Finite Group Databasebeta

← C2×#23617 #6432138

Subgroups — C2×#23617 #6432138

Order 512.

3734 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C22×#423C211yesmaximal, characteristic, complemented
3C22×#417C211yesmaximal, characteristic
4C22×#417C211yesmaximal, characteristic
5C2×#2297C211yesmaximal, characteristic
6C22×#389C211yesmaximal, characteristic, complemented
7#23617C2116yesmaximal, direct product factor
8C2×#2068C212yesmaximal, complemented
9C2×#2066C212yesmaximal, complemented
10C2×#2067C212yesmaximal, complemented
11C2×#2065C212yesmaximal, complemented
12C2×#1736C211yesmaximal, characteristic, complemented
13C23×#41C2211yescharacteristic
14C23×M16C2211yescharacteristic
15C23×M16C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4—21no—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21C2×#428—28no—
22C2×#428C2218yescomplemented
23C22×#107C2212yes—
24C22×#106C2212yes—
25C2×#423—28no—
26C2×#423C2218yescomplemented
27C22×C4⋊C8C2212yes—
28C22×C4⋊C8C2212yes—
29C2×#417—28no—
30C2×#417—28no—
31C2×#417C2218yes—
32C2×#417C2218yes—
33C22×#103C2212yes—
34C22×#103C2212yes—
35C22×#102C2212yes—
36C22×#102C2212yes—
37#2297—24no—
38#2297C2218yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389—21no—
44C2×#389—21no—
45C2×#389—24no—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#389C2218yescomplemented
49C2×#386—21no—
50C2×#386—21no—
51C2×C4×C22⋊C4—21no—
52C2×C4×C22⋊C4—21no—
53C22×#95—21no—
54#2068C22116yescomplemented
55#2067C22116yescomplemented
56#2066C22116yescomplemented
57#2065C22116yescomplemented
58#1736C2218yescomplemented
59C2×#340C2212yes—
60C2×#340C2212yes—
61C22×#41—22no—
62C22×#41—24no—
63C22×#41—24no—
64C22×#41—24no—
65C22×#41—24no—
66C22×#41—24no—
67C22×#41Q818yes—
68C22×#41C2314yes—
69C23×Q8C2311yescharacteristic
70C23×D4—21no—
71C23×D4C2311yescharacteristic
72C24×C4—21no—
73C24×C4—21no—
74C24×C4—21no—
75C24×C4D411yescharacteristic
76C24×C4D411yescharacteristic
77C24×C4C2311yescharacteristic
78C22×M16—24no—
79C22×M16—24no—
80C22×M16—24no—
81C22×M16—24no—
82C22×M16—24no—
83C22×M16D418yes—
84C22×M16C2314yes—
85C22×M16C2314yes—
86C23×C8C2312yes—
87C23×C8C2312yes—
88C2×#117—22no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117C2314yes—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—48no—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—24no—
116C22×C42—24no—
117C22×C42—21no—
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D414yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124#428—232no—
125C2×#107—28no—
126C2×#107C2318yes—
127C2×#106—28no—
128C2×#106C2318yes—
129#423—232no—
130C2×C4⋊C8—28no—
131C2×C4⋊C8—28no—
132C2×C4⋊C8C2318yes—
133C2×C4⋊C8C2318yes—
134#417—232no—
135#417—232no—
136C2×#103—28no—
137C2×#103—28no—
138C2×#103C2318yes—
139C2×#103C2318yes—
140C2×#102—28no—
141C2×#102—28no—
142C2×#102C2318yes—
143C2×#102C2318yes—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#391C2314yes—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#389—24no—
153#389—24no—
154#389—28no—
155#389—28no—
156#389C2318yes—
157#389C2318yes—
158#386—28no—
159#386—28no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C2×#95—22no—
163C2×#95—22no—
164C2×#95—28no—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169#340C2318yes—
170#340C2318yes—
171C25—21no—
172C25C2×D411yescharacteristic
173C2×#41—216no—
174C2×#41—28no—
175C2×#41—232no—
176C2×#41—232no—
177C2×#41—28no—
178C2×#41—28no—
179C2×#41—28no—
180C2×#41—28no—
181C2×#41—48no—
182C2×#41—48no—
183C2×#41—416no—
184C2×#41—416no—
185C2×#41C2×Q8116yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8C2×Q814yes—
193C22×Q8C2412yes—
194C22×D4—28no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4C2×Q814yes—
205C22×D4C2412yes—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×Q814yes—
240C23×C4#4114yes—
241C23×C4#4114yes—
242C23×C4#4114yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C2×M16—232no—
246C2×M16—28no—
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—28no—
250C2×M16—28no—
251C2×M16—416no—
252C2×M16C2×D4116yes—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8C2×D418yes—
259C22×C8C2414yes—
260C22×C8C2414yes—
261#117—28no—
262#117—28no—
263#117—48no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—432no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D4116yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—216no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—216no—
323#106—216no—
324C4⋊C8—216no—
325C4⋊C8—216no—
326#103—216no—
327#103—216no—
328#102—216no—
329#102—216no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24#12114yes—
349C24#12114yes—
350C24C22⋊Q814yes—
351C24C22×D412yes—
352C24C22×D411yescharacteristic
353#41—232no—
354#41—232no—
355#41—416no—
356#41—416no—
357#41—464no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—44no—
370C2×Q8C22×Q814yes—
371C2×D4—216no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—432no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4C22×Q814yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—48no—
416C22×C4—432no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—432no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—816no—
438C22×C4#121116yes—
439C22×C4#12114yes—
440C22×C4#12118yes—
441C22×C4#12114yes—
442C22×C4#12118yes—
443C22×C4C22⋊Q8116yes—
444C22×C4C22⋊Q8116yes—
445C22×C4C22⋊Q814yes—
446C22×C4C22⋊Q818yes—
447C22×C4#126116yes—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q814yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4114yes—
468C22×C4C2511yescharacteristic
469M16—232no—
470M16—416no—
471C2×C8—216no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—48no—
478C2×C8C22×D418yes—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—416no—
494C4⋊C4C22×D418yes—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C42—28no—
501C42—24no—
502C42—24no—
503C42—28no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—48no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23#396116yes—
534C23C2×#12114yes—
535C23C2×#12114yes—
536C23C2×#12112yes—
537C23C2×#12112yes—
538C23C2×C22⋊Q814yes—
539C23C2×C22⋊Q812yes—
540C23C2×AGL(1,ℤ/8)12yes—
541C23C2×#13712yes—
542C23#57512yes—
543C23#57512yes—
544C23C23×D411yescharacteristic
545Q8—24no—
546Q8—24no—
547Q8—48no—
548Q8—42no—
549Q8—42no—
550D4—24no—
551D4—24no—
552D4—432no—
553D4—48no—
554D4—42no—
555D4—42no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—432no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—432no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4#396116yes—
596C2×C4#396132yes—
597C2×C4C2×#12114yes—
598C2×C4C2×#12112yes—
599C2×C4C2×#12114yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12114yes—
602C2×C4C2×#12118yes—
603C2×C4C2×C22⋊Q814yes—
604C2×C4C2×C22⋊Q812yes—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q814yes—
607C2×C4C2×C22⋊Q818yes—
608C2×C4C2×C22⋊Q818yes—
609C2×C4C2×#12614yes—
610C2×C4C2×#12618yes—
611C2×C4C2×AGL(1,ℤ/8)14yes—
612C2×C4C2×#13714yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C23×D411yescharacteristic
615C2×C4C23×Q811yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C8—28no—
620C8—44no—
621C22—24no—
622C22—24no—
623C22—232no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—816no—
634C22—82no—
635C22—82no—
636C22C2×#39618yes—
637C22C2×#39614yes—
638C22#303614yes—
639C22#303714yes—
640C22#305514yes—
641C22#305614yes—
642C22#307914yes—
643C22#308014yes—
644C22C22×#12111yescharacteristic
645C22C22×#12111yescharacteristic
646C22C22×C22⋊Q811yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57511yescharacteristic
651C4—24no—
652C4—24no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—48no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4C2×#39618yes—
661C4C2×#39614yes—
662C4C22×#12111yescharacteristic
663C4C22×C22⋊Q811yescharacteristic
664C4C22×C22⋊Q811yescharacteristic
665C4C22×#12611yescharacteristic
666C2—28no—
667C2—84no—
668C2#2361718yesdirect product factor
669C2C22×#39611yescharacteristic
670C2C2×#303611yescharacteristic
671C2C2×#303711yescharacteristic
672C2C2×#305511yescharacteristic
673C2C2×#305611yescharacteristic
674C2C2×#307911yescharacteristic
675C2C2×#308011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#428C2semidirect
2. C22×#423C2semidirect
6. C22×#389C2semidirect
7. #23617C2direct (N × H)
8. C2×#2068C2semidirect
9. C2×#2066C2semidirect
10. C2×#2067C2semidirect
11. C2×#2065C2semidirect
12. C2×#1736C2semidirect
22. C2×#428C22semidirect
26. C2×#423C22semidirect
48. C2×#389C22semidirect
54. #2068C22semidirect
55. #2067C22semidirect
56. #2066C22semidirect
57. #2065C22semidirect
58. #1736C22semidirect
668. C2#23617direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4282
C22×#4232
C22×#4172
C2×#22972
C22×#3892
#236172
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#17362