Finite Group Databasebeta

← C2×#31653 #7505341

Subgroups — C2×#31653 #7505341

Order 512.

2160 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×#2566C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic, complemented
3#31653C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2074C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2026C211yesmaximal, characteristic, complemented
8C2×#2025C211yesmaximal, characteristic, complemented
9C2×#2012C211yesmaximal, characteristic
10C2×#1959C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1917C211yesmaximal, characteristic, complemented
13C2×#1879C211yesmaximal, characteristic
14C2×#1879C211yesmaximal, characteristic
15C2×#1877C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2566—24no—
34#2566C2218yescomplemented
35#2447—24no—
36#2447C2218yescomplemented
37C2×#397—21no—
38C2×#397—21no—
39C2×#396—21no—
40C2×#396—21no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#385—21no—
48C2×#385—21no—
49C2×#385—21no—
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#381—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2218yescomplemented
55#2074C2218yescomplemented
56#2055C2218yescomplemented
57#2026C2218yescomplemented
58#2025C2218yescomplemented
59#2012C2218yes—
60#1959C2218yescomplemented
61#1936C2218yescomplemented
62#1917C2218yescomplemented
63#1879C2218yes—
64#1879C2218yes—
65#1877C2218yescomplemented
66#1804C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4—21no—
82C22×SD16—24no—
83C22×SD16—42no—
84C22×SD16—42no—
85C22×SD16C2314yes—
86C22×D8—24no—
87C22×D8—24no—
88C22×D8—42no—
89C22×D8C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×#125—22no—
95C2×#125—42no—
96C2×#125—42no—
97C2×#125C2314yes—
98C2×#124—24no—
99C2×#124—22no—
100C2×#124—42no—
101C2×#124C2314yes—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—42no—
106C2×C4×D4—42no—
107C2×C4×D4—42no—
108C2×C4×D4C2314yes—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
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×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102—24no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#97—42no—
154C2×#97C2314yes—
155#397—28no—
156#397—28no—
157#396—28no—
158#396—28no—
159#391—22no—
160#391—22no—
161#391C2314yes—
162#391C2314yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#389C2314yes—
171#385—28no—
172#385—28no—
173#385—28no—
174#384—22no—
175#384C2314yes—
176#382—22no—
177#382C2314yes—
178#381—28no—
179C4×C4⋊C4D414yes—
180C4×C4⋊C4C2314yes—
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—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8C2412yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4#4114yes—
223C22×D4#4114yes—
224C22×D4C2412yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4—82no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×SD16—28no—
248C2×SD16—44no—
249C2×SD16—44no—
250C2×SD16—416no—
251C2×D8—28no—
252C2×D8—28no—
253C2×D8—44no—
254C2×D8—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—
271#125—44no—
272#125—44no—
273#125—48no—
274#124—28no—
275#124—44no—
276#124—48no—
277C4×D4—28no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C2×C4⋊C4—22no—
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—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
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—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C4⋊C8—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358#97—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4C2×#4114yes—
423C2×D4C2×#4114yes—
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—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—48no—
450C22×C4—48no—
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—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—816no—
473C22×C4—816no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488SD16—416no—
489D8—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—44no—
503C2×C8—44no—
504C2×C8—44no—
505C2×C8—44no—
506C4⋊C4—22no—
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—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C23—24no—
548C23—24no—
549C23—48no—
550C23—48no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—48no—
558C23—48no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#53512yes—
576C23#54612yes—
577C23#54812yes—
578C23#57512yes—
579C23#57512yes—
580C23#57612yes—
581C23#57712yes—
582C23C23×D411yescharacteristic
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—84no—
588D4—24no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—42no—
593D4—42no—
594D4—48no—
595D4—48no—
596D4—42no—
597D4—42no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
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—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
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—816no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4#53512yes—
649C2×C4#53814yes—
650C2×C4#53914yes—
651C2×C4#54612yes—
652C2×C4#54812yes—
653C2×C4#55114yes—
654C2×C4#56214yes—
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—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—816no—
678C22—816no—
679C22—82no—
680C22—82no—
681C22#269614yes—
682C22#321914yes—
683C22#322414yes—
684C22#330714yes—
685C22#332214yes—
686C22#337014yes—
687C22#337614yes—
688C22C2×#53511yescharacteristic
689C22C2×#54611yescharacteristic
690C22C2×#54811yescharacteristic
691C22C2×#57511yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57611yescharacteristic
694C22C2×#57711yescharacteristic
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#269614yes—
708C4C2×#53811yescharacteristic
709C4C2×#53911yescharacteristic
710C4C2×#55111yescharacteristic
711C4C2×#56211yescharacteristic
712C2—84no—
713C2—84no—
714C2#3165318yesdirect product factor
715C2C2×#269611yescharacteristic
716C2C2×#321911yescharacteristic
717C2C2×#322411yescharacteristic
718C2C2×#330711yescharacteristic
719C2C2×#332211yescharacteristic
720C2C2×#337011yescharacteristic
721C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#2566C2semidirect
2. C2×#2447C2semidirect
3. #31653C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2074C2semidirect
6. C2×#2055C2semidirect
7. C2×#2026C2semidirect
8. C2×#2025C2semidirect
10. C2×#1959C2semidirect
11. C2×#1936C2semidirect
12. C2×#1917C2semidirect
15. C2×#1877C2semidirect
16. C2×#1804C2semidirect
34. #2566C22semidirect
36. #2447C22semidirect
54. #2099C22semidirect
55. #2074C22semidirect
56. #2055C22semidirect
57. #2026C22semidirect
58. #2025C22semidirect
60. #1959C22semidirect
61. #1936C22semidirect
62. #1917C22semidirect
65. #1877C22semidirect
66. #1804C22semidirect
714. C2#31653direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#24472
#316532
C2×#20992
C2×#20742
C2×#20552
C2×#20262
C2×#20252
C2×#20122
C2×#19592
C2×#19362
C2×#19172
C2×#18792
C2×#18772
C2×#18042