Finite Group Databasebeta

← C2×#28117 #6459388

Subgroups — C2×#28117 #6459388

Order 512.

4214 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic, complemented
2C22×#447C211yesmaximal, characteristic
3C2×C4×#121C211yesmaximal, characteristic, complemented
4C22×#390C211yesmaximal, characteristic, complemented
5#28117C2116yesmaximal, direct product factor
6C2×#2065C212yesmaximal, complemented
7C2×#2054C212yesmaximal, complemented
8C2×#1926C212yesmaximal, complemented
9C2×#1879C212yesmaximal, complemented
10C2×#1824C211yesmaximal, characteristic, complemented
11C2×#1797C211yesmaximal, characteristic, complemented
12C2×#1786C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C23×D8C2212yes—
15C22×#121C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4C2211yescharacteristic, complemented
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4—21no—
20C23×C22⋊C4—21no—
21C23×C42C2211yescharacteristic
22C2×#481—28no—
23C2×#481C2218yescomplemented
24C2×#447—28no—
25C2×#447C2218yes—
26C22×#106C2212yescomplemented
27C22×C4⋊C8C2212yescomplemented
28C22×#102C2212yes—
29C22×#102C2212yes—
30C22×#102C2212yes—
31C22×#102C2212yescomplemented
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C4×#121—24no—
35C4×#121C2218yescomplemented
36C2×#390—21no—
37C2×#390—21no—
38C2×#390—24no—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#390C2218yescomplemented
42C2×#390C2211yescharacteristic, complemented
43C2×#389C2211yescharacteristic, complemented
44C2×#388—21no—
45C2×#386—21no—
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic, complemented
48C2×C4×C22⋊C4—21no—
49#2065C22116yescomplemented
50#2054C22116yescomplemented
51#1926C22116yescomplemented
52#1879C22116yescomplemented
53#1824C2218yescomplemented
54#1797C2218yescomplemented
55#1786C2218yescomplemented
56C2×#340C2212yescomplemented
57C2×#340C2212yescomplemented
58C2×#336C2212yescomplemented
59C26—21no—
60C23×D4—21no—
61C23×D4—28no—
62C23×D4—22no—
63C23×D4—21no—
64C23×D4—24no—
65C23×D4—24no—
66C23×D4—44no—
67C23×D4C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2314yes—
70C24×C4—21no—
71C24×C4—21no—
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4C2311yescharacteristic
75C22×D8—28no—
76C22×D8—28no—
77C22×D8—44no—
78C22×D8C2318yes—
79C23×C8C2312yes—
80C23×C8C2312yes—
81C2×#121—22no—
82C2×#121—24no—
83C2×#121D418yes—
84C2×#121C2314yes—
85C2×C4×D4—28no—
86C2×C4×D4—22no—
87C2×C4×D4—44no—
88C2×C4×D4—48no—
89C2×C4×D4C2314yescomplemented
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—48no—
112C22×C22⋊C4—48no—
113C22×C42—21no—
114C22×C42—44no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D414yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121#481—232no—
122#447—232no—
123C2×#106—28no—
124C2×#106C2318yescomplemented
125C2×C4⋊C8—44no—
126C2×C4⋊C8C2318yescomplemented
127C2×#102—28no—
128C2×#102—28no—
129C2×#102—28no—
130C2×#102—44no—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C2×#102C2318yes—
134C2×#102C2318yescomplemented
135C2×C22⋊C8—24no—
136C2×C22⋊C8—24no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139#390—24no—
140#390—24no—
141#390—28no—
142#390—28no—
143#390—22no—
144#390C2318yes—
145#390C2318yes—
146#390C2314yescomplemented
147#389—22no—
148#389C2314yescomplemented
149#388—28no—
150#386—28no—
151#380—22no—
152#380C2314yes—
153C4×C4⋊C4—22no—
154C4×C4⋊C4C2314yescomplemented
155C4×C22⋊C4—28no—
156C2×#95—21no—
157C2×#95—21no—
158C2×#95—21no—
159#340C2318yescomplemented
160#340C2318yescomplemented
161#336C2318yescomplemented
162C25—28no—
163C25—24no—
164C25—21no—
165C25—22no—
166C25—44no—
167C25—44no—
168C25—44no—
169C25—44no—
170C25—44no—
171C25—44no—
172C25C2×D411yescharacteristic
173C22×D4—28no—
174C22×D4—216no—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—216no—
178C22×D4—24no—
179C22×D4—21no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—48no—
192C22×D4—432no—
193C22×D4—432no—
194C22×D4—44no—
195C22×D4—48no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—48no—
200C22×D4C2×D414yes—
201C22×D4#4118yes—
202C22×D4C2412yes—
203C22×D4C2414yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—21no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D414yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4#4114yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C2×D8—216no—
237C2×D8—216no—
238C2×D8—48no—
239C2×D8—432no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8C2414yes—
247C22×C8C2414yes—
248#121—216no—
249#121—28no—
250#121C2×D4116yes—
251C4×D4—48no—
252C4×D4—48no—
253C4×D4—432no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—432no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—48no—
299C2×C42—48no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2412yes—
308C2×C42C2412yes—
309#106—216no—
310C4⋊C8—48no—
311#102—216no—
312#102—216no—
313#102—216no—
314#102—48no—
315C22⋊C8—28no—
316C22⋊C8—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—216no—
325C24—24no—
326C24—28no—
327C24—22no—
328C24—24no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—432no—
333C24—432no—
334C24—432no—
335C24—48no—
336C24—44no—
337C24—42no—
338C24—42no—
339C24—44no—
340C24—42no—
341C24—48no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—48no—
349C24—44no—
350C24—816no—
351C24—816no—
352C24—816no—
353C24—816no—
354C24#12114yes—
355C24#13514yes—
356C24AGL(1,ℤ/8)14yes—
357C24C22×D412yes—
358C24C22×D411yescharacteristic
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—432no—
375C2×D4—432no—
376C2×D4—48no—
377C2×D4—432no—
378C2×D4—464no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—432no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4C22×D414yes—
391C2×D4C2×#4118yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—816no—
441C22×C4—816no—
442C22×C4#12114yes—
443C22×C4#12118yes—
444C22×C4AGL(1,ℤ/8)14yes—
445C22×C4AGL(1,ℤ/8)14yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4C2511yescharacteristic
459D8—432no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—48no—
467C2×C8—48no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4C22×D414yes—
480C22⋊C4—216no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—44no—
494C42—44no—
495C42—48no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—28no—
500C23—22no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—28no—
508C23—432no—
509C23—432no—
510C23—432no—
511C23—416no—
512C23—416no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—416no—
520C23—48no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—816no—
530C23—816no—
531C23—8128no—
532C23—816no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—88no—
537C23—84no—
538C23—88no—
539C23—816no—
540C23—88no—
541C23—88no—
542C23#463116yes—
543C23C2×#12114yes—
544C23C2×#12112yes—
545C23#52512yes—
546C23#53412yes—
547C23C2×#13514yes—
548C23C2×#13512yes—
549C23C2×AGL(1,ℤ/8)14yes—
550C23C2×AGL(1,ℤ/8)12yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23C2×AGL(1,ℤ/8)12yes—
553C23C23×D411yescharacteristic
554D4—28no—
555D4—28no—
556D4—28no—
557D4—432no—
558D4—432no—
559D4—44no—
560D4—44no—
561D4—44no—
562D4—48no—
563D4—416no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—816no—
600C2×C4—816no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4C2×#12114yes—
605C2×C4C2×#12112yes—
606C2×C4#52512yes—
607C2×C4#53412yes—
608C2×C4#54618yes—
609C2×C4#55118yes—
610C2×C4C2×AGL(1,ℤ/8)14yes—
611C2×C4C2×AGL(1,ℤ/8)14yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C22×#4112yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)12yes—
616C8—44no—
617C8—44no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—232no—
624C22—24no—
625C22—48no—
626C22—44no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—864no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22—816no—
639C22—832no—
640C22—84no—
641C22—84no—
642C22#253414yes—
643C22C2×#46318yes—
644C22C2×#46314yes—
645C22#303814yes—
646C22#310514yes—
647C22#310914yes—
648C22#318814yes—
649C22#319214yes—
650C22C22×#12111yescharacteristic
651C22C2×#52511yescharacteristic
652C22C2×#53411yescharacteristic
653C22C22×#13511yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C22×AGL(1,ℤ/8)11yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C4—24no—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4#253414yes—
667C4C2×#54612yes—
668C4C2×#55112yes—
669C2—28no—
670C2—84no—
671C2—88no—
672C2#2811718yesdirect product factor
673C2C2×#253411yescharacteristic
674C2C22×#46311yescharacteristic
675C2C2×#303811yescharacteristic
676C2C2×#310511yescharacteristic
677C2C2×#310911yescharacteristic
678C2C2×#318811yescharacteristic
679C2C2×#319211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#481C2semidirect
3. C2×C4×#121C2semidirect
4. C22×#390C2semidirect
5. #28117C2direct (N × H)
6. C2×#2065C2semidirect
7. C2×#2054C2semidirect
8. C2×#1926C2semidirect
9. C2×#1879C2semidirect
10. C2×#1824C2semidirect
11. C2×#1797C2semidirect
12. C2×#1786C2semidirect
17. C22×C4×D4C22semidirect
23. C2×#481C22semidirect
26. C22×#106C22semidirect
27. C22×C4⋊C8C22semidirect
31. C22×#102C22semidirect
35. C4×#121C22semidirect
41. C2×#390C22semidirect
42. C2×#390C22semidirect
43. C2×#389C22semidirect
47. C2×C4×C4⋊C4C22semidirect
49. #2065C22semidirect
50. #2054C22semidirect
51. #1926C22semidirect
52. #1879C22semidirect
53. #1824C22semidirect
54. #1797C22semidirect
55. #1786C22semidirect
56. C2×#340C22semidirect
57. C2×#340C22semidirect
58. C2×#336C22semidirect
89. C2×C4×D4C23semidirect
124. C2×#106C23semidirect
126. C2×C4⋊C8C23semidirect
134. C2×#102C23semidirect
146. #390C23semidirect
148. #389C23semidirect
154. C4×C4⋊C4C23semidirect
159. #340C23semidirect
160. #340C23semidirect
161. #336C23semidirect
672. C2#28117direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4812
C22×#4472
C2×C4×#1212
C22×#3902
#281172
C2×#20652
C2×#20542
C2×#19262
C2×#18792
C2×#18242
C2×#17972
C2×#17862