Finite Group Databasebeta

← C2×#22781 #6380484

Subgroups — C2×#22781 #6380484

Order 512.

2532 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic, complemented
4#22781C2116yesmaximal, direct product factor
5C2×#1936C211yesmaximal, characteristic
6C2×#1934C211yesmaximal, characteristic
7C2×#1921C211yesmaximal, characteristic, complemented
8C2×#1918C211yesmaximal, characteristic, complemented
9C2×#1880C211yesmaximal, characteristic
10C2×#1880C211yesmaximal, characteristic
11C2×#1836C211yesmaximal, characteristic, complemented
12C2×#1803C211yesmaximal, characteristic, complemented
13C2×#1804C211yesmaximal, characteristic
14C2×#1771C211yesmaximal, characteristic, complemented
15C2×#1763C211yesmaximal, characteristic, complemented
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117—21no—
23C2×#440C4116yescomplemented
24C2×#440C2218yes—
25C22×#107C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2354—24no—
37#2354C2218yes—
38#2351—24no—
39#2351C2218yescomplemented
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#388—21no—
48C2×#384C2211yescharacteristic
49C2×#382—21no—
50C2×#382C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#1936C2218yes—
57#1934C2218yes—
58#1921C2218yescomplemented
59#1918C2218yescomplemented
60#1880C2218yes—
61#1880C2218yes—
62#1836C2218yescomplemented
63#1804C2218yes—
64#1803C2218yescomplemented
65#1771C2218yescomplemented
66#1763C2218yescomplemented
67C4×#103C2218yescomplemented
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—
81C22×SD16—24no—
82C22×SD16—24no—
83C22×SD16C2×C418yes—
84C22×SD16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—42no—
96C2×C4×Q8—42no—
97C2×C4×Q8C2×C418yes—
98C2×C4×Q8C2314yes—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—24no—
103C2×C4×D4C2×C418yes—
104C2×C4×D4C2314yes—
105C2×#117—28no—
106C2×#117—48no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#440C2×C4164yescomplemented
136C2×#107C2×C418yes—
137C2×#107C2314yes—
138C2×C4⋊C8—24no—
139C2×C4⋊C8C2314yes—
140C2×#103—24no—
141C2×#103—24no—
142C2×#103C2×C418yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102C2×C418yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#97—42no—
153C2×#97C2×C418yes—
154C2×#97C2314yes—
155C2×#97C2314yes—
156C2×C4×C8—42no—
157C2×C4×C8C2314yes—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#390C2314yes—
162#389—28no—
163#389—28no—
164#389—22no—
165#389—22no—
166#389C2314yes—
167#389C2314yes—
168#388—28no—
169#388—28no—
170#384—22no—
171#384C2314yes—
172#382—22no—
173#382—28no—
174#382C2314yes—
175#380—28no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—28no—
179C4×C4⋊C4D414yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95D411yescharacteristic
190C2×#95D411yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8C22×C414yes—
207C22×Q8#4114yes—
208C22×Q8#4114yes—
209C22×Q8C2412yes—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4C22×C414yes—
217C22×D4C2412yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C2×SD16—232no—
237C2×SD16—28no—
238C2×SD16—28no—
239C2×SD16C22×C4116yes—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8C22×C414yes—
251C22×C8C22×C414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4×Q8—216no—
257C4×Q8—28no—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×Q8C22×C4116yes—
265C4×D4—216no—
266C4×D4—28no—
267C4×D4C22×C4116yes—
268#117—432no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C22×C414yes—
302C2×C4⋊C4C22×C414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4—82no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C22×C414yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42#4114yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107C22×C4116yes—
344C4⋊C8—28no—
345#103—28no—
346#103—28no—
347#103C22×C4116yes—
348#102—44no—
349#102—44no—
350#102C22×C4116yes—
351#97—44no—
352#97C22×C4116yes—
353C4×C8—44no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24C22×D411yescharacteristic
371C2×Q8—24no—
372C2×Q8—28no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8#117116yescomplemented
381C2×Q8C23×C414yes—
382C2×Q8C2×#4114yes—
383C2×Q8C2×#4114yes—
384C2×D4—28no—
385C2×D4—24no—
386C2×D4—28no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4C23×C414yes—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—432no—
427C22×C4—48no—
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—42no—
443C22×C4—42no—
444C22×C4C4×D414yes—
445C22×C4C4×D414yes—
446C22×C4AGL(1,ℤ/8)14yes—
447C22×C4C23×C412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461SD16—232no—
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8C23×C414yes—
477C2×C8C23×C414yes—
478C4⋊C4—216no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4C23×C414yes—
504C4⋊C4C23×C414yes—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C42—216no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—48no—
519C42—48no—
520C42—48no—
521C42C4×D418yes—
522C42C23×C414yes—
523C42C2×#4114yes—
524C23—24no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23C2×C4×D412yes—
539C23#54612yes—
540C23#54812yes—
541C23C2×AGL(1,ℤ/8)12yes—
542C23#57512yes—
543C23#57712yes—
544C23#57812yes—
545C23C23×D411yescharacteristic
546Q8—216no—
547Q8—24no—
548Q8—24no—
549Q8—24no—
550Q8C2×#11718yescomplemented
551D4—24no—
552D4—24no—
553D4—48no—
554D4—42no—
555D4—42no—
556C2×C4—216no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
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—432no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
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—42no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4C2×#11714yes—
607C2×C4C2×C4×D414yes—
608C2×C4C2×C4×D414yes—
609C2×C4C2×C4×D412yes—
610C2×C4#52014yes—
611C2×C4#54612yes—
612C2×C4#54812yes—
613C2×C4#55214yes—
614C2×C4#55314yes—
615C2×C4C2×AGL(1,ℤ/8)14yes—
616C2×C4C24×C411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C8—24no—
623C8—24no—
624C8—42no—
625C8—42no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—816no—
634C22#235214yes—
635C22C4×AGL(1,ℤ/8)14yes—
636C22#294714yes—
637C22#330814yes—
638C22#332114yes—
639C22#336814yes—
640C22#337714yes—
641C22C22×C4×D411yescharacteristic
642C22C2×#54611yescharacteristic
643C22C2×#54811yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C2×#57511yescharacteristic
646C22C2×#57711yescharacteristic
647C22C2×#57811yescharacteristic
648C4—24no—
649C4—28no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#235214yes—
662C4C22×#11711yescharacteristic
663C4C2×#52011yescharacteristic
664C4C2×#55211yescharacteristic
665C4C2×#55311yescharacteristic
666C2—84no—
667C2#2278118yesdirect product factor
668C2C2×#235211yescharacteristic
669C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
670C2C2×#294711yescharacteristic
671C2C2×#330811yescharacteristic
672C2C2×#332111yescharacteristic
673C2C2×#336811yescharacteristic
674C2C2×#337711yescharacteristic
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
3. C2×#2351C2semidirect
4. #22781C2direct (N × H)
7. C2×#1921C2semidirect
8. C2×#1918C2semidirect
11. C2×#1836C2semidirect
12. C2×#1803C2semidirect
14. C2×#1771C2semidirect
15. C2×#1763C2semidirect
16. C2×C4×#103C2semidirect
23. C2×#440C4semidirect
39. #2351C22semidirect
58. #1921C22semidirect
59. #1918C22semidirect
62. #1836C22semidirect
64. #1803C22semidirect
65. #1771C22semidirect
66. #1763C22semidirect
67. C4×#103C22semidirect
135. #440C2×C4semidirect
380. C2×Q8#117semidirect
550. Q8C2×#117semidirect
667. C2#22781direct (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×#23542
C2×#23512
#227812
C2×#19362
C2×#19342
C2×#19212
C2×#19182
C2×#18802
C2×#18362
C2×#18032
C2×#18042
C2×#17712
C2×#17632
C2×C4×#1032