Finite Group Databasebeta

← C2×#62105 #10579351

Subgroups — C2×#62105 #10579351

Order 512.

6316 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3518C211yesmaximal, characteristic, complemented
2C2×#3482C211yesmaximal, characteristic, complemented
3#62105C2132yesmaximal, direct product factor
4C2×Q8×SD16C211yesmaximal, characteristic, complemented
5C2×Q8×D8C211yesmaximal, characteristic, complemented
6C2×#3363C211yesmaximal, characteristic, complemented
7C2×#3359C211yesmaximal, characteristic, complemented
8C2×#3342C213yesmaximal, complemented
9C2×#3340C213yesmaximal, complemented
10C2×#3303C213yesmaximal, complemented
11C2×#3302C213yesmaximal, complemented
12C2×#3248C213yesmaximal, complemented
13C2×#3231C213yesmaximal, complemented
14C2×#3102C213yesmaximal, complemented
15C2×#3076C213yesmaximal, complemented
16C2×#2988C211yesmaximal, characteristic, complemented
17#3518—28no—
18#3518C22116yescomplemented
19#3482—28no—
20#3482C22116yescomplemented
21C2×Q82C2211yescharacteristic
22C2×#557C2211yescharacteristic
23C2×#556—23no—
24C2×#556C2213yes—
25C2×#556C2213yes—
26C2×#555C2213yes—
27C2×#552—23no—
28C2×D4×Q8C2211yescharacteristic
29C2×D4×Q8C2211yescharacteristic
30C2×#548—21no—
31C2×#548—21no—
32C2×#547C2213yes—
33C2×#542—23no—
34C2×#533C2213yes—
35C2×#533C2213yes—
36C2×#529—23no—
37C2×#529C2213yes—
38C2×#519—21no—
39C2×C4×#41C2213yes—
40C2×C4×#41C2211yescharacteristic
41C22×C4×Q8—21no—
42Q8×SD16C22116yescomplemented
43Q8×D8C22116yescomplemented
44#3363C22116yescomplemented
45#3359C22116yescomplemented
46#3342C22148yescomplemented
47#3340C22148yescomplemented
48#3303C22148yescomplemented
49#3302C22148yescomplemented
50#3248C22148yescomplemented
51#3231C22148yescomplemented
52C2×#500C2213yes—
53C2×#498C2213yes—
54C2×#487C2213yes—
55C2×#487C2213yes—
56#3102C22148yescomplemented
57C2×#485C2216yes—
58C2×#485C2213yes—
59C2×#485C2216yes—
60C2×#482C2213yes—
61#3076C22148yescomplemented
62C2×#479C2216yes—
63C2×#479C2213yes—
64C2×#477C2213yes—
65C2×#475C2213yes—
66C2×#475C2213yes—
67C2×#478C2213yes—
68C2×#476C2213yes—
69C2×#474C2213yes—
70C2×#474C2213yes—
71C2×#467C2216yes—
72C2×#465C2216yes—
73C2×#464C2213yes—
74C2×#462C2213yes—
75C2×#463C2211yescharacteristic
76C2×#461C2213yes—
77C2×#452C2213yes—
78C2×#449C2213yes—
79#2988C22116yescomplemented
80C2×C8×Q8C2211yescharacteristic
81C2×C8×Q8C2211yescharacteristic
82C2×C4×Q16C2213yes—
83C2×C4×SD16C2213yes—
84C2×C4×SD16C2213yes—
85C2×C4×D8C2213yes—
86C2×C8×D4C2213yes—
87C2×C8×D4C2213yes—
88C2×#431C2213yes—
89C2×#427C2216yes—
90C22×C4⋊C8C2213yes—
91C2×#416C2216yes—
92C22×#41D411yescharacteristic
93C22×#41C2313yes—
94C23×Q8—21no—
95C23×Q8—21no—
96C22×Q16C2313yes—
97C22×SD16C2311yescharacteristic
98C22×SD16C2313yes—
99C22×D8C2311yescharacteristic
100C23×C8C2316yes—
101Q82—24no—
102Q82C2318yes—
103#557—24no—
104#557C2318yes—
105#556—212no—
106#556—212no—
107#556—248no—
108#556C23124yes—
109#556C23124yes—
110#555—212no—
111#555C23124yes—
112#552—248no—
113D4×Q8—24no—
114D4×Q8—24no—
115D4×Q8C2318yes—
116D4×Q8C2318yes—
117#548—216no—
118#548—216no—
119#547—212no—
120#547C23124yes—
121#542—248no—
122#533—212no—
123#533—212no—
124#533C23124yes—
125#533C23124yes—
126C2×C4⋊Q8—23no—
127C2×C4⋊Q8C2313yes—
128C2×C4⋊Q8C2313yes—
129C2×C4⋊Q8C2313yes—
130C2×C4⋊Q8C2313yes—
131#529—248no—
132#529—212no—
133#529C23124yes—
134C2×#126—26no—
135C2×#126—26no—
136C2×#126—23no—
137C2×#125—23no—
138C2×#125—23no—
139C2×#125C2313yes—
140C2×#124—23no—
141C2×#124—23no—
142C2×#124—26no—
143C2×#124C2311yescharacteristic
144C2×#123—23no—
145C2×#123—26no—
146C2×C22⋊Q8—23no—
147C2×C22⋊Q8—23no—
148C2×C22⋊Q8—23no—
149C2×C22⋊Q8—23no—
150C2×C22⋊Q8—23no—
151C2×C22⋊Q8C2316yes—
152C2×C22⋊Q8C2313yes—
153C2×#121—23no—
154C2×#121—23no—
155C2×#121C2313yes—
156#519—216no—
157C4×#41—212no—
158C4×#41D418yes—
159C4×#41C23124yes—
160C4×#41C2318yes—
161C2×C4×Q8—21no—
162C2×C4×Q8—21no—
163C2×C4×Q8—216no—
164C2×C4×Q8—23no—
165C2×C4×Q8—23no—
166C2×C4×Q8—23no—
167C2×C4×Q8—23no—
168C2×C4×Q8—23no—
169C2×C4×Q8D411yescharacteristic
170C2×C4×Q8C2313yes—
171C2×C4×Q8C2313yes—
172C2×C4×Q8C2313yes—
173C2×C4×Q8C2311yescharacteristic
174C2×C4×D4—23no—
175C2×C4×D4—23no—
176C2×C4×D4—23no—
177C2×C4×D4D413yes—
178C2×C4×D4C2311yescharacteristic
179C2×C4×D4C2313yes—
180C2×C4×D4C2313yes—
181C2×C4×D4C2313yes—
182C2×#117—23no—
183C2×#117—23no—
184C2×#117—23no—
185C2×#117—23no—
186C2×#117D413yes—
187C2×#117C2313yes—
188C2×#117C2316yes—
189C22×C4⋊C4—23no—
190C22×C42—23no—
191C22×C42C2313yes—
192#500C23124yes—
193#498C23124yes—
194#487C23124yes—
195#487C23124yes—
196#485C23148yes—
197#485C23124yes—
198#485C23148yes—
199#482C23124yes—
200#479C23148yes—
201#479C23124yes—
202#478C23124yes—
203#477C23124yes—
204#476C23124yes—
205#475C23124yes—
206#475C23124yes—
207#474C23124yes—
208#474C23124yes—
209#467C23148yes—
210#465C23148yes—
211#464C23124yes—
212#463C2318yes—
213#462C23124yes—
214#461C23124yes—
215#452C23124yes—
216#449C23124yes—
217C8×Q8C2318yes—
218C8×Q8C2318yes—
219C4×Q16C23124yes—
220C4×SD16C23124yes—
221C4×SD16C23124yes—
222C4×D8C23124yes—
223C8×D4C23124yes—
224C8×D4C23124yes—
225#431C23124yes—
226#427C23148yes—
227C2×#107C2313yes—
228C2×#107C2313yes—
229C2×#107C2316yes—
230C2×#106C2313yes—
231C2×#106C2316yes—
232C2×#106C2313yes—
233C2×C4⋊C8C2313yes—
234C2×C4⋊C8C2313yes—
235C2×C4⋊C8C2313yes—
236C2×C4⋊C8C2311yescharacteristic
237C2×C4⋊C8C23124yes—
238#416C23148yes—
239C2×#103C2313yes—
240C2×#103C2316yes—
241C2×#103C2316yes—
242C2×#103C2313yes—
243C2×#103C2313yes—
244C2×#103C2311yescharacteristic
245C2×#102C2313yes—
246C2×#102C2313yes—
247C2×#102C2311yescharacteristic
248C2×#102C2313yes—
249C2×C22⋊C8C2313yes—
250C2×C22⋊C8C2313yes—
251C2×C4×C8C2313yes—
252C2×C4×C8C2313yes—
253C2×#41—28no—
254C2×#41—26no—
255C2×#41—212no—
256C2×#41—46no—
257C2×#41C2×D418yes—
258C2×#41C24112yes—
259C22×Q8—28no—
260C22×Q8—28no—
261C22×Q8—23no—
262C22×Q8—23no—
263C22×Q8—23no—
264C22×Q8—48no—
265C22×Q8—48no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2411yescharacteristic
268C22×Q8C2413yes—
269C22×D4—23no—
270C22×D4—23no—
271C22×D4C2×D413yes—
272C22×D4C2411yescharacteristic
273C22×D4C2411yescharacteristic
274C23×C4—23no—
275C23×C4—23no—
276C23×C4—23no—
277C23×C4—21no—
278C23×C4C2×D413yes—
279C23×C4C2×D413yes—
280C23×C4C2413yes—
281C2×Q16—46no—
282C2×Q16C24112yes—
283C2×SD16—24no—
284C2×SD16—46no—
285C2×SD16C2414yes—
286C2×SD16C24112yes—
287C2×D8—24no—
288C2×D8C2414yes—
289C22×C8—412no—
290C22×C8C2413yes—
291C22×C8C2413yes—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C24124yes—
295C4⋊Q8—224no—
296C4⋊Q8—26no—
297C4⋊Q8—26no—
298C4⋊Q8—26no—
299C4⋊Q8—26no—
300C4⋊Q8C24112yes—
301C4⋊Q8C24112yes—
302C4⋊Q8C24112yes—
303C4⋊Q8C24112yes—
304#126—248no—
305#126—248no—
306#126—224no—
307#125—224no—
308#125—224no—
309#125—26no—
310#125C24112yes—
311#124—224no—
312#124—224no—
313#124—248no—
314#124—22no—
315#124C2414yes—
316#123—224no—
317#123—248no—
318C22⋊Q8—224no—
319C22⋊Q8—224no—
320C22⋊Q8—224no—
321C22⋊Q8—26no—
322C22⋊Q8—212no—
323C22⋊Q8—224no—
324C22⋊Q8—224no—
325C22⋊Q8C24112yes—
326C22⋊Q8C24124yes—
327#121—224no—
328#121—224no—
329#121—26no—
330#121C24112yes—
331C4×Q8—28no—
332C4×Q8—28no—
333C4×Q8—224no—
334C4×Q8—224no—
335C4×Q8—224no—
336C4×Q8—224no—
337C4×Q8—224no—
338C4×Q8—26no—
339C4×Q8—26no—
340C4×Q8—26no—
341C4×Q8C2×D418yes—
342C4×Q8C2×D414yes—
343C4×Q8C24112yes—
344C4×Q8C24112yes—
345C4×Q8C24112yes—
346C4×Q8C2414yes—
347C4×D4—224no—
348C4×D4—224no—
349C4×D4—224no—
350C4×D4—22no—
351C4×D4—26no—
352C4×D4—26no—
353C4×D4C2×D4124yes—
354C4×D4C2×D4112yes—
355C4×D4C2414yes—
356C4×D4C24112yes—
357C4×D4C24112yes—
358C4×D4C24112yes—
359#117—224no—
360#117—26no—
361#117—212no—
362#117—224no—
363#117—224no—
364#117—224no—
365#117C2×D4124yes—
366#117C24112yes—
367#117C24124yes—
368C2×C4⋊C4—23no—
369C2×C4⋊C4—26no—
370C2×C4⋊C4—23no—
371C2×C4⋊C4—23no—
372C2×C4⋊C4—23no—
373C2×C4⋊C4—23no—
374C2×C4⋊C4—23no—
375C2×C4⋊C4—23no—
376C2×C4⋊C4—23no—
377C2×C4⋊C4—23no—
378C2×C4⋊C4—23no—
379C2×C4⋊C4—23no—
380C2×C4⋊C4—224no—
381C2×C4⋊C4C2×D413yes—
382C2×C4⋊C4C2×D413yes—
383C2×C4⋊C4C2411yescharacteristic
384C2×C4⋊C4C2413yes—
385C2×C4⋊C4C2413yes—
386C2×C4⋊C4C2413yes—
387C2×C22⋊C4—26no—
388C2×C22⋊C4—23no—
389C2×C22⋊C4—26no—
390C2×C22⋊C4—23no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—23no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—23no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—23no—
397C2×C22⋊C4C2×D416yes—
398C2×C42—23no—
399C2×C42—26no—
400C2×C42—23no—
401C2×C42—23no—
402C2×C42—23no—
403C2×C42—224no—
404C2×C42C2×D413yes—
405C2×C42C2×D413yes—
406C2×C42C2×D4112yes—
407C2×C42C2413yes—
408C2×C42C2411yescharacteristic
409C2×C42C24112yes—
410#107C24112yes—
411#107C24124yes—
412#107C24112yes—
413#106C24112yes—
414#106C24124yes—
415#106C24112yes—
416C4⋊C8C24112yes—
417C4⋊C8C24112yes—
418C4⋊C8C24112yes—
419C4⋊C8C2414yes—
420#103C24124yes—
421#103C24124yes—
422#103C24112yes—
423#103C24112yes—
424#103C2414yes—
425#103C24112yes—
426#102C24112yes—
427#102C24112yes—
428#102C24112yes—
429#102C2414yes—
430C22⋊C8C24112yes—
431C22⋊C8C24112yes—
432C4×C8C24112yes—
433C4×C8C24112yes—
434C24—21no—
435C24—21no—
436C24C22×D413yes—
437#41—232no—
438#41—224no—
439#41—412no—
440#41—424no—
441C2×Q8—21no—
442C2×Q8—212no—
443C2×Q8—212no—
444C2×Q8—23no—
445C2×Q8—212no—
446C2×Q8—22no—
447C2×Q8—26no—
448C2×Q8—26no—
449C2×Q8—22no—
450C2×Q8—432no—
451C2×Q8—432no—
452C2×Q8—412no—
453C2×Q8—412no—
454C2×Q8—86no—
455C2×Q8#13518yes—
456C2×Q8C22×D414yes—
457C2×Q8C2516yes—
458C2×Q8C2512yes—
459C2×D4—212no—
460C2×D4—212no—
461C2×D4—212no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—412no—
468C2×D4—86no—
469C2×D4C22×D4112yes—
470C2×D42−1+414yes—
471C2×D4C2512yes—
472C2×D4C2512yes—
473C22×C4—212no—
474C22×C4—26no—
475C22×C4—26no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—23no—
479C22×C4—23no—
480C22×C4—23no—
481C22×C4—23no—
482C22×C4—24no—
483C22×C4—212no—
484C22×C4—212no—
485C22×C4—212no—
486C22×C4—46no—
487C22×C4—412no—
488C22×C4—44no—
489C22×C4—412no—
490C22×C4—412no—
491C22×C4C22×D416yes—
492C22×C4C22×D4112yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D413yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D413yes—
497C22×C4C22×D4112yes—
498C22×C4C22×D413yes—
499C22×C4C2516yes—
500C22×C4C2511yescharacteristic
501Q16—412no—
502SD16—28no—
503SD16—412no—
504D8—28no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—26no—
508C2×C8—26no—
509C2×C8—424no—
510C2×C8C2512yes—
511C2×C8C2512yes—
512C2×C8C2516yes—
513C2×C8C2516yes—
514C4⋊C4—212no—
515C4⋊C4—224no—
516C4⋊C4—212no—
517C4⋊C4—212no—
518C4⋊C4—212no—
519C4⋊C4—212no—
520C4⋊C4—212no—
521C4⋊C4—212no—
522C4⋊C4—212no—
523C4⋊C4—212no—
524C4⋊C4—212no—
525C4⋊C4—212no—
526C4⋊C4—21no—
527C4⋊C4—23no—
528C4⋊C4—23no—
529C4⋊C4—23no—
530C4⋊C4C22×D4112yes—
531C4⋊C4C22×D4112yes—
532C4⋊C4C2516yes—
533C4⋊C4C2516yes—
534C4⋊C4C2512yes—
535C4⋊C4C2516yes—
536C22⋊C4—224no—
537C22⋊C4—212no—
538C22⋊C4—224no—
539C22⋊C4—212no—
540C22⋊C4—24no—
541C22⋊C4—212no—
542C22⋊C4—24no—
543C22⋊C4—212no—
544C22⋊C4—24no—
545C22⋊C4—212no—
546C22⋊C4C22×D4124yes—
547C42—212no—
548C42—212no—
549C42—212no—
550C42—224no—
551C42—212no—
552C42C22×D4112yes—
553C42C22×D412yes—
554C42C22×D4112yes—
555C42C22×D416yes—
556C42C2516yes—
557C42C2512yes—
558C23—26no—
559C23—22no—
560C23—26no—
561C23—24no—
562C23—26no—
563C23—22no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23C23×D416yes—
570C23C23×D411yescharacteristic
571Q8—22no—
572Q8—26no—
573Q8—26no—
574Q8—22no—
575Q8—42no—
576Q8—46no—
577Q8—424no—
578Q8—424no—
579Q8—812no—
580Q8C2×#135116yes—
581D4—22no—
582D4—22no—
583D4—22no—
584D4—24no—
585D4—224no—
586D4—424no—
587D4—42no—
588D4—812no—
589D4C2×2−1+414yes—
590C2×C4—22no—
591C2×C4—26no—
592C2×C4—26no—
593C2×C4—22no—
594C2×C4—26no—
595C2×C4—26no—
596C2×C4—212no—
597C2×C4—224no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—26no—
601C2×C4—26no—
602C2×C4—26no—
603C2×C4—26no—
604C2×C4—412no—
605C2×C4—424no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—46no—
609C2×C4—46no—
610C2×C4—42no—
611C2×C4—46no—
612C2×C4—46no—
613C2×C4—424no—
614C2×C4—424no—
615C2×C4C2×#135112yes—
616C2×C4C23×D412yes—
617C2×C4C23×D416yes—
618C2×C4C23×D412yes—
619C2×C4C23×D416yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C23×D416yes—
622C2×C4C2×2−1+412yes—
623C2×C4#58512yes—
624C2×C4C2611yescharacteristic
625C8—22no—
626C8—22no—
627C8—26no—
628C8—26no—
629C22—26no—
630C22—24no—
631C22—26no—
632C22—26no—
633C22—42no—
634C22—42no—
635C22—412no—
636C22—48no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—84no—
641C22#348912yes—
642C22C22×#13512yes—
643C22#358612yes—
644C22C24×D411yescharacteristic
645C4—26no—
646C4—22no—
647C4—22no—
648C4—26no—
649C4—46no—
650C4—42no—
651C4—46no—
652C4—46no—
653C4—42no—
654C4—46no—
655C4#348912yes—
656C4C22×#135112yes—
657C4C22×2−1+411yescharacteristic
658C4C2×#58511yescharacteristic
659C2—46no—
660C2—44no—
661C2—82no—
662C2#6210514yesdirect product factor
663C2C2×#348911yescharacteristic
664C2C23×#13511yescharacteristic
665C2C2×#358611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3518C2semidirect
2. C2×#3482C2semidirect
3. #62105C2direct (N × H)
4. C2×Q8×SD16C2semidirect
5. C2×Q8×D8C2semidirect
6. C2×#3363C2semidirect
7. C2×#3359C2semidirect
8. C2×#3342C2semidirect
9. C2×#3340C2semidirect
10. C2×#3303C2semidirect
11. C2×#3302C2semidirect
12. C2×#3248C2semidirect
13. C2×#3231C2semidirect
14. C2×#3102C2semidirect
15. C2×#3076C2semidirect
16. C2×#2988C2semidirect
18. #3518C22semidirect
20. #3482C22semidirect
42. Q8×SD16C22semidirect
43. Q8×D8C22semidirect
44. #3363C22semidirect
45. #3359C22semidirect
46. #3342C22semidirect
47. #3340C22semidirect
48. #3303C22semidirect
49. #3302C22semidirect
50. #3248C22semidirect
51. #3231C22semidirect
56. #3102C22semidirect
61. #3076C22semidirect
79. #2988C22semidirect
662. C2#62105direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35182
C2×#34822
#621052
C2×Q8×SD162
C2×Q8×D82
C2×#33632
C2×#33592
C2×#33422
C2×#33402
C2×#33032
C2×#33022
C2×#32482
C2×#32312
C2×#31022
C2×#30762
C2×#29882