Finite Group Databasebeta

← C2×#62534 #10579947

Subgroups — C2×#62534 #10579947

Order 512.

6402 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3518C211yesmaximal, characteristic, complemented
2C2×#3517C211yesmaximal, characteristic, complemented
3#62534C2132yesmaximal, direct product factor
4C2×#3363C211yesmaximal, characteristic, complemented
5C2×#3362C211yesmaximal, characteristic, complemented
6C2×#3359C211yesmaximal, characteristic, complemented
7C2×#3358C211yesmaximal, characteristic, complemented
8C2×#3305C213yesmaximal, complemented
9C2×#3304C213yesmaximal, complemented
10C2×#3303C213yesmaximal, complemented
11C2×#3302C213yesmaximal, complemented
12C2×#3233C213yesmaximal, complemented
13C2×#3212C213yesmaximal, complemented
14C2×#3168C213yesmaximal, complemented
15C2×#3131C213yesmaximal, complemented
16C2×C8×#41C211yesmaximal, characteristic, complemented
17#3518—28no—
18#3518C22116yescomplemented
19#3517—28no—
20#3517C22116yescomplemented
21C2×Q82C2211yescharacteristic
22C2×#557C2211yescharacteristic
23C2×#556—23no—
24C2×#556C2213yes—
25C2×#555C2213yes—
26C2×#553—23no—
27C2×#550—21no—
28C2×#550C2213yes—
29C2×#550C2211yescharacteristic
30C2×D4×Q8C2211yescharacteristic
31C2×#548—21no—
32C2×#547C2213yes—
33C2×#542—23no—
34C2×#540—23no—
35C2×#533C2213yes—
36C2×#532C2213yes—
37C2×#529C2213yes—
38C2×#529C2213yes—
39C2×#520—21no—
40C2×#519—21no—
41C2×C4×#41C2211yescharacteristic
42#3363C22116yescomplemented
43#3362C22116yescomplemented
44#3359C22116yescomplemented
45#3358C22116yescomplemented
46#3305C22148yescomplemented
47#3304C22148yescomplemented
48#3303C22148yescomplemented
49#3302C22148yescomplemented
50#3233C22148yescomplemented
51#3212C22148yescomplemented
52#3168C22148yescomplemented
53C2×#499C2213yes—
54C2×#495C2213yes—
55#3131C22148yescomplemented
56C2×#488C2216yes—
57C2×#488C2213yes—
58C2×#488C2211yescharacteristic
59C2×#487C2213yes—
60C2×#487C2213yes—
61C2×#486C2213yes—
62C2×#486C2213yes—
63C2×#485C2216yes—
64C2×#482C2216yes—
65C2×#477C2213yes—
66C2×#475C2213yes—
67C2×#476C2213yes—
68C2×#474C2213yes—
69C2×#467C2213yes—
70C2×#466C2213yes—
71C2×#465C2213yes—
72C2×#465C2213yes—
73C2×#462C2213yes—
74C2×#460C2213yes—
75C2×#461C2213yes—
76C2×#459C2213yes—
77C2×#452C2216yes—
78C2×#449C2216yes—
79C8×#41C22116yescomplemented
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×#416C2216yes—
90C2×#416C2216yes—
91C22×C4×C8C2213yes—
92C22×#41—21no—
93C22×#41D411yescharacteristic
94C22×#41C2316yes—
95C23×Q8—21no—
96C22×Q16C2313yes—
97C22×SD16C2313yes—
98C22×SD16C2313yes—
99C22×D8C2313yes—
100C23×C8C2313yes—
101C23×C8C2313yes—
102Q82—24no—
103Q82C2318yes—
104#557—24no—
105#557C2318yes—
106#556—212no—
107#556—248no—
108#556C23124yes—
109#555—212no—
110#555C23124yes—
111#553—248no—
112#550—216no—
113#550—24no—
114#550—212no—
115#550C23124yes—
116#550C2318yes—
117D4×Q8—24no—
118D4×Q8C2318yes—
119#548—216no—
120#547—212no—
121#547C23124yes—
122#542—248no—
123#540—248no—
124#533—212no—
125#533C23124yes—
126#532—212no—
127#532C23124yes—
128C2×C4⋊Q8—23no—
129C2×C4⋊Q8C2313yes—
130C2×C4⋊Q8C2313yes—
131C2×C4⋊Q8C2313yes—
132C2×#127—23no—
133C2×#127C2313yes—
134#529—212no—
135#529—212no—
136#529C23124yes—
137#529C23124yes—
138C2×#126—26no—
139C2×#126—23no—
140C2×#126—23no—
141C2×#125—23no—
142C2×#125C2313yes—
143C2×#125C2311yescharacteristic
144C2×#124—23no—
145C2×#124—26no—
146C2×#124C2313yes—
147C2×#124C2311yescharacteristic
148C2×#123—23no—
149C2×#123—23no—
150C2×#123—26no—
151C2×C22⋊Q8—23no—
152C2×C22⋊Q8—23no—
153C2×C22⋊Q8—23no—
154C2×C22⋊Q8C2313yes—
155C2×C22⋊Q8C2313yes—
156C2×#121—23no—
157C2×#121—23no—
158C2×#121—23no—
159C2×#121—23no—
160C2×#121—23no—
161C2×#121—26no—
162C2×#121C2313yes—
163C2×#121C2313yes—
164#520—216no—
165#519—216no—
166C4×#41D418yes—
167C4×#41C2318yes—
168C2×C4×Q8—21no—
169C2×C4×Q8—21no—
170C2×C4×Q8—23no—
171C2×C4×Q8—23no—
172C2×C4×Q8—23no—
173C2×C4×Q8D411yescharacteristic
174C2×C4×Q8C2313yes—
175C2×C4×Q8C2313yes—
176C2×C4×Q8C2311yescharacteristic
177C2×C4×D4—23no—
178C2×C4×D4—23no—
179C2×C4×D4—23no—
180C2×C4×D4—23no—
181C2×C4×D4—23no—
182C2×C4×D4D413yes—
183C2×C4×D4C2313yes—
184C2×C4×D4C2313yes—
185C2×C4×D4C2313yes—
186C2×#117—23no—
187C2×#117—23no—
188C2×#117—23no—
189C2×#117D413yes—
190C2×#117C2316yes—
191C22×C4⋊C4—23no—
192C22×C42C2313yes—
193#499C23124yes—
194#495C23124yes—
195#488C23124yes—
196#488C23148yes—
197#488C2318yes—
198#487C23124yes—
199#487C23124yes—
200#486C23124yes—
201#486C23124yes—
202#485C23148yes—
203#482C23148yes—
204#477C23124yes—
205#476C23124yes—
206#475C23124yes—
207#474C23124yes—
208#467C23124yes—
209#466C23124yes—
210#465C23124yes—
211#465C23124yes—
212#462C23124yes—
213#461C23124yes—
214#460C23124yes—
215#459C23124yes—
216#452C23148yes—
217#449C23148yes—
218C8×Q8C2318yes—
219C8×Q8C2318yes—
220C4×Q16C23124yes—
221C4×SD16C23124yes—
222C4×SD16C23124yes—
223C4×D8C23124yes—
224C8×D4C23124yes—
225C8×D4C23124yes—
226#431C23124yes—
227C2×#107C2313yes—
228C2×#107C2313yes—
229C2×#106C2313yes—
230C2×#106C2313yes—
231C2×C4⋊C8C2313yes—
232C2×C4⋊C8C2313yes—
233#416C23148yes—
234#416C23148yes—
235C2×#103C2316yes—
236C2×#103C2316yes—
237C2×#103C2313yes—
238C2×#103C2313yes—
239C2×#103C2311yescharacteristic
240C2×#103C2311yescharacteristic
241C2×#102C2316yes—
242C2×#102C2316yes—
243C2×#102C2313yes—
244C2×#102C2313yes—
245C2×#102C2311yescharacteristic
246C2×#102C2311yescharacteristic
247C2×C22⋊C8C2316yes—
248C2×C4×C8C2311yescharacteristic
249C2×C4×C8C2313yes—
250C2×C4×C8C2313yes—
251C2×C4×C8C2313yes—
252C2×C4×C8C23124yes—
253C2×#41—28no—
254C2×#41—28no—
255C2×#41—212no—
256C2×#41—412no—
257C2×#41—412no—
258C2×#41—48no—
259C2×#41C2×D418yes—
260C2×#41C24124yes—
261C22×Q8—28no—
262C22×Q8—23no—
263C22×Q8—23no—
264C22×Q8—48no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×Q8C2413yes—
268C22×D4—23no—
269C22×D4—23no—
270C22×D4—26no—
271C22×D4—23no—
272C22×D4—23no—
273C22×D4C2×D413yes—
274C22×D4C2411yescharacteristic
275C22×D4C2413yes—
276C23×C4—23no—
277C23×C4—23no—
278C23×C4—26no—
279C23×C4C2×D413yes—
280C23×C4C2×D413yes—
281C23×C4C2413yes—
282C2×Q16—46no—
283C2×Q16C24112yes—
284C2×SD16—46no—
285C2×SD16—46no—
286C2×SD16C24112yes—
287C2×SD16C24112yes—
288C2×D8—46no—
289C2×D8C24112yes—
290C22×C8—46no—
291C22×C8—46no—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2413yes—
295C22×C8C2413yes—
296C22×C8C24112yes—
297C22×C8C24112yes—
298C4⋊Q8—224no—
299C4⋊Q8—26no—
300C4⋊Q8—26no—
301C4⋊Q8—26no—
302C4⋊Q8C24112yes—
303C4⋊Q8C24112yes—
304C4⋊Q8C24112yes—
305#127—224no—
306#127—26no—
307#127C24112yes—
308#126—248no—
309#126—224no—
310#126—224no—
311#125—224no—
312#125—22no—
313#125—26no—
314#125C2414yes—
315#125C24112yes—
316#124—224no—
317#124—248no—
318#124—22no—
319#124—26no—
320#124C2414yes—
321#124C24112yes—
322#123—224no—
323#123—224no—
324#123—248no—
325C22⋊Q8—224no—
326C22⋊Q8—224no—
327C22⋊Q8—26no—
328C22⋊Q8—26no—
329C22⋊Q8—224no—
330C22⋊Q8C24112yes—
331C22⋊Q8C24112yes—
332#121—224no—
333#121—224no—
334#121—224no—
335#121—224no—
336#121—26no—
337#121—26no—
338#121—224no—
339#121—248no—
340#121C24112yes—
341#121C24112yes—
342C4×Q8—28no—
343C4×Q8—28no—
344C4×Q8—224no—
345C4×Q8—224no—
346C4×Q8—224no—
347C4×Q8—26no—
348C4×Q8—26no—
349C4×Q8C2×D418yes—
350C4×Q8C2×D414yes—
351C4×Q8C24112yes—
352C4×Q8C24112yes—
353C4×Q8C2414yes—
354C4×D4—224no—
355C4×D4—224no—
356C4×D4—224no—
357C4×D4—224no—
358C4×D4—224no—
359C4×D4—26no—
360C4×D4—26no—
361C4×D4C2×D4124yes—
362C4×D4C2×D4112yes—
363C4×D4C24112yes—
364C4×D4C24112yes—
365C4×D4C24112yes—
366#117—212no—
367#117—224no—
368#117—224no—
369#117—224no—
370#117C2×D4124yes—
371#117C24124yes—
372C2×C4⋊C4—23no—
373C2×C4⋊C4—23no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—23no—
376C2×C4⋊C4—23no—
377C2×C4⋊C4—23no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—23no—
380C2×C4⋊C4—23no—
381C2×C4⋊C4—23no—
382C2×C4⋊C4—23no—
383C2×C4⋊C4—23no—
384C2×C4⋊C4—224no—
385C2×C4⋊C4C2×D413yes—
386C2×C4⋊C4C2×D413yes—
387C2×C4⋊C4C2413yes—
388C2×C4⋊C4C2413yes—
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C22⋊C4—26no—
392C2×C22⋊C4—23no—
393C2×C22⋊C4—23no—
394C2×C22⋊C4—23no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—23no—
397C2×C22⋊C4—23no—
398C2×C22⋊C4—23no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—23no—
401C2×C22⋊C4—23no—
402C2×C22⋊C4C2×D416yes—
403C2×C42—26no—
404C2×C42—23no—
405C2×C42—23no—
406C2×C42C2×D413yes—
407C2×C42C2×D413yes—
408C2×C42C2×D4112yes—
409C2×C42C2413yes—
410C2×C42C24112yes—
411C2×C42C2411yescharacteristic
412#107C24112yes—
413#107C24112yes—
414#106C24112yes—
415#106C24112yes—
416C4⋊C8C24112yes—
417C4⋊C8C24112yes—
418#103C24112yes—
419#103C24112yes—
420#103C24124yes—
421#103C24124yes—
422#103C2414yes—
423#103C2414yes—
424#102C24112yes—
425#102C24112yes—
426#102C24124yes—
427#102C24124yes—
428#102C2414yes—
429#102C2414yes—
430C22⋊C8C24124yes—
431C4×C8C2414yes—
432C4×C8C24112yes—
433C4×C8C24112yes—
434C4×C8C24112yes—
435C24—21no—
436C24—23no—
437C24C22×D413yes—
438#41—232no—
439#41—424no—
440#41—424no—
441#41—432no—
442#41—824no—
443C2×Q8—21no—
444C2×Q8—212no—
445C2×Q8—23no—
446C2×Q8—212no—
447C2×Q8—22no—
448C2×Q8—26no—
449C2×Q8—22no—
450C2×Q8—26no—
451C2×Q8—432no—
452C2×Q8—412no—
453C2×Q8—86no—
454C2×Q8#13518yes—
455C2×Q8C22×D414yes—
456C2×Q8C2516yes—
457C2×Q8C2512yes—
458C2×D4—212no—
459C2×D4—212no—
460C2×D4—224no—
461C2×D4—212no—
462C2×D4—21no—
463C2×D4—212no—
464C2×D4—23no—
465C2×D4—212no—
466C2×D4—22no—
467C2×D4—26no—
468C2×D4—22no—
469C2×D4—26no—
470C2×D4—412no—
471C2×D4—86no—
472C2×D4—86no—
473C2×D4—86no—
474C2×D4—812no—
475C2×D4C22×D4112yes—
476C2×D4C2516yes—
477C2×D4C2512yes—
478C22×C4—212no—
479C22×C4—26no—
480C22×C4—26no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—23no—
484C22×C4—23no—
485C22×C4—23no—
486C22×C4—21no—
487C22×C4—212no—
488C22×C4—212no—
489C22×C4—224no—
490C22×C4—46no—
491C22×C4—412no—
492C22×C4—412no—
493C22×C4—812no—
494C22×C4C22×D416yes—
495C22×C4C22×D4112yes—
496C22×C4C22×D413yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D413yes—
500C22×C4C22×D4112yes—
501C22×C4C22×D413yes—
502C22×C4C2516yes—
503C22×C4C2511yescharacteristic
504Q16—412no—
505SD16—412no—
506SD16—412no—
507D8—412no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—26no—
511C2×C8—26no—
512C2×C8—412no—
513C2×C8—412no—
514C2×C8C2512yes—
515C2×C8C2512yes—
516C2×C8C2516yes—
517C2×C8C2516yes—
518C4⋊C4—212no—
519C4⋊C4—212no—
520C4⋊C4—24no—
521C4⋊C4—212no—
522C4⋊C4—212no—
523C4⋊C4—212no—
524C4⋊C4—24no—
525C4⋊C4—212no—
526C4⋊C4—212no—
527C4⋊C4—212no—
528C4⋊C4—212no—
529C4⋊C4—212no—
530C4⋊C4—23no—
531C4⋊C4—23no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4C22×D4112yes—
535C4⋊C4C22×D4112yes—
536C4⋊C4C2516yes—
537C4⋊C4C2516yes—
538C4⋊C4C2512yes—
539C4⋊C4C2512yes—
540C22⋊C4—224no—
541C22⋊C4—212no—
542C22⋊C4—212no—
543C22⋊C4—212no—
544C22⋊C4—24no—
545C22⋊C4—212no—
546C22⋊C4—212no—
547C22⋊C4—212no—
548C22⋊C4—24no—
549C22⋊C4—212no—
550C22⋊C4—212no—
551C22⋊C4C22×D4124yes—
552C42—224no—
553C42—212no—
554C42—212no—
555C42C22×D412yes—
556C42C22×D4112yes—
557C42C22×D4112yes—
558C42C22×D416yes—
559C42C2512yes—
560C42C2516yes—
561C23—26no—
562C23—26no—
563C23—26no—
564C23—22no—
565C23—26no—
566C23—46no—
567C23—42no—
568C23—46no—
569C23—42no—
570C23—42no—
571C23—46no—
572C23C23×D416yes—
573C23C23×D411yescharacteristic
574Q8—22no—
575Q8—26no—
576Q8—22no—
577Q8—26no—
578Q8—42no—
579Q8—46no—
580Q8—424no—
581Q8—812no—
582Q8C2×#135116yes—
583D4—22no—
584D4—26no—
585D4—22no—
586D4—26no—
587D4—224no—
588D4—42no—
589D4—46no—
590D4—424no—
591D4—812no—
592D4—812no—
593D4—812no—
594D4—824no—
595C2×C4—22no—
596C2×C4—26no—
597C2×C4—26no—
598C2×C4—22no—
599C2×C4—26no—
600C2×C4—26no—
601C2×C4—212no—
602C2×C4—224no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—26no—
606C2×C4—26no—
607C2×C4—26no—
608C2×C4—22no—
609C2×C4—412no—
610C2×C4—42no—
611C2×C4—46no—
612C2×C4—46no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—46no—
616C2×C4—424no—
617C2×C4—424no—
618C2×C4—824no—
619C2×C4C2×#135112yes—
620C2×C4C23×D416yes—
621C2×C4C23×D412yes—
622C2×C4C23×D412yes—
623C2×C4C23×D416yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C23×D416yes—
626C2×C4#58512yes—
627C2×C4#58512yes—
628C2×C4C2611yescharacteristic
629C8—22no—
630C8—22no—
631C8—26no—
632C8—26no—
633C22—26no—
634C22—26no—
635C22—26no—
636C22—412no—
637C22—46no—
638C22—42no—
639C22—42no—
640C22—46no—
641C22—46no—
642C22—42no—
643C22—812no—
644C22—84no—
645C22#350012yes—
646C22C22×#13512yes—
647C22#358412yes—
648C22C24×D411yescharacteristic
649C4—26no—
650C4—22no—
651C4—22no—
652C4—26no—
653C4—46no—
654C4—42no—
655C4—42no—
656C4—46no—
657C4—42no—
658C4—46no—
659C4#350012yes—
660C4C22×#135112yes—
661C4C2×#58511yescharacteristic
662C4C2×#58511yescharacteristic
663C2—46no—
664C2—86no—
665C2—82no—
666C2#6253414yesdirect product factor
667C2C2×#350011yescharacteristic
668C2C23×#13511yescharacteristic
669C2C2×#358411yescharacteristic
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×#3517C2semidirect
3. #62534C2direct (N × H)
4. C2×#3363C2semidirect
5. C2×#3362C2semidirect
6. C2×#3359C2semidirect
7. C2×#3358C2semidirect
8. C2×#3305C2semidirect
9. C2×#3304C2semidirect
10. C2×#3303C2semidirect
11. C2×#3302C2semidirect
12. C2×#3233C2semidirect
13. C2×#3212C2semidirect
14. C2×#3168C2semidirect
15. C2×#3131C2semidirect
16. C2×C8×#41C2semidirect
18. #3518C22semidirect
20. #3517C22semidirect
42. #3363C22semidirect
43. #3362C22semidirect
44. #3359C22semidirect
45. #3358C22semidirect
46. #3305C22semidirect
47. #3304C22semidirect
48. #3303C22semidirect
49. #3302C22semidirect
50. #3233C22semidirect
51. #3212C22semidirect
52. #3168C22semidirect
55. #3131C22semidirect
79. C8×#41C22semidirect
666. C2#62534direct (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×#35172
#625342
C2×#33632
C2×#33622
C2×#33592
C2×#33582
C2×#33052
C2×#33042
C2×#33032
C2×#33022
C2×#32332
C2×#32122
C2×#31682
C2×#31312
C2×C8×#412