Finite Group Databasebeta

← C2×#20197 #6359549

Subgroups — C2×#20197 #6359549

Order 512.

1781 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic, complemented
2C2×#3372C211yesmaximal, characteristic, complemented
3C2×#3335C211yesmaximal, characteristic
4C2×#3276C211yesmaximal, characteristic
5#20197C2116yesmaximal, direct product factor
6C2×#1723C211yesmaximal, characteristic, complemented
7C2×#1692C211yesmaximal, characteristic, complemented
8C2×#1690C211yesmaximal, characteristic, complemented
9C2×#1703C211yesmaximal, characteristic, complemented
10C2×#1673C211yesmaximal, characteristic, complemented
11C2×#1648C211yesmaximal, characteristic, complemented
12C2×#1647C211yesmaximal, characteristic
13C2×#1637C211yesmaximal, characteristic, complemented
14C2×#1643C211yesmaximal, characteristic, complemented
15C2×#1592C211yesmaximal, characteristic, complemented
16C2×#1582C211yesmaximal, characteristic, complemented
17C22×#137—21no—
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×D4×Q8C2211yescharacteristic
21C2×#547C2211yescharacteristic
22#3386—24no—
23#3386C2218yescomplemented
24#3372—24no—
25#3372C2218yescomplemented
26#3335—24no—
27#3335C2218yes—
28#3276—24no—
29#3276C2218yes—
30C2×#501—21no—
31C2×#500C2211yescharacteristic
32C2×#500C2211yescharacteristic
33C2×#498C2211yescharacteristic
34C2×#494C2211yescharacteristic
35C2×#491—21no—
36C2×#487C2211yescharacteristic
37C2×#487C2211yescharacteristic
38C2×#481—21no—
39C2×#483—21no—
40C2×#479—21no—
41C2×#477—21no—
42C2×#477C2211yescharacteristic
43C2×#475C2211yescharacteristic
44C2×#475C2211yescharacteristic
45C2×#478—21no—
46C2×#476C2211yescharacteristic
47C2×#474C2211yescharacteristic
48C2×#470—21no—
49C2×#465—21no—
50C2×#462C2211yescharacteristic
51C2×#461—21no—
52C2×#461C2211yescharacteristic
53C2×#461C2211yescharacteristic
54C2×#452—21no—
55C2×#451—21no—
56C2×#446C2211yescharacteristic
57C2×#446C2211yescharacteristic
58C2×#440—21no—
59C2×#440—21no—
60C2×C4×SD16C2211yescharacteristic
61C2×C4×SD16C2211yescharacteristic
62C2×#436C2211yescharacteristic
63C2×C8×D4C2211yescharacteristic
64C22×#106—21no—
65#1723C2218yescomplemented
66#1703C2218yescomplemented
67#1692C2218yescomplemented
68#1690C2218yescomplemented
69#1673C2218yescomplemented
70#1648C2218yescomplemented
71#1647C2218yes—
72#1643C2218yescomplemented
73#1637C2218yescomplemented
74#1592C2218yescomplemented
75#1582C2218yescomplemented
76C2×#335C2211yescharacteristic
77C2×#334C2211yescharacteristic
78C2×#333C2211yescharacteristic
79C2×#332C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×Q8⋊C8C2211yescharacteristic
83C2×Q8⋊C8C2211yescharacteristic
84C2×D4⋊C8C2211yescharacteristic
85C2×D4⋊C8C2211yescharacteristic
86C2×#322C2211yescharacteristic
87C22×#41—21no—
88C23×Q8—21no—
89C2×#137—28no—
90C2×#137—48no—
91C22×Q16—21no—
92C22×Q16—21no—
93C22×Q16—21no—
94C22×SD16—21no—
95C22×SD16—21no—
96C22×SD16D411yescharacteristic
97C22×SD16D411yescharacteristic
98C22×M16—21no—
99C23×C8—21no—
100#557—22no—
101#557—22no—
102#557—41no—
103#557C2314yes—
104#554—22no—
105#554—22no—
106#554—41no—
107#554C2314yes—
108D4×Q8—22no—
109D4×Q8—22no—
110D4×Q8—41no—
111D4×Q8C2314yes—
112#547—22no—
113#547—22no—
114#547—41no—
115#547C2314yes—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×#125—21no—
123C2×#125—21no—
124C2×#125—41no—
125C2×#123—41no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—41no—
130C2×C22⋊Q8—41no—
131C2×#121—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142#501—28no—
143#500—22no—
144#500—41no—
145#500C2314yes—
146#500C2314yes—
147#498—22no—
148#498C2314yes—
149#494—22no—
150#494C2314yes—
151#491—28no—
152#487—22no—
153#487—41no—
154#487C2314yes—
155#487C2314yes—
156#483—28no—
157#481—28no—
158#479—28no—
159#478—28no—
160#477—22no—
161#477—28no—
162#477C2314yes—
163#476—22no—
164#476C2314yes—
165#475—22no—
166#475—22no—
167#475C2314yes—
168#475C2314yes—
169#474—22no—
170#474C2314yes—
171#470—28no—
172#465—28no—
173#462—22no—
174#462C2314yes—
175#461—28no—
176#461—22no—
177#461—22no—
178#461C2314yes—
179#461C2314yes—
180#452—28no—
181#451—28no—
182#446—22no—
183#446—22no—
184#446C2314yes—
185#446C2314yes—
186#440—28no—
187#440—28no—
188C4×SD16D414yes—
189C4×SD16D414yes—
190C4×SD16C2314yes—
191C4×SD16C2314yes—
192#436—22no—
193#436C2314yes—
194C8×D4—22no—
195C8×D4C2314yes—
196C2×#107—21no—
197C2×#107—21no—
198C2×#107—21no—
199C2×#107—21no—
200C2×#107—41no—
201C2×#106—21no—
202C2×#106—21no—
203C2×#106—21no—
204C2×#106—21no—
205C2×#106—21no—
206C2×#106—28no—
207C2×#106D411yescharacteristic
208C2×#106D411yescharacteristic
209C2×C4⋊C8—21no—
210C2×C4⋊C8—21no—
211C2×C4⋊C8C2311yescharacteristic
212C2×C4⋊C8C2311yescharacteristic
213C2×C4⋊C8C2311yescharacteristic
214C2×C4⋊C8C2311yescharacteristic
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#103—21no—
220C2×#103—21no—
221C2×#103—21no—
222C2×#103—21no—
223C2×#103—41no—
224C2×#103D411yescharacteristic
225C2×#103D411yescharacteristic
226C2×#102—21no—
227C2×#102—21no—
228C2×#102D411yescharacteristic
229C2×#102D411yescharacteristic
230C2×C22⋊C8—21no—
231C2×C22⋊C8—21no—
232C2×#97—21no—
233C2×#97—21no—
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236C2×C4×C8C2311yescharacteristic
237#335C2314yes—
238#334C2314yes—
239#333C2314yes—
240#332C2314yes—
241#332C2314yes—
242#332C2314yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2314yes—
245D4⋊C8C2314yes—
246D4⋊C8C2314yes—
247#322C2314yes—
248C2×#41—24no—
249C2×#41—42no—
250C2×#41—44no—
251C2×#41—82no—
252C22×Q8—24no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—41no—
258C22×Q8—42no—
259C22×Q8—44no—
260C22×Q8—82no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×D4—41no—
264C22×D4C2×D411yescharacteristic
265C22×D4C2×D411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—41no—
269C23×C4—41no—
270C23×C4—41no—
271#137—432no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×Q16—82no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—44no—
283C2×SD16—44no—
284C2×SD16C2×D414yes—
285C2×SD16C2×D414yes—
286C2×M16—24no—
287C2×M16—44no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—21no—
293C22×C8—24no—
294C22×C8—44no—
295C22×C8C2×D411yescharacteristic
296C22×C8C2×D411yescharacteristic
297C22×C8C2×D411yescharacteristic
298C22×C8C2×D411yescharacteristic
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—42no—
314C4⋊Q8—42no—
315C4⋊Q8C2412yes—
316C4⋊Q8C2412yes—
317C4⋊Q8C2412yes—
318C4⋊Q8C2412yes—
319#125—24no—
320#125—24no—
321#125—42no—
322#125—42no—
323#125—44no—
324#125—44no—
325#123—44no—
326#123—44no—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330C22⋊Q8—44no—
331C22⋊Q8—44no—
332C22⋊Q8—44no—
333C22⋊Q8—44no—
334C22⋊Q8—42no—
335C22⋊Q8—42no—
336C22⋊Q8—42no—
337#121—24no—
338#121—42no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8—24no—
344C4×Q8—24no—
345C4×Q8—42no—
346C4×Q8—42no—
347C4×Q8C2×D412yes—
348C4×Q8C2×D412yes—
349C4×Q8C2412yes—
350C4×Q8C2412yes—
351C4×D4—24no—
352C4×D4—24no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—42no—
358C4×D4—42no—
359C4×D4C2×D412yes—
360C4×D4C2×D412yes—
361C4×D4C2412yes—
362C4×D4C2412yes—
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—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—41no—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D411yescharacteristic
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—41no—
391C2×C22⋊C4—41no—
392C2×C22⋊C4—41no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—41no—
396C2×C42C2411yescharacteristic
397#107—24no—
398#107—24no—
399#107—24no—
400#107—24no—
401#107—44no—
402#106—24no—
403#106—24no—
404#106—24no—
405#106—24no—
406#106—24no—
407#106C2×D414yes—
408#106C2×D414yes—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—21no—
415C4⋊C8C2412yes—
416C4⋊C8C2412yes—
417C4⋊C8C2412yes—
418C4⋊C8C2412yes—
419#103—24no—
420#103—24no—
421#103—24no—
422#103—24no—
423#103—24no—
424#103—24no—
425#103—24no—
426#103—24no—
427#103—44no—
428#103C2×D414yes—
429#103C2×D414yes—
430#102—24no—
431#102—24no—
432#102C2×D414yes—
433#102C2×D414yes—
434C22⋊C8—24no—
435C22⋊C8—24no—
436#97—24no—
437#97—24no—
438C4×C8—21no—
439C4×C8C2×D412yes—
440C4×C8C2×D412yes—
441C4×C8C2412yes—
442C4×C8C2412yes—
443C4×C8C2412yes—
444C24—21no—
445C24—21no—
446#41—48no—
447#41—84no—
448#41—88no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—44no—
460C2×Q8—41no—
461C2×Q8—41no—
462C2×Q8—41no—
463C2×Q8—41no—
464C2×Q8—48no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—82no—
472C2×Q8—84no—
473C2×Q8—84no—
474C2×Q8—88no—
475C2×Q8C22×D412yes—
476C2×Q8C22×D412yes—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—44no—
483C2×D4—84no—
484C2×D4AGL(1,ℤ/8)14yes—
485C2×D4C22×D412yes—
486C2×D4C22×D412yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—162no—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4H(5,𝔽2)11yescharacteristic
514Q16—48no—
515Q16—48no—
516Q16—84no—
517SD16—28no—
518SD16—28no—
519SD16—48no—
520SD16—48no—
521M16—48no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—42no—
531C2×C8—42no—
532C2×C8—42no—
533C2×C8—42no—
534C2×C8—42no—
535C2×C8—48no—
536C2×C8#13714yescomplemented
537C2×C8C22×D412yes—
538C2×C8C22×D412yes—
539C2×C8C22×D412yes—
540C2×C8C22×D412yes—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
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—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—41no—
565C4⋊C4—41no—
566C4⋊C4—41no—
567C4⋊C4—41no—
568C4⋊C4—41no—
569C4⋊C4—41no—
570C4⋊C4—41no—
571C4⋊C4—41no—
572C4⋊C4C22×D412yes—
573C4⋊C4C22×D412yes—
574C4⋊C4C22×D412yes—
575C4⋊C4C22×D412yes—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—44no—
588C42C22×D411yescharacteristic
589C42C22×D411yescharacteristic
590C42H(5,𝔽2)11yescharacteristic
591C42C2511yescharacteristic
592C23—22no—
593C23—24no—
594C23—22no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23D4211yescharacteristic
601Q8—22no—
602Q8—22no—
603Q8—22no—
604Q8—22no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—82no—
614Q8—88no—
615Q8—82no—
616Q8—82no—
617Q8—82no—
618Q8—161no—
619D4—24no—
620D4—22no—
621D4—22no—
622D4—22no—
623D4—42no—
624D4—88no—
625D4C2×AGL(1,ℤ/8)14yes—
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—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—88no—
650C2×C4—88no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—164no—
656C2×C4D4211yescharacteristic
657C2×C4D4211yescharacteristic
658C2×C4D4211yescharacteristic
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×#13712yes—
661C2×C4#57712yes—
662C2×C4#57812yes—
663C2×C4C23×D411yescharacteristic
664C2×C4C23×D411yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C8—22no—
667C8—22no—
668C8—22no—
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C8C2×#13714yescomplemented
675C22—24no—
676C22—42no—
677C22—42no—
678C22—48no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—84no—
683C22#326512yes—
684C22#327612yes—
685C22#328912yes—
686C22C2×D4211yescharacteristic
687C4—21no—
688C4—21no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#326512yes—
700C4#327612yes—
701C4C22×AGL(1,ℤ/8)11yescharacteristic
702C4C22×#13711yescharacteristic
703C4C2×#57711yescharacteristic
704C4C2×#57811yescharacteristic
705C2—44no—
706C2—82no—
707C2#2019714yesdirect product factor
708C2C2×#326511yescharacteristic
709C2C2×#327611yescharacteristic
710C2C2×#328911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC42C25
prime radicalC2×#547C22
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
1. C2×#3386C2semidirect
2. C2×#3372C2semidirect
5. #20197C2direct (N × H)
6. C2×#1723C2semidirect
7. C2×#1692C2semidirect
8. C2×#1690C2semidirect
9. C2×#1703C2semidirect
10. C2×#1673C2semidirect
11. C2×#1648C2semidirect
13. C2×#1637C2semidirect
14. C2×#1643C2semidirect
15. C2×#1592C2semidirect
16. C2×#1582C2semidirect
23. #3386C22semidirect
25. #3372C22semidirect
65. #1723C22semidirect
66. #1703C22semidirect
67. #1692C22semidirect
68. #1690C22semidirect
69. #1673C22semidirect
70. #1648C22semidirect
72. #1643C22semidirect
73. #1637C22semidirect
74. #1592C22semidirect
75. #1582C22semidirect
536. C2×C8#137semidirect
674. C8C2×#137semidirect
707. C2#20197direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33862
C2×#33722
C2×#33352
C2×#32762
#201972
C2×#17232
C2×#16922
C2×#16902
C2×#17032
C2×#16732
C2×#16482
C2×#16472
C2×#16372
C2×#16432
C2×#15922
C2×#15822