Finite Group Databasebeta

← C2×#15669 #609305

Subgroups — C2×#15669 #609305

Order 512.

6670 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic, complemented
3C22×#397C211yesmaximal, characteristic, complemented
4C22×#382C211yesmaximal, characteristic, complemented
5#15669C2116yesmaximal, direct product factor
6C2×#1445C211yesmaximal, characteristic, complemented
7C2×#1441C212yesmaximal, complemented
8C2×#1439C212yesmaximal, complemented
9C2×#1438C212yesmaximal
10C2×#1437C211yesmaximal, characteristic, complemented
11C2×#1437C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C212yesmaximal, complemented
13C24×D4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×#123—28no—
16C22×#123—28no—
17C22×#123—28no—
18C22×#123C2218yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2212yes—
30C23×C42C2211yescharacteristic
31C2×#399—28no—
32C2×#399C2218yescomplemented
33C2×#397—28no—
34C2×#397C2218yescomplemented
35C2×#382—28no—
36C2×#382C2218yescomplemented
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1445C2218yescomplemented
47#1441C22116yescomplemented
48#1439C22116yescomplemented
49#1438C22116yes—
50#1437C2218yescomplemented
51#1437C2218yescomplemented
52C22⋊#95C22116yescomplemented
53C26C2311yescharacteristic
54C26C2311yescharacteristic
55C23×D4—24no—
56C23×D4—24no—
57C23×D4—28no—
58C23×D4—24no—
59C23×D4—44no—
60C23×D4C2314yes—
61C24×C4—24no—
62C24×C4—24no—
63C24×C4—24no—
64C24×C4—28no—
65C24×C4—44no—
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
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—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2318yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4D418yes—
116C22×C22⋊C4D418yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2318yes—
124C22×C42—24no—
125C22×C42—28no—
126C22×C42C2314yes—
127#399—232no—
128#397—232no—
129#382—232no—
130C2×#95—28no—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—28no—
134C2×#95—28no—
135C2×#95—44no—
136C2×#95—44no—
137C2×#95—44no—
138C2×#95—44no—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C25—22no—
149C25—22no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25—22no—
156C25—22no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—24no—
162C25—24no—
163C25C2×D414yes—
164C25C2×D414yes—
165C25#4118yes—
166C25C2412yes—
167C25C2412yes—
168C25C2411yescharacteristic
169C22×D4—232no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—216no—
173C22×D4—28no—
174C22×D4—48no—
175C22×D4—432no—
176C22×D4—432no—
177C22×D4—416no—
178C22×D4—416no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—232no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—216no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—48no—
229C23×C4—432no—
230C23×C4—432no—
231C23×C4—416no—
232C23×C4—416no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4#4118yes—
241C23×C4#4118yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251#123—2256no—
252C2×C4⋊C4—232no—
253C2×C4⋊C4—232no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—432no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4C2×D4116yes—
268C2×C22⋊C4—232no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—232no—
271C2×C22⋊C4—232no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—432no—
291C2×C22⋊C4—432no—
292C2×C22⋊C4—432no—
293C2×C22⋊C4—432no—
294C2×C22⋊C4C2×D4116yes—
295C2×C22⋊C4C2×D4116yes—
296C2×C42—232no—
297C2×C42—28no—
298C2×C42—216no—
299#95—216no—
300#95—216no—
301#95—28no—
302#95—216no—
303#95—28no—
304#95—48no—
305#95—48no—
306#95—48no—
307#95—48no—
308C24—28no—
309C24—216no—
310C24—28no—
311C24—216no—
312C24—216no—
313C24—216no—
314C24—216no—
315C24—216no—
316C24—216no—
317C24—216no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—28no—
326C24—28no—
327C24—28no—
328C24—216no—
329C24—216no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—44no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—48no—
351C24—44no—
352C24—44no—
353C24—48no—
354C24—48no—
355C24—48no—
356C24—48no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24#124116yes—
365C24C22×D414yes—
366C24C22×D414yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C2×#4118yes—
371C24C2×#4114yes—
372C24C2×#4114yes—
373C24C2×#4114yes—
374C24H(5,𝔽2)14yes—
375C24H(5,𝔽2)14yes—
376C242−1+414yes—
377C24C2511yescharacteristic
378C2×D4—232no—
379C2×D4—432no—
380C2×D4—432no—
381C2×D4—416no—
382C2×D4—4128no—
383C2×D4—416no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—232no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—432no—
448C22×C4—432no—
449C22×C4—416no—
450C22×C4—4128no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4#120116yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C2×#4118yes—
495C22×C4C2×#4118yes—
496C4⋊C4—232no—
497C4⋊C4—232no—
498C4⋊C4—432no—
499C4⋊C4—416no—
500C22⋊C4—232no—
501C22⋊C4—232no—
502C22⋊C4—232no—
503C22⋊C4—432no—
504C22⋊C4—432no—
505C22⋊C4—432no—
506C22⋊C4—432no—
507C42—232no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—28no—
512C23—232no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—264no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—416no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—416no—
533C23—432no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—432no—
542C23—416no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—416no—
552C23—416no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—432no—
561C23—44no—
562C23—432no—
563C23—816no—
564C23—816no—
565C23—816no—
566C23—816no—
567C23C2×#12014yes—
568C23#52514yes—
569C23C2×#12418yes—
570C23C2×#12414yes—
571C23#52918yes—
572C23#53214yes—
573C23#53514yes—
574C23#53514yes—
575C23#53614yes—
576C23#53918yes—
577C23#53918yes—
578C23#54018yes—
579C23#54618yes—
580C23#54618yes—
581C23#54618yes—
582C23#54718yes—
583C23#54718yes—
584C23#55018yes—
585C23#55118yes—
586C23#55118yes—
587C23#55218yes—
588C23#56218yes—
589C23C23×D411yescharacteristic
590C23C23×D411yescharacteristic
591C23C23×D411yescharacteristic
592C23C22×#4112yes—
593C23C22×#4112yes—
594C23C22×#4112yes—
595C23C2×H(5,𝔽2)12yes—
596C23C2×H(5,𝔽2)12yes—
597C23C2×2−1+412yes—
598D4—464no—
599C2×C4—232no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—464no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—432no—
643C2×C4—432no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—816no—
660C2×C4—816no—
661C2×C4C2×#12018yes—
662C22—28no—
663C22—216no—
664C22—48no—
665C22—464no—
666C22—48no—
667C22—44no—
668C22—48no—
669C22—48no—
670C22—48no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—44no—
675C22—48no—
676C22—44no—
677C22—832no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22#240618yes—
683C22#243118yes—
684C22#243518yes—
685C22#2471116yes—
686C22#2488116yes—
687C22#2553116yes—
688C22#2694116yes—
689C22#2706116yes—
690C22#2706116yes—
691C22C22×#12011yescharacteristic
692C22C2×#52511yescharacteristic
693C22C22×#12411yescharacteristic
694C22C2×#52912yes—
695C22C2×#53211yescharacteristic
696C22C2×#53511yescharacteristic
697C22C2×#53511yescharacteristic
698C22C2×#53611yescharacteristic
699C22C2×#53912yes—
700C22C2×#53912yes—
701C22C2×#54012yes—
702C22C2×#54612yes—
703C22C2×#54612yes—
704C22C2×#54612yes—
705C22C2×#54712yes—
706C22C2×#54712yes—
707C22C2×#55012yes—
708C22C2×#55112yes—
709C22C2×#55112yes—
710C22C2×#55212yes—
711C22C2×#56212yes—
712C4—28no—
713C4—44no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—48no—
718C4—48no—
719C4—84no—
720C4—84no—
721C2—48no—
722C2—84no—
723C2#15669116yesdirect product factor
724C2C2×#243511yescharacteristic
725C2C2×#243111yescharacteristic
726C2C2×#247112yes—
727C2C2×#240611yescharacteristic
728C2C2×#248812yes—
729C2C2×#255312yes—
730C2C2×#269412yes—
731C2C2×#270612yes—
732C2C2×#270612yes—
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×#399C2semidirect
3. C22×#397C2semidirect
4. C22×#382C2semidirect
5. #15669C2direct (N × H)
6. C2×#1445C2semidirect
7. C2×#1441C2semidirect
8. C2×#1439C2semidirect
10. C2×#1437C2semidirect
11. C2×#1437C2semidirect
12. C2×C22⋊#95C2semidirect
32. C2×#399C22semidirect
34. C2×#397C22semidirect
36. C2×#382C22semidirect
46. #1445C22semidirect
47. #1441C22semidirect
48. #1439C22semidirect
50. #1437C22semidirect
51. #1437C22semidirect
52. C22⋊#95C22semidirect
723. C2#15669direct (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×#3992
C22×#3972
C22×#3822
#156692
C2×#14452
C2×#14412
C2×#14392
C2×#14382
C2×#14372
C2×C22⋊#952