Finite Group Databasebeta

← C2×#24717 #6449893

Subgroups — C2×#24717 #6449893

Order 512.

2286 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic, complemented
2C22×#433C211yesmaximal, characteristic, complemented
3C2×#2629C211yesmaximal, characteristic, complemented
4C2×#2324C211yesmaximal, characteristic
5#24717C2116yesmaximal, direct product factor
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic, complemented
8C2×#2034C211yesmaximal, characteristic
9C2×#2034C211yesmaximal, characteristic
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1948C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1936C211yesmaximal, characteristic, complemented
15C2×#1934C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#478—28no—
22C2×#478C2218yescomplemented
23C2×#433—28no—
24C2×#433C2218yescomplemented
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36C22×C22⋊C8C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#2629—24no—
40#2629C2218yescomplemented
41C2×#398—21no—
42C2×#398—21no—
43C2×#396—21no—
44C2×#396—21no—
45#2324—24no—
46#2324C2218yes—
47C2×#390—21no—
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#386—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2077C2218yescomplemented
61#2075C2218yescomplemented
62#2034C2218yes—
63#2034C2218yes—
64#1989C2218yescomplemented
65#1948C2218yescomplemented
66#1944C2218yescomplemented
67#1941C2218yescomplemented
68#1936C2218yescomplemented
69#1934C2218yescomplemented
70#1802C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#125—22no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125C2314yes—
91C2×#121—24no—
92C2×#121—22no—
93C2×#121—24no—
94C2×#121C2314yes—
95C2×C4×D4—22no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4C2314yes—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117D414yes—
102C2×#117C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#478—232no—
132#433—232no—
133C2×#107—24no—
134C2×#107C2314yes—
135C2×#106—24no—
136C2×#106—24no—
137C2×#106—24no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8—24no—
142C2×C4⋊C8D418yescomplemented
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×#97—24no—
158C2×#97C2314yes—
159C2×C4×C8—24no—
160C2×C4×C8C2314yes—
161#398—28no—
162#398—28no—
163#396—28no—
164#396—28no—
165#390—28no—
166#390—22no—
167#390C2314yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#388—28no—
175#386—28no—
176#385—28no—
177#384—22no—
178#384C2314yes—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184#380C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202C25—21no—
203C25C2×D411yescharacteristic
204C22×D4—28no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—82no—
213C22×D4—82no—
214C22×D4#4114yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—44no—
231C23×C4—82no—
232C23×C4—82no—
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
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×C8—22no—
251C22×C8C2×D414yes—
252C22×C8C2×D414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#125—28no—
258#125—28no—
259#125—48no—
260#121—216no—
261#121—28no—
262#121—28no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—48no—
266#117—216no—
267#117—28no—
268#117—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
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—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—82no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D414yes—
340C2×C42#4114yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—28no—
344#106—28no—
345#106—28no—
346#106—28no—
347C4⋊C8—28no—
348C4⋊C8C2×D4116yescomplemented
349#102—28no—
350#102—28no—
351#102—28no—
352#102—28no—
353C22⋊C8—28no—
354C22⋊C8—28no—
355#97—28no—
356C4×C8—28no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24C22×D412yes—
380C24C22×D411yescharacteristic
381C2×D4—216no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—816no—
397C2×D4C2×#4114yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
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×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—816no—
456C22×C4#12114yes—
457C22×C4#12114yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C22×C4C2511yescharacteristic
471C2×C8—28no—
472C2×C8—28no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8C22×D414yes—
486C2×C8C22×D414yes—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—24no—
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—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C2×#4114yes—
517C22⋊C4—216no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—84no—
530C42#12118yes—
531C42C22×D414yes—
532C42C2×#4114yes—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—44no—
542C23—48no—
543C23—44no—
544C23—48no—
545C23—42no—
546C23—44no—
547C23—48no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23C2×#12112yes—
562C23#52512yes—
563C23#53612yes—
564C23#57612yes—
565C23#57712yes—
566C23#57712yes—
567C23#57712yes—
568C23C23×D411yescharacteristic
569D4—24no—
570D4—24no—
571D4—48no—
572D4—42no—
573D4—42no—
574D4—816no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—816no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4C2×#12114yes—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4#52512yes—
631C2×C4#53612yes—
632C2×C4#54714yes—
633C2×C4#54814yes—
634C2×C4#55214yes—
635C2×C4#55614yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C8—24no—
643C8—24no—
644C8—42no—
645C8—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—82no—
657C22—82no—
658C22—88no—
659C22—816no—
660C22—82no—
661C22—82no—
662C22#241514yes—
663C22#306014yes—
664C22#306214yes—
665C22#311914yes—
666C22#312314yes—
667C22#324114yes—
668C22#324414yes—
669C22C22×#12111yescharacteristic
670C22C2×#52511yescharacteristic
671C22C2×#53611yescharacteristic
672C22C2×#57611yescharacteristic
673C22C2×#57711yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57711yescharacteristic
676C4—24no—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#241514yes—
689C4C2×#54711yescharacteristic
690C4C2×#54811yescharacteristic
691C4C2×#55211yescharacteristic
692C4C2×#55611yescharacteristic
693C2—82no—
694C2—84no—
695C2#2471718yesdirect product factor
696C2C2×#241511yescharacteristic
697C2C2×#306011yescharacteristic
698C2C2×#306211yescharacteristic
699C2C2×#311911yescharacteristic
700C2C2×#312311yescharacteristic
701C2C2×#324111yescharacteristic
702C2C2×#324411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#478C2semidirect
2. C22×#433C2semidirect
3. C2×#2629C2semidirect
5. #24717C2direct (N × H)
6. C2×#2077C2semidirect
7. C2×#2075C2semidirect
10. C2×#1989C2semidirect
11. C2×#1948C2semidirect
12. C2×#1944C2semidirect
13. C2×#1941C2semidirect
14. C2×#1936C2semidirect
15. C2×#1934C2semidirect
16. C2×#1802C2semidirect
22. C2×#478C22semidirect
24. C2×#433C22semidirect
40. #2629C22semidirect
60. #2077C22semidirect
61. #2075C22semidirect
64. #1989C22semidirect
65. #1948C22semidirect
66. #1944C22semidirect
67. #1941C22semidirect
68. #1936C22semidirect
69. #1934C22semidirect
70. #1802C22semidirect
142. C2×C4⋊C8D4semidirect
348. C4⋊C8C2×D4semidirect
695. C2#24717direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C22×#4332
C2×#26292
C2×#23242
#247172
C2×#20772
C2×#20752
C2×#20342
C2×#19892
C2×#19482
C2×#19442
C2×#19412
C2×#19362
C2×#19342
C2×#18022