Finite Group Databasebeta

← C2×#33084 #6629578

Subgroups — C2×#33084 #6629578

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic, complemented
2C2×C42×D4C211yesmaximal, characteristic
3#33084C2116yesmaximal, direct product factor
4C2×#1928C211yesmaximal, characteristic, complemented
5C2×#1928C211yesmaximal, characteristic, complemented
6C2×#1927C211yesmaximal, characteristic, complemented
7C2×#1921C211yesmaximal, characteristic, complemented
8C2×#1919C211yesmaximal, characteristic, complemented
9C2×#1918C211yesmaximal, characteristic, complemented
10C2×#1877C211yesmaximal, characteristic
11C2×#1877C211yesmaximal, characteristic
12C2×#1877C211yesmaximal, characteristic
13C2×#1864C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic
16C2×#1839C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C42—21no—
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31#2863—24no—
32#2863C2218yescomplemented
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#387—21no—
40C2×#387—21no—
41C2×#387—21no—
42C2×#383—21no—
43C42×D4—24no—
44C42×D4C2218yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4—21no—
50C2×C4×C22⋊C4—21no—
51C2×C43C2211yescharacteristic
52#1928C2218yescomplemented
53#1928C2218yescomplemented
54#1927C2218yescomplemented
55#1921C2218yescomplemented
56#1919C2218yescomplemented
57#1918C2218yescomplemented
58#1877C2218yes—
59#1877C2218yes—
60#1877C2218yes—
61#1864C2218yes—
62#1862C2218yes—
63#1862C2218yes—
64#1839C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4—21no—
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4C2314yes—
100C2×C4×D4C2314yes—
101C2×C4×D4C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C42—21no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—28no—
118C22×C42—48no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—24no—
127C2×#107—24no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—24no—
131C2×#106C2314yes—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144#391—22no—
145#391—22no—
146#391—22no—
147#391C2314yes—
148#391C2314yes—
149#391C2314yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#387—28no—
157#387—28no—
158#387—28no—
159#383—28no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—28no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C43D414yes—
182C43C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8C2412yes—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4#4114yes—
209C22×D4#4114yes—
210C22×D4#4114yes—
211C22×D4C2412yes—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×SD16—28no—
239C2×SD16—28no—
240C2×SD16—28no—
241C2×SD16—416no—
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×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4×D4—216no—
259C4×D4—216no—
260C4×D4—216no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—28no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
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—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—432no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330#107—28no—
331#107—28no—
332#106—28no—
333#103—28no—
334#103—28no—
335#103—28no—
336#102—28no—
337#102—28no—
338#102—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C22×D411yescharacteristic
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4#126116yes—
383C2×D4C2×#4114yes—
384C2×D4C2×#4114yes—
385C2×D4C2×#4114yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
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—22no—
413C22×C4—22no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—432no—
417C22×C4—432no—
418C22×C4—432no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
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#13514yes—
442C22×C4AGL(1,ℤ/8)14yes—
443C22×C4#13714yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2511yescharacteristic
458SD16—416no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4C2×#4114yes—
494C4⋊C4C2×#4114yes—
495C4⋊C4C2×#4114yes—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—28no—
503C42—28no—
504C42—28no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—44no—
512C42—48no—
513C42—48no—
514C42—48no—
515C23—216no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23#52512yes—
530C23#52512yes—
531C23#52512yes—
532C23C2×#13512yes—
533C23C2×AGL(1,ℤ/8)12yes—
534C23C2×#13712yes—
535C23#57512yes—
536C23C23×D411yescharacteristic
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—84no—
542D4—216no—
543D4—24no—
544D4—24no—
545D4—24no—
546D4C2×#12618yes—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—432no—
569C2×C4—432no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
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—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4C2×#12614yes—
603C2×C4#52914yes—
604C2×C4#52914yes—
605C2×C4#52914yes—
606C2×C4C2×#13514yes—
607C2×C4C2×AGL(1,ℤ/8)14yes—
608C2×C4C2×#13714yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C8—42no—
616C8—42no—
617C8—42no—
618C8—42no—
619C22—28no—
620C22—432no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22#285214yes—
628C22#310214yes—
629C22#310514yes—
630C22#310714yes—
631C22#311014yes—
632C22#311414yes—
633C22#311614yes—
634C22C2×#52511yescharacteristic
635C22C2×#52511yescharacteristic
636C22C2×#52511yescharacteristic
637C22C22×#13511yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57511yescharacteristic
641C4—24no—
642C4—24no—
643C4—24no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4#285214yes—
655C4C22×#12611yescharacteristic
656C4C2×#52911yescharacteristic
657C4C2×#52911yescharacteristic
658C4C2×#52911yescharacteristic
659C2—48no—
660C2#3308418yesdirect product factor
661C2C2×#285211yescharacteristic
662C2C2×#310211yescharacteristic
663C2C2×#310511yescharacteristic
664C2C2×#310711yescharacteristic
665C2C2×#311411yescharacteristic
666C2C2×#311611yescharacteristic
667C2C2×#311011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2863C2semidirect
3. #33084C2direct (N × H)
4. C2×#1928C2semidirect
5. C2×#1928C2semidirect
6. C2×#1927C2semidirect
7. C2×#1921C2semidirect
8. C2×#1919C2semidirect
9. C2×#1918C2semidirect
16. C2×#1839C2semidirect
32. #2863C22semidirect
52. #1928C22semidirect
53. #1928C22semidirect
54. #1927C22semidirect
55. #1921C22semidirect
56. #1919C22semidirect
57. #1918C22semidirect
64. #1839C22semidirect
660. C2#33084direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28632
C2×C42×D42
#330842
C2×#19282
C2×#19272
C2×#19212
C2×#19192
C2×#19182
C2×#18772
C2×#18642
C2×#18622
C2×#18392