Finite Group Databasebeta

← C2×#45683 #7701989

Subgroups — C2×#45683 #7701989

Order 512.

6534 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2778C214yesmaximal, complemented
2C2×#2760C212yesmaximal, complemented
3C2×#2737C214yesmaximal, complemented
4C2×#2709C212yesmaximal, complemented
5#45683C2132yesmaximal, direct product factor
6C2×#2824C212yesmaximal, complemented
7C2×#2850C212yesmaximal, complemented
8C2×#2541C211yesmaximal, characteristic
9C2×#2544C212yesmaximal, complemented
10C2×#2514C212yesmaximal, complemented
11C2×#2436C212yesmaximal, complemented
12C2×#2427C212yesmaximal, complemented
13C2×#2420C212yesmaximal, complemented
14C2×#2409C211yesmaximal, characteristic, complemented
15C2×#2380C212yesmaximal, complemented
16C2×C4×#123C211yesmaximal, characteristic, complemented
17C24×Q8C2211yescharacteristic
18C22×#127C2212yes—
19C22×#127C2211yescharacteristic
20C22×#124C2212yes—
21C22×#124C2212yes—
22C22×#123C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×#121C2212yes—
25C22×#121C2212yes—
26C22×C4×D4C2212yes—
27C22×C4×D4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2850C22132yescomplemented
30#2824C22132yescomplemented
31#2778C22164yescomplemented
32#2760C22132yescomplemented
33#2737C22164yescomplemented
34#2709C22132yescomplemented
35#2544C22132yescomplemented
36#2541C22116yes—
37#2514C22132yescomplemented
38#2436C22132yescomplemented
39#2427C22132yescomplemented
40#2420C22132yescomplemented
41#2409C22116yescomplemented
42C2×#401C2214yes—
43C2×#400C2214yes—
44C2×#397C2214yes—
45C2×#397C2214yes—
46C2×#395C2212yes—
47C2×#394C2214yes—
48C2×#394C2214yes—
49C2×#394C2214yes—
50C2×#394C2214yes—
51C2×#393C2212yes—
52C2×#393C2214yes—
53C2×#393C2214yes—
54C2×#392C2212yes—
55C2×#392C2212yes—
56#2380C22132yescomplemented
57C4×#123C22116yescomplemented
58C2×#391C2212yes—
59C2×#391C2212yes—
60C2×#391C2212yes—
61C2×#390C2212yes—
62C2×#390C2212yes—
63C2×#389C2212yes—
64C2×#388C2214yes—
65C2×#388C2214yes—
66C2×#388C2212yes—
67C2×#387C2214yes—
68C2×#387C2212yes—
69C2×#386C2214yes—
70C2×#386C2214yes—
71C2×#386C2211yescharacteristic
72C2×#386C2214yes—
73C2×#386C2212yes—
74C2×#386C2214yes—
75C2×#386C2214yes—
76C2×#386C2214yes—
77C2×#386C2214yes—
78C2×#386C2214yes—
79C2×#385C2211yescharacteristic
80C2×#385C2212yes—
81C2×#383C2214yes—
82C2×#382C2212yes—
83C2×#381C2214yes—
84C2×#381C2211yescharacteristic
85C2×#380C2211yescharacteristic
86C2×#380C2214yes—
87C2×C4×C4⋊C4C2212yes—
88C2×C4×C4⋊C4C2211yescharacteristic
89C2×C4×C22⋊C4C2212yes—
90C2×C4×C22⋊C4C2211yescharacteristic
91C23×Q8—44no—
92C23×Q8—48no—
93C23×Q8C2312yes—
94C23×Q8C2312yes—
95C23×Q8C2318yes—
96C23×D4C2312yes—
97C23×D4C2314yes—
98C23×D4C2314yes—
99C23×D4C2312yes—
100C23×D4C2312yes—
101C23×D4C2311yescharacteristic
102C24×C4C2311yescharacteristic
103C24×C4C2311yescharacteristic
104C24×C4C2311yescharacteristic
105C24×C4C2312yes—
106C24×C4C2311yescharacteristic
107C24×C4C2314yes—
108C2×#127—216no—
109C2×#127—44no—
110C2×#127C23116yes—
111C2×#127C2318yes—
112C2×#124—48no—
113C2×#124—48no—
114C2×#124C23116yes—
115C2×#124C23116yes—
116C2×#123—28no—
117C2×#123C2318yes—
118C2×C22⋊Q8—48no—
119C2×C22⋊Q8C23116yes—
120C2×#121—48no—
121C2×#121—48no—
122C2×#121C23116yes—
123C2×#121C23116yes—
124C2×C4×D4—48no—
125C2×C4×D4—44no—
126C2×C4×D4C2318yes—
127C2×C4×D4C23116yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2314yes—
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2311yescharacteristic
151C22×C22⋊C4C2312yes—
152C22×C42—44no—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2311yescharacteristic
156C22×C42C2312yes—
157C22×C42C2318yes—
158C22×C42C2312yes—
159#401C23132yes—
160#400C23132yes—
161#397C23132yes—
162#397C23132yes—
163#395C23116yes—
164#394C23132yes—
165#394C23132yes—
166#394C23132yes—
167#394C23132yes—
168#393C23116yes—
169#393C23132yes—
170#393C23132yes—
171#392C23116yes—
172#392C23116yes—
173#391C23116yes—
174#391C23116yes—
175#391C23116yes—
176#390C23116yes—
177#390C23116yes—
178#389C23116yes—
179#388C23132yes—
180#388C23132yes—
181#388C23116yes—
182#387C23132yes—
183#387C23116yes—
184#386C23132yes—
185#386C23132yes—
186#386C2318yes—
187#386C23116yes—
188#386C23132yes—
189#386C23132yes—
190#386C23132yes—
191#386C23132yes—
192#386C23132yes—
193#386C23132yes—
194#385C2318yes—
195#385C23116yes—
196#383C23132yes—
197#382C23116yes—
198#381C23132yes—
199#381C2318yes—
200#380C2318yes—
201#380C23132yes—
202C4×C4⋊C4C2318yes—
203C4×C4⋊C4C23116yes—
204C4×C22⋊C4C23116yes—
205C4×C22⋊C4C2318yes—
206C2×#95C2312yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2312yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2314yes—
216C2×#95C2314yes—
217C2×#95C2312yes—
218C2×#95C2314yes—
219C2×#95C2314yes—
220C2×#95C2312yes—
221C2×#95C2312yes—
222C25C2414yes—
223C25C2411yescharacteristic
224C25C2411yescharacteristic
225C22×Q8—28no—
226C22×Q8—28no—
227C22×Q8—28no—
228C22×Q8—48no—
229C22×Q8—44no—
230C22×Q8—416no—
231C22×Q8—432no—
232C22×Q8—464no—
233C22×Q8C2418yes—
234C22×Q8C2418yes—
235C22×D4—216no—
236C22×D4—216no—
237C22×D4—28no—
238C22×D4—28no—
239C22×D4—28no—
240C22×D4—28no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4—48no—
244C22×D4—48no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—48no—
249C22×D4—48no—
250C22×D4—48no—
251C22×D4#4118yes—
252C22×D4C2418yes—
253C22×D4C24116yes—
254C22×D4C24116yes—
255C22×D4C2418yes—
256C22×D4C2418yes—
257C22×D4C2414yes—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—28no—
262C23×C4—28no—
263C23×C4—42no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4—48no—
267C23×C4—42no—
268C23×C4—42no—
269C23×C4—48no—
270C23×C4—48no—
271C23×C4—48no—
272C23×C4C2414yes—
273C23×C4C2414yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C23×C4C2414yes—
277C23×C4C2411yescharacteristic
278C23×C4C2414yes—
279C23×C4C2412yes—
280C23×C4C2412yes—
281C23×C4C2412yes—
282C23×C4C2414yes—
283C23×C4C2411yescharacteristic
284C23×C4C2411yescharacteristic
285C23×C4C2414yes—
286C23×C4C24116yes—
287C23×C4C2418yes—
288C23×C4C2414yes—
289C23×C4C2411yescharacteristic
290#127—264no—
291#127—416no—
292#124—432no—
293#124—432no—
294#123—232no—
295C22⋊Q8—432no—
296#121—432no—
297#121—432no—
298C4×D4—432no—
299C4×D4—416no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4C2418yes—
308C2×C4⋊C4C2418yes—
309C2×C4⋊C4C2418yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C24116yes—
312C2×C4⋊C4C2418yes—
313C2×C4⋊C4C24116yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4—48no—
331C2×C22⋊C4C2414yes—
332C2×C22⋊C4C2418yes—
333C2×C22⋊C4C2414yes—
334C2×C22⋊C4C2414yes—
335C2×C22⋊C4C24116yes—
336C2×C22⋊C4C24116yes—
337C2×C22⋊C4C24116yes—
338C2×C22⋊C4C24116yes—
339C2×C22⋊C4C24116yes—
340C2×C22⋊C4C24116yes—
341C2×C22⋊C4C24116yes—
342C2×C22⋊C4C24116yes—
343C2×C22⋊C4C2414yes—
344C2×C22⋊C4C2418yes—
345C2×C22⋊C4C2418yes—
346C2×C22⋊C4C24116yes—
347C2×C22⋊C4C24116yes—
348C2×C42—28no—
349C2×C42—28no—
350C2×C42—24no—
351C2×C42—28no—
352C2×C42—416no—
353C2×C42—44no—
354C2×C42C2418yes—
355C2×C42C2414yes—
356C2×C42C2418yes—
357C2×C42C2418yes—
358C2×C42C2418yes—
359#95C2418yes—
360#95C2418yes—
361#95C2418yes—
362#95C2418yes—
363#95C2418yes—
364#95C2418yes—
365#95C2418yes—
366#95C24116yes—
367#95C24116yes—
368#95C24116yes—
369#95C24116yes—
370#95C24116yes—
371#95C24116yes—
372#95C24116yes—
373#95C24116yes—
374#95C24116yes—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—28no—
384C24—28no—
385C24—28no—
386C24—28no—
387C24—28no—
388C24—28no—
389C24—28no—
390C24C2×#4114yes—
391C24C2×#4114yes—
392C24C2518yes—
393C24C2512yes—
394C24C2512yes—
395C24C2511yescharacteristic
396C2×Q8—216no—
397C2×Q8—216no—
398C2×Q8—216no—
399C2×Q8—416no—
400C2×Q8—48no—
401C2×Q8—4128no—
402C2×Q8—432no—
403C2×Q8—432no—
404C2×D4—232no—
405C2×D4—232no—
406C2×D4—216no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—44no—
411C2×D4—416no—
412C2×D4—44no—
413C2×D4—416no—
414C2×D4—416no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—432no—
422C2×D4—816no—
423C2×D4—832no—
424C2×D4—832no—
425C2×D4—816no—
426C2×D4C2×#41116yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—216no—
460C22×C4—216no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—432no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—832no—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)18yes—
479C22×C4C2514yes—
480C22×C4C2518yes—
481C22×C4C2512yes—
482C22×C4C2514yes—
483C22×C4C2514yes—
484C22×C4C2518yes—
485C22×C4C2512yes—
486C22×C4C2518yes—
487C22×C4C2514yes—
488C22×C4C2512yes—
489C22×C4C2512yes—
490C22×C4C2512yes—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—416no—
498C22⋊C4—28no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—416no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C42—216no—
516C42—216no—
517C42—216no—
518C42—28no—
519C42—48no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—216no—
531C23—216no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—48no—
544C23—48no—
545C23—416no—
546C23—416no—
547C23—416no—
548C23—416no—
549C23—416no—
550C23—416no—
551C23—416no—
552C23C22×#4114yes—
553C23C22×#4114yes—
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)14yes—
557C23#58512yes—
558C23#58514yes—
559C23C2611yescharacteristic
560Q8—432no—
561Q8—432no—
562D4—416no—
563D4—432no—
564D4—816no—
565D4—832no—
566D4—832no—
567D4—816no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—432no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—416no—
610C2×C4—48no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—832no—
614C2×C4#560116yes—
615C2×C4C22×#4114yes—
616C2×C4C2×H(5,𝔽2)14yes—
617C2×C4C2×H(5,𝔽2)18yes—
618C22—24no—
619C22—24no—
620C22—28no—
621C22—28no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—416no—
627C22—416no—
628C22—42no—
629C22—42no—
630C22—48no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—832no—
638C22#350514yes—
639C22#351418yes—
640C22C2×#56014yes—
641C22#354014yes—
642C22#356218yes—
643C22C23×#4111yescharacteristic
644C22C22×H(5,𝔽2)11yescharacteristic
645C22C22×H(5,𝔽2)12yes—
646C22C2×#58512yes—
647C22C2×#58511yescharacteristic
648C4—24no—
649C4—44no—
650C4—42no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—48no—
656C4—42no—
657C4—42no—
658C4—48no—
659C4C2×#56018yes—
660C2—44no—
661C2—44no—
662C2—88no—
663C2#4568318yesdirect product factor
664C2C22×#56011yescharacteristic
665C2C2×#354011yescharacteristic
666C2C2×#356212yes—
667C2C2×#350511yescharacteristic
668C2C2×#351412yes—
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC2×#2541C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2778C2semidirect
2. C2×#2760C2semidirect
3. C2×#2737C2semidirect
4. C2×#2709C2semidirect
5. #45683C2direct (N × H)
6. C2×#2824C2semidirect
7. C2×#2850C2semidirect
9. C2×#2544C2semidirect
10. C2×#2514C2semidirect
11. C2×#2436C2semidirect
12. C2×#2427C2semidirect
13. C2×#2420C2semidirect
14. C2×#2409C2semidirect
15. C2×#2380C2semidirect
16. C2×C4×#123C2semidirect
29. #2850C22semidirect
30. #2824C22semidirect
31. #2778C22semidirect
32. #2760C22semidirect
33. #2737C22semidirect
34. #2709C22semidirect
35. #2544C22semidirect
37. #2514C22semidirect
38. #2436C22semidirect
39. #2427C22semidirect
40. #2420C22semidirect
41. #2409C22semidirect
56. #2380C22semidirect
57. C4×#123C22semidirect
663. C2#45683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27782
C2×#27602
C2×#27372
C2×#27092
#456832
C2×#28242
C2×#28502
C2×#25412
C2×#25442
C2×#25142
C2×#24362
C2×#24272
C2×#24202
C2×#24092
C2×#23802
C2×C4×#1232