Finite Group Databasebeta

← C2×#22776 #6380481

Subgroups — C2×#22776 #6380481

Order 512.

2532 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×#440C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic
4#22776C2116yesmaximal, direct product factor
5C2×#1935C211yesmaximal, characteristic, complemented
6C2×#1933C211yesmaximal, characteristic, complemented
7C2×#1921C211yesmaximal, characteristic, complemented
8C2×#1918C211yesmaximal, characteristic, complemented
9C2×#1877C211yesmaximal, characteristic
10C2×#1877C211yesmaximal, characteristic
11C2×#1836C211yesmaximal, characteristic, complemented
12C2×#1805C211yesmaximal, characteristic, complemented
13C2×#1802C211yesmaximal, characteristic
14C2×#1771C211yesmaximal, characteristic, complemented
15C2×#1762C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#440C4116yescomplemented
23C2×#440C2218yes—
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2355—24no—
36#2355C2218yescomplemented
37#2349—24no—
38#2349C2218yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387—21no—
44C2×#387—21no—
45C2×#385—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×#382C2211yescharacteristic
50C2×#380—21no—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C22×#95—21no—
56#1935C2218yescomplemented
57#1933C2218yescomplemented
58#1921C2218yescomplemented
59#1918C2218yescomplemented
60#1877C2218yes—
61#1877C2218yes—
62#1836C2218yescomplemented
63#1805C2218yescomplemented
64#1802C2218yes—
65#1771C2218yescomplemented
66#1762C2218yes—
67C4×#102C2218yes—
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4—21no—
82C24×C4—21no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16C2×C418yes—
86C22×SD16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C4×Q8—22no—
92C2×C4×Q8—24no—
93C2×C4×Q8C2×C418yescomplemented
94C2×C4×Q8C2314yes—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C2×C4×D4C2×C418yes—
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C2×C4×D4C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#440C2×C4164yescomplemented
130C2×#107C2×C418yescomplemented
131C2×#107C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2314yes—
134C2×#103—42no—
135C2×#103—42no—
136C2×#103C2×C418yescomplemented
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102C2×C418yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97—42no—
147C2×#97C2×C418yescomplemented
148C2×#97C2314yes—
149C2×#97C2314yes—
150C2×C4×C8—42no—
151C2×C4×C8C2314yes—
152#391—22no—
153#391—22no—
154#391C2314yes—
155#391C2314yes—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2314yes—
160#387—28no—
161#387—28no—
162#385—28no—
163#385—28no—
164#384—22no—
165#384C2314yes—
166#382—28no—
167#382—22no—
168#382C2314yes—
169#380—28no—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D414yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—28no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C22×C414yes—
208C22×Q8C2412yes—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4C22×C414yes—
214C22×D4#4114yes—
215C22×D4#4114yes—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—82no—
233C23×C4—82no—
234C23×C4—82no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×SD16—232no—
244C2×SD16—28no—
245C2×SD16—28no—
246C2×SD16C22×C4116yes—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8C22×C414yes—
258C22×C8C22×C414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4×Q8—216no—
264C4×Q8—28no—
265C4×Q8C22×C4116yescomplemented
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—216no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4C22×C4116yes—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C22×C414yes—
304C2×C4⋊C4C22×C414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—82no—
328C2×C42—82no—
329C2×C42C22×C414yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42#4114yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107C22×C4116yescomplemented
338C4⋊C8—28no—
339#103—44no—
340#103—44no—
341#103C22×C4116yescomplemented
342#102—28no—
343#102—28no—
344#102C22×C4116yes—
345#97—44no—
346#97C22×C4116yescomplemented
347C4×C8—44no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8C23×C414yes—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4#117116yes—
392C2×D4C23×C414yes—
393C2×D4C2×#4114yes—
394C2×D4C2×#4114yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4—816no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—816no—
449C22×C4—816no—
450C22×C4C4×D414yes—
451C22×C4C4×D414yes—
452C22×C4#13714yes—
453C22×C4C23×C412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467SD16—232no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8C23×C414yes—
483C2×C8C23×C414yes—
484C4⋊C4—216no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4C23×C414yes—
508C4⋊C4C23×C414yes—
509C22⋊C4—216no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C42—216no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—84no—
520C42—84no—
521C42C4×D418yes—
522C42C23×C414yes—
523C42C2×#4114yes—
524C23—216no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C2×C4×D412yes—
539C23#54612yes—
540C23#54812yes—
541C23C2×#13712yes—
542C23#57512yes—
543C23#57612yes—
544C23#57712yes—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—48no—
549Q8—42no—
550Q8—42no—
551D4—216no—
552D4—24no—
553D4—24no—
554D4—24no—
555D4C2×#11718yes—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4—816no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4C2×#11714yes—
609C2×C4C2×C4×D414yes—
610C2×C4C2×C4×D414yes—
611C2×C4C2×C4×D412yes—
612C2×C4#52014yes—
613C2×C4#54612yes—
614C2×C4#54812yes—
615C2×C4#55214yes—
616C2×C4#55314yes—
617C2×C4C2×#13714yes—
618C2×C4C24×C411yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C8—24no—
625C8—24no—
626C8—42no—
627C8—42no—
628C22—28no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—432no—
634C22—44no—
635C22—44no—
636C22#235214yes—
637C22C4×#13714yes—
638C22#294614yes—
639C22#331114yes—
640C22#331414yes—
641C22#337014yes—
642C22#337214yes—
643C22C22×C4×D411yescharacteristic
644C22C2×#54611yescharacteristic
645C22C2×#54811yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57511yescharacteristic
648C22C2×#57611yescharacteristic
649C22C2×#57711yescharacteristic
650C4—24no—
651C4—24no—
652C4—24no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#235214yes—
664C4C22×#11711yescharacteristic
665C4C2×#52011yescharacteristic
666C4C2×#55211yescharacteristic
667C4C2×#55311yescharacteristic
668C2—48no—
669C2#2277618yesdirect product factor
670C2C2×#235211yescharacteristic
671C2C2×C4×#13711yescharacteristic
672C2C2×#294611yescharacteristic
673C2C2×#331411yescharacteristic
674C2C2×#331111yescharacteristic
675C2C2×#337211yescharacteristic
676C2C2×#337011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
4. #22776C2direct (N × H)
5. C2×#1935C2semidirect
6. C2×#1933C2semidirect
7. C2×#1921C2semidirect
8. C2×#1918C2semidirect
11. C2×#1836C2semidirect
12. C2×#1805C2semidirect
14. C2×#1771C2semidirect
22. C2×#440C4semidirect
36. #2355C22semidirect
56. #1935C22semidirect
57. #1933C22semidirect
58. #1921C22semidirect
59. #1918C22semidirect
62. #1836C22semidirect
63. #1805C22semidirect
65. #1771C22semidirect
93. C2×C4×Q8C2×C4semidirect
129. #440C2×C4semidirect
130. C2×#107C2×C4semidirect
136. C2×#103C2×C4semidirect
147. C2×#97C2×C4semidirect
265. C4×Q8C22×C4semidirect
337. #107C22×C4semidirect
341. #103C22×C4semidirect
346. #97C22×C4semidirect
669. C2#22776direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#23552
C2×#23492
#227762
C2×#19352
C2×#19332
C2×#19212
C2×#19182
C2×#18772
C2×#18362
C2×#18052
C2×#18022
C2×#17712
C2×#17622
C2×C4×#1022