Finite Group Databasebeta

← C2×#19897 #6358264

Subgroups — C2×#19897 #6358264

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3392C211yesmaximal, characteristic
2C2×#3387C211yesmaximal, characteristic, complemented
3C2×Q8×SD16C211yesmaximal, characteristic, complemented
4C2×#3278C211yesmaximal, characteristic
5#19897C2116yesmaximal, direct product factor
6C2×#1705C211yesmaximal, characteristic, complemented
7C2×#1686C211yesmaximal, characteristic, complemented
8C2×#1691C211yesmaximal, characteristic, complemented
9C2×#1672C211yesmaximal, characteristic, complemented
10C2×#1666C211yesmaximal, characteristic, complemented
11C2×#1623C211yesmaximal, characteristic, complemented
12C2×#1631C211yesmaximal, characteristic
13C2×#1653C211yesmaximal, characteristic, complemented
14C2×#1651C211yesmaximal, characteristic, complemented
15C2×#1590C211yesmaximal, characteristic, complemented
16C2×C8×Q16C211yesmaximal, characteristic, complemented
17C22×#137—21no—
18C2×Q82C2211yescharacteristic
19C2×#557C2211yescharacteristic
20C2×#550C2211yescharacteristic
21C2×D4×Q8C2211yescharacteristic
22#3392—24no—
23#3392C2218yes—
24#3387—24no—
25#3387C2218yescomplemented
26Q8×SD16—24no—
27Q8×SD16C2218yescomplemented
28#3278—24no—
29#3278C2218yes—
30C2×#501—21no—
31C2×#498—21no—
32C2×#498C2211yescharacteristic
33C2×#498C2211yescharacteristic
34C2×#498C2211yescharacteristic
35C2×#492—21no—
36C2×#492C2211yescharacteristic
37C2×#492C2211yescharacteristic
38C2×#492C2211yescharacteristic
39C2×#479—21no—
40C2×#477C2211yescharacteristic
41C2×#475—21no—
42C2×#475—21no—
43C2×#475C2211yescharacteristic
44C2×#475C2211yescharacteristic
45C2×#476—21no—
46C2×#476C2211yescharacteristic
47C2×#468—21no—
48C2×#462C2211yescharacteristic
49C2×#460—21no—
50C2×#460C2211yescharacteristic
51C2×#460C2211yescharacteristic
52C2×#461—21no—
53C2×#461C2211yescharacteristic
54C2×#452—21no—
55C2×#448—21no—
56C2×#446C2211yescharacteristic
57C2×C8×Q8C2211yescharacteristic
58C2×C8×Q8C2211yescharacteristic
59C2×#441—21no—
60C2×C4×Q16C2211yescharacteristic
61C2×C4×SD16—21no—
62C2×C4×SD16—21no—
63C2×C4×SD16C2211yescharacteristic
64C2×#436C2211yescharacteristic
65#1705C2218yescomplemented
66#1691C2218yescomplemented
67#1686C2218yescomplemented
68#1672C2218yescomplemented
69#1666C2218yescomplemented
70#1653C2218yescomplemented
71#1651C2218yescomplemented
72#1631C2218yes—
73#1623C2218yescomplemented
74#1590C2218yescomplemented
75C8×Q16C2218yescomplemented
76C2×#335C2211yescharacteristic
77C2×#334C2211yescharacteristic
78C2×#333C2211yescharacteristic
79C2×#332C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#331C2211yescharacteristic
82C2×Q8⋊C8C2211yescharacteristic
83C2×Q8⋊C8C2211yescharacteristic
84C2×Q8⋊C8C2211yescharacteristic
85C2×D4⋊C8C2211yescharacteristic
86C2×C82C2211yescharacteristic
87C22×#41—21no—
88C23×Q8—21no—
89C2×#137—28no—
90C2×#137—48no—
91C22×Q16—21no—
92C22×Q16D411yescharacteristic
93C22×SD16—21no—
94C22×SD16—21no—
95C22×SD16—21no—
96C22×SD16—21no—
97C22×SD16—21no—
98C22×SD16—21no—
99C22×SD16—21no—
100C22×SD16—41no—
101C22×SD16D411yescharacteristic
102C22×M16—21no—
103Q82—22no—
104Q82—22no—
105Q82—41no—
106Q82C2314yes—
107#557—22no—
108#557—22no—
109#557—41no—
110#557C2314yes—
111#550—22no—
112#550—22no—
113#550—41no—
114#550C2314yes—
115D4×Q8—22no—
116D4×Q8—22no—
117D4×Q8—41no—
118D4×Q8C2314yes—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—21no—
123C2×C4⋊Q8—41no—
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2311yescharacteristic
127C2×#127—21no—
128C2×#127C2311yescharacteristic
129C2×#125—21no—
130C2×#125—41no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—41no—
133C2×#121—21no—
134C2×#121—41no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4C2311yescharacteristic
145#501—28no—
146#498—28no—
147#498—22no—
148#498—22no—
149#498—41no—
150#498C2314yes—
151#498C2314yes—
152#498C2314yes—
153#492—28no—
154#492—22no—
155#492—22no—
156#492—41no—
157#492C2314yes—
158#492C2314yes—
159#492C2314yes—
160#479—28no—
161#477—22no—
162#477C2314yes—
163#476—22no—
164#476—28no—
165#476C2314yes—
166#475—22no—
167#475—22no—
168#475—28no—
169#475—28no—
170#475C2314yes—
171#475C2314yes—
172#468—28no—
173#462—22no—
174#462C2314yes—
175#461—22no—
176#461—28no—
177#461C2314yes—
178#460—22no—
179#460—22no—
180#460—28no—
181#460C2314yes—
182#460C2314yes—
183#452—28no—
184#448—28no—
185#446—22no—
186#446C2314yes—
187C8×Q8—22no—
188C8×Q8—22no—
189C8×Q8C2314yes—
190C8×Q8C2314yes—
191#441—28no—
192C4×Q16D414yes—
193C4×Q16C2314yes—
194C4×SD16—28no—
195C4×SD16—28no—
196C4×SD16D414yes—
197C4×SD16C2314yes—
198#436—22no—
199#436C2314yes—
200C2×#107—21no—
201C2×#107D411yescharacteristic
202C2×#106—21no—
203C2×#106—21no—
204C2×#106—21no—
205C2×#106—21no—
206C2×#106—21no—
207C2×#106—21no—
208C2×#106—21no—
209C2×#106—41no—
210C2×#106D411yescharacteristic
211C2×C4⋊C8—21no—
212C2×C4⋊C8—21no—
213C2×C4⋊C8—21no—
214C2×C4⋊C8C2311yescharacteristic
215C2×C4⋊C8C2311yescharacteristic
216C2×C4⋊C8C2311yescharacteristic
217C2×C4⋊C8C2311yescharacteristic
218C2×#103—21no—
219C2×#103—21no—
220C2×#103—21no—
221C2×#103—21no—
222C2×#103—21no—
223C2×#103D411yescharacteristic
224C2×#103D411yescharacteristic
225C2×#103D411yescharacteristic
226C2×#102—21no—
227C2×#102—21no—
228C2×#102—21no—
229C2×#102D411yescharacteristic
230C2×C22⋊C8—21no—
231C2×#97—21no—
232C2×C4×C8—21no—
233C2×C4×C8—21no—
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236C2×C4×C8C2311yescharacteristic
237#335C2314yes—
238#334C2314yes—
239#333C2314yes—
240#332C2314yes—
241#332C2314yes—
242#331C2314yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2314yes—
245Q8⋊C8C2314yes—
246D4⋊C8C2314yes—
247C82C2314yes—
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—41no—
257C22×Q8—41no—
258C22×Q8—42no—
259C22×Q8—44no—
260C22×Q8—82no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2×D411yescharacteristic
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4—41no—
268C22×D4—41no—
269C22×D4C2×D411yescharacteristic
270C23×C4—21no—
271C23×C4—41no—
272C23×C4—41no—
273#137—432no—
274C2×Q16—24no—
275C2×Q16—44no—
276C2×Q16SD1618yescomplemented
277C2×Q16C2×D414yes—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—24no—
285C2×SD16—24no—
286C2×SD16—44no—
287C2×SD16—44no—
288C2×SD16—44no—
289C2×SD16—44no—
290C2×SD16—82no—
291C2×SD16—82no—
292C2×SD16—84no—
293C2×SD16—82no—
294C2×SD16—82no—
295C2×SD16C2×D414yes—
296C2×M16—24no—
297C2×M16—44no—
298C22×C8—21no—
299C22×C8—21no—
300C22×C8—21no—
301C22×C8—21no—
302C22×C8—21no—
303C22×C8C2×D411yescharacteristic
304C22×C8C2×D411yescharacteristic
305C22×C8C2×D411yescharacteristic
306C22×C8C2×D411yescharacteristic
307C4⋊Q8—24no—
308C4⋊Q8—24no—
309C4⋊Q8—24no—
310C4⋊Q8—24no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—42no—
321C4⋊Q8—42no—
322C4⋊Q8—42no—
323C4⋊Q8—42no—
324C4⋊Q8—44no—
325C4⋊Q8—44no—
326C4⋊Q8C2412yes—
327C4⋊Q8C2412yes—
328C4⋊Q8C2412yes—
329#127—24no—
330#127—21no—
331#127—21no—
332#127—21no—
333#127—42no—
334#127C2412yes—
335#125—24no—
336#125—42no—
337#125—44no—
338#125—44no—
339C22⋊Q8—24no—
340C22⋊Q8—44no—
341C22⋊Q8—44no—
342C22⋊Q8—42no—
343#121—24no—
344#121—44no—
345#121—44no—
346#121—42no—
347C4×Q8—21no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—24no—
354C4×Q8—24no—
355C4×Q8—24no—
356C4×Q8—24no—
357C4×Q8—42no—
358C4×Q8—42no—
359C4×Q8—42no—
360C4×Q8—42no—
361C4×Q8C2×D412yes—
362C4×Q8C2×D412yes—
363C4×Q8C2×D412yes—
364C4×Q8C2412yes—
365C4×Q8C2412yes—
366C4×Q8C2412yes—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—42no—
372C4×D4—42no—
373C4×D4C2×D412yes—
374C4×D4C2412yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—41no—
385C2×C4⋊C4—41no—
386C2×C4⋊C4—41no—
387C2×C4⋊C4—41no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—41no—
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2×D411yescharacteristic
392C2×C4⋊C4C2×D411yescharacteristic
393C2×C4⋊C4C2×D411yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—41no—
396C2×C22⋊C4—41no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—41no—
401C2×C42—41no—
402C2×C42C2411yescharacteristic
403#107—24no—
404#107C2×D414yes—
405#106—24no—
406#106—24no—
407#106—24no—
408#106—24no—
409#106—24no—
410#106—24no—
411#106—24no—
412#106—44no—
413#106C2×D414yes—
414C4⋊C8—24no—
415C4⋊C8—24no—
416C4⋊C8—24no—
417C4⋊C8—21no—
418C4⋊C8—21no—
419C4⋊C8—21no—
420C4⋊C8—21no—
421C4⋊C8C2412yes—
422C4⋊C8C2412yes—
423C4⋊C8C2412yes—
424C4⋊C8C2412yes—
425#103—24no—
426#103—24no—
427#103—24no—
428#103—24no—
429#103—24no—
430#103C2×D414yes—
431#103C2×D414yes—
432#103C2×D414yes—
433#102—24no—
434#102—24no—
435#102—24no—
436#102C2×D414yes—
437C22⋊C8—24no—
438#97—24no—
439C4×C8—24no—
440C4×C8—24no—
441C4×C8—21no—
442C4×C8C2×D412yes—
443C4×C8C2×D412yes—
444C4×C8C2412yes—
445C4×C8C2412yes—
446C4×C8C2412yes—
447C24—21no—
448C24—41no—
449#41—48no—
450#41—84no—
451#41—88no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—41no—
465C2×Q8—41no—
466C2×Q8—41no—
467C2×Q8—48no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—84no—
473C2×Q8—82no—
474C2×Q8—84no—
475C2×Q8—88no—
476C2×Q8—162no—
477C2×Q8C2×SD1614yes—
478C2×Q8C2×SD1614yes—
479C2×Q8C22×D412yes—
480C2×Q8C22×D412yes—
481C2×Q8C22×D412yes—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—41no—
491C2×D4—41no—
492C2×D4—41no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—84no—
498C2×D4—82no—
499C2×D4—162no—
500C2×D4C22×D412yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—41no—
515C22×C4—41no—
516C22×C4—41no—
517C22×C4—84no—
518C22×C4—84no—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4H(5,𝔽2)11yescharacteristic
522Q16—48no—
523Q16C2×SD16116yescomplemented
524SD16—28no—
525SD16—48no—
526SD16—48no—
527SD16—48no—
528SD16—84no—
529SD16—84no—
530SD16—88no—
531SD16—84no—
532SD16—84no—
533M16—48no—
534C2×C8—24no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—42no—
543C2×C8—42no—
544C2×C8—42no—
545C2×C8—42no—
546C2×C8C2×SD1614yes—
547C2×C8#13714yescomplemented
548C2×C8C22×D412yes—
549C2×C8C22×D412yes—
550C2×C8C22×D412yes—
551C2×C8C22×D412yes—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—44no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—41no—
575C4⋊C4—41no—
576C4⋊C4—41no—
577C4⋊C4—41no—
578C4⋊C4—41no—
579C4⋊C4—41no—
580C4⋊C4C22×D412yes—
581C4⋊C4C22×D412yes—
582C4⋊C4C22×D412yes—
583C4⋊C4C22×D412yes—
584C22⋊C4—22no—
585C22⋊C4—22no—
586C22⋊C4—44no—
587C22⋊C4—44no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—44no—
595C42—44no—
596C42C22×D411yescharacteristic
597C42C22×D411yescharacteristic
598C42H(5,𝔽2)11yescharacteristic
599C42C2511yescharacteristic
600C23—22no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23D4211yescharacteristic
609Q8—24no—
610Q8—24no—
611Q8—22no—
612Q8—22no—
613Q8—22no—
614Q8—22no—
615Q8—42no—
616Q8—42no—
617Q8—42no—
618Q8—42no—
619Q8—42no—
620Q8—88no—
621Q8—82no—
622Q8—82no—
623Q8—82no—
624Q8—161no—
625Q8—164no—
626Q8C22×SD1614yes—
627Q8C22×SD1614yes—
628D4—22no—
629D4—22no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—42no—
635D4—88no—
636D4—82no—
637D4—82no—
638D4—82no—
639D4—161no—
640D4—164no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4—82no—
665C2×C4—82no—
666C2×C4—82no—
667C2×C4D4211yescharacteristic
668C2×C4D4211yescharacteristic
669C2×C4D4211yescharacteristic
670C2×C4C22×SD1612yes—
671C2×C4C2×#13712yes—
672C2×C4#57712yes—
673C2×C4#57712yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C23×D411yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C8—24no—
678C8—22no—
679C8—22no—
680C8—42no—
681C8—42no—
682C8—42no—
683C8—42no—
684C8C22×SD1614yes—
685C8C2×#13714yescomplemented
686C22—42no—
687C22—42no—
688C22—42no—
689C22—84no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—164no—
694C22#327812yes—
695C22D4×SD1612yes—
696C22#329112yes—
697C22C2×D4211yescharacteristic
698C4—24no—
699C4—24no—
700C4—21no—
701C4—21no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4#327812yes—
711C4D4×SD1612yes—
712C4C23×SD1611yescharacteristic
713C4C22×#13711yescharacteristic
714C4C2×#57711yescharacteristic
715C4C2×#57711yescharacteristic
716C2—82no—
717C2—162no—
718C2#1989714yesdirect product factor
719C2C2×#327811yescharacteristic
720C2C2×D4×SD1611yescharacteristic
721C2C2×#329111yescharacteristic
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×Q8×SD16C2semidirect
5. #19897C2direct (N × H)
6. C2×#1705C2semidirect
7. C2×#1686C2semidirect
8. C2×#1691C2semidirect
9. C2×#1672C2semidirect
10. C2×#1666C2semidirect
11. C2×#1623C2semidirect
13. C2×#1653C2semidirect
14. C2×#1651C2semidirect
15. C2×#1590C2semidirect
16. C2×C8×Q16C2semidirect
25. #3387C22semidirect
27. Q8×SD16C22semidirect
65. #1705C22semidirect
66. #1691C22semidirect
67. #1686C22semidirect
68. #1672C22semidirect
69. #1666C22semidirect
70. #1653C22semidirect
71. #1651C22semidirect
73. #1623C22semidirect
74. #1590C22semidirect
75. C8×Q16C22semidirect
276. C2×Q16SD16semidirect
523. Q16C2×SD16semidirect
547. C2×C8#137semidirect
685. C8C2×#137semidirect
718. C2#19897direct (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×Q8×SD162
C2×#32782
#198972
C2×#17052
C2×#16862
C2×#16912
C2×#16722
C2×#16662
C2×#16232
C2×#16312
C2×#16532
C2×#16512
C2×#15902
C2×C8×Q162