Finite Group Databasebeta

← C2×#14921 #607962

Subgroups — C2×#14921 #607962

Order 512.

7586 conjugacy classes of subgroups.

The subgroup lattice has 804 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#386C211yesmaximal, characteristic
2C22×#386C211yesmaximal, characteristic
3C22×#386C211yesmaximal, characteristic
4C22×#385C211yesmaximal, characteristic, complemented
5C22×#380C211yesmaximal, characteristic, complemented
6C23×#95C211yesmaximal, characteristic, complemented
7#14921C2116yesmaximal, direct product factor
8C2×#1443C212yesmaximal, complemented
9C2×#1441C212yesmaximal, complemented
10C2×#1439C212yesmaximal, complemented
11C2×#1439C212yesmaximal, complemented
12C2×C22⋊#95C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C2×#386—28no—
27C2×#386—28no—
28C2×#386—28no—
29C2×#386C2218yes—
30C2×#386C2218yes—
31C2×#386C2218yes—
32C2×#385—28no—
33C2×#385C2218yescomplemented
34C2×#380—28no—
35C2×#380C2218yescomplemented
36C22×#95—28no—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2212yes—
47C22×#95C2218yescomplemented
48#1443C22116yescomplemented
49#1441C22116yescomplemented
50#1439C22116yescomplemented
51#1439C22116yescomplemented
52C22⋊#95C2218yescomplemented
53C26C2311yescharacteristic
54C26C2311yescharacteristic
55C23×D4—24no—
56C23×D4—24no—
57C23×D4—24no—
58C23×D4—24no—
59C23×D4—24no—
60C23×D4—24no—
61C23×D4Q818yes—
62C23×D4C2314yes—
63C24×C4—24no—
64C24×C4—24no—
65C24×C4—24no—
66C24×C4—24no—
67C24×C4—24no—
68C24×C4—28no—
69C24×C4—28no—
70C24×C4—24no—
71C24×C4—24no—
72C24×C4—44no—
73C24×C4—44no—
74C24×C4—44no—
75C24×C4—44no—
76C24×C4D418yes—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2314yes—
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2318yes—
98C22×C22⋊C4—24no—
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—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
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⋊C4D418yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2318yes—
124#386—232no—
125#386—232no—
126#386—232no—
127#385—232no—
128#380—232no—
129C2×#95—28no—
130C2×#95—28no—
131C2×#95—232no—
132C2×#95—28no—
133C2×#95—28no—
134C2×#95—28no—
135C2×#95—28no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—28no—
139C2×#95—44no—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—22no—
155C25—22no—
156C25—22no—
157C25—24no—
158C25—24no—
159C25—22no—
160C25—22no—
161C25—22no—
162C25—24no—
163C25—24no—
164C25C2×D414yes—
165C25C2×D414yes—
166C25C2×D414yes—
167C25C2×Q814yes—
168C25C2×Q814yes—
169C25#4114yes—
170C25#4114yes—
171C25#4114yes—
172C25C2412yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×D4—232no—
176C22×D4—232no—
177C22×D4—232no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—416no—
185C22×D4—416no—
186C22×D4—416no—
187C22×D4—416no—
188C22×D4C2×Q8116yes—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—232no—
193C23×C4—232no—
194C23×C4—232no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
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—232no—
229C23×C4—216no—
230C23×C4—216no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—28no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—28no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—416no—
251C23×C4—432no—
252C23×C4—48no—
253C23×C4—48no—
254C23×C4—48no—
255C23×C4—48no—
256C23×C4—432no—
257C23×C4—432no—
258C23×C4—432no—
259C23×C4—432no—
260C23×C4—416no—
261C23×C4—416no—
262C23×C4—416no—
263C23×C4—432no—
264C23×C4—416no—
265C23×C4C2×D418yes—
266C23×C4C2×D418yes—
267C23×C4C2×D414yes—
268C23×C4C2×D4116yes—
269C23×C4C2×D414yes—
270C23×C4C2×D4116yes—
271C23×C4C2×Q814yes—
272C23×C4#4118yes—
273C23×C4#4118yes—
274C23×C4C2412yes—
275C23×C4C2414yes—
276C23×C4C2412yes—
277C23×C4C2414yes—
278C23×C4C2414yes—
279C23×C4C2412yes—
280C23×C4C2412yes—
281C23×C4C2414yes—
282C23×C4C2412yes—
283C2×C4⋊C4—216no—
284C2×C4⋊C4—216no—
285C2×C4⋊C4—216no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—432no—
290C2×C4⋊C4—432no—
291C2×C22⋊C4—232no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—216no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—432no—
311C2×C22⋊C4—432no—
312C2×C22⋊C4—416no—
313C2×C22⋊C4—416no—
314C2×C22⋊C4—416no—
315C2×C22⋊C4—416no—
316C2×C22⋊C4C2×D4116yes—
317#95—216no—
318#95—216no—
319#95—216no—
320#95—216no—
321#95—216no—
322#95—28no—
323#95—216no—
324#95—28no—
325#95—216no—
326#95—48no—
327C24—28no—
328C24—28no—
329C24—216no—
330C24—216no—
331C24—28no—
332C24—216no—
333C24—28no—
334C24—216no—
335C24—216no—
336C24—216no—
337C24—216no—
338C24—216no—
339C24—216no—
340C24—216no—
341C24—216no—
342C24—216no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—28no—
348C24—28no—
349C24—28no—
350C24—216no—
351C24—216no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—44no—
363C24—44no—
364C24—44no—
365C24—44no—
366C24—44no—
367C24—44no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—48no—
378C24—48no—
379C24—48no—
380C24#121116yes—
381C24#121116yes—
382C24#121116yes—
383C24C22⋊Q8116yes—
384C24C22⋊Q8116yes—
385C24C22⋊Q8116yes—
386C24#126116yes—
387C24C22×D414yes—
388C24C22×D414yes—
389C24C22×D414yes—
390C24C22×D412yes—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C22×Q814yes—
394C24C22×Q814yes—
395C24C22×Q812yes—
396C24C2×#4114yes—
397C24C2×#4114yes—
398C24C2×#4114yes—
399C24C2×#4112yes—
400C24C2×#4112yes—
401C24C2×#4112yes—
402C24C2×#4114yes—
403C24C2×#4114yes—
404C24H(5,𝔽2)14yes—
405C24H(5,𝔽2)14yes—
406C24C2511yescharacteristic
407C2×D4—232no—
408C2×D4—232no—
409C2×D4—232no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—4128no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—232no—
423C22×C4—216no—
424C22×C4—232no—
425C22×C4—232no—
426C22×C4—216no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—28no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—232no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—416no—
487C22×C4—432no—
488C22×C4—4128no—
489C22×C4—432no—
490C22×C4—432no—
491C22×C4—432no—
492C22×C4—432no—
493C22×C4—4128no—
494C22×C4—4128no—
495C22×C4—416no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—432no—
499C22×C4—416no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—44no—
519C22×C4—44no—
520C22×C4—48no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4—48no—
524C22×C4—48no—
525C22×C4—48no—
526C22×C4—44no—
527C22×C4—48no—
528C22×C4—48no—
529C22×C4—44no—
530C22×C4—48no—
531C22×C4—48no—
532C22×C4—48no—
533C22×C4—48no—
534C22×C4—48no—
535C22×C4#123116yes—
536C22×C4C22×D414yes—
537C22×C4C22×D418yes—
538C22×C4C22×D418yes—
539C22×C4C22×D414yes—
540C22×C4C22×Q814yes—
541C22×C4C2×#4118yes—
542C22×C4C2×#4118yes—
543C4⋊C4—432no—
544C4⋊C4—432no—
545C22⋊C4—232no—
546C22⋊C4—432no—
547C22⋊C4—416no—
548C22⋊C4—432no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—416no—
552C23—264no—
553C23—232no—
554C23—232no—
555C23—232no—
556C23—24no—
557C23—232no—
558C23—24no—
559C23—232no—
560C23—28no—
561C23—28no—
562C23—24no—
563C23—28no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—28no—
570C23—28no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—264no—
575C23—232no—
576C23—264no—
577C23—264no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—28no—
582C23—28no—
583C23—432no—
584C23—416no—
585C23—416no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—42no—
595C23—44no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—432no—
605C23—416no—
606C23—88no—
607C23—88no—
608C23—88no—
609C23—88no—
610C23—816no—
611C23—816no—
612C23#396164yes—
613C23C2×#12118yes—
614C23C2×#12118yes—
615C23C2×#12118yes—
616C23C2×#12114yes—
617C23C2×#12114yes—
618C23C2×#12114yes—
619C23C2×C22⋊Q818yes—
620C23C2×C22⋊Q818yes—
621C23C2×C22⋊Q818yes—
622C23C2×C22⋊Q814yes—
623C23C2×C22⋊Q814yes—
624C23C2×C22⋊Q814yes—
625C23C2×#12314yes—
626C23#52514yes—
627C23C2×#12618yes—
628C23C2×#12614yes—
629C23#52918yes—
630C23#53218yes—
631C23#53418yes—
632C23#53414yes—
633C23#53514yes—
634C23#53818yes—
635C23#54618yes—
636C23#54618yes—
637C23#55118yes—
638C23#55118yes—
639C23#55318yes—
640C23#55318yes—
641C23#55418yes—
642C23#55418yes—
643C23C23×D411yescharacteristic
644C23C23×D411yescharacteristic
645C23C23×D411yescharacteristic
646C23C23×Q811yescharacteristic
647C23C22×#4111yescharacteristic
648C23C22×#4111yescharacteristic
649C23C22×#4111yescharacteristic
650C23C22×#4112yes—
651C23C22×#4112yes—
652C23C2×H(5,𝔽2)12yes—
653C23C2×H(5,𝔽2)12yes—
654D4—464no—
655C2×C4—24no—
656C2×C4—232no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—28no—
660C2×C4—28no—
661C2×C4—24no—
662C2×C4—28no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—28no—
668C2×C4—24no—
669C2×C4—28no—
670C2×C4—28no—
671C2×C4—28no—
672C2×C4—28no—
673C2×C4—28no—
674C2×C4—28no—
675C2×C4—464no—
676C2×C4—464no—
677C2×C4—464no—
678C2×C4—432no—
679C2×C4—416no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—432no—
685C2×C4—416no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—432no—
701C2×C4—44no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—42no—
711C2×C4—44no—
712C2×C4—42no—
713C2×C4—44no—
714C2×C4—44no—
715C2×C4—44no—
716C2×C4—44no—
717C2×C4—44no—
718C2×C4—88no—
719C2×C4—88no—
720C2×C4—816no—
721C2×C4C2×#12318yes—
722C22—216no—
723C22—28no—
724C22—28no—
725C22—216no—
726C22—2128no—
727C22—28no—
728C22—216no—
729C22—28no—
730C22—216no—
731C22—28no—
732C22—28no—
733C22—44no—
734C22—48no—
735C22—48no—
736C22—44no—
737C22—44no—
738C22—84no—
739C22—82no—
740C22—832no—
741C22—82no—
742C22—82no—
743C22—84no—
744C22—82no—
745C22C2×#396116yes—
746C22C2×#39618yes—
747C22#241118yes—
748C22#241218yes—
749C22#2450116yes—
750C22#248718yes—
751C22#2510116yes—
752C22#2543116yes—
753C22#2564116yes—
754C22#257918yes—
755C22#261718yes—
756C22#285418yes—
757C22C22×#12111yescharacteristic
758C22C22×#12111yescharacteristic
759C22C22×#12111yescharacteristic
760C22C22×C22⋊Q811yescharacteristic
761C22C22×C22⋊Q811yescharacteristic
762C22C22×C22⋊Q811yescharacteristic
763C22C22×#12311yescharacteristic
764C22C2×#52511yescharacteristic
765C22C22×#12611yescharacteristic
766C22C2×#52912yes—
767C22C2×#53212yes—
768C22C2×#53412yes—
769C22C2×#53411yescharacteristic
770C22C2×#53511yescharacteristic
771C22C2×#53812yes—
772C22C2×#54612yes—
773C22C2×#54612yes—
774C22C2×#55112yes—
775C22C2×#55112yes—
776C22C2×#55312yes—
777C22C2×#55312yes—
778C22C2×#55412yes—
779C22C2×#55412yes—
780C4—28no—
781C4—48no—
782C4—44no—
783C4—48no—
784C4—44no—
785C4—48no—
786C4—84no—
787C4—82no—
788C4—82no—
789C2—216no—
790C2—84no—
791C2#14921116yesdirect product factor
792C2C22×#39611yescharacteristic
793C2C2×#261711yescharacteristic
794C2C2×#241211yescharacteristic
795C2C2×#245012yes—
796C2C2×#241111yescharacteristic
797C2C2×#251012yes—
798C2C2×#248711yescharacteristic
799C2C2×#254312yes—
800C2C2×#257911yescharacteristic
801C2C2×#256412yes—
802C2C2×#285411yescharacteristic
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
4. C22×#385C2semidirect
5. C22×#380C2semidirect
6. C23×#95C2semidirect
7. #14921C2direct (N × H)
8. C2×#1443C2semidirect
9. C2×#1441C2semidirect
10. C2×#1439C2semidirect
11. C2×#1439C2semidirect
12. C2×C22⋊#95C2semidirect
33. C2×#385C22semidirect
35. C2×#380C22semidirect
47. C22×#95C22semidirect
48. #1443C22semidirect
49. #1441C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. C22⋊#95C22semidirect
791. C2#14921direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3862
C22×#3852
C22×#3802
C23×#952
#149212
C2×#14432
C2×#14412
C2×#14392
C2×C22⋊#952