Finite Group Databasebeta

← C2×#16262 #610740

Subgroups — C2×#16262 #610740

Order 512.

6182 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#397C212yesmaximal
3C22×#395C211yesmaximal, characteristic, complemented
4C22×#385C211yesmaximal, characteristic
5C22×#381C211yesmaximal, characteristic
6#16262C2116yesmaximal, direct product factor
7C2×#1446C211yesmaximal, characteristic, complemented
8C2×#1444C212yesmaximal, complemented
9C2×#1442C211yesmaximal, characteristic, complemented
10C2×#1442C212yesmaximal, complemented
11C2×#1441C211yesmaximal, characteristic, complemented
12C2×#1440C212yesmaximal
13C24×Q8C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8—28no—
16C22×C22⋊Q8—216no—
17C22×C22⋊Q8C2218yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C2×#397—216no—
28C2×#397C22116yes—
29C2×#395—28no—
30C2×#395C2218yescomplemented
31C2×#385—28no—
32C2×#385C2218yes—
33C2×#381—28no—
34C2×#381C2218yes—
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2212yes—
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47#1446C2218yescomplemented
48#1444C22116yescomplemented
49#1442C2218yescomplemented
50#1442C22116yescomplemented
51#1441C2218yescomplemented
52#1440C22116yes—
53C26C2311yescharacteristic
54C23×Q8—28no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—44no—
59C23×Q8C2314yes—
60C24×C4—28no—
61C24×C4—24no—
62C24×C4—24no—
63C24×C4—44no—
64C24×C4D418yes—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C2×C22⋊Q8—2128no—
77C2×C22⋊Q8—232no—
78C2×C22⋊Q8—264no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2318yes—
95C22×C4⋊C4C2318yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4D4116yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112#397—264no—
113#395—232no—
114#385—232no—
115#381—232no—
116C2×#95—28no—
117C2×#95—28no—
118C2×#95—28no—
119C2×#95—28no—
120C2×#95—28no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—44no—
127C2×#95D418yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C2×#95C2314yes—
140C25—22no—
141C25—22no—
142C25—22no—
143C25—22no—
144C25—24no—
145C25—24no—
146C25—22no—
147C25—22no—
148C25—24no—
149C25C2×D414yes—
150C25C2×D418yes—
151C25C2412yes—
152C25C2411yescharacteristic
153C22×Q8—232no—
154C22×Q8—216no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—28no—
158C22×Q8—48no—
159C22×Q8—432no—
160C22×Q8—432no—
161C22×Q8—416no—
162C22×Q8—416no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—232no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—216no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—232no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—416no—
220C23×C4—48no—
221C23×C4—432no—
222C23×C4—432no—
223C23×C4C2×D418yes—
224C23×C4C2×D418yes—
225C23×C4C2×D418yes—
226C23×C4C2×D414yes—
227C23×C4C2×D4116yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4#4114yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C22⋊Q8—2256no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—264no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—416no—
262C2×C22⋊C4—232no—
263C2×C22⋊C4—264no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—432no—
275C2×C22⋊C4—416no—
276C2×C22⋊C4C2×D4132yes—
277#95—216no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—216no—
282#95—216no—
283#95—216no—
284#95—216no—
285#95—216no—
286#95—216no—
287#95—48no—
288#95C2×D4116yes—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—28no—
293C24—28no—
294C24—216no—
295C24—216no—
296C24—28no—
297C24—216no—
298C24—216no—
299C24—216no—
300C24—216no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24—48no—
319C24—48no—
320C24#120116yes—
321C24C22×D414yes—
322C24C22×D418yes—
323C24C22×D412yes—
324C24C22×D414yes—
325C24C22×D412yes—
326C24C2×#4112yes—
327C24C2×#4112yes—
328C24C2×#4112yes—
329C24H(5,𝔽2)14yes—
330C242−1+414yes—
331C242−1+412yes—
332C242−1+412yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—232no—
336C2×Q8—432no—
337C2×Q8—432no—
338C2×Q8—4128no—
339C2×Q8—416no—
340C2×Q8—416no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—28no—
350C22×C4—232no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—232no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—416no—
416C22×C4—4128no—
417C22×C4—432no—
418C22×C4—432no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4#121116yes—
458C22×C4C22×D418yes—
459C22×C4C22×D418yes—
460C22×C4C22×D418yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C4⋊C4—232no—
468C4⋊C4—264no—
469C4⋊C4—432no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C22⋊C4—232no—
473C22⋊C4—264no—
474C22⋊C4—432no—
475C22⋊C4—416no—
476C23—232no—
477C23—232no—
478C23—24no—
479C23—232no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—416no—
492C23—416no—
493C23—416no—
494C23—416no—
495C23—432no—
496C23—44no—
497C23—432no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—432no—
502C23—42no—
503C23—42no—
504C23—432no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23C2×#12018yes—
510C23C2×#12014yes—
511C23C2×#12114yes—
512C23#52514yes—
513C23#53414yes—
514C23#53614yes—
515C23#53614yes—
516C23#53614yes—
517C23#53618yes—
518C23#53618yes—
519C23#53618yes—
520C23#53918yes—
521C23#54118yes—
522C23#54218yes—
523C23D4218yes—
524C23#54618yes—
525C23#54718yes—
526C23#54718yes—
527C23#54714yes—
528C23#54814yes—
529C23#55214yes—
530C23#55614yes—
531C23#56318yes—
532C23#56318yes—
533C23C23×D411yescharacteristic
534C23C23×D412yes—
535C23C23×D411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+412yes—
544Q8—464no—
545C2×C4—24no—
546C2×C4—232no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—464no—
568C2×C4—416no—
569C2×C4—432no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—416no—
573C2×C4—432no—
574C2×C4—44no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—432no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—816no—
615C2×C4C2×#12118yes—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—48no—
623C22—48no—
624C22—464no—
625C22—48no—
626C22—48no—
627C22—44no—
628C22#240618yes—
629C22#241018yes—
630C22#241518yes—
631C22#261418yes—
632C22#262118yes—
633C22#263818yes—
634C22#2674116yes—
635C22#2680116yes—
636C22#2689116yes—
637C22#2698116yes—
638C22#281718yes—
639C22C22×#12011yescharacteristic
640C22C22×#12111yescharacteristic
641C22C2×#52511yescharacteristic
642C22C2×#53411yescharacteristic
643C22C2×#53611yescharacteristic
644C22C2×#53611yescharacteristic
645C22C2×#53612yes—
646C22C2×#53612yes—
647C22C2×#53611yescharacteristic
648C22C2×#53612yes—
649C22C2×#53912yes—
650C22C2×#54112yes—
651C22C2×#54212yes—
652C22C2×D4212yes—
653C22C2×#54612yes—
654C22C2×#54712yes—
655C22C2×#54711yescharacteristic
656C22C2×#54712yes—
657C22C2×#54811yescharacteristic
658C22C2×#55211yescharacteristic
659C22C2×#55611yescharacteristic
660C22C2×#56312yes—
661C22C2×#56312yes—
662C4—28no—
663C4—44no—
664C4—44no—
665C4—48no—
666C4—48no—
667C4—44no—
668C4—48no—
669C4—82no—
670C4—82no—
671C4—84no—
672C2—48no—
673C2#16262116yesdirect product factor
674C2C2×#263811yescharacteristic
675C2C2×#261411yescharacteristic
676C2C2×#262111yescharacteristic
677C2C2×#241011yescharacteristic
678C2C2×#241511yescharacteristic
679C2C2×#240611yescharacteristic
680C2C2×#281711yescharacteristic
681C2C2×#267412yes—
682C2C2×#268012yes—
683C2C2×#268912yes—
684C2C2×#269812yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#395C2semidirect
6. #16262C2direct (N × H)
7. C2×#1446C2semidirect
8. C2×#1444C2semidirect
9. C2×#1442C2semidirect
10. C2×#1442C2semidirect
11. C2×#1441C2semidirect
30. C2×#395C22semidirect
47. #1446C22semidirect
48. #1444C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
673. C2#16262direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C22×#3972
C22×#3952
C22×#3852
C22×#3812
#162622
C2×#14462
C2×#14442
C2×#14422
C2×#14412
C2×#14402