Finite Group Databasebeta

← C2×#46016 #7702678

Subgroups — C2×#46016 #7702678

Order 512.

6986 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C212yesmaximal, complemented
2C2×#2753C214yesmaximal, complemented
3C2×#2733C212yesmaximal, complemented
4C2×#2712C214yesmaximal, complemented
5#46016C2132yesmaximal, direct product factor
6C2×#2819C212yesmaximal, complemented
7C2×#2862C211yesmaximal, characteristic, complemented
8C2×#2858C211yesmaximal, characteristic, complemented
9C2×#2592C212yesmaximal, complemented
10C2×#2492C212yesmaximal, complemented
11C2×#2491C212yesmaximal, complemented
12C2×#2432C212yesmaximal, complemented
13C2×#2431C212yesmaximal, complemented
14C2×#2407C212yesmaximal, complemented
15C2×#2379C212yesmaximal, complemented
16C2×C4×#124C211yesmaximal, characteristic, complemented
17C23×#41C2212yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#127C2211yescharacteristic
22C22×#124C2212yes—
23C22×#124C2211yescharacteristic
24C22×#124C2212yes—
25C22×#124C2212yes—
26C22×#124C2214yes—
27C22×C22⋊Q8C2212yes—
28C22×#121C2212yes—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31#2862C22116yescomplemented
32#2858C22116yescomplemented
33#2819C22132yescomplemented
34#2804C22132yescomplemented
35#2753C22164yescomplemented
36#2733C22132yescomplemented
37#2712C22164yescomplemented
38#2592C22132yescomplemented
39#2492C22132yescomplemented
40#2491C22132yescomplemented
41#2432C22132yescomplemented
42#2431C22132yescomplemented
43#2407C22132yescomplemented
44C2×#400C2214yes—
45C2×#397C2214yes—
46C2×#394C2214yes—
47C2×#394C2214yes—
48C2×#394C2214yes—
49C2×#394C2214yes—
50C2×#393C2214yes—
51C2×#393C2214yes—
52C2×#393C2214yes—
53C2×#392C2214yes—
54C2×#392C2214yes—
55C2×#392C2214yes—
56#2379C22132yescomplemented
57C4×#124C22116yescomplemented
58C2×#391C2212yes—
59C2×#391C2212yes—
60C2×#391C2212yes—
61C2×#391C2212yes—
62C2×#391C2214yes—
63C2×#391C2211yescharacteristic
64C2×#391C2212yes—
65C2×#390C2212yes—
66C2×#390C2212yes—
67C2×#390C2212yes—
68C2×#390C2212yes—
69C2×#390C2212yes—
70C2×#390C2211yescharacteristic
71C2×#388C2214yes—
72C2×#388C2214yes—
73C2×#388C2214yes—
74C2×#388C2212yes—
75C2×#388C2214yes—
76C2×#388C2212yes—
77C2×#386C2214yes—
78C2×#386C2214yes—
79C2×#386C2214yes—
80C2×#385C2214yes—
81C2×#385C2214yes—
82C2×#385C2214yes—
83C2×#382C2212yes—
84C2×#382C2212yes—
85C2×#382C2211yescharacteristic
86C2×#380C2212yes—
87C2×C4×C4⋊C4C2212yes—
88C2×C4×C22⋊C4C2212yes—
89C2×C4×C22⋊C4C2212yes—
90C2×C43C2211yescharacteristic
91C22×#41—48no—
92C22×#41—48no—
93C22×#41—48no—
94C22×#41C23116yes—
95C23×Q8C2312yes—
96C23×Q8C2311yescharacteristic
97C23×Q8C2312yes—
98C23×Q8C2312yes—
99C23×D4C2312yes—
100C23×D4C2311yescharacteristic
101C23×D4C2312yes—
102C23×D4C2312yes—
103C23×D4C2314yes—
104C23×D4C2314yes—
105C24×C4C2312yes—
106C24×C4C2314yes—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8—44no—
110C2×C4⋊Q8C2318yes—
111C2×C4⋊Q8C2318yes—
112C2×C4⋊Q8C2318yes—
113C2×#127—44no—
114C2×#127C2318yes—
115C2×#124—28no—
116C2×#124—216no—
117C2×#124—48no—
118C2×#124—48no—
119C2×#124—416no—
120C2×#124C2318yes—
121C2×#124C23116yes—
122C2×#124C23116yes—
123C2×#124C23116yes—
124C2×#124C23132yes—
125C2×C22⋊Q8—48no—
126C2×C22⋊Q8C23116yes—
127C2×#121—48no—
128C2×#121C23116yes—
129C2×C4×Q8—44no—
130C2×C4×Q8C2318yes—
131C2×C4×D4—44no—
132C2×C4×D4C2318yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2314yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2314yes—
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2314yes—
153C22×C22⋊C4C2314yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2312yes—
158C22×C22⋊C4C2314yes—
159C22×C42C2311yescharacteristic
160C22×C42C2312yes—
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2314yes—
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168#400C23132yes—
169#397C23132yes—
170#394C23132yes—
171#394C23132yes—
172#394C23132yes—
173#394C23132yes—
174#393C23132yes—
175#393C23132yes—
176#393C23132yes—
177#392C23132yes—
178#392C23132yes—
179#392C23132yes—
180#391C23116yes—
181#391C23116yes—
182#391C23116yes—
183#391C23116yes—
184#391C23116yes—
185#391C23132yes—
186#391C2318yes—
187#390C23116yes—
188#390C23116yes—
189#390C23116yes—
190#390C23116yes—
191#390C23116yes—
192#390C2318yes—
193#388C23132yes—
194#388C23132yes—
195#388C23132yes—
196#388C23116yes—
197#388C23132yes—
198#388C23116yes—
199#386C23132yes—
200#386C23132yes—
201#386C23132yes—
202#385C23132yes—
203#385C23132yes—
204#385C23132yes—
205#382C23116yes—
206#382C23116yes—
207#382C2318yes—
208#380C23116yes—
209C4×C4⋊C4C23116yes—
210C4×C22⋊C4C23116yes—
211C4×C22⋊C4C23116yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2314yes—
216C2×#95C2314yes—
217C2×#95C2312yes—
218C2×#95C2314yes—
219C2×#95C2314yes—
220C2×#95C2314yes—
221C2×#95C2312yes—
222C2×#95C2312yes—
223C2×#95C2312yes—
224C43C2318yes—
225C25C2412yes—
226C25C2414yes—
227C2×#41—432no—
228C2×#41—432no—
229C2×#41—432no—
230C2×#41—4128no—
231C22×Q8—24no—
232C22×Q8—28no—
233C22×Q8—28no—
234C22×Q8—28no—
235C22×Q8—28no—
236C22×Q8—28no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2418yes—
244C22×Q8C2414yes—
245C22×Q8C2418yes—
246C22×Q8C2418yes—
247C22×D4—24no—
248C22×D4—28no—
249C22×D4—28no—
250C22×D4—28no—
251C22×D4—28no—
252C22×D4—28no—
253C22×D4—42no—
254C22×D4—42no—
255C22×D4—48no—
256C22×D4—48no—
257C22×D4—48no—
258C22×D4—48no—
259C22×D4—44no—
260C22×D4—48no—
261C22×D4—44no—
262C22×D4—44no—
263C22×D4—44no—
264C22×D4—48no—
265C22×D4C2418yes—
266C22×D4C2414yes—
267C22×D4C2418yes—
268C22×D4C2418yes—
269C22×D4C24116yes—
270C22×D4C24116yes—
271C23×C4—28no—
272C23×C4—28no—
273C23×C4—28no—
274C23×C4—48no—
275C23×C4—48no—
276C23×C4—48no—
277C23×C4C2418yes—
278C23×C4C2414yes—
279C23×C4C2411yescharacteristic
280C23×C4C2412yes—
281C23×C4C2412yes—
282C23×C4C2414yes—
283C23×C4C2412yes—
284C23×C4C2414yes—
285C23×C4C2411yescharacteristic
286C23×C4C2411yescharacteristic
287C23×C4C2411yescharacteristic
288C23×C4C2411yescharacteristic
289C23×C4C2411yescharacteristic
290C23×C4C24116yes—
291C23×C4C2411yescharacteristic
292C4⋊Q8—416no—
293C4⋊Q8—416no—
294C4⋊Q8—416no—
295#127—416no—
296#124—232no—
297#124—264no—
298#124—432no—
299#124—432no—
300#124—464no—
301C22⋊Q8—432no—
302#121—432no—
303C4×Q8—416no—
304C4×D4—416no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2418yes—
316C2×C4⋊C4C2418yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2418yes—
320C2×C4⋊C4C24116yes—
321C2×C4⋊C4C2418yes—
322C2×C4⋊C4C2418yes—
323C2×C4⋊C4C2418yes—
324C2×C4⋊C4C2418yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—216no—
327C2×C22⋊C4—216no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—216no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—48no—
335C2×C22⋊C4—48no—
336C2×C22⋊C4—48no—
337C2×C22⋊C4—48no—
338C2×C22⋊C4—48no—
339C2×C22⋊C4—48no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4C2418yes—
342C2×C22⋊C4C24116yes—
343C2×C22⋊C4C2418yes—
344C2×C22⋊C4C2418yes—
345C2×C22⋊C4C24116yes—
346C2×C22⋊C4C24116yes—
347C2×C22⋊C4C2418yes—
348C2×C22⋊C4C2418yes—
349C2×C22⋊C4C24116yes—
350C2×C22⋊C4C24116yes—
351C2×C22⋊C4C24116yes—
352C2×C22⋊C4C24116yes—
353C2×C22⋊C4C24116yes—
354C2×C22⋊C4C24116yes—
355C2×C22⋊C4C24116yes—
356C2×C22⋊C4C24116yes—
357C2×C42—24no—
358C2×C42—28no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—48no—
364C2×C42—42no—
365C2×C42#4118yescomplemented
366C2×C42C2414yes—
367C2×C42C2418yes—
368C2×C42C2414yes—
369C2×C42C2414yes—
370C2×C42C2414yes—
371C2×C42C2414yes—
372C2×C42C24116yes—
373C2×C42C2414yes—
374C2×C42C2414yes—
375#95C2418yes—
376#95C2418yes—
377#95C2418yes—
378#95C2418yes—
379#95C24116yes—
380#95C24116yes—
381#95C24116yes—
382#95C24116yes—
383#95C24116yes—
384#95C24116yes—
385#95C24116yes—
386#95C24116yes—
387C24—24no—
388C24—28no—
389C24—28no—
390C24—24no—
391C24—24no—
392C24—28no—
393C24—24no—
394C24—28no—
395C24—24no—
396C24—28no—
397C24—24no—
398C24—28no—
399C24—28no—
400C24—24no—
401C24C2514yes—
402C24C2518yes—
403C24C2511yescharacteristic
404#41—4256no—
405C2×Q8—28no—
406C2×Q8—216no—
407C2×Q8—216no—
408C2×Q8—216no—
409C2×Q8—216no—
410C2×Q8—216no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—416no—
418C2×Q8—432no—
419C2×Q8—816no—
420C2×Q8—816no—
421C2×D4—28no—
422C2×D4—216no—
423C2×D4—216no—
424C2×D4—216no—
425C2×D4—216no—
426C2×D4—216no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—416no—
432C2×D4—416no—
433C2×D4—416no—
434C2×D4—48no—
435C2×D4—416no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—416no—
440C2×D4—464no—
441C2×D4—432no—
442C2×D4—816no—
443C2×D4—816no—
444C2×D4—832no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—24no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—216no—
477C22×C4—216no—
478C22×C4—216no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—464no—
483C22×C4—432no—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+414yes—
492C22×C4C2514yes—
493C22×C4C2518yes—
494C22×C4C2514yes—
495C22×C4C2518yes—
496C22×C4C2512yes—
497C22×C4C2514yes—
498C22×C4C2518yes—
499C22×C4C2512yes—
500C22×C4C2512yes—
501C22×C4C2512yes—
502C22×C4C2512yes—
503C22×C4C2512yes—
504C22×C4C2512yes—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C22⋊C4—216no—
516C22⋊C4—232no—
517C22⋊C4—232no—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—232no—
521C22⋊C4—216no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C22⋊C4—416no—
531C42—28no—
532C42—216no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—28no—
537C42—44no—
538C42—416no—
539C42C2×#41116yescomplemented
540C23—24no—
541C23—28no—
542C23—24no—
543C23—28no—
544C23—24no—
545C23—28no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—28no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—24no—
554C23—48no—
555C23—416no—
556C23—416no—
557C23—48no—
558C23—416no—
559C23—48no—
560C23—416no—
561C23—48no—
562C23—416no—
563C23—48no—
564C23—48no—
565C23—416no—
566C23—416no—
567C23—48no—
568C23C22×#4112yes—
569C23C2×H(5,𝔽2)12yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×2−1+412yes—
574C23#58512yes—
575C23C2611yescharacteristic
576Q8—416no—
577Q8—432no—
578Q8—816no—
579Q8—816no—
580D4—416no—
581D4—464no—
582D4—432no—
583D4—816no—
584D4—816no—
585D4—832no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—464no—
623C2×C4—48no—
624C2×C4—416no—
625C2×C4—48no—
626C2×C4—416no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—416no—
630C2×C4—432no—
631C2×C4H(3,𝔽4)116yes—
632C2×C4C22×#4114yes—
633C2×C4C22×#4114yes—
634C2×C4C22×#4114yes—
635C2×C4C2×H(5,𝔽2)14yes—
636C2×C4C2×H(5,𝔽2)14yes—
637C2×C4C2×H(5,𝔽2)14yes—
638C2×C4C2×H(5,𝔽2)14yes—
639C2×C4C2×2−1+414yes—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—48no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—48no—
649C22—44no—
650C22—48no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—816no—
655C22—832no—
656C22#350614yes—
657C22#350814yes—
658C22#351114yes—
659C22C2×H(3,𝔽4)14yes—
660C22#354814yes—
661C22#355814yes—
662C22#356114yes—
663C22C23×#4111yescharacteristic
664C22C22×H(5,𝔽2)11yescharacteristic
665C22C22×H(5,𝔽2)11yescharacteristic
666C22C22×H(5,𝔽2)11yescharacteristic
667C22C22×H(5,𝔽2)11yescharacteristic
668C22C22×2−1+411yescharacteristic
669C22C2×#58511yescharacteristic
670C4—24no—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—48no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—44no—
681C4—48no—
682C4C2×H(3,𝔽4)18yes—
683C2—84no—
684C2—88no—
685C2#4601618yesdirect product factor
686C2C22×H(3,𝔽4)11yescharacteristic
687C2C2×#355811yescharacteristic
688C2C2×#356111yescharacteristic
689C2C2×#354811yescharacteristic
690C2C2×#350611yescharacteristic
691C2C2×#350811yescharacteristic
692C2C2×#351111yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
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×#2804C2semidirect
2. C2×#2753C2semidirect
3. C2×#2733C2semidirect
4. C2×#2712C2semidirect
5. #46016C2direct (N × H)
6. C2×#2819C2semidirect
7. C2×#2862C2semidirect
8. C2×#2858C2semidirect
9. C2×#2592C2semidirect
10. C2×#2492C2semidirect
11. C2×#2491C2semidirect
12. C2×#2432C2semidirect
13. C2×#2431C2semidirect
14. C2×#2407C2semidirect
15. C2×#2379C2semidirect
16. C2×C4×#124C2semidirect
31. #2862C22semidirect
32. #2858C22semidirect
33. #2819C22semidirect
34. #2804C22semidirect
35. #2753C22semidirect
36. #2733C22semidirect
37. #2712C22semidirect
38. #2592C22semidirect
39. #2492C22semidirect
40. #2491C22semidirect
41. #2432C22semidirect
42. #2431C22semidirect
43. #2407C22semidirect
56. #2379C22semidirect
57. C4×#124C22semidirect
365. C2×C42#41semidirect
539. C42C2×#41semidirect
685. C2#46016direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#27532
C2×#27332
C2×#27122
#460162
C2×#28192
C2×#28622
C2×#28582
C2×#25922
C2×#24922
C2×#24912
C2×#24322
C2×#24312
C2×#24072
C2×#23792
C2×C4×#1242