Finite Group Databasebeta

← C2×#31321 #7504420

Subgroups — C2×#31321 #7504420

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2414C211yesmaximal, characteristic
3#31321C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2059C211yesmaximal, characteristic, complemented
6C2×#2034C211yesmaximal, characteristic, complemented
7C2×#2034C211yesmaximal, characteristic, complemented
8C2×#2033C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic, complemented
10C2×#1925C211yesmaximal, characteristic, complemented
11C2×#1919C211yesmaximal, characteristic, complemented
12C2×#1893C211yesmaximal, characteristic, complemented
13C2×#1869C211yesmaximal, characteristic, complemented
14C2×#1854C211yesmaximal, characteristic, complemented
15C2×#1825C211yesmaximal, characteristic, complemented
16C2×#1808C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2562—24no—
35#2562C2218yescomplemented
36#2414—24no—
37#2414C2218yes—
38C2×#400—21no—
39C2×#394—21no—
40C2×#394—21no—
41C2×#392—21no—
42C2×#390—21no—
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#388—21no—
47C2×#388—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2077C2218yescomplemented
55#2059C2218yescomplemented
56#2034C2218yescomplemented
57#2034C2218yescomplemented
58#2033C2218yescomplemented
59#2025C2218yescomplemented
60#1925C2218yescomplemented
61#1919C2218yescomplemented
62#1893C2218yescomplemented
63#1869C2218yescomplemented
64#1854C2218yescomplemented
65#1825C2218yescomplemented
66#1808C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4—21no—
79C23×D4D411yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8—24no—
84C23×C8—42no—
85C23×C8—42no—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2314yes—
91C2×#125—22no—
92C2×#125—42no—
93C2×#125—42no—
94C2×#125C2314yes—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121—24no—
98C2×#121—42no—
99C2×#121—42no—
100C2×#121—42no—
101C2×#121C2314yes—
102C2×#121C2314yes—
103C2×C4×D4—42no—
104C2×C4×D4—42no—
105C2×C4×D4D414yes—
106C2×C4×D4C2314yes—
107C2×#117—22no—
108C2×#117—24no—
109C2×#117—42no—
110C2×#117C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C2×#107—42no—
138C2×#107—42no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106—24no—
142C2×#106—42no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#102—24no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—42no—
155C2×C22⋊C8—42no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#400—28no—
160#394—28no—
161#394—28no—
162#392—28no—
163#390—28no—
164#390—22no—
165#390C2314yes—
166#389—22no—
167#389C2314yes—
168#388—28no—
169#388—28no—
170#388—28no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179#380C2314yes—
180#380C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25C2×D411yescharacteristic
202C25C2×D411yescharacteristic
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—82no—
219C22×D4—82no—
220C22×D4—82no—
221C22×D4C2×D414yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—28no—
263C22×C8—44no—
264C22×C8—44no—
265C22×C8—416no—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#125—44no—
272#125—44no—
273#125—48no—
274#121—28no—
275#121—44no—
276#121—44no—
277#121—44no—
278#121—48no—
279#121—48no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283#117—28no—
284#117—44no—
285#117—48no—
286C2×C4⋊C4—21no—
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—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4#4114yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
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—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4C2×D414yes—
345C2×C22⋊C4C2×D414yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—44no—
351C2×C42—82no—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354#107—44no—
355#107—44no—
356#106—28no—
357#106—44no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—28no—
363C22⋊C8—44no—
364C22⋊C8—44no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24#13514yes—
394C24C22×D412yes—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—816no—
421C2×D4—816no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
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—22no—
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—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
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—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500C2×C8—28no—
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—22no—
512C2×C8—416no—
513C2×C8—44no—
514C2×C8—44no—
515C2×C8—44no—
516C2×C8C2×#4114yes—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
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—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—84no—
541C4⋊C4C2×#4114yes—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—48no—
558C42—84no—
559C23—28no—
560C23—22no—
561C23—28no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23#53412yes—
601C23#54612yes—
602C23#54712yes—
603C23C2×#13514yes—
604C23C2×#13512yes—
605C23#57512yes—
606C23#57612yes—
607C23#57712yes—
608C23C23×D411yescharacteristic
609D4—416no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—816no—
615D4—816no—
616D4—84no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—48no—
640C2×C4—48no—
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—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4#53412yes—
672C2×C4#53814yes—
673C2×C4#54214yes—
674C2×C4#54612yes—
675C2×C4#54712yes—
676C2×C4#55514yes—
677C2×C4#56214yes—
678C2×C4C22×#4111yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C2×C4C2×2−1+411yescharacteristic
684C8—24no—
685C8—42no—
686C8—42no—
687C8—42no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—42no—
692C22—42no—
693C22—416no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—88no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—168no—
712C22#268614yes—
713C22#319114yes—
714C22#319814yes—
715C22#329614yes—
716C22#330914yes—
717C22#333914yes—
718C22#334814yes—
719C22C2×#53411yescharacteristic
720C22C2×#54611yescharacteristic
721C22C2×#54711yescharacteristic
722C22C22×#13511yescharacteristic
723C22C2×#57511yescharacteristic
724C22C2×#57611yescharacteristic
725C22C2×#57711yescharacteristic
726C4—42no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#268614yes—
738C4C2×#53811yescharacteristic
739C4C2×#54211yescharacteristic
740C4C2×#55511yescharacteristic
741C4C2×#56211yescharacteristic
742C2—44no—
743C2—82no—
744C2—162no—
745C2#3132118yesdirect product factor
746C2C2×#268611yescharacteristic
747C2C2×#319111yescharacteristic
748C2C2×#319811yescharacteristic
749C2C2×#329611yescharacteristic
750C2C2×#330911yescharacteristic
751C2C2×#333911yescharacteristic
752C2C2×#334811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2414C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2562C2semidirect
3. #31321C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2059C2semidirect
6. C2×#2034C2semidirect
7. C2×#2034C2semidirect
8. C2×#2033C2semidirect
9. C2×#2025C2semidirect
10. C2×#1925C2semidirect
11. C2×#1919C2semidirect
12. C2×#1893C2semidirect
13. C2×#1869C2semidirect
14. C2×#1854C2semidirect
15. C2×#1825C2semidirect
16. C2×#1808C2semidirect
35. #2562C22semidirect
54. #2077C22semidirect
55. #2059C22semidirect
56. #2034C22semidirect
57. #2034C22semidirect
58. #2033C22semidirect
59. #2025C22semidirect
60. #1925C22semidirect
61. #1919C22semidirect
62. #1893C22semidirect
63. #1869C22semidirect
64. #1854C22semidirect
65. #1825C22semidirect
66. #1808C22semidirect
745. C2#31321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#24142
#313212
C2×#20772
C2×#20592
C2×#20342
C2×#20332
C2×#20252
C2×#19252
C2×#19192
C2×#18932
C2×#18692
C2×#18542
C2×#18252
C2×#18082