Finite Group Databasebeta

← C2×#24506 #6451965

Subgroups — C2×#24506 #6451965

Order 512.

4052 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#447C211yesmaximal, characteristic
2C22×#432C211yesmaximal, characteristic, complemented
3C2×#2854C211yesmaximal, characteristic, complemented
4C22×#390C211yesmaximal, characteristic, complemented
5#24506C2116yesmaximal, direct product factor
6C2×#2065C212yesmaximal, complemented
7C2×#2054C212yesmaximal, complemented
8C2×#1972C212yesmaximal, complemented
9C2×#1936C212yesmaximal, complemented
10C2×#1874C211yesmaximal, characteristic, complemented
11C2×#1853C211yesmaximal, characteristic, complemented
12C2×#1786C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C23×D8C2212yes—
15C22×#123—21no—
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4—21no—
19C23×C22⋊C4—21no—
20C23×C42C2211yescharacteristic
21C2×#447—28no—
22C2×#447C2218yes—
23C2×#432—28no—
24C2×#432C2218yescomplemented
25C22×#106C2212yescomplemented
26C22×C4⋊C8C2212yescomplemented
27C22×#102C2212yes—
28C22×#102C2212yes—
29C22×#102C2212yes—
30C22×#102C2212yescomplemented
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2212yescomplemented
34#2854—24no—
35#2854C2218yescomplemented
36C2×#390—21no—
37C2×#390—21no—
38C2×#390—24no—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#390C2218yescomplemented
42C2×#390C2211yescharacteristic, complemented
43C2×#390C2211yescharacteristic, complemented
44C2×#389C2211yescharacteristic, complemented
45C2×#389C2211yescharacteristic, complemented
46C2×#388—21no—
47C2×#388—21no—
48C2×#386—21no—
49C2×#380C2211yescharacteristic
50#2065C22116yescomplemented
51#2054C22116yescomplemented
52#1972C22116yescomplemented
53#1936C22116yescomplemented
54#1874C2218yescomplemented
55#1853C2218yescomplemented
56#1786C2218yescomplemented
57C2×#340C2212yescomplemented
58C2×#340C2212yescomplemented
59C26—21no—
60C23×D4—28no—
61C23×D4—21no—
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×C4D411yescharacteristic
72C24×C4D411yescharacteristic
73C24×C4C2311yescharacteristic
74C22×D8—28no—
75C22×D8—28no—
76C22×D8—44no—
77C22×D8C2318yes—
78C23×C8C2312yes—
79C23×C8C2312yes—
80C2×#123—28no—
81C2×#123—48no—
82C2×#121—22no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121C2314yes—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—48no—
109C22×C22⋊C4—48no—
110C22×C42—24no—
111C22×C42—24no—
112C22×C42D414yes—
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#447—232no—
119#432—232no—
120C2×#106—44no—
121C2×#106C2318yescomplemented
122C2×C4⋊C8—28no—
123C2×C4⋊C8C2318yescomplemented
124C2×#102—28no—
125C2×#102—28no—
126C2×#102—44no—
127C2×#102—44no—
128C2×#102C2318yes—
129C2×#102C2318yes—
130C2×#102C2318yes—
131C2×#102C2318yescomplemented
132C2×C22⋊C8—24no—
133C2×C22⋊C8D418yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97—28no—
137C2×#97C2318yescomplemented
138#390—22no—
139#390—22no—
140#390—24no—
141#390—24no—
142#390—28no—
143#390—28no—
144#390C2318yes—
145#390C2318yes—
146#390C2314yescomplemented
147#390C2314yescomplemented
148#389—22no—
149#389—22no—
150#389C2314yescomplemented
151#389C2314yescomplemented
152#388—28no—
153#388—28no—
154#386—28no—
155#380—22no—
156#380C2314yes—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95—21no—
160#340C2318yescomplemented
161#340C2318yescomplemented
162C25—28no—
163C25—24no—
164C25—22no—
165C25—21no—
166C25—44no—
167C25—44no—
168C25—44no—
169C25—44no—
170C25—44no—
171C25—44no—
172C25C2×D411yescharacteristic
173C22×D4—28no—
174C22×D4—216no—
175C22×D4—216no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—28no—
182C22×D4—21no—
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—44no—
200C22×D4—48no—
201C22×D4#4118yes—
202C22×D4C2412yes—
203C22×D4C2414yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D414yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D414yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D414yes—
228C23×C4C2×D411yescharacteristic
229C23×C4#4114yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C2×D8—216no—
233C2×D8—216no—
234C2×D8—48no—
235C2×D8—432no—
236C22×C8—24no—
237C22×C8—24no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8C2×D418yes—
242C22×C8C2414yes—
243C22×C8C2414yes—
244#123—432no—
245#121—28no—
246#121—28no—
247#121—48no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—432no—
285C2×C22⋊C4—432no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—48no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—216no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—22no—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306C2×C42C2412yes—
307#106—48no—
308C4⋊C8—216no—
309#102—216no—
310#102—216no—
311#102—48no—
312#102—48no—
313C22⋊C8—28no—
314C22⋊C8C2×D4116yes—
315#97—216no—
316#95—24no—
317#95—24no—
318#95—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—216no—
324C24—28no—
325C24—24no—
326C24—22no—
327C24—48no—
328C24—432no—
329C24—48no—
330C24—432no—
331C24—432no—
332C24—48no—
333C24—48no—
334C24—44no—
335C24—44no—
336C24—48no—
337C24—44no—
338C24—44no—
339C24—48no—
340C24—44no—
341C24—44no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—816no—
350C24—816no—
351C24—816no—
352C24—816no—
353C24#12114yes—
354C24AGL(1,ℤ/8)14yes—
355C24AGL(1,ℤ/8)14yes—
356C24C22×D412yes—
357C24C22×D411yescharacteristic
358C2×D4—216no—
359C2×D4—28no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—432no—
372C2×D4—432no—
373C2×D4—432no—
374C2×D4—48no—
375C2×D4—464no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—816no—
390C2×D4C2×#4118yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—816no—
438C22×C4#12114yes—
439C22×C4#12118yes—
440C22×C4AGL(1,ℤ/8)14yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4C2511yescharacteristic
454D8—432no—
455C2×C8—216no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—48no—
462C2×C8C22×D418yes—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C42—28no—
487C42—24no—
488C42—28no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—432no—
506C23—432no—
507C23—432no—
508C23—416no—
509C23—416no—
510C23—44no—
511C23—416no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—816no—
525C23—816no—
526C23—816no—
527C23—8128no—
528C23—88no—
529C23—88no—
530C23—816no—
531C23—88no—
532C23—88no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23#469116yes—
541C23C2×#12114yes—
542C23C2×#12112yes—
543C23#52512yes—
544C23#53512yes—
545C23C2×AGL(1,ℤ/8)14yes—
546C23C2×AGL(1,ℤ/8)14yes—
547C23C2×AGL(1,ℤ/8)12yes—
548C23C2×AGL(1,ℤ/8)12yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23#57512yes—
551C23C23×D411yescharacteristic
552D4—28no—
553D4—28no—
554D4—432no—
555D4—416no—
556D4—44no—
557D4—44no—
558D4—44no—
559D4—44no—
560D4—816no—
561D4—84no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—216no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—816no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—88no—
601C2×C4—84no—
602C2×C4C2×#12114yes—
603C2×C4C2×#12112yes—
604C2×C4#52512yes—
605C2×C4#53512yes—
606C2×C4#54618yes—
607C2×C4#55118yes—
608C2×C4C2×AGL(1,ℤ/8)14yes—
609C2×C4C23×D411yescharacteristic
610C2×C4C22×#4112yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)12yes—
613C8—28no—
614C8—44no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—232no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—864no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—832no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—168no—
639C22#241114yes—
640C22C2×#46918yes—
641C22C2×#46914yes—
642C22#305514yes—
643C22#310914yes—
644C22#311414yes—
645C22#321314yes—
646C22#321714yes—
647C22C22×#12111yescharacteristic
648C22C2×#52511yescharacteristic
649C22C2×#53511yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C22×AGL(1,ℤ/8)11yescharacteristic
652C22C22×AGL(1,ℤ/8)11yescharacteristic
653C22C2×#57511yescharacteristic
654C4—24no—
655C4—24no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4#241114yes—
664C4C2×#54612yes—
665C4C2×#55112yes—
666C2—28no—
667C2—88no—
668C2—162no—
669C2#2450618yesdirect product factor
670C2C2×#241111yescharacteristic
671C2C22×#46911yescharacteristic
672C2C2×#305511yescharacteristic
673C2C2×#311411yescharacteristic
674C2C2×#310911yescharacteristic
675C2C2×#321311yescharacteristic
676C2C2×#321711yescharacteristic
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
2. C22×#432C2semidirect
3. C2×#2854C2semidirect
4. C22×#390C2semidirect
5. #24506C2direct (N × H)
6. C2×#2065C2semidirect
7. C2×#2054C2semidirect
8. C2×#1972C2semidirect
9. C2×#1936C2semidirect
10. C2×#1874C2semidirect
11. C2×#1853C2semidirect
12. C2×#1786C2semidirect
24. C2×#432C22semidirect
25. C22×#106C22semidirect
26. C22×C4⋊C8C22semidirect
30. C22×#102C22semidirect
33. C22×#97C22semidirect
35. #2854C22semidirect
41. C2×#390C22semidirect
42. C2×#390C22semidirect
43. C2×#390C22semidirect
44. C2×#389C22semidirect
45. C2×#389C22semidirect
50. #2065C22semidirect
51. #2054C22semidirect
52. #1972C22semidirect
53. #1936C22semidirect
54. #1874C22semidirect
55. #1853C22semidirect
56. #1786C22semidirect
57. C2×#340C22semidirect
58. C2×#340C22semidirect
121. C2×#106C23semidirect
123. C2×C4⋊C8C23semidirect
131. C2×#102C23semidirect
137. C2×#97C23semidirect
146. #390C23semidirect
147. #390C23semidirect
150. #389C23semidirect
151. #389C23semidirect
160. #340C23semidirect
161. #340C23semidirect
669. C2#24506direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4472
C22×#4322
C2×#28542
C22×#3902
#245062
C2×#20652
C2×#20542
C2×#19722
C2×#19362
C2×#18742
C2×#18532
C2×#17862