Finite Group Databasebeta

← C2×#31559 #7505074

Subgroups — C2×#31559 #7505074

Order 512.

2264 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic, complemented
2C2×#2422C211yesmaximal, characteristic
3#31559C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2075C211yesmaximal, characteristic, complemented
7C2×#2068C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, 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×#1804C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×#123—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2422—24no—
35#2422C2218yes—
36#2421—24no—
37#2421C2218yescomplemented
38C2×#400—21no—
39C2×#397—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×#382C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2100C2218yescomplemented
56#2075C2218yescomplemented
57#2068C2218yescomplemented
58#2067C2218yescomplemented
59#2000C2218yes—
60#1893C2218yes—
61#1892C2218yescomplemented
62#1880C2218yescomplemented
63#1877C2218yescomplemented
64#1850C2218yescomplemented
65#1804C2218yescomplemented
66#1803C2218yescomplemented
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⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C2×#107—42no—
137C2×#107C2314yes—
138C2×C4⋊C8—42no—
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8—42no—
155C2×C22⋊C8C2314yes—
156#400—28no—
157#397—28no—
158#397—28no—
159#394—28no—
160#391—22no—
161#391C2314yes—
162#390—22no—
163#390C2314yes—
164#389—22no—
165#389—22no—
166#389C2314yes—
167#389C2314yes—
168#388—28no—
169#387—28no—
170#385—28no—
171#384—22no—
172#384C2314yes—
173#382—22no—
174#382C2314yes—
175#380—22no—
176#380—22no—
177#380C2314yes—
178#380C2314yes—
179C4×C22⋊C4D414yes—
180C4×C22⋊C4C2314yes—
181C2×#95—21no—
182C2×#95—21no—
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—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
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—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—44no—
350C4⋊C8—44no—
351C4⋊C8—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358C22⋊C8—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95C2×D414yes—
368#95C2×D414yes—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—22no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24AGL(1,ℤ/8)14yes—
384C24C22×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8C2×#4114yes—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—84no—
414C2×D4—816no—
415C2×D4C2×#4114yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
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—48no—
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—84no—
474C22×C4—816no—
475C22×C4#13514yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489SD16—416no—
490C2×C8—22no—
491C2×C8—22no—
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—416no—
503C2×C8—44no—
504C2×C8—44no—
505C2×C8—44no—
506C2×C8—44no—
507C4⋊C4—22no—
508C4⋊C4—22no—
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—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—84no—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—48no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23#53712yes—
575C23#54612yes—
576C23#54612yes—
577C23C2×#13512yes—
578C23C2×AGL(1,ℤ/8)14yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23#57712yes—
581C23#57812yes—
582C23C23×D411yescharacteristic
583Q8—24no—
584Q8—24no—
585Q8—42no—
586Q8—48no—
587Q8—42no—
588D4—24no—
589D4—24no—
590D4—42no—
591D4—48no—
592D4—42no—
593D4—816no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
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—816no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#53712yes—
648C2×C4#53914yes—
649C2×C4#53914yes—
650C2×C4#54612yes—
651C2×C4#54612yes—
652C2×C4#55114yes—
653C2×C4#56014yes—
654C2×C4C2×#13514yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—42no—
662C8—42no—
663C8—42no—
664C8—42no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—416no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—816no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22#269314yes—
682C22#324714yes—
683C22#325814yes—
684C22#330114yes—
685C22#330214yes—
686C22#330814yes—
687C22#331014yes—
688C22C2×#53711yescharacteristic
689C22C2×#54611yescharacteristic
690C22C2×#54611yescharacteristic
691C22C22×#13511yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C2×#57711yescharacteristic
694C22C2×#57811yescharacteristic
695C4—24no—
696C4—42no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#269314yes—
708C4C2×#53911yescharacteristic
709C4C2×#53911yescharacteristic
710C4C2×#55111yescharacteristic
711C4C2×#56011yescharacteristic
712C2—44no—
713C2—84no—
714C2#3155918yesdirect product factor
715C2C2×#269311yescharacteristic
716C2C2×#324711yescharacteristic
717C2C2×#325811yescharacteristic
718C2C2×#330111yescharacteristic
719C2C2×#330211yescharacteristic
720C2C2×#331011yescharacteristic
721C2C2×#330811yescharacteristic
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×#2421C2semidirect
3. #31559C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2101C2semidirect
6. C2×#2075C2semidirect
7. C2×#2068C2semidirect
8. C2×#2067C2semidirect
11. C2×#1892C2semidirect
12. C2×#1880C2semidirect
13. C2×#1877C2semidirect
14. C2×#1850C2semidirect
15. C2×#1803C2semidirect
16. C2×#1804C2semidirect
37. #2421C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2075C22semidirect
57. #2068C22semidirect
58. #2067C22semidirect
61. #1892C22semidirect
62. #1880C22semidirect
63. #1877C22semidirect
64. #1850C22semidirect
65. #1804C22semidirect
66. #1803C22semidirect
714. C2#31559direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24212
C2×#24222
#315592
C2×#21002
C2×#21012
C2×#20752
C2×#20682
C2×#20672
C2×#20002
C2×#18932
C2×#18922
C2×#18802
C2×#18772
C2×#18502
C2×#18032
C2×#18042