Finite Group Databasebeta

← C2×#16800 #611568

Subgroups — C2×#16800 #611568

Order 512.

5166 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic, complemented
2C22×#394C211yesmaximal, characteristic
3C22×#390C211yesmaximal, characteristic
4C22×#387C211yesmaximal, characteristic, complemented
5#16800C2116yesmaximal, direct product factor
6C2×#1445C211yesmaximal, characteristic, complemented
7C2×#1440C212yesmaximal, complemented
8C2×#1439C212yesmaximal, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1439C212yesmaximal, complemented
11C2×#1438C211yesmaximal, characteristic
12C2×#1434C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C42C2211yescharacteristic
25C2×#398—28no—
26C2×#398C2218yescomplemented
27C2×#394—28no—
28C2×#394C2218yes—
29C2×#390—28no—
30C2×#390C2218yes—
31C2×#387—28no—
32C2×#387C2218yescomplemented
33C22×#95C2211yescharacteristic
34C22×#95C2211yescharacteristic
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44#1445C2218yescomplemented
45#1440C22116yescomplemented
46#1439C22116yescomplemented
47#1439C22116yescomplemented
48#1439C22116yescomplemented
49#1438C2218yes—
50#1434C2218yescomplemented
51C26C2312yes—
52C23×D4—28no—
53C23×D4—24no—
54C23×D4—24no—
55C23×D4—24no—
56C23×D4—44no—
57C23×D4C2314yes—
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4D418yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2318yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C42—44no—
108C22×C42D418yes—
109C22×C42C2314yes—
110#398—232no—
111#394—232no—
112#390—232no—
113#387—232no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—28no—
117C2×#95—24no—
118C2×#95—24no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95—28no—
122C2×#95—44no—
123C2×#95—44no—
124C2×#95—44no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C25—24no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—24no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25#4118yes—
151C25C2414yes—
152C25C2411yescharacteristic
153C22×D4—232no—
154C22×D4—216no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—48no—
159C22×D4—432no—
160C22×D4—432no—
161C22×D4—416no—
162C22×D4—416no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4#4118yes—
220C23×C4#4114yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—432no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4C2×D4116yes—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—432no—
265C2×C22⋊C4—432no—
266C2×C22⋊C4—432no—
267C2×C22⋊C4—432no—
268C2×C22⋊C4—432no—
269C2×C22⋊C4—432no—
270C2×C42—416no—
271C2×C42—48no—
272C2×C42C2×D4116yes—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—216no—
280#95—216no—
281#95—48no—
282#95—48no—
283#95—48no—
284C24—216no—
285C24—216no—
286C24—216no—
287C24—216no—
288C24—216no—
289C24—216no—
290C24—216no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24C22×D412yes—
334C24C22×D412yes—
335C24C2×#4118yes—
336C24C2×#4112yes—
337C24C2×#4114yes—
338C24C2×#4114yes—
339C24H(5,𝔽2)14yes—
340C24H(5,𝔽2)12yes—
341C24H(5,𝔽2)12yes—
342C24H(5,𝔽2)12yes—
343C24H(5,𝔽2)12yes—
344C242−1+414yes—
345C24C2511yescharacteristic
346C2×D4—232no—
347C2×D4—432no—
348C2×D4—432no—
349C2×D4—416no—
350C2×D4—4128no—
351C2×D4—416no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4#121116yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C4⋊C4—232no—
469C4⋊C4—416no—
470C4⋊C4—432no—
471C4⋊C4—416no—
472C22⋊C4—432no—
473C22⋊C4—432no—
474C22⋊C4—432no—
475C22⋊C4—432no—
476C22⋊C4—432no—
477C22⋊C4—432no—
478C42—416no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—432no—
487C23—432no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—432no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—432no—
514C23—432no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—432no—
520C23—432no—
521C23—816no—
522C23—816no—
523C23—816no—
524C23—816no—
525C23—816no—
526C23—816no—
527C23—816no—
528C23—816no—
529C23C2×#12114yes—
530C23#52514yes—
531C23#53214yes—
532C23#53414yes—
533C23#53514yes—
534C23#53514yes—
535C23#53514yes—
536C23#53514yes—
537C23#53614yes—
538C23#53818yes—
539C23#53918yes—
540C23#53918yes—
541C23#54014yes—
542C23#54014yes—
543C23#54018yes—
544C23#54618yes—
545C23#54818yes—
546C23#55118yes—
547C23#55118yes—
548C23#55318yes—
549C23#55618yes—
550C23#56218yes—
551C23#56218yes—
552C23C23×D411yescharacteristic
553C23C23×D411yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4112yes—
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×2−1+412yes—
563D4—464no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—232no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—432no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—416no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—88no—
628C2×C4—816no—
629C2×C4—88no—
630C2×C4—816no—
631C2×C4C2×#12118yes—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—864no—
646C22—84no—
647C22—84no—
648C22#253418yes—
649C22#254518yes—
650C22#260318yes—
651C22#261918yes—
652C22#263318yes—
653C22#266118yes—
654C22#2696116yes—
655C22#2711116yes—
656C22#2740116yes—
657C22#2786116yes—
658C22#281618yes—
659C22C22×#12111yescharacteristic
660C22C2×#52511yescharacteristic
661C22C2×#53211yescharacteristic
662C22C2×#53411yescharacteristic
663C22C2×#53511yescharacteristic
664C22C2×#53511yescharacteristic
665C22C2×#53511yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53611yescharacteristic
668C22C2×#53812yes—
669C22C2×#53912yes—
670C22C2×#53912yes—
671C22C2×#54011yescharacteristic
672C22C2×#54012yes—
673C22C2×#54011yescharacteristic
674C22C2×#54612yes—
675C22C2×#54812yes—
676C22C2×#55112yes—
677C22C2×#55112yes—
678C22C2×#55312yes—
679C22C2×#55612yes—
680C22C2×#56212yes—
681C22C2×#56212yes—
682C4—28no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—48no—
688C4—82no—
689C4—82no—
690C4—84no—
691C4—84no—
692C2—88no—
693C2#16800116yesdirect product factor
694C2C2×#266111yescharacteristic
695C2C2×#263311yescharacteristic
696C2C2×#261911yescharacteristic
697C2C2×#253411yescharacteristic
698C2C2×#260311yescharacteristic
699C2C2×#254511yescharacteristic
700C2C2×#281611yescharacteristic
701C2C2×#269612yes—
702C2C2×#271112yes—
703C2C2×#274012yes—
704C2C2×#278612yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#398C2semidirect
4. C22×#387C2semidirect
5. #16800C2direct (N × H)
6. C2×#1445C2semidirect
7. C2×#1440C2semidirect
8. C2×#1439C2semidirect
9. C2×#1439C2semidirect
10. C2×#1439C2semidirect
12. C2×#1434C2semidirect
26. C2×#398C22semidirect
32. C2×#387C22semidirect
44. #1445C22semidirect
45. #1440C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
50. #1434C22semidirect
693. C2#16800direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3942
C22×#3902
C22×#3872
#168002
C2×#14452
C2×#14402
C2×#14392
C2×#14382
C2×#14342