊Q8—42no—
309C4⋊Q8—44no—
310C4⋊Q8—44no—
311C4⋊Q8C2412yes—
312C4⋊Q8C2412yes—
313C4⋊Q8C2412yes—
314#127—24no—
315#127—21no—
316#127—21no—
317#127—21no—
318#127—42no—
319#127C2412yes—
320#125—24no—
321#125—24no—
322#125—42no—
323#125—42no—
324#125—44no—
325#125—44no—
326C22⋊Q8—24no—
327C22⋊Q8—44no—
328C22⋊Q8—44no—
329C22⋊Q8—42no—
330#121—24no—
331#121—44no—
332#121—44no—
333#121—42no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—24no—
341C4×Q8—24no—
342C4×Q8—24no—
343C4×Q8—24no—
344C4×Q8—42no—
345C4×Q8—42no—
346C4×Q8—42no—
347C4×Q8—42no—
348C4×Q8C2×D412yes—
349C4×Q8C2×D412yes—
350C4×Q8C2×D412yes—
351C4×Q8C2412yes—
352C4×Q8C2412yes—
353C4×Q8C2412yes—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—42no—
359C4×D4—42no—
360C4×D4C2×D412yes—
361C4×D4C2412yes—
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—24no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4—41no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D411yescharacteristic
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4—41no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—41no—
391C2×C42—41no—
392C2×C42C2411yescharacteristic
393#107—24no—
394#107—24no—
395#107—24no—
396#107—24no—
397#107—24no—
398#107C2×D414yes—
399#106—24no—
400#106—24no—
401#106—24no—
402#106—24no—
403#106C2×D414yes—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8C2412yes—
412C4⋊C8C2412yes—
413C4⋊C8C2412yes—
414C4⋊C8C2412yes—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—44no—
421#103C2×D414yes—
422#103C2×D414yes—
423#103C2×D414yes—
424#102—24no—
425#102—24no—
426#102—24no—
427#102—24no—
428#102—24no—
429#102C2×D414yes—
430C22⋊C8—24no—
431#97—24no—
432#97—24no—
433C4×C8—24no—
434C4×C8—21no—
435C4×C8C2×D412yes—
436C4×C8C2×D412yes—
437C4×C8C2412yes—
438C4×C8C2412yes—
439C4×C8C2412yes—
440C24—21no—
441C24—41no—
442#41—48no—
443#41—84no—
444#41—88no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—44no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—41no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—82no—
463C2×Q8—82no—
464C2×Q8—162no—
465C2×Q8#13714yes—
466C2×Q8C22×D412yes—
467C2×Q8C22×D412yes—
468C2×Q8C22×D412yes—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—84no—
485C2×D4—82no—
486C2×D4—162no—
487C2×D4C22×D412yes—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—84no—
505C22×C4—162no—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4H(5,𝔽2)11yescharacteristic
509Q16—28no—
510SD16—28no—
511SD16—48no—
512SD16—84no—
513SD16—84no—
514SD16—84no—
515SD16—84no—
516SD16—88no—
517M16—84no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—42no—
527C2×C8—42no—
528C2×C8—42no—
529C2×C8—42no—
530C2×C8—42no—
531C2×C8#13714yes—
532C2×C8C22×D412yes—
533C2×C8C22×D412yes—
534C2×C8C22×D412yes—
535C2×C8C22×D412yes—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—41no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4—41no—
563C4⋊C4—41no—
564C4⋊C4—41no—
565C4⋊C4C22×D412yes—
566C4⋊C4C22×D412yes—
567C4⋊C4C22×D412yes—
568C4⋊C4C22×D412yes—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—44no—
580C42—44no—
581C42C22×D411yescharacteristic
582C42C22×D411yescharacteristic
583C42H(5,𝔽2)11yescharacteristic
584C42C2511yescharacteristic
585C23—22no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—82no—
591C23—82no—
592C23—82no—
593C23D4211yescharacteristic
594Q8—24no—
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8—22no—
599Q8—22no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—82no—
605Q8—82no—
606Q8—82no—
607Q8—161no—
608Q8—161no—
609Q8—164no—
610Q8C2×#13714yes—
611D4—22no—
612D4—22no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—88no—
619D4—82no—
620D4—82no—
621D4—82no—
622D4—161no—
623D4—164no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—88no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—164no—
650C2×C4D4211yescharacteristic
651C2×C4D4211yescharacteristic
652C2×C4D4211yescharacteristic
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4#57712yes—
656C2×C4#57712yes—
657C2×C4C23×D411yescharacteristic
658C2×C4C23×D411yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C8—22no—
661C8—22no—
662C8—22no—
663C8—42no—
664C8—42no—
665C8—42no—
666C8—42no—
667C8—42no—
668C8C2×#13714yes—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—164no—
677C22#326612yes—
678C22#327812yes—
679C22#328812yes—
680C22C2×D4211yescharacteristic
681C4—24no—
682C4—21no—
683C4—21no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#326612yes—
694C4#327812yes—
695C4C22×#13711yescharacteristic
696C4C22×#13711yescharacteristic
697C4C2×#57711yescharacteristic
698C4C2×#57711yescharacteristic
699C2—82no—
700C2—162no—
701C2#2018814yesdirect product factor
702C2C2×#326611yescharacteristic
703C2C2×#327811yescharacteristic
704C2C2×#328811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC42C25
prime radicalC2×#550C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3387C2semidirect
3. C2×#3385C2semidirect
5. #20188C2direct (N × H)
6. C2×#1714C2semidirect
7. C2×#1698C2semidirect
8. C2×#1686C2semidirect
9. C2×#1691C2semidirect
10. C2×#1676C2semidirect
11. C2×#1623C2semidirect
13. C2×#1636C2semidirect
14. C2×#1643C2semidirect
15. C2×#1592C2semidirect
16. C2×#1584C2semidirect
24. #3387C22semidirect
26. #3385C22semidirect
65. #1714C22semidirect
66. #1698C22semidirect
67. #1691C22semidirect
68. #1686C22semidirect
69. #1676C22semidirect
70. #1643C22semidirect
71. #1636C22semidirect
73. #1623C22semidirect
74. #1592C22semidirect
75. #1584C22semidirect
701. C2#20188direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33922
C2×#33872
C2×#33852
C2×#33332
#201882
C2×#17142
C2×#16982
C2×#16862
C2×#16912
C2×#16762
C2×#16232
C2×#16312
C2×#16362
C2×#16432
C2×#15922
C2×#15842