Finite Group Databasebeta

← C2×#32985 #6629129

Subgroups — C2×#32985 #6629129

Order 512.

4990 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C22×#485C212yesmaximal, complemented
3C2×#2848C211yesmaximal, characteristic
4#32985C2116yesmaximal, direct product factor
5C2×#1873C212yesmaximal, complemented
6C2×#1872C211yesmaximal, characteristic, complemented
7C2×#1872C211yesmaximal, characteristic, complemented
8C2×#1854C212yesmaximal, complemented
9C2×#1853C211yesmaximal, characteristic, complemented
10C2×#1852C211yesmaximal, characteristic, complemented
11C2×#1825C212yesmaximal, complemented
12C2×#1779C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117—24no—
18C22×#117—21no—
19C22×#117—21no—
20C22×#117—216no—
21C22×#117C2212yes—
22C22×#117C2218yes—
23C22×#117C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4—21no—
27C23×C42—21no—
28C2×#485—216no—
29C2×#485C22116yescomplemented
30C22×#107C2212yescomplemented
31C22×#106C2212yescomplemented
32C22×#103C2212yescomplemented
33C22×#103C2212yescomplemented
34C22×C22⋊C8C2212yes—
35C22×C22⋊C8C2211yescharacteristic
36C22×C22⋊C8C2211yescharacteristic
37C22×C22⋊C8C2212yes—
38#2848—24no—
39#2848C2218yes—
40C2×#385—21no—
41C2×#385—21no—
42C2×#385—21no—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#379—21no—
47#1873C22116yescomplemented
48#1872C2218yescomplemented
49#1872C2218yescomplemented
50#1854C22116yescomplemented
51#1853C2218yescomplemented
52#1852C2218yescomplemented
53#1825C22116yescomplemented
54#1779C2218yes—
55C2×#340C2212yescomplemented
56C2×#340C2212yescomplemented
57C2×#340C2212yescomplemented
58C2×#340C2212yescomplemented
59C26D411yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—28no—
62C24×C4—28no—
63C24×C4—24no—
64C24×C4—24no—
65C24×C4—21no—
66C24×C4—21no—
67C24×C4D414yes—
68C24×C4D411yescharacteristic
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4C2314yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C23×C8C2312yes—
76C23×C8C2312yes—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8C2314yes—
85C2×C22⋊Q8C2314yes—
86C2×C22⋊Q8C2314yes—
87C2×#117—264no—
88C2×#117—216no—
89C2×#117—216no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—264no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117D4116yes—
99C2×#117C2318yes—
100C2×#117C2318yes—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—216no—
118C22×C22⋊C4—28no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—216no—
128C22×C42—28no—
129C22×C42—22no—
130C22×C42—22no—
131#485—264no—
132C2×#107—28no—
133C2×#107C2318yescomplemented
134C2×#106—28no—
135C2×#106C2318yescomplemented
136C2×#103—28no—
137C2×#103—28no—
138C2×#103C2318yescomplemented
139C2×#103C2318yescomplemented
140C2×C22⋊C8—28no—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8—28no—
144C2×C22⋊C8C2318yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2318yes—
148#385—28no—
149#385—28no—
150#385—28no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380C2314yes—
155#380C2314yes—
156#380C2314yes—
157#379—28no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161#340C2318yescomplemented
162#340C2318yescomplemented
163#340C2318yescomplemented
164#340C2318yescomplemented
165C25—24no—
166C25—24no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—42no—
171C25—42no—
172C25C2×D411yescharacteristic
173C25C2×D411yescharacteristic
174C25C2×D414yes—
175C25C2×D411yescharacteristic
176C22×Q8—21no—
177C22×Q8—24no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8C2412yes—
182C23×C4—264no—
183C23×C4—264no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—216no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—216no—
195C23×C4—216no—
196C23×C4—24no—
197C23×C4—216no—
198C23×C4—24no—
199C23×C4—216no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D418yes—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4#4118yes—
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C22⋊Q8—216no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250C22⋊Q8—216no—
251C22⋊Q8—48no—
252C22⋊Q8—48no—
253C22⋊Q8—48no—
254#117—232no—
255#117—264no—
256#117—264no—
257#117—232no—
258#117—216no—
259#117—216no—
260#117—216no—
261#117C2×D4132yes—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2414yes—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—264no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C42—216no—
312C2×C42—264no—
313C2×C42—216no—
314C2×C42—28no—
315C2×C42—28no—
316#107—216no—
317#106—216no—
318#103—216no—
319#103—216no—
320C22⋊C8—216no—
321C22⋊C8—28no—
322C22⋊C8—28no—
323C22⋊C8—216no—
324#95—24no—
325#95—24no—
326#95—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—232no—
332C24—28no—
333C24—22no—
334C24—28no—
335C24—28no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—28no—
348C24—28no—
349C24—44no—
350C24—44no—
351C24—416no—
352C24—416no—
353C24—416no—
354C24—416no—
355C24—416no—
356C24—416no—
357C24#12014yes—
358C24#12314yes—
359C24#12314yes—
360C24#13514yes—
361C24#13714yes—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—48no—
374C2×Q8—48no—
375C22×C4—232no—
376C22×C4—232no—
377C22×C4—232no—
378C22×C4—2128no—
379C22×C4—264no—
380C22×C4—264no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—432no—
427C22×C4—432no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
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—42no—
447C22×C4—42no—
448C22×C4#12014yes—
449C22×C4#12018yes—
450C22×C4#12314yes—
451C22×C4#12314yes—
452C22×C4#12318yes—
453C22×C4#12318yes—
454C22×C4C22×D412yes—
455C22×C4C22×D418yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D414yes—
460C22×C4C2×#4118yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4C2511yescharacteristic
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—48no—
474C2×C8—48no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4C22×D418yes—
492C4⋊C4C22×D418yes—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C42—232no—
502C42—232no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—28no—
511C23—22no—
512C23—22no—
513C23—232no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—416no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—416no—
537C23—4128no—
538C23—44no—
539C23—44no—
540C23—416no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23#485116yescomplemented
548C23C2×#12014yes—
549C23C2×#12012yes—
550C23C2×#12314yes—
551C23C2×#12314yes—
552C23C2×#12312yes—
553C23C2×#12312yes—
554C23C2×#13514yes—
555C23C2×#13512yes—
556C23C2×#13714yes—
557C23C2×#13712yes—
558C23#57512yes—
559C23#57512yes—
560C23C23×D411yescharacteristic
561Q8—44no—
562Q8—44no—
563Q8—84no—
564C2×C4—232no—
565C2×C4—232no—
566C2×C4—232no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—216no—
577C2×C4—216no—
578C2×C4—216no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—432no—
588C2×C4—432no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4C2×#12014yes—
612C2×C4C2×#12012yes—
613C2×C4C2×#12314yes—
614C2×C4C2×#12314yes—
615C2×C4C2×#12312yes—
616C2×C4C2×#12312yes—
617C2×C4#52518yes—
618C2×C4#52518yes—
619C2×C4C23×D412yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4112yes—
623C8—44no—
624C8—44no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—232no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—464no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—416no—
648C22—416no—
649C22#284914yes—
650C22#300214yes—
651C22#300714yes—
652C22C2×#48514yes—
653C22C2×#48518yescomplemented
654C22#309114yes—
655C22#309314yes—
656C22#309514yes—
657C22C22×#12011yescharacteristic
658C22C22×#12311yescharacteristic
659C22C22×#12311yescharacteristic
660C22C22×#13511yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57511yescharacteristic
663C22C2×#57511yescharacteristic
664C4—24no—
665C4—24no—
666C4—24no—
667C4—48no—
668C4—48no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#284914yes—
674C4C2×#52512yes—
675C4C2×#52512yes—
676C2—28no—
677C2—44no—
678C2—44no—
679C2#3298518yesdirect product factor
680C2C2×#284911yescharacteristic
681C2C2×#300211yescharacteristic
682C2C2×#300711yescharacteristic
683C2C22×#48511yescharacteristic
684C2C2×#309111yescharacteristic
685C2C2×#309311yescharacteristic
686C2C2×#309511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#485C2semidirect
4. #32985C2direct (N × H)
5. C2×#1873C2semidirect
6. C2×#1872C2semidirect
7. C2×#1872C2semidirect
8. C2×#1854C2semidirect
9. C2×#1853C2semidirect
10. C2×#1852C2semidirect
11. C2×#1825C2semidirect
29. C2×#485C22semidirect
30. C22×#107C22semidirect
31. C22×#106C22semidirect
32. C22×#103C22semidirect
33. C22×#103C22semidirect
47. #1873C22semidirect
48. #1872C22semidirect
49. #1872C22semidirect
50. #1854C22semidirect
51. #1853C22semidirect
52. #1852C22semidirect
53. #1825C22semidirect
55. C2×#340C22semidirect
56. C2×#340C22semidirect
57. C2×#340C22semidirect
58. C2×#340C22semidirect
133. C2×#107C23semidirect
135. C2×#106C23semidirect
138. C2×#103C23semidirect
139. C2×#103C23semidirect
161. #340C23semidirect
162. #340C23semidirect
163. #340C23semidirect
164. #340C23semidirect
547. C23#485semidirect
653. C22C2×#485semidirect
679. C2#32985direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1172
C22×#4852
C2×#28482
#329852
C2×#18732
C2×#18722
C2×#18542
C2×#18532
C2×#18522
C2×#18252
C2×#17792