Finite Group Databasebeta

← C2×#32600 #7510450

Subgroups — C2×#32600 #7510450

Order 512.

2230 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic
3#32600C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic, complemented
5C2×#2071C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic, complemented
8C2×#2041C211yesmaximal, characteristic
9C2×#2014C211yesmaximal, characteristic
10C2×#1984C211yesmaximal, characteristic, complemented
11C2×#1968C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1880C211yesmaximal, characteristic, complemented
16C2×#1880C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#124—21no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2582—24no—
36#2582C2218yescomplemented
37#2548—24no—
38#2548C2218yes—
39C2×#398—21no—
40C2×#398—21no—
41C2×#394—21no—
42C2×#394—21no—
43C2×#391—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390—21no—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2094C2218yescomplemented
57#2071C2218yescomplemented
58#2065C2218yescomplemented
59#2055C2218yescomplemented
60#2041C2218yes—
61#2014C2218yes—
62#1984C2218yescomplemented
63#1968C2218yescomplemented
64#1934C2218yescomplemented
65#1927C2218yescomplemented
66#1920C2218yescomplemented
67#1880C2218yescomplemented
68#1880C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16—42no—
89C22×SD16C2314yes—
90C22×SD16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#127—22no—
96C2×#127—42no—
97C2×#127—42no—
98C2×#127C2314yes—
99C2×#124—28no—
100C2×#124—48no—
101C2×C4×Q8—22no—
102C2×C4×Q8—42no—
103C2×C4×Q8—42no—
104C2×C4×Q8—42no—
105C2×C4×Q8—42no—
106C2×C4×Q8D414yes—
107C2×C4×Q8C2314yes—
108C2×C4×Q8C2314yes—
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⋊C4D411yescharacteristic
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×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C2×#106—24no—
136C2×#106—42no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×#103—24no—
142C2×#103—42no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102—24no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C4×C8—42no—
158C2×C4×C8C2314yes—
159#398—28no—
160#398—28no—
161#394—28no—
162#394—28no—
163#391—28no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#390—28no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#390C2314yes—
176#390C2314yes—
177#389—28no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#384—22no—
183#384C2314yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8SD1618yescomplemented
212C22×Q8C2×D414yes—
213C22×Q8#4114yes—
214C22×Q8C2412yes—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—82no—
236C22×D4—82no—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16—416no—
259C2×SD16—416no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8#4114yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276#127—44no—
277#127—44no—
278#127—48no—
279#124—432no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—48no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#106—28no—
355#106—44no—
356C4⋊C8—44no—
357#103—28no—
358#103—44no—
359#102—28no—
360#102—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365C4×C8—44no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373C24—22no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24C22×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—44no—
399C2×Q8—416no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8C2×SD16116yescomplemented
406C2×Q8C2×#4114yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—816no—
440C2×D4—816no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C2×SD1614yes—
481C22×C4C2×SD1614yes—
482C22×C4C2×SD1614yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C42−1+412yes—
496C22×C4C2511yescharacteristic
497SD16—416no—
498SD16—416no—
499C2×C8—28no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8C2×#4114yes—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—48no—
556C42—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23#53412yes—
586C23#54812yes—
587C23#55012yes—
588C23C22×SD1612yes—
589C23#57512yes—
590C23#57612yes—
591C23#57712yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—416no—
596Q8—42no—
597Q8—48no—
598Q8—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—816no—
608D4—816no—
609D4—84no—
610D4—84no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4#53412yes—
657C2×C4#53814yes—
658C2×C4#54214yes—
659C2×C4#54812yes—
660C2×C4#55012yes—
661C2×C4#55114yes—
662C2×C4#56214yes—
663C2×C4C22×SD1614yes—
664C2×C4C22×SD1614yes—
665C2×C4C22×SD1614yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—24no—
673C8—42no—
674C8—42no—
675C8—42no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—168no—
691C22—168no—
692C22#273014yes—
693C22#318714yes—
694C22#319814yes—
695C22#336214yes—
696C22#337114yes—
697C22#339214yes—
698C22#340314yes—
699C22C2×#53411yescharacteristic
700C22C2×#54811yescharacteristic
701C22C2×#55011yescharacteristic
702C22C23×SD1611yescharacteristic
703C22C2×#57511yescharacteristic
704C22C2×#57611yescharacteristic
705C22C2×#57711yescharacteristic
706C4—24no—
707C4—24no—
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#273014yes—
719C4C2×#53811yescharacteristic
720C4C2×#54211yescharacteristic
721C4C2×#55111yescharacteristic
722C4C2×#56211yescharacteristic
723C2—162no—
724C2—162no—
725C2#3260018yesdirect product factor
726C2C2×#273011yescharacteristic
727C2C2×#318711yescharacteristic
728C2C2×#319811yescharacteristic
729C2C2×#336211yescharacteristic
730C2C2×#337111yescharacteristic
731C2C2×#339211yescharacteristic
732C2C2×#340311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
3. #32600C2direct (N × H)
4. C2×#2094C2semidirect
5. C2×#2071C2semidirect
6. C2×#2065C2semidirect
7. C2×#2055C2semidirect
10. C2×#1984C2semidirect
11. C2×#1968C2semidirect
12. C2×#1934C2semidirect
13. C2×#1927C2semidirect
14. C2×#1920C2semidirect
15. C2×#1880C2semidirect
16. C2×#1880C2semidirect
36. #2582C22semidirect
56. #2094C22semidirect
57. #2071C22semidirect
58. #2065C22semidirect
59. #2055C22semidirect
62. #1984C22semidirect
63. #1968C22semidirect
64. #1934C22semidirect
65. #1927C22semidirect
66. #1920C22semidirect
67. #1880C22semidirect
68. #1880C22semidirect
211. C22×Q8SD16semidirect
405. C2×Q8C2×SD16semidirect
725. C2#32600direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25482
#326002
C2×#20942
C2×#20712
C2×#20652
C2×#20552
C2×#20412
C2×#20142
C2×#19842
C2×#19682
C2×#19342
C2×#19272
C2×#19202
C2×#18802