Finite Group Databasebeta

← C2×#24055 #6440397

Subgroups — C2×#24055 #6440397

Order 512.

4568 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3C22×#417C211yesmaximal, characteristic, complemented
4C2×#2487C211yesmaximal, characteristic, complemented
5#24055C2116yesmaximal, direct product factor
6C2×#2043C212yesmaximal, complemented
7C2×#2035C212yesmaximal, complemented
8C2×#1894C211yesmaximal, characteristic, complemented
9C2×#1870C211yesmaximal, characteristic, complemented
10C2×#1864C212yesmaximal, complemented
11C2×#1788C212yesmaximal, complemented
12C2×#1788C211yesmaximal, characteristic, complemented
13C23×#41C2211yescharacteristic
14C23×#41C2211yescharacteristic
15C23×M16C2211yescharacteristic
16C2×#547—28no—
17C2×#547—28no—
18C2×#547—24no—
19C2×#547C2218yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#123—21no—
22C22×#123—21no—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121—21no—
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4—21no—
29C23×C4⋊C4C2211yescharacteristic
30C2×#483—28no—
31C2×#483C2218yescomplemented
32C22×#106C2212yes—
33C2×#417—28no—
34C2×#417C2218yescomplemented
35C22×#103C2212yes—
36C22×#103C2212yes—
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#102C2212yes—
40C22×#102C2212yes—
41C22×C22⋊C8C2211yescharacteristic
42C22×C22⋊C8C2212yes—
43#2487—24no—
44#2487C2218yescomplemented
45C2×#400—21no—
46C2×#399—21no—
47C2×#389C2212yes—
48C2×#386—21no—
49C2×#385—21no—
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52#2043C22116yescomplemented
53#2035C22116yescomplemented
54#1894C2218yescomplemented
55#1870C2218yescomplemented
56#1864C22116yescomplemented
57#1788C22116yescomplemented
58#1788C2218yescomplemented
59C2×#340C2212yes—
60C2×#336C2212yes—
61C22×#41—24no—
62C22×#41—24no—
63C22×#41—24no—
64C22×#41—22no—
65C22×#41—22no—
66C22×#41—28no—
67C22×#41—28no—
68C22×#41—44no—
69C22×#41—44no—
70C22×#41D418yes—
71C22×#41C2314yes—
72C22×#41C2314yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2311yescharacteristic
75C23×D4—21no—
76C23×D4—21no—
77C23×D4D411yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4—21no—
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×M16—24no—
86C22×M16—44no—
87C22×M16C2314yes—
88C23×C8C2312yes—
89C23×C8C2312yes—
90#547—264no—
91#547—232no—
92#547—232no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2314yes—
97C2×#123—216no—
98C2×#123—216no—
99C2×#123—28no—
100C2×#123—28no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8D418yes—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C2×#121—216no—
110C2×#121—28no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4D414yes—
117C2×C4×D4C2314yes—
118C2×C4×D4C2314yes—
119C22×C4⋊C4—216no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—28no—
125C22×C4⋊C4—24no—
126C22×C4⋊C4—24no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4D411yescharacteristic
141C22×C22⋊C4D411yescharacteristic
142C22×C42C2311yescharacteristic
143#483—232no—
144C2×#106—28no—
145C2×#106C2318yes—
146#417—232no—
147C2×#103—28no—
148C2×#103—28no—
149C2×#103—44no—
150C2×#103—44no—
151C2×#103C2318yes—
152C2×#103C2318yes—
153C2×#103C2318yes—
154C2×#103C2318yes—
155C2×#102—28no—
156C2×#102—28no—
157C2×#102C2318yes—
158C2×#102C2318yes—
159C2×C22⋊C8—24no—
160C2×C22⋊C8—44no—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2318yes—
163#400—28no—
164#399—28no—
165#389—24no—
166#389C2318yes—
167#386—28no—
168#385—28no—
169#380—24no—
170#380—22no—
171#380C2314yes—
172#380C2318yes—
173C2×#95—21no—
174C2×#95—22no—
175C2×#95—22no—
176#340C2318yes—
177#336C2318yes—
178C25—21no—
179C25C2×D411yescharacteristic
180C25C2×D411yescharacteristic
181C2×#41—216no—
182C2×#41—28no—
183C2×#41—28no—
184C2×#41—216no—
185C2×#41—28no—
186C2×#41—232no—
187C2×#41—216no—
188C2×#41—216no—
189C2×#41—416no—
190C2×#41—48no—
191C2×#41—48no—
192C2×#41—432no—
193C2×#41—416no—
194C2×#41—416no—
195C2×#41—432no—
196C2×#41—416no—
197C2×#41—416no—
198C2×#41C2×D4116yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8C2×D414yes—
208C22×Q8C2×D414yes—
209C22×Q8C2412yes—
210C22×Q8C2412yes—
211C22×D4—28no—
212C22×D4—24no—
213C22×D4—28no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—42no—
221C22×D4—48no—
222C22×D4—48no—
223C22×D4C2×D414yes—
224C22×D4C2×D414yes—
225C22×D4#4118yes—
226C22×D4C2412yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—28no—
230C23×C4—28no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—28no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—48no—
246C23×C4—48no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—84no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4C2×D414yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—28no—
263C2×M16—48no—
264C2×M16—416no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—216no—
274C4⋊Q8—28no—
275C4⋊Q8—28no—
276#123—264no—
277#123—264no—
278C22⋊Q8—216no—
279C22⋊Q8—28no—
280C22⋊Q8—216no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8C2×D4116yes—
284#121—264no—
285C4×D4—216no—
286C4×D4—28no—
287C4×D4—216no—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—264no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—84no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339#106—216no—
340#103—216no—
341#103—216no—
342#103—48no—
343#103—48no—
344#102—216no—
345#102—216no—
346C22⋊C8—28no—
347C22⋊C8—48no—
348#95—28no—
349#95—24no—
350#95—28no—
351C24—24no—
352C24—24no—
353C24—28no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—42no—
364C24—42no—
365C24—44no—
366C24#12014yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370#41—232no—
371#41—416no—
372#41—464no—
373#41—432no—
374#41—432no—
375#41—464no—
376#41—416no—
377#41—416no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—216no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8C22×D414yes—
395C2×Q8C22×D414yes—
396C2×D4—216no—
397C2×D4—28no—
398C2×D4—216no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—216no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—44no—
407C2×D4—416no—
408C2×D4—432no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—432no—
412C2×D4#123116yes—
413C2×D4C22×D414yes—
414C2×D4C2×#4118yes—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—432no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—432no—
449C22×C4—48no—
450C22×C4—432no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—88no—
467C22×C4—816no—
468C22×C4#12014yes—
469C22×C4#12018yes—
470C22×C4#13714yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4C2511yescharacteristic
483M16—416no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—48no—
491C2×C8—48no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—88no—
507C4⋊C4C22×D414yes—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—88no—
515C42—24no—
516C42—22no—
517C42—22no—
518C23—28no—
519C23—216no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—48no—
534C23—416no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—416no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—48no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—48no—
547C23C2×#12014yes—
548C23C2×#12012yes—
549C23#54614yes—
550C23C2×#13712yes—
551C23#57512yes—
552C23#57714yes—
553C23C23×D411yescharacteristic
554Q8—24no—
555Q8—24no—
556Q8—28no—
557Q8—48no—
558Q8—48no—
559Q8—42no—
560Q8—42no—
561Q8—44no—
562D4—216no—
563D4—24no—
564D4—28no—
565D4—416no—
566D4—432no—
567D4—432no—
568D4C2×#12318yes—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—216no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—432no—
586C2×C4—432no—
587C2×C4—432no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—816no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4C2×#12014yes—
611C2×C4C2×#12012yes—
612C2×C4C2×#12314yes—
613C2×C4#53414yes—
614C2×C4#54614yes—
615C2×C4#54618yes—
616C2×C4C2×#13714yes—
617C2×C4C23×D411yescharacteristic
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4112yes—
620C2×C4C2×H(5,𝔽2)12yes—
621C8—44no—
622C8—44no—
623C22—24no—
624C22—28no—
625C22—24no—
626C22—24no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—48no—
633C22—416no—
634C22—432no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—88no—
641C22#240514yes—
642C22#300714yes—
643C22#301014yes—
644C22#330718yes—
645C22#333118yes—
646C22C22×#12011yescharacteristic
647C22C2×#54612yes—
648C22C22×#13711yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57712yes—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—84no—
659C4—84no—
660C4#240514yes—
661C4C22×#12311yescharacteristic
662C4C2×#53411yescharacteristic
663C4C2×#54612yes—
664C2—44no—
665C2—48no—
666C2—82no—
667C2#2405518yesdirect product factor
668C2C2×#240511yescharacteristic
669C2C2×#300711yescharacteristic
670C2C2×#301011yescharacteristic
671C2C2×#330712yes—
672C2C2×#333112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#483C2semidirect
3. C22×#417C2semidirect
4. C2×#2487C2semidirect
5. #24055C2direct (N × H)
6. C2×#2043C2semidirect
7. C2×#2035C2semidirect
8. C2×#1894C2semidirect
9. C2×#1870C2semidirect
10. C2×#1864C2semidirect
11. C2×#1788C2semidirect
12. C2×#1788C2semidirect
31. C2×#483C22semidirect
34. C2×#417C22semidirect
44. #2487C22semidirect
52. #2043C22semidirect
53. #2035C22semidirect
54. #1894C22semidirect
55. #1870C22semidirect
56. #1864C22semidirect
57. #1788C22semidirect
58. #1788C22semidirect
667. C2#24055direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5472
C22×#4832
C22×#4172
C2×#24872
#240552
C2×#20432
C2×#20352
C2×#18942
C2×#18702
C2×#18642
C2×#17882