Finite Group Databasebeta

← C2×#24116 #6448564

Subgroups — C2×#24116 #6448564

Order 512.

4898 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic, complemented
3C22×C4×D8C211yesmaximal, characteristic, complemented
4C2×#2537C211yesmaximal, characteristic, complemented
5#24116C2116yesmaximal, direct product factor
6C2×#2033C212yesmaximal, complemented
7C2×#1999C212yesmaximal, complemented
8C2×#1990C211yesmaximal, characteristic, complemented
9C2×#1954C211yesmaximal, characteristic, complemented
10C2×#1874C212yesmaximal, complemented
11C2×#1852C212yesmaximal, complemented
12C2×#1802C211yesmaximal, characteristic, complemented
13C23×#41—21no—
14C23×D8C2211yescharacteristic
15C2×#547—28no—
16C2×#547—24no—
17C2×#547—28no—
18C2×#547C2218yescomplemented
19C22×C4⋊Q8C2211yescharacteristic
20C22×#123—22no—
21C22×C22⋊Q8—21no—
22C22×#121—21no—
23C22×#121C2212yes—
24C22×#121C2212yes—
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29C2×#474—28no—
30C2×#474C2218yescomplemented
31C2×C4×D8—28no—
32C2×C4×D8C2218yescomplemented
33C22×#107C2212yes—
34C22×#107C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C22×#102C2212yes—
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2212yes—
40C22×C22⋊C8C2212yes—
41C22×C22⋊C8C2212yes—
42C22×C4×C8C2211yescharacteristic
43#2537—24no—
44#2537C2218yescomplemented
45C2×#394—22no—
46C2×#384C2211yescharacteristic
47C2×#380—21no—
48C2×#380C2212yes—
49#2033C22116yescomplemented
50#1999C22116yescomplemented
51#1990C2218yescomplemented
52#1954C2218yescomplemented
53#1874C22116yescomplemented
54#1852C22116yescomplemented
55#1802C2218yescomplemented
56C2×#340C2212yes—
57C2×#340C2212yes—
58C2×#340C2212yes—
59C22×#41—216no—
60C22×#41—28no—
61C22×#41—48no—
62C22×#41—48no—
63C23×Q8—21no—
64C23×D4—22no—
65C23×D4—22no—
66C23×D4D411yescharacteristic
67C23×D4C2311yescharacteristic
68C23×D4C2311yescharacteristic
69C24×C4—22no—
70C24×C4—21no—
71C24×C4—21no—
72C24×C4C2312yes—
73C22×D8—24no—
74C22×D8—24no—
75C22×D8—24no—
76C22×D8C2314yes—
77C23×C8C2311yescharacteristic
78C23×C8C2312yes—
79C23×C8C2311yescharacteristic
80#547—264no—
81#547—232no—
82#547—232no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8D418yescomplemented
86C2×C4⋊Q8C2314yes—
87C2×#123—232no—
88C2×#123—216no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—28no—
91C2×#121—28no—
92C2×#121—24no—
93C2×#121—28no—
94C2×#121—24no—
95C2×#121—28no—
96C2×#121—28no—
97C2×#121—44no—
98C2×#121—48no—
99C2×#121C2318yes—
100C2×#121C2318yes—
101C2×C4×D4—22no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4D414yes—
109C2×C4×D4D418yescomplemented
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C2×C4×D4C2314yes—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—28no—
117C22×C4⋊C4—28no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2318yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4D412yes—
132C22×C42C2311yescharacteristic
133#474—232no—
134C4×D8—232no—
135C2×#107—24no—
136C2×#107—28no—
137C2×#107C2314yes—
138C2×#107C2318yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8C2314yes—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—28no—
144C2×#102—44no—
145C2×#102C2314yes—
146C2×#102C2318yes—
147C2×#102C2314yes—
148C2×#102C2318yes—
149C2×C22⋊C8—28no—
150C2×C22⋊C8—44no—
151C2×C22⋊C8C2318yes—
152C2×C22⋊C8C2318yes—
153C2×C4×C8—24no—
154C2×C4×C8C2314yes—
155#394—216no—
156#384—22no—
157#384C2314yes—
158#380—24no—
159#380—28no—
160#380C2318yes—
161C2×#95—22no—
162#340C2318yes—
163#340C2318yes—
164#340C2318yes—
165C25—21no—
166C25—21no—
167C25C2×D412yes—
168C2×#41—264no—
169C2×#41—432no—
170C2×#41—4128no—
171C2×#41—432no—
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×D4—216no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—48no—
194C22×D4—48no—
195C22×D4—48no—
196C22×D4—48no—
197C22×D4—84no—
198C22×D4D818yescomplemented
199C22×D4C2×D414yes—
200C22×D4C2×D414yes—
201C22×D4#4114yescomplemented
202C22×D4C2412yes—
203C22×D4C2412yes—
204C23×C4—216no—
205C23×C4—28no—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—48no—
220C23×C4—48no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C2×D8—232no—
233C2×D8—28no—
234C2×D8—28no—
235C2×D8—28no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—22no—
242C22×C8—24no—
243C22×C8—22no—
244C22×C8#4114yescomplemented
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—216no—
249C4⋊Q8—28no—
250C4⋊Q8C2×D4116yescomplemented
251#123—2128no—
252C22⋊Q8—264no—
253#121—232no—
254#121—216no—
255#121—216no—
256#121—216no—
257#121—48no—
258#121—416no—
259#121—432no—
260C4×D4—216no—
261C4×D4—216no—
262C4×D4—28no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—48no—
268C4×D4C2×D4116yescomplemented
269C2×C4⋊C4—216no—
270C2×C4⋊C4—232no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4C2×D418yes—
310C2×C42C2×D412yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2412yes—
314#107—28no—
315#107—216no—
316C4⋊C8—28no—
317#102—28no—
318#102—28no—
319#102—216no—
320#102—48no—
321C22⋊C8—216no—
322C22⋊C8—48no—
323C4×C8—28no—
324#95—28no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24C2×D818yes—
348C24C22×D414yes—
349C24C22×D411yescharacteristic
350#41—4256no—
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—432no—
355C2×D4—232no—
356C2×D4—28no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—232no—
375C2×D4—464no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—88no—
389C2×D4—832no—
390C2×D4C2×D8116yescomplemented
391C2×D4C22×D414yes—
392C2×D4C2×#4114yescomplemented
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—232no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—464no—
416C22×C4—416no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—416no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—432no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—816no—
439C22×C4#12014yes—
440C22×C4#12014yes—
441C22×C4C2×D814yes—
442C22×C4AGL(1,ℤ/8)14yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454D8—232no—
455C2×C8—28no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—44no—
465C2×C8—48no—
466C2×C8C2×#4114yescomplemented
467C4⋊C4—232no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—28no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4C22×D414yes—
483C4⋊C4C22×D418yes—
484C22⋊C4—232no—
485C22⋊C4—216no—
486C22⋊C4—216no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C42#12018yes—
491C42C22×D414yes—
492C42C22×D414yes—
493C23—216no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—216no—
498C23—216no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23C2×#12012yes—
535C23#54612yes—
536C23#54712yes—
537C23C22×D818yes—
538C23C22×D812yes—
539C23C2×AGL(1,ℤ/8)12yes—
540C23#57712yes—
541C23#57812yes—
542C23C23×D411yescharacteristic
543Q8—432no—
544D4—24no—
545D4—24no—
546D4—24no—
547D4—24no—
548D4—232no—
549D4—464no—
550D4—48no—
551D4—48no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—832no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—464no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—416no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—432no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—84no—
586C2×C4—88no—
587C2×C4—816no—
588C2×C4#447116yes—
589C2×C4C2×#12014yes—
590C2×C4C2×#12012yes—
591C2×C4C2×#12014yes—
592C2×C4#52514yes—
593C2×C4#53714yes—
594C2×C4#54612yes—
595C2×C4#54712yes—
596C2×C4#54814yes—
597C2×C4#55014yes—
598C2×C4C22×D814yes—
599C2×C4C2×AGL(1,ℤ/8)14yes—
600C2×C4C23×D411yescharacteristic
601C2×C4C23×D411yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C8—24no—
607C8—42no—
608C8—44no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—432no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—816no—
626C22—816no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#240714yes—
634C22C2×#44714yes—
635C22#301314yes—
636C22#329414yes—
637C22#333014yes—
638C22#333414yes—
639C22#335414yes—
640C22C22×#12011yescharacteristic
641C22C2×#54611yescharacteristic
642C22C2×#54711yescharacteristic
643C22C23×D811yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57811yescharacteristic
647C4—24no—
648C4—44no—
649C4—44no—
650C4—48no—
651C4—84no—
652C4—82no—
653C4#240714yes—
654C4C2×#44718yes—
655C4C2×#52511yescharacteristic
656C4C2×#53711yescharacteristic
657C4C2×#54811yescharacteristic
658C4C2×#55011yescharacteristic
659C2—48no—
660C2—84no—
661C2—84no—
662C2#2411618yesdirect product factor
663C2C2×#240711yescharacteristic
664C2C22×#44711yescharacteristic
665C2C2×#301311yescharacteristic
666C2C2×#329411yescharacteristic
667C2C2×#333011yescharacteristic
668C2C2×#333411yescharacteristic
669C2C2×#335411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
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
1. C22×#547C2semidirect
2. C22×#474C2semidirect
3. C22×C4×D8C2semidirect
4. C2×#2537C2semidirect
5. #24116C2direct (N × H)
6. C2×#2033C2semidirect
7. C2×#1999C2semidirect
8. C2×#1990C2semidirect
9. C2×#1954C2semidirect
10. C2×#1874C2semidirect
11. C2×#1852C2semidirect
12. C2×#1802C2semidirect
18. C2×#547C22semidirect
30. C2×#474C22semidirect
32. C2×C4×D8C22semidirect
44. #2537C22semidirect
49. #2033C22semidirect
50. #1999C22semidirect
51. #1990C22semidirect
52. #1954C22semidirect
53. #1874C22semidirect
54. #1852C22semidirect
55. #1802C22semidirect
85. C2×C4⋊Q8D4semidirect
109. C2×C4×D4D4semidirect
198. C22×D4D8semidirect
201. C22×D4#41semidirect
244. C22×C8#41semidirect
250. C4⋊Q8C2×D4semidirect
268. C4×D4C2×D4semidirect
390. C2×D4C2×D8semidirect
392. C2×D4C2×#41semidirect
466. C2×C8C2×#41semidirect
662. C2#24116direct (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×#4742
C22×C4×D82
C2×#25372
#241162
C2×#20332
C2×#19992
C2×#19902
C2×#19542
C2×#18742
C2×#18522
C2×#18022