Finite Group Databasebeta

← C2×#31638 #7505314

Subgroups — C2×#31638 #7505314

Order 512.

2360 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2445C211yesmaximal, characteristic
3#31638C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2071C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2023C211yesmaximal, characteristic, complemented
8C2×#2023C211yesmaximal, characteristic, complemented
9C2×#2014C211yesmaximal, characteristic
10C2×#1957C211yesmaximal, characteristic
11C2×#1933C211yesmaximal, characteristic, complemented
12C2×#1916C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic, complemented
14C2×#1880C211yesmaximal, characteristic, complemented
15C2×#1877C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#127C2211yescharacteristic
21C22×#123—21no—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2567—24no—
35#2567C2218yescomplemented
36#2445—24no—
37#2445C2218yes—
38C2×#395—21no—
39C2×#395—21no—
40C2×#394—21no—
41C2×#394—21no—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389—21no—
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#386—21no—
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2218yescomplemented
55#2071C2218yescomplemented
56#2055C2218yescomplemented
57#2023C2218yescomplemented
58#2023C2218yescomplemented
59#2014C2218yes—
60#1957C2218yes—
61#1933C2218yescomplemented
62#1916C2218yescomplemented
63#1880C2218yescomplemented
64#1880C2218yescomplemented
65#1877C2218yescomplemented
66#1802C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—42no—
88C22×SD16—42no—
89C22×SD16—42no—
90C22×SD16C2314yes—
91C22×SD16C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8C2314yes—
100C2×#127—24no—
101C2×#127—22no—
102C2×#127—42no—
103C2×#127C2314yes—
104C2×#123—28no—
105C2×#123—48no—
106C2×C4×Q8—22no—
107C2×C4×Q8—42no—
108C2×C4×Q8—42no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—24no—
111C2×C4×D4—22no—
112C2×C4×D4—42no—
113C2×C4×D4C2314yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×#103—24no—
142C2×#103—42no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102—42no—
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#395—28no—
160#395—28no—
161#394—28no—
162#394—28no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#391C2314yes—
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—28no—
174#389—28no—
175#389—22no—
176#389C2314yes—
177#386—28no—
178#384—22no—
179#384—22no—
180#384C2314yes—
181#384C2314yes—
182C4×C4⋊C4D414yes—
183C4×C4⋊C4C2314yes—
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#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—44no—
211C22×Q8—82no—
212C22×Q8—82no—
213C22×Q8#4114yes—
214C22×Q8C2412yes—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—24no—
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—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4—82no—
235C22×D4—82no—
236C22×D4#4114yes—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×SD16—416no—
263C2×SD16—416no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—48no—
283#127—28no—
284#127—44no—
285#127—48no—
286#123—432no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—48no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4SD1618yescomplemented
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4—82no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C4⋊C8—44no—
360#103—28no—
361#103—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366#102—44no—
367#102—44no—
368C4×C8—44no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95C2×D414yes—
375#95C2×D414yes—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24C22×D411yescharacteristic
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8—816no—
409C2×Q8C2×#4114yes—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—816no—
442C2×D4—816no—
443C2×D4C2×#4114yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
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—22no—
464C22×C4—22no—
465C22×C4—48no—
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×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—48no—
484C22×C4—84no—
485C22×C4—816no—
486C22×C4C2×SD1614yes—
487C22×C4C2×SD1614yes—
488C22×C4C2×SD1614yes—
489C22×C4#13714yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504SD16—416no—
505SD16—416no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—44no—
519C2×C8—44no—
520C2×C8—44no—
521C2×C8—44no—
522C4⋊C4—28no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—48no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4C2×SD16116yescomplemented
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C42—24no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—48no—
563C23—24no—
564C23—42no—
565C23—48no—
566C23—42no—
567C23—48no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23#53512yes—
592C23#54612yes—
593C23#54812yes—
594C23C22×SD1612yes—
595C23C2×#13712yes—
596C23#57612yes—
597C23#57612yes—
598C23C23×D411yescharacteristic
599Q8—24no—
600Q8—24no—
601Q8—42no—
602Q8—48no—
603Q8—42no—
604Q8—816no—
605D4—24no—
606D4—24no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—48no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—816no—
615D4—816no—
616D4—84no—
617C2×C4—22no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—44no—
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—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—816no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4#53512yes—
663C2×C4#53814yes—
664C2×C4#53914yes—
665C2×C4#54612yes—
666C2×C4#54812yes—
667C2×C4#55114yes—
668C2×C4#56214yes—
669C2×C4C22×SD1614yes—
670C2×C4C22×SD1614yes—
671C2×C4C22×SD1614yes—
672C2×C4C2×#13714yes—
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×H(5,𝔽2)11yescharacteristic
678C2×C4C2×2−1+411yescharacteristic
679C8—42no—
680C8—42no—
681C8—42no—
682C8—42no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—816no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—168no—
699C22#269614yes—
700C22#320614yes—
701C22#322014yes—
702C22#329514yes—
703C22#331414yes—
704C22#336214yes—
705C22#337514yes—
706C22C2×#53511yescharacteristic
707C22C2×#54611yescharacteristic
708C22C2×#54811yescharacteristic
709C22C23×SD1611yescharacteristic
710C22C22×#13711yescharacteristic
711C22C2×#57611yescharacteristic
712C22C2×#57611yescharacteristic
713C4—24no—
714C4—24no—
715C4—24no—
716C4—24no—
717C4—42no—
718C4—44no—
719C4—42no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4#269614yes—
726C4C2×#53811yescharacteristic
727C4C2×#53911yescharacteristic
728C4C2×#55111yescharacteristic
729C4C2×#56211yescharacteristic
730C2—84no—
731C2—162no—
732C2#3163818yesdirect product factor
733C2C2×#269611yescharacteristic
734C2C2×#320611yescharacteristic
735C2C2×#322011yescharacteristic
736C2C2×#329511yescharacteristic
737C2C2×#331411yescharacteristic
738C2C2×#336211yescharacteristic
739C2C2×#337511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×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
1. C2×#2567C2semidirect
3. #31638C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2071C2semidirect
6. C2×#2055C2semidirect
7. C2×#2023C2semidirect
8. C2×#2023C2semidirect
11. C2×#1933C2semidirect
12. C2×#1916C2semidirect
13. C2×#1880C2semidirect
14. C2×#1880C2semidirect
15. C2×#1877C2semidirect
16. C2×#1802C2semidirect
35. #2567C22semidirect
54. #2100C22semidirect
55. #2071C22semidirect
56. #2055C22semidirect
57. #2023C22semidirect
58. #2023C22semidirect
61. #1933C22semidirect
62. #1916C22semidirect
63. #1880C22semidirect
64. #1880C22semidirect
65. #1877C22semidirect
66. #1802C22semidirect
323. C2×C4⋊C4SD16semidirect
545. C4⋊C4C2×SD16semidirect
732. C2#31638direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#24452
#316382
C2×#21002
C2×#20712
C2×#20552
C2×#20232
C2×#20142
C2×#19572
C2×#19332
C2×#19162
C2×#18802
C2×#18772
C2×#18022