Finite Group Databasebeta

← C2×#9111 #196356

Subgroups — C2×#9111 #196356

Order 512.

3182 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C2×#2848C211yesmaximal, characteristic
3C22×#328C212yesmaximal, complemented
4C2×C22⋊#95C211yesmaximal, characteristic
5#9111C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C24×C8—21no—
9C22×C22⋊Q8C411yescharacteristic, complemented
10C22×C22⋊Q8C411yescharacteristic, complemented
11C22×C22⋊Q8C2211yescharacteristic
12C23×C22⋊C4—21no—
13C22×C22⋊C8—28no—
14C22×C22⋊C8—22no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8C2214yes—
17C22×C22⋊C8C2212yes—
18#2848—22no—
19#2848—22no—
20#2848C414yescomplemented
21#2848C2214yes—
22C2×#385—21no—
23C2×#385—21no—
24C2×#385—21no—
25C2×#380C411yescharacteristic, complemented
26C2×#380C411yescharacteristic, complemented
27C2×#380C2211yescharacteristic
28C2×#379—21no—
29C22×#95—21no—
30C22×#95C2212yes—
31C2×#328—28no—
32C2×#328C2218yescomplemented
33C22⋊#95—22no—
34C22⋊#95C2214yes—
35#1294C2218yescomplemented
36C26D411yescharacteristic
37C23×Q8C2×C411yescharacteristic
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—22no—
41C24×C4—44no—
42C24×C4—44no—
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D412yes—
49C24×C4D412yes—
50C24×C4D411yescharacteristic
51C24×C4C2312yes—
52C24×C4C2311yescharacteristic
53C23×C8—28no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—44no—
57C23×C8—44no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—44no—
65C2×C22⋊Q8—44no—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C414yescomplemented
68C2×C22⋊Q8C2×C414yescomplemented
69C2×C22⋊Q8C2312yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—28no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—44no—
85C22×C22⋊C4—44no—
86C2×C22⋊C8—28no—
87C2×C22⋊C8—216no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—216no—
92C2×C22⋊C8D418yes—
93C2×C22⋊C8C2314yes—
94#385—24no—
95#385—24no—
96#385—24no—
97#385—24no—
98#385—24no—
99#385—24no—
100#380—22no—
101#380—22no—
102#380—21no—
103#380—21no—
104#380C2×C412yes—
105#380C2×C414yescomplemented
106#380C2×C414yescomplemented
107#380C2312yes—
108#379—24no—
109#379—24no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95—22no—
116C2×#95—24no—
117C2×#95—44no—
118C2×#95C2314yes—
119#328—216no—
120C25—24no—
121C25—24no—
122C25—24no—
123C25—24no—
124C25—42no—
125C25—42no—
126C25C22⋊C411yescharacteristic
127C25C22⋊C412yes—
128C25C22⋊C411yescharacteristic
129C25SD1614yes—
130C25Q1614yes—
131C25C2×D412yes—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—24no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C22×C412yes—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—28no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—432no—
178C23×C4—432no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4C22⋊C412yes—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C412yes—
188C23×C4C22⋊C412yes—
189C23×C4C22⋊C412yes—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C412yes—
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D414yes—
203C23×C4#4114yes—
204C23×C4C2411yescharacteristic
205C22×C8—216no—
206C22×C8—28no—
207C22×C8—24no—
208C22×C8—432no—
209C22×C8—48no—
210C22×C8—44no—
211C22×C8—44no—
212C22×C8—48no—
213C22⋊Q8—28no—
214C22⋊Q8—24no—
215C22⋊Q8—24no—
216C22⋊Q8—48no—
217C22⋊Q8—48no—
218C22⋊Q8—42no—
219C22⋊Q8—44no—
220C22⋊Q8—42no—
221C22⋊Q8—48no—
222C22⋊Q8—48no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—432no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—44no—
258C22⋊C8—216no—
259C22⋊C8—28no—
260C22⋊C8—24no—
261C22⋊C8C2×D418yes—
262#95—22no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—24no—
267#95—24no—
268#95—48no—
269#95—44no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—24no—
280C24—216no—
281C24—216no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—216no—
295C24—216no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—48no—
300C24—48no—
301C24—42no—
302C24—416no—
303C24—416no—
304C24—416no—
305C24—416no—
306C24#9914yes—
307C24#10314yes—
308C24#10318yes—
309C24#10314yes—
310C24#10414yes—
311C24C2×C22⋊C412yes—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C412yes—
314C24C2×C22⋊C411yescharacteristic
315C24#12114yes—
316C24C2×SD1614yes—
317C24C2×SD1612yes—
318C24C2×Q1614yes—
319C24C2×Q1612yes—
320C24#13514yes—
321C24C22×D411yescharacteristic
322C2×Q8—24no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—88no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—416no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—432no—
393C22×C4—432no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—44no—
403C22×C4—416no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4#9914yes—
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C412yes—
428C22×C4C4×D414yes—
429C22×C4C4×D414yes—
430C22×C4#12012yes—
431C22×C4#12012yes—
432C22×C4#12114yes—
433C22×C4#12312yes—
434C22×C4#12312yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C2×#4112yes—
439C2×C8—28no—
440C2×C8—432no—
441C2×C8—44no—
442C2×C8—44no—
443C2×C8—416no—
444C4⋊C4—24no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—432no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—28no—
471C23—28no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—22no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—216no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—216no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—22no—
504C23—28no—
505C23—48no—
506C23—44no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—464no—
513C23—464no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23#328116yescomplemented
523C23#38612yes—
524C23#38612yes—
525C23C2×#9914yes—
526C23C2×#9912yes—
527C23C2×#9912yes—
528C23C2×#10314yes—
529C23C2×#10312yes—
530C23C2×#10314yes—
531C23C2×#10312yes—
532C23#41612yes—
533C23#41612yes—
534C23C2×#10414yes—
535C23C2×#10412yes—
536C23#42112yes—
537C23#46518yes—
538C23#46718yes—
539C23C22×C22⋊C411yescharacteristic
540C23C2×#12112yes—
541C23C22×SD1611yescharacteristic
542C23C22×Q1611yescharacteristic
543C23C2×#13512yes—
544Q8—84no—
545Q8—84no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—216no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
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—832no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4#38612yes—
607C2×C4#38612yes—
608C2×C4C2×#9914yes—
609C2×C4C2×C4×D412yes—
610C2×C4C2×#12011yescharacteristic
611C2×C4C2×#12311yescharacteristic
612C8—48no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—216no—
620C22—232no—
621C22—216no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—24no—
626C22—22no—
627C22—24no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—416no—
640C22—432no—
641C22—416no—
642C22—42no—
643C22—416no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C2×#32814yes—
648C22C2×#32818yescomplemented
649C22#150114yes—
650C22#151614yes—
651C22#152014yes—
652C22#187114yes—
653C22#189214yes—
654C22#189214yes—
655C22#189814yes—
656C22C2×#38611yescharacteristic
657C22C22×#9911yescharacteristic
658C22C22×#9911yescharacteristic
659C22C22×#10311yescharacteristic
660C22C2×#41611yescharacteristic
661C22C22×#10411yescharacteristic
662C22C2×#42111yescharacteristic
663C22C2×#46512yes—
664C22C2×#46712yes—
665C4—21no—
666C4—21no—
667C4—22no—
668C4—24no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—88no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C2—22no—
678C2—24no—
679C2—28no—
680C2—22no—
681C2—44no—
682C2#911118yesdirect product factor
683C2C22×#32811yescharacteristic
684C2C2×#150111yescharacteristic
685C2C2×#152011yescharacteristic
686C2C2×#151611yescharacteristic
687C2C2×#187111yescharacteristic
688C2C2×#189211yescharacteristic
689C2C2×#189811yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#328C2semidirect
5. #9111C2direct (N × H)
6. C2×#1294C2semidirect
9. C22×C22⋊Q8C4semidirect
10. C22×C22⋊Q8C4semidirect
20. #2848C4semidirect
25. C2×#380C4semidirect
26. C2×#380C4semidirect
32. C2×#328C22semidirect
35. #1294C22semidirect
67. C2×C22⋊Q8C2×C4semidirect
68. C2×C22⋊Q8C2×C4semidirect
105. #380C2×C4semidirect
106. #380C2×C4semidirect
522. C23#328semidirect
648. C22C2×#328semidirect
682. C2#9111direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C2×#28482
C22×#3282
C2×C22⋊#952
#91112
C2×#12942