Finite Group Databasebeta

← C2×#31649 #7505327

Subgroups — C2×#31649 #7505327

Order 512.

2056 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic
3#31649C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2074C211yesmaximal, characteristic, complemented
6C2×#2056C211yesmaximal, characteristic, complemented
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#2023C211yesmaximal, characteristic
9C2×#2015C211yesmaximal, characteristic
10C2×#1958C211yesmaximal, characteristic
11C2×#1933C211yesmaximal, characteristic, complemented
12C2×#1916C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1877C211yesmaximal, characteristic
16C2×#1804C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2568—24no—
34#2568C2218yescomplemented
35#2447—24no—
36#2447C2218yes—
37C2×#400—21no—
38C2×#400—21no—
39C2×#397—21no—
40C2×#397—21no—
41C2×#391—21no—
42C2×#391—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#385—21no—
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#381—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2074C2218yescomplemented
56#2056C2218yescomplemented
57#2024C2218yescomplemented
58#2023C2218yes—
59#2015C2218yes—
60#1958C2218yes—
61#1933C2218yescomplemented
62#1916C2218yescomplemented
63#1880C2218yes—
64#1878C2218yescomplemented
65#1877C2218yes—
66#1804C2218yes—
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×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×Q16—24no—
83C22×Q16—42no—
84C22×Q16—42no—
85C22×Q16C2314yes—
86C22×SD16—24no—
87C22×SD16—24no—
88C22×SD16—42no—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8C2314yes—
98C2×#124—24no—
99C2×#124—22no—
100C2×#124—42no—
101C2×#124C2314yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—42no—
104C2×C4×Q8—42no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—42no—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×C4⋊C8—42no—
130C2×C4⋊C8C2314yes—
131C2×#103—24no—
132C2×#103—42no—
133C2×#103—42no—
134C2×#103—42no—
135C2×#103—42no—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102—42no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C4×C8—42no—
148C2×C4×C8C2314yes—
149#400—28no—
150#400—28no—
151#397—28no—
152#397—28no—
153#391—28no—
154#391—28no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#391C2314yes—
163#390—22no—
164#390C2314yes—
165#389—22no—
166#389C2314yes—
167#385—28no—
168#382—22no—
169#382—22no—
170#382C2314yes—
171#382C2314yes—
172#381—28no—
173C4×C4⋊C4D414yes—
174C4×C4⋊C4C2314yes—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8#4114yes—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4#4114yes—
228C22×D4C2412yes—
229C23×C4—21no—
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×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—28no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—416no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—416no—
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—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—48no—
274#124—28no—
275#124—44no—
276#124—48no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×D4—28no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
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—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C4⋊C8—44no—
338#103—28no—
339#103—44no—
340#103—44no—
341#103—44no—
342#103—44no—
343#102—44no—
344#102—44no—
345#102—44no—
346C4×C8—44no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24C22×D411yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—816no—
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—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4C2×#4114yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
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—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—84no—
462C22×C4—816no—
463C22×C4#13514yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478Q16—416no—
479SD16—416no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—44no—
493C2×C8—44no—
494C2×C8—44no—
495C2×C8—44no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—84no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C42—24no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—48no—
531C23—24no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—48no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23#53512yes—
546C23#54612yes—
547C23#54812yes—
548C23C2×#13512yes—
549C23#57512yes—
550C23#57712yes—
551C23#57812yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—42no—
556Q8—48no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—816no—
563Q8—84no—
564D4—24no—
565D4—24no—
566D4—42no—
567D4—48no—
568D4—42no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—816no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#53512yes—
622C2×C4#53814yes—
623C2×C4#53914yes—
624C2×C4#54612yes—
625C2×C4#54812yes—
626C2×C4#55114yes—
627C2×C4#56214yes—
628C2×C4C2×#13514yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—816no—
647C22#269614yes—
648C22#321114yes—
649C22#322514yes—
650C22#330214yes—
651C22#331114yes—
652C22#336514yes—
653C22#337714yes—
654C22C2×#53511yescharacteristic
655C22C2×#54611yescharacteristic
656C22C2×#54811yescharacteristic
657C22C22×#13511yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22C2×#57811yescharacteristic
661C4—24no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4#269614yes—
675C4C2×#53811yescharacteristic
676C4C2×#53911yescharacteristic
677C4C2×#55111yescharacteristic
678C4C2×#56211yescharacteristic
679C2—84no—
680C2#3164918yesdirect product factor
681C2C2×#269611yescharacteristic
682C2C2×#321111yescharacteristic
683C2C2×#322511yescharacteristic
684C2C2×#330211yescharacteristic
685C2C2×#331111yescharacteristic
686C2C2×#336511yescharacteristic
687C2C2×#337711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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×#2568C2semidirect
3. #31649C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2074C2semidirect
6. C2×#2056C2semidirect
7. C2×#2024C2semidirect
11. C2×#1933C2semidirect
12. C2×#1916C2semidirect
14. C2×#1878C2semidirect
34. #2568C22semidirect
54. #2101C22semidirect
55. #2074C22semidirect
56. #2056C22semidirect
57. #2024C22semidirect
61. #1933C22semidirect
62. #1916C22semidirect
64. #1878C22semidirect
680. C2#31649direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24472
#316492
C2×#21012
C2×#20742
C2×#20562
C2×#20242
C2×#20232
C2×#20152
C2×#19582
C2×#19332
C2×#19162
C2×#18802
C2×#18782
C2×#18772
C2×#18042