Finite Group Databasebeta

← C2×#20212 #6359565

Subgroups — C2×#20212 #6359565

Order 512.

1613 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic
2C2×#3374C211yesmaximal, characteristic, complemented
3C2×#3372C211yesmaximal, characteristic
4C2×#3299C211yesmaximal, characteristic
5#20212C2116yesmaximal, direct product factor
6C2×#1718C211yesmaximal, characteristic, complemented
7C2×#1704C211yesmaximal, characteristic, complemented
8C2×#1704C211yesmaximal, characteristic, complemented
9C2×#1694C211yesmaximal, characteristic, complemented
10C2×#1679C211yesmaximal, characteristic
11C2×#1653C211yesmaximal, characteristic, complemented
12C2×#1651C211yesmaximal, characteristic
13C2×#1636C211yesmaximal, characteristic, complemented
14C2×#1644C211yesmaximal, characteristic
15C2×#1593C211yesmaximal, characteristic, complemented
16C2×#1582C211yesmaximal, characteristic
17C2×#557C2211yescharacteristic
18C2×#557C2211yescharacteristic
19C2×#554C2211yescharacteristic
20C2×D4×Q8C2211yescharacteristic
21#3386—24no—
22#3386C2218yes—
23#3374—24no—
24#3374C2218yescomplemented
25#3372—24no—
26#3372C2218yes—
27#3299—24no—
28#3299C2218yes—
29C2×#501—21no—
30C2×#500C2211yescharacteristic
31C2×#498C2211yescharacteristic
32C2×#491—21no—
33C2×#491—21no—
34C2×#487C2211yescharacteristic
35C2×#487C2211yescharacteristic
36C2×#487C2211yescharacteristic
37C2×#487C2211yescharacteristic
38C2×#484—21no—
39C2×#479—21no—
40C2×#479—21no—
41C2×#477—21no—
42C2×#477C2211yescharacteristic
43C2×#477C2211yescharacteristic
44C2×#475C2211yescharacteristic
45C2×#475C2211yescharacteristic
46C2×#478—21no—
47C2×#474C2211yescharacteristic
48C2×#467—21no—
49C2×#462—21no—
50C2×#462C2211yescharacteristic
51C2×#462C2211yescharacteristic
52C2×#461—21no—
53C2×#461C2211yescharacteristic
54C2×#451—21no—
55C2×#446C2211yescharacteristic
56C2×#446C2211yescharacteristic
57C2×#446C2211yescharacteristic
58C2×#441—21no—
59C2×#440—21no—
60C2×#440—21no—
61C2×C4×Q16C2211yescharacteristic
62C2×C4×SD16C2211yescharacteristic
63C2×C8×D4C2211yescharacteristic
64C22×#103—21no—
65#1718C2218yescomplemented
66#1704C2218yescomplemented
67#1704C2218yescomplemented
68#1694C2218yescomplemented
69#1679C2218yes—
70#1653C2218yescomplemented
71#1651C2218yes—
72#1644C2218yes—
73#1636C2218yescomplemented
74#1593C2218yescomplemented
75#1582C2218yes—
76C2×#334C2211yescharacteristic
77C2×#334C2211yescharacteristic
78C2×#333C2211yescharacteristic
79C2×#332C2211yescharacteristic
80C2×#332C2211yescharacteristic
81C2×#332C2211yescharacteristic
82C2×Q8⋊C8C2211yescharacteristic
83C2×Q8⋊C8C2211yescharacteristic
84C2×Q8⋊C8C2211yescharacteristic
85C2×D4⋊C8C2211yescharacteristic
86C2×#322C2211yescharacteristic
87C23×Q8—21no—
88C22×Q16—21no—
89C22×Q16—21no—
90C22×Q16D411yescharacteristic
91C22×SD16—21no—
92C22×SD16D411yescharacteristic
93C23×C8—21no—
94#557—22no—
95#557—22no—
96#557—22no—
97#557—22no—
98#557—41no—
99#557—41no—
100#557C2314yes—
101#557C2314yes—
102#554—22no—
103#554—22no—
104#554—41no—
105#554C2314yes—
106D4×Q8—22no—
107D4×Q8—22no—
108D4×Q8—41no—
109D4×Q8C2314yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#125—21no—
120C2×#125—41no—
121C2×#125—41no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—41no—
125C2×C22⋊Q8—41no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×Q8C2311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2311yescharacteristic
136C22×C4⋊C4—21no—
137#501—28no—
138#500—22no—
139#500C2314yes—
140#498—41no—
141#498C2314yes—
142#491—28no—
143#491—28no—
144#487—22no—
145#487—22no—
146#487—22no—
147#487—41no—
148#487C2314yes—
149#487C2314yes—
150#487C2314yes—
151#487C2314yes—
152#484—28no—
153#479—28no—
154#479—28no—
155#478—28no—
156#477—22no—
157#477—22no—
158#477—28no—
159#477C2314yes—
160#477C2314yes—
161#475—22no—
162#475—22no—
163#475C2314yes—
164#475C2314yes—
165#474—22no—
166#474C2314yes—
167#467—28no—
168#462—28no—
169#462—22no—
170#462—22no—
171#462C2314yes—
172#462C2314yes—
173#461—28no—
174#461—22no—
175#461C2314yes—
176#451—28no—
177#446—22no—
178#446—22no—
179#446—22no—
180#446C2314yes—
181#446C2314yes—
182#446C2314yes—
183#441—28no—
184#440—28no—
185#440—28no—
186C4×Q16D414yes—
187C4×Q16C2314yes—
188C4×SD16D414yes—
189C4×SD16C2314yes—
190C8×D4—22no—
191C8×D4C2314yes—
192C2×#107—21no—
193C2×#107—21no—
194C2×#107—21no—
195C2×#107—21no—
196C2×#107—21no—
197C2×#107D411yescharacteristic
198C2×#106—21no—
199C2×#106—21no—
200C2×#106—41no—
201C2×#106D411yescharacteristic
202C2×C4⋊C8—21no—
203C2×C4⋊C8—21no—
204C2×C4⋊C8—21no—
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207C2×C4⋊C8C2311yescharacteristic
208C2×C4⋊C8C2311yescharacteristic
209C2×#103—21no—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#103—21no—
220C2×#103—21no—
221C2×#103—21no—
222C2×#103—28no—
223C2×#103—41no—
224C2×#103D411yescharacteristic
225C2×#103D411yescharacteristic
226C2×#103D411yescharacteristic
227C2×#102—21no—
228C2×#102D411yescharacteristic
229C2×C22⋊C8—21no—
230C2×#97—21no—
231C2×#97—21no—
232C2×#97—21no—
233C2×C4×C8C2311yescharacteristic
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236#334C2314yes—
237#334C2314yes—
238#333C2314yes—
239#332C2314yes—
240#332C2314yes—
241#332C2314yes—
242Q8⋊C8C2314yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2314yes—
245D4⋊C8C2314yes—
246#322C2314yes—
247C22×Q8—24no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—41no—
253C22×Q8—42no—
254C22×Q8—44no—
255C22×Q8—82no—
256C22×Q8C2×D411yescharacteristic
257C22×Q8C2×D411yescharacteristic
258C22×Q8C2×D411yescharacteristic
259C22×D4C2×D411yescharacteristic
260C23×C4—21no—
261C23×C4—41no—
262C23×C4—41no—
263C2×Q16—24no—
264C2×Q16—24no—
265C2×Q16—42no—
266C2×Q16—44no—
267C2×Q16—82no—
268C2×Q16C2×D414yes—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—44no—
272C2×SD16C2×D414yes—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—24no—
279C22×C8—82no—
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8C2412yes—
303C4⋊Q8C2412yes—
304C4⋊Q8C2412yes—
305C4⋊Q8C2412yes—
306#125—24no—
307#125—24no—
308#125—24no—
309#125—42no—
310#125—42no—
311#125—42no—
312#125—44no—
313#125—44no—
314#125—44no—
315#125—44no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—44no—
319C22⋊Q8—44no—
320C22⋊Q8—44no—
321C22⋊Q8—44no—
322C22⋊Q8—42no—
323C22⋊Q8—42no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—24no—
331C4×Q8—24no—
332C4×Q8—24no—
333C4×Q8—24no—
334C4×Q8—42no—
335C4×Q8—42no—
336C4×Q8—42no—
337C4×Q8—42no—
338C4×Q8C2×D412yes—
339C4×Q8C2×D412yes—
340C4×Q8C2×D412yes—
341C4×Q8C2412yes—
342C4×Q8C2412yes—
343C4×Q8C2412yes—
344C4×D4—24no—
345C4×D4—24no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4C2×D412yes—
351C4×D4C2412yes—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—41no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—41no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—41no—
385C2×C42—41no—
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—24no—
389#107—24no—
390#107—24no—
391#107—24no—
392#107C2×D414yes—
393#106—24no—
394#106—24no—
395#106—44no—
396#106C2×D414yes—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8C2412yes—
405C4⋊C8C2412yes—
406C4⋊C8C2412yes—
407C4⋊C8C2412yes—
408#103—24no—
409#103—24no—
410#103—24no—
411#103—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421#103—44no—
422#103C2×D414yes—
423#103C2×D414yes—
424#103C2×D414yes—
425#102—24no—
426#102C2×D414yes—
427C22⋊C8—24no—
428#97—24no—
429#97—24no—
430#97—24no—
431C4×C8—21no—
432C4×C8C2×D412yes—
433C4×C8C2×D412yes—
434C4×C8C2412yes—
435C4×C8C2412yes—
436C4×C8C2412yes—
437C24—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—44no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—41no—
456C2×Q8—48no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—82no—
464C2×Q8—84no—
465C2×Q8—84no—
466C2×Q8—88no—
467C2×Q8C22×D412yes—
468C2×Q8C22×D412yes—
469C2×Q8C22×D412yes—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4#13714yes—
473C2×D4C22×D412yes—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—84no—
492C22×C4—162no—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4H(5,𝔽2)11yescharacteristic
496Q16—44no—
497Q16—48no—
498Q16—84no—
499SD16—28no—
500SD16—48no—
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—42no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—84no—
516C2×C8C22×D412yes—
517C2×C8C22×D412yes—
518C2×C8C22×D412yes—
519C2×C8C22×D412yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4C22×D412yes—
556C4⋊C4C22×D412yes—
557C4⋊C4C22×D412yes—
558C4⋊C4C22×D412yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—44no—
570C42—44no—
571C42C22×D411yescharacteristic
572C42C22×D411yescharacteristic
573C42H(5,𝔽2)11yescharacteristic
574C42C2511yescharacteristic
575C23—22no—
576C23—24no—
577C23—42no—
578C23—42no—
579C23D4211yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—82no—
596Q8—88no—
597Q8—82no—
598Q8—82no—
599Q8—82no—
600Q8—161no—
601D4—24no—
602D4—22no—
603D4C2×#13714yes—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—88no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—164no—
632C2×C4D4211yescharacteristic
633C2×C4D4211yescharacteristic
634C2×C4D4211yescharacteristic
635C2×C4C2×#13712yes—
636C2×C4#57512yes—
637C2×C4#57712yes—
638C2×C4#57812yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C23×D411yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C8—22no—
643C8—22no—
644C8—22no—
645C8—22no—
646C8—42no—
647C8—42no—
648C8—42no—
649C8—42no—
650C8—42no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—48no—
655C22#327012yes—
656C22#327512yes—
657C22#328912yes—
658C22C2×D4211yescharacteristic
659C4—21no—
660C4—21no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#327012yes—
673C4#327512yes—
674C4C22×#13711yescharacteristic
675C4C2×#57511yescharacteristic
676C4C2×#57711yescharacteristic
677C4C2×#57811yescharacteristic
678C2—44no—
679C2#2021214yesdirect product factor
680C2C2×#327011yescharacteristic
681C2C2×#327511yescharacteristic
682C2C2×#328911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC42C25
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3374C2semidirect
5. #20212C2direct (N × H)
6. C2×#1718C2semidirect
7. C2×#1704C2semidirect
8. C2×#1704C2semidirect
9. C2×#1694C2semidirect
11. C2×#1653C2semidirect
13. C2×#1636C2semidirect
15. C2×#1593C2semidirect
24. #3374C22semidirect
65. #1718C22semidirect
66. #1704C22semidirect
67. #1704C22semidirect
68. #1694C22semidirect
70. #1653C22semidirect
73. #1636C22semidirect
74. #1593C22semidirect
679. C2#20212direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33862
C2×#33742
C2×#33722
C2×#32992
#202122
C2×#17182
C2×#17042
C2×#16942
C2×#16792
C2×#16532
C2×#16512
C2×#16362
C2×#16442
C2×#15932
C2×#15822