Finite Group Databasebeta

← C2×#31552 #7505070

Subgroups — C2×#31552 #7505070

Order 512.

2286 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic, complemented
2C2×#2422C211yesmaximal, characteristic
3#31552C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2000C211yesmaximal, characteristic
10C2×#1893C211yesmaximal, characteristic
11C2×#1892C211yesmaximal, characteristic, complemented
12C2×#1880C211yesmaximal, characteristic, complemented
13C2×#1877C211yesmaximal, characteristic, complemented
14C2×#1850C211yesmaximal, characteristic, complemented
15C2×#1803C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×#123—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2423—24no—
35#2423C2218yescomplemented
36#2422—24no—
37#2422C2218yes—
38C2×#399—21no—
39C2×#398—21no—
40C2×#397—21no—
41C2×#394—21no—
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#387—21no—
48C2×#385—21no—
49C2×#384C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2100C2218yescomplemented
55#2099C2218yescomplemented
56#2077C2218yescomplemented
57#2066C2218yescomplemented
58#2065C2218yescomplemented
59#2000C2218yes—
60#1893C2218yes—
61#1892C2218yescomplemented
62#1880C2218yescomplemented
63#1877C2218yescomplemented
64#1850C2218yescomplemented
65#1803C2218yescomplemented
66#1802C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—42no—
85C22×SD16C2314yes—
86C23×C8—42no—
87C23×C8—42no—
88C23×C8—42no—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2314yes—
94C2×#123—28no—
95C2×#123—48no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—42no—
99C2×C22⋊Q8C2314yes—
100C2×#121—22no—
101C2×#121—24no—
102C2×#121—42no—
103C2×#121C2314yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—24no—
106C2×C4×Q8—42no—
107C2×C4×Q8C2314yes—
108C2×C4×D4—24no—
109C2×C4×D4—22no—
110C2×C4×D4—42no—
111C2×C4×D4C2314yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C2×#106—42no—
139C2×#106C2314yes—
140C2×C4⋊C8—42no—
141C2×C4⋊C8—42no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103—42no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102—42no—
151C2×#102—42no—
152C2×#102—42no—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8—42no—
157C2×C22⋊C8C2314yes—
158#399—28no—
159#398—28no—
160#397—28no—
161#394—28no—
162#391—22no—
163#391C2314yes—
164#390—22no—
165#390C2314yes—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#388—28no—
171#387—28no—
172#385—28no—
173#384—22no—
174#384—22no—
175#384C2314yes—
176#384C2314yes—
177#380—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181C4×C22⋊C4D414yes—
182C4×C22⋊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—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8#4114yes—
210C22×Q8C2412yes—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—82no—
221C22×D4#4114yes—
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—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—416no—
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—22no—
263C22×C8—22no—
264C22×C8—44no—
265C22×C8—44no—
266C22×C8—44no—
267C22×C8—416no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272#123—432no—
273C22⋊Q8—28no—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276#121—28no—
277#121—44no—
278#121—48no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×D4—28no—
283C4×D4—44no—
284C4×D4—48no—
285C2×C4⋊C4—22no—
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—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
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—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
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—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4SD1618yescomplemented
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353#106—44no—
354C4⋊C8—44no—
355C4⋊C8—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—44no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—22no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C2×SD1614yes—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8C2×#4114yes—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—84no—
416C2×D4—816no—
417C2×D4C2×#4114yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
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—48no—
474C22×C4—84no—
475C22×C4—816no—
476C22×C4C2×SD1614yes—
477C22×C4C2×SD1614yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491SD16—416no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—416no—
505C2×C8—44no—
506C2×C8—44no—
507C2×C8—44no—
508C2×C8—44no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C22⋊C4C2×SD16116yescomplemented
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—48no—
567C23—42no—
568C23—48no—
569C23—48no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23#53712yes—
579C23#54612yes—
580C23#54612yes—
581C23C22×SD1614yes—
582C23C22×SD1612yes—
583C23#57512yes—
584C23#57612yes—
585C23#57812yes—
586C23C23×D411yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—42no—
590Q8—48no—
591Q8—42no—
592D4—24no—
593D4—24no—
594D4—42no—
595D4—48no—
596D4—42no—
597D4—816no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—816no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4#53712yes—
651C2×C4#53914yes—
652C2×C4#53914yes—
653C2×C4#54612yes—
654C2×C4#54612yes—
655C2×C4#55114yes—
656C2×C4#56014yes—
657C2×C4C22×SD1614yes—
658C2×C4C22×SD1614yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C8—42no—
666C8—42no—
667C8—42no—
668C8—42no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—416no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—816no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22#269314yes—
686C22#324614yes—
687C22#326214yes—
688C22#329514yes—
689C22#329814yes—
690C22#330914yes—
691C22#331514yes—
692C22C2×#53711yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#54611yescharacteristic
695C22C23×SD1611yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57611yescharacteristic
698C22C2×#57811yescharacteristic
699C4—24no—
700C4—24no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#269314yes—
712C4C2×#53911yescharacteristic
713C4C2×#53911yescharacteristic
714C4C2×#55111yescharacteristic
715C4C2×#56011yescharacteristic
716C2—44no—
717C2—84no—
718C2#3155218yesdirect product factor
719C2C2×#269311yescharacteristic
720C2C2×#324611yescharacteristic
721C2C2×#326211yescharacteristic
722C2C2×#329511yescharacteristic
723C2C2×#329811yescharacteristic
724C2C2×#331511yescharacteristic
725C2C2×#330911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2423C2semidirect
3. #31552C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2099C2semidirect
6. C2×#2077C2semidirect
7. C2×#2066C2semidirect
8. C2×#2065C2semidirect
11. C2×#1892C2semidirect
12. C2×#1880C2semidirect
13. C2×#1877C2semidirect
14. C2×#1850C2semidirect
15. C2×#1803C2semidirect
16. C2×#1802C2semidirect
35. #2423C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2077C22semidirect
57. #2066C22semidirect
58. #2065C22semidirect
61. #1892C22semidirect
62. #1880C22semidirect
63. #1877C22semidirect
64. #1850C22semidirect
65. #1803C22semidirect
66. #1802C22semidirect
340. C2×C22⋊C4SD16semidirect
543. C22⋊C4C2×SD16semidirect
718. C2#31552direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#24222
#315522
C2×#21002
C2×#20992
C2×#20772
C2×#20662
C2×#20652
C2×#20002
C2×#18932
C2×#18922
C2×#18802
C2×#18772
C2×#18502
C2×#18032
C2×#18022