Finite Group Databasebeta

← C2×#16359 #610865

Subgroups — C2×#16359 #610865

Order 512.

6570 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic, complemented
3C22×#397C211yesmaximal, characteristic, complemented
4C22×#380C211yesmaximal, characteristic, complemented
5#16359C2116yesmaximal, direct product factor
6C2×#1442C212yesmaximal, complemented
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1439C212yesmaximal, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1439C211yesmaximal, characteristic, complemented
11C2×#1438C212yesmaximal
12C2×C22⋊#95C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×#123—28no—
16C22×#123—28no—
17C22×#123—28no—
18C22×#123C2218yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2212yes—
29C23×C22⋊C4C2212yes—
30C23×C22⋊C4C2212yes—
31C2×#398—28no—
32C2×#398C2218yescomplemented
33C2×#397—28no—
34C2×#397C2218yescomplemented
35C2×#380—28no—
36C2×#380C2218yescomplemented
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1442C22116yescomplemented
47#1441C2218yescomplemented
48#1439C22116yescomplemented
49#1439C22116yescomplemented
50#1439C2218yescomplemented
51#1438C22116yes—
52C22⋊#95C2218yescomplemented
53C26C2311yescharacteristic
54C26C2311yescharacteristic
55C23×D4—24no—
56C23×D4—24no—
57C23×D4—28no—
58C23×D4—24no—
59C23×D4—44no—
60C23×D4C2314yes—
61C24×C4—28no—
62C24×C4—24no—
63C24×C4—24no—
64C24×C4—44no—
65C24×C4D418yes—
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C2×#123—2128no—
77C2×#123—232no—
78C2×#123—232no—
79C2×#123—232no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
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⋊C4—44no—
93C22×C4⋊C4D418yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4—44no—
116C22×C22⋊C4—44no—
117C22×C22⋊C4—44no—
118C22×C22⋊C4—44no—
119C22×C22⋊C4D418yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2318yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2318yes—
127#398—232no—
128#397—232no—
129#380—232no—
130C2×#95—28no—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—28no—
134C2×#95—28no—
135C2×#95—28no—
136C2×#95—44no—
137C2×#95—44no—
138C2×#95—44no—
139C2×#95C2314yes—
140C2×#95C2318yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C25—22no—
149C25—22no—
150C25—24no—
151C25—22no—
152C25—22no—
153C25—22no—
154C25—24no—
155C25—24no—
156C25—22no—
157C25—22no—
158C25—22no—
159C25—24no—
160C25—24no—
161C25—22no—
162C25—22no—
163C25—22no—
164C25—24no—
165C25—22no—
166C25—24no—
167C25C2×D414yes—
168C25C2×D414yes—
169C25#4118yes—
170C25C2412yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×D4—232no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—216no—
177C22×D4—28no—
178C22×D4—48no—
179C22×D4—432no—
180C22×D4—432no—
181C22×D4—416no—
182C22×D4—416no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—232no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—216no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—232no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—416no—
236C23×C4—48no—
237C23×C4—432no—
238C23×C4—432no—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D4116yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4#4118yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2414yes—
256C23×C4C2412yes—
257#123—2256no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—432no—
275C2×C4⋊C4—416no—
276C2×C4⋊C4C2×D4116yes—
277C2×C22⋊C4—232no—
278C2×C22⋊C4—232no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—232no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—216no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—432no—
301C2×C22⋊C4—432no—
302C2×C22⋊C4—432no—
303C2×C22⋊C4—432no—
304C2×C22⋊C4C2×D4116yes—
305#95—216no—
306#95—28no—
307#95—216no—
308#95—28no—
309#95—216no—
310#95—216no—
311#95—48no—
312#95—48no—
313#95—48no—
314C24—28no—
315C24—28no—
316C24—28no—
317C24—28no—
318C24—28no—
319C24—28no—
320C24—216no—
321C24—216no—
322C24—216no—
323C24—216no—
324C24—216no—
325C24—216no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—28no—
336C24—28no—
337C24—28no—
338C24—216no—
339C24—216no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24—48no—
355C24—48no—
356C24—48no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—44no—
364C24—44no—
365C24—44no—
366C24—44no—
367C24—44no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—48no—
378C24#123116yes—
379C24C22×D414yes—
380C24C22×D414yes—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C2×#4118yes—
385C24C2×#4114yes—
386C24C2×#4114yes—
387C24H(5,𝔽2)14yes—
388C24H(5,𝔽2)12yes—
389C24H(5,𝔽2)12yes—
390C24H(5,𝔽2)12yes—
391C242−1+414yes—
392C242−1+412yes—
393C24C2511yescharacteristic
394C2×D4—232no—
395C2×D4—432no—
396C2×D4—432no—
397C2×D4—4128no—
398C2×D4—416no—
399C2×D4—416no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—232no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—232no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—416no—
467C22×C4—4128no—
468C22×C4—432no—
469C22×C4—432no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4#120116yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4C22×D414yes—
511C22×C4C22×D414yes—
512C22×C4C22×D414yes—
513C22×C4C22×D414yes—
514C22×C4C22×D414yes—
515C22×C4C2×#4118yes—
516C4⋊C4—232no—
517C4⋊C4—232no—
518C4⋊C4—416no—
519C4⋊C4—432no—
520C4⋊C4—416no—
521C22⋊C4—232no—
522C22⋊C4—232no—
523C22⋊C4—232no—
524C22⋊C4—432no—
525C22⋊C4—432no—
526C22⋊C4—432no—
527C22⋊C4—432no—
528C23—232no—
529C23—232no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—232no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—28no—
547C23—28no—
548C23—416no—
549C23—416no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—416no—
557C23—432no—
558C23—432no—
559C23—44no—
560C23—44no—
561C23—432no—
562C23—432no—
563C23—44no—
564C23—44no—
565C23—416no—
566C23—416no—
567C23—416no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—416no—
581C23—432no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—432no—
587C23—44no—
588C23—44no—
589C23—88no—
590C23—88no—
591C23—816no—
592C23—816no—
593C23—88no—
594C23—88no—
595C23C2×#12014yes—
596C23C2×#12318yes—
597C23C2×#12314yes—
598C23#53214yes—
599C23#53414yes—
600C23#53514yes—
601C23#53514yes—
602C23#53514yes—
603C23#53614yes—
604C23#53714yes—
605C23#53714yes—
606C23#53714yes—
607C23#53818yes—
608C23#53918yes—
609C23#53918yes—
610C23#54018yes—
611C23#54618yes—
612C23#54618yes—
613C23#54718yes—
614C23#54818yes—
615C23#55118yes—
616C23#55218yes—
617C23#56218yes—
618C23#56318yes—
619C23C23×D411yescharacteristic
620C23C23×D411yescharacteristic
621C23C23×D411yescharacteristic
622C23C22×#4112yes—
623C23C22×#4112yes—
624C23C2×H(5,𝔽2)12yes—
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)11yescharacteristic
627C23C2×H(5,𝔽2)11yescharacteristic
628C23C2×2−1+412yes—
629C23C2×2−1+411yescharacteristic
630D4—464no—
631C2×C4—232no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—28no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—464no—
650C2×C4—416no—
651C2×C4—416no—
652C2×C4—416no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—416no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—432no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—816no—
693C2×C4—816no—
694C2×C4—816no—
695C2×C4C2×#12018yes—
696C22—28no—
697C22—28no—
698C22—28no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—48no—
703C22—48no—
704C22—44no—
705C22—48no—
706C22—48no—
707C22—464no—
708C22—44no—
709C22—44no—
710C22—48no—
711C22—44no—
712C22—48no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—84no—
717C22—84no—
718C22—832no—
719C22—82no—
720C22#240518yes—
721C22#243218yes—
722C22#249818yes—
723C22#261518yes—
724C22#262318yes—
725C22#266118yes—
726C22#2695116yes—
727C22#2696116yes—
728C22#2706116yes—
729C22#2707116yes—
730C22#281418yes—
731C22C22×#12011yescharacteristic
732C22C22×#12311yescharacteristic
733C22C2×#53211yescharacteristic
734C22C2×#53411yescharacteristic
735C22C2×#53511yescharacteristic
736C22C2×#53511yescharacteristic
737C22C2×#53511yescharacteristic
738C22C2×#53611yescharacteristic
739C22C2×#53711yescharacteristic
740C22C2×#53711yescharacteristic
741C22C2×#53711yescharacteristic
742C22C2×#53812yes—
743C22C2×#53912yes—
744C22C2×#53912yes—
745C22C2×#54012yes—
746C22C2×#54612yes—
747C22C2×#54612yes—
748C22C2×#54712yes—
749C22C2×#54812yes—
750C22C2×#55112yes—
751C22C2×#55212yes—
752C22C2×#56212yes—
753C22C2×#56312yes—
754C4—28no—
755C4—44no—
756C4—44no—
757C4—44no—
758C4—48no—
759C4—44no—
760C4—84no—
761C4—84no—
762C4—84no—
763C2—48no—
764C2—84no—
765C2#16359116yesdirect product factor
766C2C2×#266111yescharacteristic
767C2C2×#262311yescharacteristic
768C2C2×#261511yescharacteristic
769C2C2×#240511yescharacteristic
770C2C2×#243211yescharacteristic
771C2C2×#249811yescharacteristic
772C2C2×#281411yescharacteristic
773C2C2×#269512yes—
774C2C2×#269612yes—
775C2C2×#270612yes—
776C2C2×#270712yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#398C2semidirect
3. C22×#397C2semidirect
4. C22×#380C2semidirect
5. #16359C2direct (N × H)
6. C2×#1442C2semidirect
7. C2×#1441C2semidirect
8. C2×#1439C2semidirect
9. C2×#1439C2semidirect
10. C2×#1439C2semidirect
12. C2×C22⋊#95C2semidirect
32. C2×#398C22semidirect
34. C2×#397C22semidirect
36. C2×#380C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
52. C22⋊#95C22semidirect
765. C2#16359direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1232
C22×#3982
C22×#3972
C22×#3802
#163592
C2×#14422
C2×#14412
C2×#14392
C2×#14382
C2×C22⋊#952