Finite Group Databasebeta

← C2×#28651 #6464055

Subgroups — C2×#28651 #6464055

Order 512.

2142 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C22×#446C211yesmaximal, characteristic, complemented
3C2×#2567C211yesmaximal, characteristic, complemented
4C2×#2592C211yesmaximal, characteristic
5#28651C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1983C211yesmaximal, characteristic
12C2×#1936C211yesmaximal, characteristic
13C2×#1935C211yesmaximal, characteristic, complemented
14C2×#1934C211yesmaximal, characteristic
15C2×#1933C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#489—28no—
21C2×#489C2218yes—
22C2×#446—28no—
23C2×#446C2218yescomplemented
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×#97C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2592—24no—
41#2592C2218yes—
42#2567—24no—
43#2567C2218yescomplemented
44C2×#397—21no—
45C2×#395—21no—
46C2×#395—21no—
47C2×#393—21no—
48C2×#391—21no—
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#388—21no—
58C2×#388—21no—
59C2×#384—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62#2068C2218yescomplemented
63#2067C2218yes—
64#2066C2218yescomplemented
65#2065C2218yes—
66#1988C2218yescomplemented
67#1983C2218yes—
68#1936C2218yes—
69#1935C2218yescomplemented
70#1934C2218yes—
71#1933C2218yescomplemented
72#1776C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8D411yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8C2314yes—
91C2×#124—22no—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124C2314yes—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8D414yes—
98C2×C4×Q8C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#489—232no—
125#446—232no—
126C2×#107—24no—
127C2×#107C2314yes—
128C2×#106—24no—
129C2×#106C2314yes—
130C2×C4⋊C8—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103—24no—
137C2×#103—24no—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#97—24no—
153C2×#97D418yes—
154C2×#97C2314yes—
155C2×#97C2314yes—
156C2×C4×C8—24no—
157C2×C4×C8C2314yes—
158#397—28no—
159#395—28no—
160#395—28no—
161#393—28no—
162#391—28no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#391C2314yes—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#388—28no—
180#388—28no—
181#384—28no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202C25—21no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—42no—
214C22×Q8—44no—
215C22×Q8—82no—
216C22×Q8—82no—
217C22×Q8C2×D414yes—
218C22×Q8C2412yes—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8C2×D414yes—
252C22×C8C2×D414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—48no—
260#124—28no—
261#124—28no—
262#124—48no—
263C4×Q8—216no—
264C4×Q8—28no—
265C4×Q8—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42—82no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D414yes—
330C2×C42#4114yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334#107—28no—
335#106—28no—
336C4⋊C8—28no—
337C4⋊C8—28no—
338C4⋊C8—28no—
339#103—28no—
340#103—28no—
341#103—28no—
342#103—28no—
343#102—28no—
344#102—28no—
345#102—28no—
346#102—28no—
347#97—28no—
348#97C2×D4116yes—
349C4×C8—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—44no—
378C2×Q8—416no—
379C2×Q8—42no—
380C2×Q8—48no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×Q8—816no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—44no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4#13714yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463C2×C8—28no—
464C2×C8—28no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8C22×D414yes—
478C2×C8C22×D414yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4C2×#4114yes—
504C4⋊C4C2×#4114yes—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—84no—
519C42#12118yes—
520C42C22×D414yes—
521C42C2×#4114yes—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23C2×#12112yes—
537C23#54812yes—
538C23#54812yes—
539C23C2×#13712yes—
540C23#57512yes—
541C23#57612yes—
542C23#57712yes—
543C23C23×D411yescharacteristic
544Q8—416no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—816no—
550Q8—84no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4C2×#12114yes—
606C2×C4C2×#12114yes—
607C2×C4C2×#12112yes—
608C2×C4#53214yes—
609C2×C4#53614yes—
610C2×C4#54812yes—
611C2×C4#54812yes—
612C2×C4#55614yes—
613C2×C4#55614yes—
614C2×C4C2×#13714yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C8—24no—
622C8—24no—
623C8—42no—
624C8—42no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—168no—
633C22#254514yes—
634C22#305614yes—
635C22#306214yes—
636C22#336914yes—
637C22#337014yes—
638C22#337514yes—
639C22#337614yes—
640C22C22×#12111yescharacteristic
641C22C2×#54811yescharacteristic
642C22C2×#54811yescharacteristic
643C22C22×#13711yescharacteristic
644C22C2×#57511yescharacteristic
645C22C2×#57611yescharacteristic
646C22C2×#57711yescharacteristic
647C4—24no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#254514yes—
661C4C2×#53211yescharacteristic
662C4C2×#53611yescharacteristic
663C4C2×#55611yescharacteristic
664C4C2×#55611yescharacteristic
665C2—162no—
666C2#2865118yesdirect product factor
667C2C2×#254511yescharacteristic
668C2C2×#305611yescharacteristic
669C2C2×#306211yescharacteristic
670C2C2×#336911yescharacteristic
671C2C2×#337511yescharacteristic
672C2C2×#337011yescharacteristic
673C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#446C2semidirect
3. C2×#2567C2semidirect
5. #28651C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2066C2semidirect
10. C2×#1988C2semidirect
13. C2×#1935C2semidirect
15. C2×#1933C2semidirect
16. C2×#1776C2semidirect
23. C2×#446C22semidirect
43. #2567C22semidirect
62. #2068C22semidirect
64. #2066C22semidirect
66. #1988C22semidirect
69. #1935C22semidirect
71. #1933C22semidirect
72. #1776C22semidirect
666. C2#28651direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4462
C2×#25672
C2×#25922
#286512
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#19882
C2×#19832
C2×#19362
C2×#19352
C2×#19342
C2×#19332
C2×#17762